\chapter*{Part II. Hyperk\"ahler structures on total spaces of
holomorphic cotangent bundles}

\markboth{COTANGENT BUNDLES}{INTRODUCTION}

\vspace{5pt}

\centerline{\Huge D. Kaledin}

\hfill

\refstepcounter{chapter}
\addcontentsline{toc}{chapter}{\numberline{}Part II. Hyperk\"ahler
structures on total spaces\protect\\ of holomorphic cotangent
bundles. {\protect\normalfont D. Kaledin}}

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\index{terms}{adic completion|see{formal completion}}
\index{terms}{augmentation bidegree|see{bidegree, augmentation}}
\index{terms}{augmentation degree|see{degree, augmentation}}
\index{terms}{augmentation bigrading|see{bigrading, augmentation}}
\index{terms}{augmentation grading|see{grading, augmentation}}
\index{terms}{total degree|see{degree, total}}
\index{terms}{total grading|see{grading, total}}
\index{terms}{Hodge bidegree|see{bidegree, Hodge}}
\index{terms}{Hodge degree|see{degree, Hodge}}
\index{terms}{Hodge type bigrading|see{bigrading, Hodge type}}
\index{terms}{connections!extended|see{extended connection}}
\index{terms}{connections!Hodge|see{Hodge connection}}
\index{terms}{Euclidean metric|see{metric, Euclidean}}
\index{terms}{Hermitian metric|see{metric, Hermitian}}
\index{terms}{Riemannian metric|see{metric, Riemannian}}
\index{terms}{hyperk\"ahler metric|see{metric, hyperk\"ahler}}
\index{terms}{K\"ahler metric|see{metric, K\"ahler}}
\index{terms} 
{Quaternionic-Hermitian metric|see{metric, Quaternionic-Hermitian}}
\index{terms}{Hermitian-Hodge metric|see{metric, Hermitian-Hodge}}
\index{terms}{K\"ahlerian connection|see{connections, K\"ahlerian}}

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\section*{Introduction}

A hyperk\"ahler manifold\index{terms}{hyperk\"ahler manifold} is by
definition a Riemannian manifold $M$ equipped with two
anti-commuting almost complex structures $I$, $J$ parallel with
respect to the Levi-Civita
connection\index{terms}{connections!Levi-Civita}.  Hyperk\"ahler
manifolds\index{terms}{hyperk\"ahler manifold} were introduced by
E. Calabi\index{names}{Calabi, E.} in
\cite{C}. Since then they have become the topic of much research. We
refer the reader to \cite{Bes} and to \cite{HKLR} for excellent
overviews of the subject.

Let $M$ be a hyperk\"ahler manifold. The almost complex structures
$I$ and $J$ generate an action of the
quaternion\index{terms}{quaternion algebra} algebra $\h$ in the
tangent bundle $\Theta(M)$ to the manifold $M$. This action is
parallel with respect to the Levi-Civita
connection\index{terms}{connections!Levi-Civita}. Every quaternion $h
\in \h$ with $h^2 = -1$, in particular, the product $K = IJ \in \h$,
defines by means of the $\h$-action an almost complex structure
$M_h$ on $M$. This almost complex structure is also parallel, hence
integrable and K\"ahler. Thus every hyperk\"ahler
manifold\index{terms}{hyperk\"ahler manifold} $M$ is canonically
K\"ahler, and in many different ways. For the sake of convenience,
we will consider $M$ as a K\"ahler manifold by means of the complex
structure $M_I$, unless indicated otherwise.

One of the basic facts about hyperk\"ahler manifolds is that the
K\"ahler manifold $M_I$ underlying a hyperk\"ahler
manifold\index{terms}{hyperk\"ahler manifold} $M$ is
canonically holomorphically symplectic. To see this, let $\omega_J$,
$\omega_K$ be the K\"ahler forms for the complex structures $M_J$,
$M_K$ on the manifold $M$, and consider the $2$-form $\Omega =
\omega_J + \sqrt{-1}\omega_K$ on $M$. It is easy to check that the
form $\Omega$ is of Hodge type\index{terms}{bigrading!Hodge type} $(2,0)$
for the complex structure $M_I$ on $M$. Since it is obviously
non-degenerate and closed, it is holomorphic, and the K\"ahler
manifold $M_I$ equipped with the form $\Omega$ is a holomorphically
symplectic manifold.

It is natural to ask whether every holomorphically symplectic
manifold $\langle M,\Omega\rangle$ underlies a hyperk\"ahler
structure\index{terms}{hyperk\"ahler structures} on $M$, and if so,
then how many such hyperk\"ahler structures are there. Note that if
such a hyperk\"ahler structure exists, it is completely defined by
the K\"ahler metric\index{terms}{metric!K\"ahler} $h$ on $M$. Indeed,
the K\"ahler forms $\omega_J$ and $\omega_K$ are by definition the
real and imaginary parts of the form $\Omega$, and the forms
$\omega_J$ and $\omega_K$ together with the metric define the
complex structures $J$ and $K$ on $M$ and, consequently, the whole
$\h$-action in the tangent bundle $\Theta(M)$. For the sake of
simplicity, we will call a metric $h$ on a holomorphically
symplectic manifold $\langle M,\Omega\rangle$ hyperk\"ahler if the
Riemannian manifold $\langle M, h \rangle$ with the quaternionic
action associated to the pair $\langle \Omega, h \rangle$ is a
hyperk\"ahler manifold.

It is known (see, e.g., \cite{Beauv}) that if the holomorphically
symplectic manifold $M$ is compact, for example, if $M$ is a
$K3$-surface, then every K\"ahler class in $H^{1,1}(M)$ contains a
unique hyperk\"ahler metric\index{terms}{metric!hyperk\"ahler}. This
is, in fact, a consequence of the famous
Calabi-Yau\index{names}{Yau, S.-T.} Theorem\index{terms}{Calabi--Yau
Theorem}, which provides the canonical Ricci-flat
metric\index{terms}{metric!Ricci-flat} on $M$ with the given cohomology
class. This Ricci-flat metric turns out to be hyperk\"ahler. Thus in
the compact case holomorphically symplectic and hyperk\"ahler
manifolds\index{terms}{hyperk\"ahler manifold} are essentially the same.

The situation is completely different in the general case. For
example, all holomorphically symplectic structures on the formal
neighborhood\index{terms}{formal neighborhood} of the origin $0 \in
\C^{2n}$ in the $2n$-dimensional complex vector space $\C^{2n}$ are
isomorphic by the appropriate version of the Darboux
Theorem\index{terms}{Darboux--Weinstein Theorem}. On the other hand,
hyperk\"ahler structures\index{terms}{hyperk\"ahler structures} on
this formal neighborhood\index{terms}{formal neighborhood} form an
infinite-dimensional family (see, e.g.,
\cite{HKLR}, where there is a construction of a smaller, but still
infinite-dimensional family of hyperk\"ahler
metrics\index{terms}{metric!hyperk\"ahler} defined on the
whole $\C^{2n}$). Thus, to obtain meaningful results, it seems
necessary to restrict our attention to holomorphically symplectic
manifolds belonging to some special class.

The simplest class of non-compact holomorphically symplectic
manifolds is formed by total spaces $T^*M$ to the cotangent bundle
to complex manifolds $M$. In fact, the first examples of
hyperk\"ahler manifolds\index{terms}{hyperk\"ahler manifold} given by
Calabi\index{names}{Calabi, E.} in \cite{C} were of this type, with $M$
being a K\"ahler manifold of constant holomorphic sectional
curvature (for example, a complex projective space). It has been
conjectured for some time that every total space $T^*M$ of the
cotangent bundle to a K\"ahler manifold admits a hyperk\"ahler
structure\index{terms}{hyperk\"ahler structures}. The goal of this
paper is to prove that this is indeed the case, if one agrees to
consider only an open neighborhood $U
\subset T^*M$ of the zero section $M \subset T^*M$.  Our main result
is the following.

\begin{thm}\label{th.1}
Let $M$ be a complex manifold equipped with a K\"ahler
metric\index{terms}{metric!K\"ahler}. The metric on $M$ extends to a
hyperk\"ahler metric\index{terms}{metric!hyperk\"ahler} $h$ defined
in the formal neighborhood\index{terms}{formal neighborhood} of the
zero section $M \subset T^*M$ in the total space $T^*M$ to the
holomorphic cotangent bundle to $M$. The extended metric $h$ is
invariant under the action of the group $U(1)$ on $T^*M$ given by
dilatations along the fibers of the canonical projection $\rho:T^*M
\to M$.  Moreover, every other $U(1)$-invariant hyperk\"ahler metric
on the holomorphically symplectic manifold $T^*M$ becomes equal to
$h$ after a holomorphic symplectic $U(1)$-equivariant automorphism
of $T^*M$ over $M$.  Finally, if the K\"ahler metric on $M$ is
real-analytic, then the formal hyperk\"ahler
metric\index{terms}{metric!formal hyperk\"ahler} $h$ converges to a
real-analytic metric\index{terms}{metric!real-analytic} in an open
neighborhood $U \subset T^*M$ of the zero section $M
\subset T^*M$.
\end{thm}

Many of the examples of hyperk\"ahler
metrics\index{terms}{metric!hyperk\"ahler} obtained by
Theorem~\ref{th.1} are already known. (See, e.g., \cite{K1},
\cite{K2}, \cite{Nak}, \cite{H}, \cite{BG}, \cite{Sw}.) In these
examples $M$ is usually a generalized flag manifold or a homogeneous
space of some kind. On the other hand, very little is known for
manifolds of general type. In particular, it seems that even for
curves of genus $g \geq 2$ Theorem~\ref{th.1} is new.

We would like to stress the importance of the $U(1)$-invariance
condition on the metric in the formulation of
Theorem~\ref{th.1}. This condition for a total space $T^*M$ of a
cotangent bundle is equivalent to a more general compatibility
condition between a $U(1)$-action and a hyperk\"ahler
structure\index{terms}{hyperk\"ahler structures} on a smooth manifold
introduced by N.J. Hitchin\index{names}{Hitchin, N. J.} (see, e.g.,
\cite{H}). Thus Theorem~\ref{th.1} can be also regarded as answering
a question of Hitchin's in \cite{H}, namely, whether every K\"ahler
manifold can be embedded as the sub-manifold of $U(1)$-fixed points
in a $U(1)$-equivariant hyperk\"ahler
manifold\index{terms}{hyperk\"ahler manifold}. On the other hand, it
is this $U(1)$-invariance that guarantees the uniqueness of the
metric $h$ claimed in Theorem~\ref{th.1}.

We also prove a version of Theorem~\ref{th.1} ``without the
metrics''. The K\"ahler metric\index{terms}{metric!K\"ahler} on $M$
in this theorem is replaced with a holomorphic
connection\index{terms}{connections!holomorphic} $\nabla$ on the
cotangent bundle to $M$ without torsion and $(2,0)$-curvature. We
call such connections\index{terms}{connections!K\"ahlerian} {\em
K\"ah\-le\-ri\-an}. The total space of the cotangent bundle $T^*M$
is replaced with the total space $\tm$ of the complex-conjugate to
the tangent bundle to $M$. (Note that {\em a priori} there is no
complex structure on $\tm$, but the $U(1)$-action by dilatations on
this space is well-defined.) The analog of the notion of a
hyperk\"ahler manifold\index{terms}{hyperk\"ahler manifold}
``without the metric'' is the notion if a hypercomplex
manifold\index{terms}{hypercomplex manifold} (see, e.g.,
\cite{Bo}). We define a version of Hitchin\index{names}{Hitchin, N. J.}'s
condition\index{terms}{Hitchin's compatibility condition} on the
$U(1)$-action for hypercomplex manifolds and prove the following.

\begin{thm}\label{th.2}
Let $M$ be a complex manifold, and let $\tm$ be the total space of
the complex-conjugate to the tangent bundle to $M$ equipped with an
action of the group $U(1)$ by dilatation along the fibers of the
projection $\tm \to M$. There exists a natural bijection between the
set of all K\"ah\-le\-ri\-an
connections\index{terms}{connections!K\"ahlerian} on the cotangent
bundle to $M$ and the set of all isomorphism classes of
$U(1)$-equivariant hypercomplex structures\index{terms}{hypercomplex
structure} on the formal neighborhood\index{terms}{formal
neighborhood} of the zero section $M \subset \tm$ in $\tm$ such that
the projection $\rho:\tm
\to M$ is holomorphic. If the K\"ah\-le\-ri\-an connection on $M$ is
real-analytic, the corresponding hypercomplex
structure\index{terms}{hypercomplex structure} is defined in an open
neighborhood $U \subset \tm$ of the zero section.
\end{thm}

Our main technical tool in this paper is the relation between
$U(1)$\--equi\-va\-ri\-ant hyperk\"ahler
manifolds\index{terms}{hyperk\"ahler manifold!$U(1)$-equivariant} and
the theory of $\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge
structure} discovered by P. Deligne\index{names}{Deligne, P.} and
C. Simpson\index{names}{Simpson, C.} (see \cite{De2}, \cite{Simpson}). To
emphasize this relation, we use the name {\em Hodge
manifolds\index{terms}{Hodge manifold}} for the hypercomplex
manifolds\index{terms}{hypercomplex manifold} equipped with a
compatible $U(1)$-action.

It must be noted that many examples of hyperk\"ahler
manifolds\index{terms}{hyperk\"ahler manifold}
equipped with a compatible $U(1)$-action are already known. Such
are, for example, many of the manifolds constructred by the
so-called hyperk\"ahler reduction\index{terms}{hyperk\"ahler
reduction} from flat hyperk\"ahler spaces (see
\cite{H} and \cite{HKLR}). An important class of such manifolds is
formed by the so-called quiver varieties, studied by
H. Nakajima\index{names}{Nakajima, H.} (\cite{Nak}). On the other hand,
the moduli spaces $\M$ of flat connections on a complex manifold
$M$, studied by Hitchin\index{names}{Hitchin, N. J.} (\cite{Hcurves}) when
$M$ is a curve and by Simpson\index{names}{Simpson, C.} (\cite{Smoduli})
in the general case, also belong to the class of Hodge
manifolds\index{terms}{Hodge manifold}, as
Simpson\index{names}{Simpson, C.} has shown in \cite{Simpson}. Some
parts of our theory, especially the uniqueness statement of
Theorem~\ref{th.1}, can be applied to these known examples.

\medskip

We now give a brief outline of the paper. Sections 1-3 are
preliminary and included mostly to fix notation and
terminology. Most of the facts in these sections are well-known.
\begin{itemize}
\item In Section 1 we have collected the necessary facts from linear
algebra about quaternionic\index{terms}{quaternionic vector space}
vector spaces and $\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge
structure}. Everything is standard, with an exception, perhaps, of
the notion of {\em weakly Hodge map\index{terms}{weakly Hodge map}},
which we introduce in Subsection~\ref{w.H.sub}.

\item We begin Section 2 with introducing a technical notion of a
{\em Hodge bundle\index{terms}{Hodge bundle}} on a smooth manifold
equipped with a $U(1)$-action. This notion will be heavily used
throughout the paper.  Then we switch to our main subject, namely,
various differential-geometric objects related to a quaternion
action in the tangent bundle. The rest of Section 2 deals with
almost quaternionic manifolds\index{terms}{quaternionic manifold}
and the compatibility conditions between an almost quaternionic
structure\index{terms}{quaternionic structure} and a $U(1)$-action
on a smooth manifold $M$.

\item In Section 3 we describe various integrability conditions on
an almost quaternionic structure\index{terms}{quaternionic
structure}. In particular, we recall the definition of a
hypercomplex manifold\index{terms}{hypercomplex manifold} and
introduce $U(1)$-equivariant hypercomplex manifolds under the name
of {\em Hodge manifolds\index{terms}{Hodge manifold}}. We then
rewrite the definition of a Hodge manifold in the more convenient
language of Hodge bundles\index{terms}{Hodge bundle}, to be used
throughout the rest of the paper. Finally, in
Subsection~\ref{polarization} we discuss metrics on hypercomplex and
Hodge manifolds. We recall the definition of a hyperk\"ahler
manifold\index{terms}{hyperk\"ahler manifold} and define the notion
of a {\em polarization\index{terms}{polarization}} of a Hodge
manifold. A polarized Hodge manifold\index{terms}{Hodge
manifold!polarized} is the same as a hyperk\"ahler
manifold\index{terms}{hyperk\"ahler manifold!$U(1)$-equivariant}
equipped with a $U(1)$-action compatible with the hyperk\"ahler
structure of the sense of
Hitchin\index{names}{Hitchin, N. J.}\index{terms}{Hitchin's compatibility
condition}, \cite{H}.

\item The main part of the paper begins in Section 4. We start
with arbitrary Hodge manifolds\index{terms}{Hodge manifold} and
prove that in a neighborhood of the subset $M^{U(1)}$ of ``regular''
$U(1)$-fixed points\item{terms}{regular fixed point} every such
manifold $M$ is canonically isomorphic to an open neighborhood of
the zero section in a total space $\overline{T}M^{U(1)}$ of the
tangent bundle to the fixed point set. A fixed point $m \in M$ is
``regular'' if the group $U(1)$ acts on the tangent space $T_mM$
with weight\index{terms}{weight!of an $U(1)$-action}s $0$ and
$1$. We call this canonical isomorphism {\em the
linearization\index{terms}{linearization}} of the regular Hodge
manifold\index{terms}{Hodge manifold!regular}.

The linearization construction can be considered as a hyperk\"ahler
analog of the Darboux-Weinstein\index{names}{Weinstein, A.}
Theorem\index{terms}{Darboux--Weinstein Theorem} in the symplectic
geometry. Apart from the cotangent bundles, it can be applied to the
Hitchin\index{names}{Hitchin, N. J.}-Simpson\index{names}{Simpson,
C.} moduli space\index{terms}{Hitchin-Simpson moduli space of flat
connections} $\M$ of flat connections on a K\"ahler manifold
$X$. The regular points in this case correspond to stable flat
connections such that the underlying holomorphic bundle is also
stable. The linearization\index{terms}{linearization} construction
provides a canonical embedding of the subspace $\M^{reg} \subset \M$
of regular points into the total space $T^*\M_0$ of the cotangent
bundle to the space $\M_0$ of stable holomorphic bundles on $X$. The
unicity statement of Theorem~\ref{th.1} guarantees that the
hyperk\"ahler metric\index{terms}{metric!hyperk\"ahler} on
$\M^{reg}$ provided by the Simpson\index{names}{Simpson, C.} theory
is the same as the canonical metric constructed in
Theorem~\ref{th.1}.

\item Starting with Section 5, we deal only with total spaces $\tm$
of the complex-conjugate to tangent bundles to complex manifolds
$M$. In Section 5 we describe Hodge manifolds\index{terms}{Hodge
manifold} structures on $\tm$ in
terms of certain connections on the locally trivial fibration $\tm
\to M$. ``Connection'' here is understood as a choice of the
subbundle of horizontal vectors, regardless of the vector bundle
structure on the fibration $\tm \to M$. We establish a
correspondence between Hodge manifold structures on $\tm$ and
connections on $\tm$ over $M$ of certain type, which we call {\em
Hodge connection\index{terms}{Hodge connection}}.

\item In Section 6 we restrict our attention to the formal
neighborhood\index{terms}{formal neighborhood} of the zero section
$M \subset \tm$. We introduce the appropriate ``formal'' versions of
all the definitions and then establish a correspondence between
formal Hodge connections\index{terms}{Hodge connection!formal} on
$\tm$ over $M$ and certain differential operators on the manifold
$M$ itself, which we call {\em extended
connections\item{terms}{extended connection}}. We also introduce a
certain canonical algebra bundle $\B^\cdot(M,\C)$ on the complex
manifold $M$, which we call {\em the Weil algebra\index{terms}{Weil
algebra}} of the manifold $M$. Extended connections give rise to
natural derivations of the Weil algebra.

\item Before we can proceed with classification of extended
connections\item{terms}{extended connection} on the manifold $M$ and
therefore of regular Hodge manifolds\index{terms}{Hodge
manifold!regular}, we need to derive some linear-algebraic facts on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$. This is the
subject of Section 7. We begin with introducing a certain version of
the de Rham complex\index{terms}{de Rham complex} of a smooth
complex manifold, which we call {\em the total de Rham
complex}\index{terms}{de Rham complex!total}. Then we combine it the
material of Section 6 to define the so-called {\em total Weil
algebra\index{terms}{Weil algebra!total}} of the manifold $M$ and
establish some of its properties. Section 7 is the most technical
part of the whole paper. The reader is advised to skip reading it
until needed.

\item Section 8 is the main section of the paper. In this section we
prove, using the technical results of Section 7, that
extended\item{terms}{extended connection} connections on $M$ are in
a natural one-to-one correspondence with K\"ah\-le\-ri\-an
connections\index{terms}{connections!K\"ahlerian} on the cotangent
bundle to $M$ (Theorem~\ref{kal=ext}). This proves the formal part
of Theorem~\ref{th.2}.

\item In Section 9 we deal with
polarizations\index{terms}{polarization}. After some preliminaries,
we use Theorem~\ref{th.2} to deduce the formal part of
Theorem~\ref{th.1} (see Theorem~\ref{metrics}).

\item Finally, in Section~\ref{convergence} we study the convergence
of our formal series and prove Theorem~\ref{th.1} and
Theorem~\ref{th.2} in the real-analytic case.

\item We have also attached a brief Appendix, where we sketch a more
conceptual approach to some of the linear algebra used in the
paper, in particular, to our notion of a weakly Hodge
map\index{terms}{weakly Hodge map}. This approach also allows us to
describe a simple and conceptual proof of Proposition~\ref{ac}, the
most technical of the facts proved in Section 7. The Appendix is
mostly based on results of Deligne\index{names}{Deligne, P.} and
Simpson\index{names}{Simpson, C.} (\cite{De2}, \cite{Simpson}).
\end{itemize}

\noindent
{\bf Acknowledgment.} I would like to thank
A. Beilinson\index{names}{Beilinson, A.},
D. Kazhdan\index{names}{Kazhdan, D.}, A. Levin\index{names}{Levin, A.},
L. Posicelsky\index{names}{Posicelsky, L.}, A. Shen\index{names}{Shen, A.}
and A. Tyurin\index{names}{Tyurin, A.} for stimulating discussions. I
am especially grateful to my friends Misha Verbitsky and Tony
Pantev\index{names}{Pantev, T.} for innumerable discussions, constant
interest and help, without which this paper most probably would not
have been written. I would also like to mention here how much I have
benefited from a course on moduli spaces and Hodge theory given by
Carlos Simpson\index{names}{Simpson, C.} at MIT in the Fall of 1993. On
a different note, I would like to express my dearest gratitude to
Julie Lynch, formerly at International Press in Cambridge, and also
to the George Soros's Foundation and to CRDF for providing me with a
source of income during the preparation of this paper.

\section{Preliminary facts from linear algebra}

\subsection{Quaternionic vector spaces}\index{terms}{quaternionic
vector space}

\punkt Throughout the paper denote by $\h$ the $\R$-algebra of quaternions. 

\defn A {\em quaternionic vector space\index{terms}{quaternionic
vector space}} is a finite-dimensional left module
over the algebra\index{terms}{quaternion algebra} $\h$. 

Let $V$ be a quaternionic vector space.  Every algebra embedding
$I:\C \to \h$ defines by restriction an action of $\C$ on
$V$. Denote the corresponding complex vector space by $V_I$. 

Fix once and for all an algebra embedding $I:\C \to \h$ and call the
complex structure $V_I$ {\em the preferred\index{terms}{preferred complex structure} complex structure} on
$V$. 

\punkt \label{u.acts.on.h}
Let the group $\C^*$ act on the algebra $\h$ by conjugation by
$I(\C^*)$.  Since $I(\R^*) \subset I(\C^*)$ lies in the center of the
algebra $\h$, this action factors through the map $N:\C^* \to
\C^*/\R^* \cong U(1)$ from $\C^*$ to the one-dimensional unitary group
defined by $N(a) = a^{-1}\overline{a}$. Call this action {\em the
standard action} of $U(1)$ on $\h$.

The standard action commutes with the multiplication and leaves
invariant the subalgebra $I(\C)$. Therefore it extends to an action
of the complex algebraic group $\C^* \supset U(1)$ on the algebra
$\h$ considered as a right complex vector space over $I(\C)$. Call
this action {\em the standard action} of $\C^*$ on $\h$.

\punkt \defn An {\em equivariant} quaternionic vector
space\index{terms}{quaternionic vector space!equivariant} is a quaternionic
vector space $V$ equipped with an action of the group $U(1)$ such that the
action map 
$$
\h \otimes_\R V \to V
$$
is $U(1)$-equivariant. 

The $U(1)$-action on $V$ extends to an action of $\C^*$ on the complex
vector space $V_I$. The action map $\h \otimes_\R V \to V$ factors through a
map 
$$
\Mult:\h \otimes_\C V_I \to V_I 
$$
of complex vector spaces. This map is $\C^*$-equivariant if and only if $V$
is an equivariant quaternionic vector space. 

\punkt The category of complex algebraic representations $V$ of the
group $\C^*$ is equivalent to the category of graded vector spaces $V
= \oplus V^\cdot$. We will say that a representation $W$ is {\em of
weight\index{terms}{weight!of an $U(1)$-action} $i$} if $W = W^i$, that is, if an element $z \in \C^*$ acts
on $W$ by multiplication by $z^k$. For every representation $W$ we
will denote by $W(k)$ the representation corresponding to the
grading
$$
W(k)^i = W^{k+i}.
$$

\punkt \label{h=c+c} 
The algebra $\h$ considered as a complex vector space by means of right 
multiplication by $I(\C)$ decomposes $\h = I(\C) \oplus \cc$ with respect
to the standard $\C^*$-action. The first summand is of weight\index{terms}{weight!of an $U(1)$-action} $0$, and the
second is of weight\index{terms}{weight!of an $U(1)$-action} $1$. This decomposition is compatible with the left
$I(\C)$-actions as well and induces for every complex vector space $W$ a
decomposition
$$
\h \otimes_\C W = W \oplus \cc \otimes_\C W.
$$
If $W$ is equipped with an $\C^*$-action, the second summand is canonically
isomorphic to $\overline{W}(1)$, where $\overline{W}$ is the vector space
complex-conjugate to $W$. 

\punkt Let $V$ be an equivariant quaternionic vector
space\index{terms}{quaternionic vector space!equivariant}. The action map 
$$
\Mult:\h \otimes_\C V_I \cong V_I \oplus \cc \otimes V_I \to V_I
$$
decomposes $\Mult = \id \oplus j$ for a certain map $j:\overline{V_I}(1)
\to V_I$. The map $j$ satisfies $j \circ \overline{j} = -\id$, and we
obviously have the following. 

\begin{lemma}\label{explicit.eqvs}
The correspondence $V \longmapsto \langle V_I, j \rangle$ is an
equivalence of categories bet\-ween the category of equivariant
quaternionic vector spaces\index{terms}{quaternionic
vector space!equivariant} and the category of pairs $\langle W, j
\rangle$ of a graded complex vector space $W$ and a map
$j:\overline{W^\cdot} \to W^{1-\cdot}$ satisfying $j \circ
\overline{j} = -\id$.
\end{lemma}

\punkt We will also need a particular class of equivariant quaternionic
vector spaces which we will call regular\index{terms}{quaternionic
vector space!regular equivariant}. 

\defn An equivariant quaternionic vector
space\index{terms}{quaternionic vector space!regular equivariant}
$V$ is called {\em regular} if every irreducible
$\C^*$-subrepresentation of $V_I$ is either trivial or of
weight\index{terms}{weight!of an $U(1)$-action} $1$.

\begin{lemma}\label{regular.quaternionic}
Let $V$ be a regular $\C^*$-equivariant quaternionic vector space and let
$V^0_I \subset V_I$ be the subspace of $\C^*$-invariant vectors. Then the
action map
$$
V^0_I \oplus \overline{V^0_I} \cong \h \otimes_\C V^0_I \to V_I
$$
is invertible. 
\end{lemma}

\proof Let $V^1_I \subset V_I$ be the weight\index{terms}{weight!of
an $U(1)$-action} $1$ subspace with respect to 
the $ gm$-action.  Since $V$ is regular\index{terms}{quaternionic
vector space!regular equivariant}, $V_I = V^0_I \oplus V^1_I$. On the
other hand, $j:\overline{V_I} \to V_I$ interchanges $V^0_I$ and
$V^1_I$. Therefore $V^1_I \cong \overline{V^0_I}$ and we are done.
\endproof

Thus every regular equivariant quaternionic vector
space\index{terms}{quaternionic vector space!regular equivariant} is a sum of
several copies of the algebra $\h$ itself. The corresponding Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} has Hodge numbers
$h^{1,0} = h^{0,1}$, $h^{p,q} = 0$ otherwise.

\subsection{The complementary complex
structure}\label{complementary}\index{terms}{complementary complex
structure} 
 
\punkt 
Let $J:\C \to \h$ be another algebra embedding. Say that embeddings $I$ and
$J$ are {\em complementary\index{terms}{complementary complex structure}} if
$$
J(\sqrt{-1}) I(\sqrt{-1}) = - I(\sqrt{-1}) J(\sqrt{-1}).
$$
Let $V$ be an equivariant quaternionic vector
space\index{terms}{quaternionic vector space!equivariant}.  The
standard $U(1)$-action on $\h$ induces an action of $U(1)$ on the
set of all algebra embeddings $\C \to \h$.  On the subset of
embeddings complementary\index{terms}{complementary complex
structure} to $I$ this action is transitive.  Therefore for every
two embeddings $J_1,J_2:\C \to \h$
complementary\index{terms}{complementary complex structure} 
to $I$ the complex vector spaces $V_{J_1}$ and $V_{J_2}$ are
canonically isomorphic. We will from now on choose for convenience
an algebra embedding $J:\C \to \h$
complementary\index{terms}{complementary complex structure} to $I$
and speak of {\em the complementary\index{terms}{complementary
complex structure} complex structure} $V_J$ on $V$; however, nothing
depends on this choice.

\punkt For every equivariant quaternionic vector space $V$ the
complementary\index{terms}{complementary complex structure} embedding $J:\C \to \h$ induces an isomorphism
$$
J \otimes \id:\C \otimes_\R V \to \h \otimes_{I(\C)} V_I 
$$
of complex vector spaces. Let $\Mult:\h \otimes_{I(\C)} V_I \to V_I$, $\Mult:\C
\otimes_\R V \to V_J$ be the action maps. Then there exists a unique
isomorphism $H:V_J \to V_I$ of complex vector spaces such that the diagram 
$$
\begin{CD}
\C \otimes_\R V @>{J \otimes \id}>> \h \otimes_{I(\C)} V_I\\
@V{\Mult}VV                          @V{\Mult}VV          \\
V_J             @>{H}>>              V_I
\end{CD}
$$
is commutative. Call the map $H:V_J \to V_I$ {\em the standard
isomorphism} between the complementary\index{terms}{complementary
complex structure} and the preferred\index{terms}{preferred complex
structure} complex structures on the equivariant quaternionic
vector\index{terms}{quaternionic vector space!regular equivariant}
space $V$.  

\punkt Note that both $V_I$ and $V_J$ are canonically isomorphic to $V$ as
real vector spaces; therefore the map $H:V_J \to V_I$ is in fact an
automorphism of the real vector space $V$. Up to a constant this
automorphism is given by the action of the element $I(\sqrt{-1}) +
J(\sqrt{-1}) \in \h$ on the $\h$-module $V$. 

\subsection{$\protect\R$-Hodge structures}\index{terms}{Hodge structure@$\R$-Hodge structure} 

\punkt Recall that {\em a pure $\R$-Hodge structure $W$ of
weight\index{terms}{weight!of a Hodge structure} 
$i$} is a pair of a graded complex vector space $W = \oplus_{p+q=i}
W^{p,q}$ and a {\em real structure} map $\conj:\overline{W^{p,q}}
\to W^{q,p}$ satisfying $\conj \circ \conj = \id$. The bigrading
$W^{p,q}$ is called {\em the Hodge type bigrading\index{terms}{bigrading!Hodge
type}}. The dimensions $h^{p,q} = \dim_\C W^{p,q}$ are 
called {\em the Hodge numbers} of the pure $\R$-Hodge structure
$W$. Maps between pure Hodge structures are by definition maps of
their underlying complex vector spaces compatible with the bigrading
and the real structure maps.

\punkt Recall also that for every $k$ the {\em Hodge-Tate} pure $\R$-Hodge
structure $\R(k)$ of weight\index{terms}{weight!of a Hodge structure} $-2k$ is by definition the
$1$-dimensional complex vector space with complex conjugation equal
to $(-1)^k$ times the usual one, and with Hodge
bigrading\index{terms}{bigrading!Hodge type}
$$
\R(k)^{p,q} = \begin{cases} \R(k), \quad p=q=-k,\\ 0, \quad
\text{otherwise}. \end{cases} 
$$
For a pure $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure}
$V$ denote, as usual, by $V(k)$ the tensor product $V(k) = V \otimes
\R(k)$.

\punkt \label{w.1} We will also need special notation for another
common $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure},
which we now introduce. Note that for every complex $V$ be a complex
vector space the complex vector space $V \otimes_\R \C$ carries a
canonical $\R$-Hodge structure of weight\index{terms}{weight!of a
Hodge structure} $1$ with Hodge 
bigrading\index{terms}{bigrading!Hodge type}
given by
$$
V^{1,0} = V \subset V \otimes_\R \C \qquad\qquad V^{0,1} =
\overline{V} \otimes_\R \C. 
$$
In particular, $\C \otimes_\R \C$ carries a natural $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} of weight\index{terms}{weight!of a Hodge structure} $1$ with
Hodge numbers $h^{1,0} = h^{0,1} = 1$. Denote this Hodge structure
by $\W_1$.

\punkt Let $\langle W, \conj \rangle$ be a pure Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure}, and denote by $W_\R
\subset W$ the $\R$-vector subspace of elements preserved by
$\conj$. Define the {\em Weil operator\index{terms}{Weil operator}}
$C:W \to W$ by
$$
C = (\sqrt{-1})^{p-q}:W^{p,q} \to W^{q,p}.
$$
The operator $C:W \to W$ preserves the $\R$-Hodge structure, in
particular, the subspace $W_\R \subset W$.  On pure $\R$-Hodge
structures of weight\index{terms}{weight!of a Hodge structure} $0$ the Weil operator\index{terms}{Weil
operator} $C$ corresponds to the action of $-1 \in U(1) \subset
\C^*$ in the corresponding representation.

\punkt \label{Weil} For a pure Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} $W$ of weight\index{terms}{weight!of a Hodge structure} $i$ let
$$
j = C \circ \conj:\overline{W^{\cdot,1-\cdot}} \to W^{1-\cdot,\cdot}. 
$$
If $i$ is odd, in particular, if $i=1$, then $j \circ \overline{j} =
-\id$. Together with Lemma~\ref{explicit.eqvs} this gives the following.

\begin{lemma}\label{eqvs.hodge}
The category of equivariant quaternionic vector
spaces\index{terms}{quaternionic vector space!regular equivariant}
is equivalent to the 
category of pure $\R$-Hodge structures of weight\index{terms}{weight!of a Hodge structure} $1$.  
\end{lemma} 

\punkt Let $V$ be an equivariant quaternionic vector space, and let
$\langle W, \conj \rangle$ be $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} of weight\index{terms}{weight!of a Hodge structure} $1$
corresponding to $V$ under the equivalence of
Lemma~\ref{eqvs.hodge}. By definition the complex vector space $W$
is canonically isomorphic to the complex vector space $V_I$ with the
preferred\index{terms}{preferred complex structure} complex structure on $V$. It will be more convenient for
us to identify $W$ with the complementary\index{terms}{complementary complex structure} complex vector space $V_J$
by means of the standard isomorphism $H:V_J \to V_I$. The
multiplication map
$$
\Mult:V_I \otimes_\R \C \cong V \otimes_\C \h \to V_J
$$
is then a map of $\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge
structure}. The complex conjugation $\conj:W \to \overline{W}$ is
given by
\begin{equation}\label{i.conj}
\conj = C \circ H \circ j \circ H^{-1} = C \circ i:V_J \to \overline{V_J}, 
\end{equation}
where $i:V_J \to \overline{V_J}$ is the action of the element
$I(\sqrt{-1}) \subset \h$. 

\subsection{Weakly Hodge maps}\label{w.H.sub}\index{terms}{weakly
Hodge map}

\punkt Recall that the category of pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} of weight\index{terms}{weight!of a Hodge structure} $i$ is
equivalent to the category of pairs $\langle V, F^\cdot\rangle$ of a
real vector space $V$ and a decreasing filtration $F^\cdot$ on the
complex vector space $V_\C = V \otimes_\R \C$ satisfying
$$
V_\C = \bigoplus_{p+q=i}F^pV_\C \cap \overline{F^qV_\C}.
$$
The filtration $F^\cdot$ is called {\em the Hodge filtration}.  The
Hodge type bigrading\index{terms}{bigrading!Hodge type} and the Hodge
filtration are related by $V^{p,q} = F^pV_\C \cap
\overline{F^qV_\C}$ and $F^pV_\C = \oplus_{k \geq p}V^{k,i-k}$.

\punkt \label{weakly.hodge} Let $\langle V,F^\cdot\rangle$ and
$\langle W,F^\cdot \rangle$ be pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} of weight\index{terms}{weight!of a Hodge structure}s
$n$ and $m$ respectively. Usually maps of pure Hodge structures are
required to preserve the weight\index{terms}{weight!of a Hodge structure}s, so that $\Hom(V,W)=0$ unless
$n=m$. In this paper we will need the following weaker notion of
maps between pure $\R$-Hodge structures.

\defn An $\R$-linear map $f:V \to W$ is said to be {\em weakly
Hodge\index{terms}{weakly Hodge map}} if it preserves the Hodge
filtrations.

Equivalently, the complexified map $f:V_\C \to W_\C$ must decompose 
\begin{equation}\label{H.type}
f = \sum_{0 \leq p \leq m-n}f^{p,m-n-p},
\end{equation}
where the map $f^{p,m-n-p}:V_\C \to W_\C$ is of Hodge
type\index{terms}{bigrading!Hodge type} $(p,m-n-p)$. Note that this condition
is indeed weaker than the usual definition of a map of Hodge
structures: a weakly Hodge map $f:V \to W$ can be non-trivial if $m$
is strictly greater than $n$. If $m < n$, then $f$ must be trivial,
and if $m=n$, then weakly Hodge maps from $V$ to $W$ are the same as
usual maps of $\R$-Hodge structures.

\punkt We will denote by $\prehodge$ the category of pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} of arbitrary weight\index{terms}{weight!of a Hodge structure}
with weakly Hodge maps\index{terms}{weakly Hodge map} as
morphisms. For every integer $n$ let $\prehodge_n$ be the full
subcategory in $\prehodge$ consisting of pure $\R$-Hodge structures
of weight\index{terms}{weight!of a Hodge structure} $n$, and let $\prehodge_{\geq n}$ be the full subcategory
of $\R$-Hodge structures of weight\index{terms}{weight!of a Hodge structure} not less than $n$. Since weakly
Hodge maps between $\R$-Hodge structures of the same weight\index{terms}{weight!of a Hodge structure} are the
same as usual maps of $\R$-Hodge structures, the category
$\prehodge_n$ is the usual category of pure $\R$-Hodge structures of
weight\index{terms}{weight!of a Hodge structure} $n$.

\punkt \label{w.k} Let $\W_1 = \C \otimes_\R \C$ be the pure Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} of weight\index{terms}{weight!of a Hodge structure} $1$ with
Hodge numbers $h^{1,0}=h^{0,1}=1$, as in \ref{w.1}.  The diagonal
embedding $\C \to \C \otimes_\R \C$ considered as a map $w_1:\R \to
\W_1$ from the trivial pure $\R$-Hodge structure $\R$ of weight\index{terms}{weight!of a Hodge structure} $0$
to $\W_1$ is obviously weakly Hodge\index{terms}{weakly Hodge
map}. It decomposes $w_1 = w_1^{1,0} + w_1^{0,1}$ as in
\eqref{H.type}, and the components $w_1^{1,0}:\C \to \W_1^{1,0}$ and
$w_1^{0,1}:\C \to \W_1^{0,1}$ are isomorphisms of complex vectors
spaces. Moreover, for every pure $\R$-Hodge structure $V$ of weight\index{terms}{weight!of a Hodge structure}
$n$ the map $w_1$ induces a weakly Hodge map $w_1:V \to \W_1 \otimes
V$, and the components $w_1^{1,0}:V_\C \to V_\C \otimes \W_1^{1,0}$
and $w_1^{0,1}:V_\C \to V_\C \otimes \W_1^{0,1}$ are again
isomorphisms of complex vector spaces.

More generally, for every $k \geq 0$ let $\W_k = S^k\W_1$ be the
$k$-th symmetric power of the Hodge structure $\W_1$.  The space
$\W_k$ is a pure $\R$-Hodge structure of weight\index{terms}{weight!of a Hodge structure} $k$, with Hodge
numbers $h^{k,0} = h^{k-1,1} =\ldots = h^{0,k} = 1$. Let $w_k:\R \to
\W_k$ be the $k$-th symmetric power of the map $w_1:\R \to
\W_1$. For every pure $\R$-Hodge structure $V$ of weight\index{terms}{weight!of a Hodge structure} $n$ the map
$w_k$ induces a weakly Hodge map $w_k:V \to \W_k \otimes V$, and the
components 
$$
w_k^{p,q}:V_\C \to V_\C \otimes \W_k^{p,k-p}, \qquad 0 \leq p \leq k
$$
are isomorphisms of complex vector spaces. 

\punkt \label{w.k.uni} The map $w_k$ is a universal weakly Hodge
map\index{terms}{weakly Hodge map!universal} from a pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} $V$ of weight\index{terms}{weight!of a Hodge structure} $n$ to a
pure $\R$-Hodge structure of weight\index{terms}{weight!of a Hodge structure} $n+k$. More precisely, every
weakly Hodge map\index{terms}{weakly Hodge map} $f:V \to V'$ from
$V$ to a pure $\R$-Hodge structure $V'$ of weight\index{terms}{weight!of a Hodge structure} $n+k$ factors
uniquely through $w_k:V \to \W_k \otimes V$ by means of a map
$f':\W_k \otimes V \to V'$ preserving the pure $\R$-Hodge
structures. Indeed, $V_\C \otimes \W_k = \bigoplus_{0 \leq p \leq k}
V_\C \otimes \W_k^{p,k-p}$, and the maps $w_k^{p,k-p}:V_\C \to V_\C
\otimes \W_k^{p,k-p}$ are invertible.  Hence to obtain the desired
factorization it is necessary and sufficient to set
$$
f' = f^{p,k-p} \circ \left(w_k^{p,k-p}\right)^{-1}:V_\C \otimes
\W_k^{p,k-p} \to V_\C \to \V'_\C, 
$$
where $f = \sum_{0 \leq p \leq k}f^{p,k-p}$ is the Hodge
type\index{terms}{bigrading!Hodge type} decomposition
\eqref{H.type}. 

\punkt It will be convenient to reformulate the universal properties
of the maps $w_k$ as follows. By definition $\W_k = S^k\W_1$,
therefore the dual $\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge
structure} equal $\W_k^* = S^k\W_1^*$, and for every $n,k \geq 0$ we
have a canonical projection $\can:\W_n^* \otimes \W_k^* \to
\W_{n+k}^*$. For every pure $\R$-Hodge structure $V$ of weight\index{terms}{weight!of a Hodge structure} $k
\geq 0$ let $\Gamma(V) = V \otimes \W_k^*$.

\begin{lemma}\label{g.ex}
The correspondence $V \mapsto \Gamma(V)$ extends to a functor
$$
\Gamma:\prehodge_{\geq 0} \to \prehodge_0
$$ 
adjoint on the right to the canonical embedding $\prehodge_0
\hookrightarrow \prehodge_{\geq 0}$.
\end{lemma}

\proof Consider a weakly Hodge map\index{terms}{weakly Hodge map}
$f:V_n \to V_{n+k}$ from $\R$-Hodge structure $V_n$ of weight\index{terms}{weight!of a Hodge structure} $n$ to
a pure $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure}
$V_{n+k}$ of weight\index{terms}{weight!of a Hodge structure} $n+k$. By the universal property the map $f$
factors through the canonical map $w_k:V_n \to V_n \otimes \W_k$ by
means of a map $f_k:V_n \otimes \W_k \to V_{n+k}$. Let $f_k':V_k \to
V_{n+k} \otimes \W_k^*$ be the adjoint map, and let
\begin{align*}
\Gamma(f) = \can \circ f_k':&\Gamma(V_n) = V_n \otimes \W_k^* \to
V_{n+k} \otimes \W_n^* \otimes \W_k^* \to\\ 
\to &\Gamma(V_{n+k}) = V \otimes \W_{n+k}^*. 
\end{align*}
This defines the desired functor $\Gamma:\prehodge_{\geq 0} \to
\prehodge_0$. The adjointness is obvious. 
\endproof

\rem See Appendix for a more geometric description of the functor
$\Gamma:\prehodge_{\geq 0} \to \prehodge_0$.

\punkt \label{l.r} 
The complex vector space $\W_1 = \C \otimes_\R \C$ is equipped with
a canonical skew-symmetric trace pairing $\W_1 \otimes_\C \W_1 \to
\C$. Let $\gamma:\C \to \W_1^*$ be the map dual to $w_1:\R \to \W_1$
under this pairing. The map $\gamma$ is not weakly
Hodge\index{terms}{weakly Hodge map}, but it decomposes $\gamma =
\gamma^{-1,0} + \gamma^{0,-1}$ with respect to the Hodge type
bigrading\index{terms}{bigrading!Hodge type}. Denote $\gamma_l =
\gamma^{-1,0}$, $\gamma_r = \gamma^{0,-1}$. For every $0 \leq p \leq
k$ the symmetric powers of the maps $\gamma_l$, $\gamma_r$ give
canonical complex-linear embeddings
$$
\gamma_l, \gamma_r:\W_p^* \to \W_k^*. 
$$

\punkt The map $\gamma_l$ if of Hodge type\index{terms}{bigrading!Hodge type}
$(p-k,0)$, while $\gamma_r$ is of Hodge type $(0, p-k)$, and the
maps $\gamma_l$, $\gamma_r$ are complex conjugate to each
other. Moreover, they are each compatible with the natural maps
$\can:\W_p^* \otimes \W_q^*
\to \W_{p+q}^*$ in the sense that $\can \circ (\gamma_l \otimes
\gamma_l) = \gamma_l \circ \can$ and $\can \circ (\gamma_r \otimes
\gamma_r) = \gamma_r \circ \can$. For every $p$, $q$, $k$ such that
$p+q \geq k$ we have a short exact sequence
\begin{equation}\label{p+q}
\begin{CD}
0 @>>> \W_{p+q-k} @>{\gamma_r - \gamma_l}>> \W_p^* \oplus \W_q^*
@>{\gamma_l + \gamma_l}>> \W_k^* @>>> 0 
\end{CD}
\end{equation}
of complex vector spaces. We will need this exact sequence in
\ref{gamma.use}.

\punkt \label{gamma.tensor} The functor $\Gamma$ is, in general, not
a tensor functor. However, the canonical maps $\can:\W_n^* \otimes
\W_k^* \to \W_{n+k}^*$ define for every two pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} $V_1$, $V_2$ of
non-negative weight\index{terms}{weight!of a Hodge structure} a surjective map
$$
\Gamma(V_1) \otimes \Gamma(V_2) \to \Gamma(V_1 \otimes V_2). 
$$
These maps are functorial in $V_1$ and $V_2$ and commute with the
associativity and the commutativity morphisms. Moreover, for every
algebra $\A$ in the tensor category $\prehodge_{\geq 0}$ they turn
$\Gamma(\A)$ into an algebra in $\prehodge_0$.

\subsection{Polarizations}

\punkt \label{hyperherm}
Consider a quaternionic vector space $V$\index{terms}{quaternionic
vector space}, and let $h$ be a Euclidean
metric\index{terms}{metric!Euclidean}
on $V$.

\defn The metric $h$ is called {\em Qua\-ter\-nionic\--Her\-mi\-tian} if
for any algebra embedding $I:\C \to \h$ the metric $h$ is the real part of
an Hermitian metric on the complex vector space $V_I$.

Equivalently, a metric is Quaternionic-Hermitian
\index{terms}{metric!Quaternionic-Hermitian} if it is invariant
under the action of the group $SU(2) \subset \h$ of unitary
quaternions.

\punkt \label{hermhodge} Assume that the quaternionic vector space
$V$ is equivariant\index{terms}{quaternionic vector
space!equivariant}. Say that a metric $h$ on $V$ is {\em 
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge}} if it is
Qua\-ter\-nionic-\-Her\-mi\-tian and, in addition, invariant under
the $U(1)$-action on $V$.

Let $V_I$ be the vector space $V$ with the preferred\index{terms}{preferred complex structure} complex structure $I$,
and let 
$$
V_I = \bigoplus V^\cdot 
$$ 
and $j:\overline{V^\cdot} \to V^{1-\cdot}$ be as in
Lemma~\ref{explicit.eqvs}. The metric $h$ is
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge} if and only if
\begin{enumerate}
\item it is the real part of an Hermitian
metric\index{terms}{metric!Hermitian} on $V_I$, 
\item $h(V^p,V^q) = 0$ whenever $p \neq q$, and 
\item $h(j(a),b) = - h(a,j(b))$ for every $a,b \in V$. 
\end{enumerate}

\punkt Recall that a {\em polarization\index{terms}{polarization}}
$S$ on a pure $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge
structure} $W$ of weight\index{terms}{weight!of a Hodge structure}
$i$ is a bilinear form $S:W \otimes W \to \R(-i)$ which is a
morphism of pure Hodge structures and satisfies
\begin{align*}
S(a,b) &= (-1)^{i}S(b,a)\\
S(a,Ca) &> 0
\end{align*}
for every $a,b \in W$. (Here $C:W \to W$ is the Weil
operator\index{terms}{Weil operator}.)

\punkt \label{pol} Let $V$ be an equivariant quaternionic vector
space\index{terms}{quaternionic vector space!equivariant} equipped
with an Euclidean metric\index{terms}{metric!Euclidean} $h$, and let
$\langle W,
\conj \rangle$ be the pure $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} of
weight\index{terms}{weight!of a Hodge structure} $1$ 
associated to $V$ by Lemma~\ref{eqvs.hodge}. Recall that $W = V_J$
as a complex vector space. Assume that $h$ extends to an Hermitian
metric\index{terms}{metric!Hermitian} $h_J$ on $V_J$, and let $S:W
\otimes W \to \R(-1)$ be the bilinear form defined by
$$
S(a,b) = h(a,C\overline{b}), \qquad a,b \in W_\R \subset W.
$$
The form $S$ is a polarization\index{terms}{polarization} if and
only if the metric $h$ is
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge}. This gives a
one-to-one correspondence between the set of Hermitian-Hodge metrics
on $V$ and the set of polarizations on the Hodge structure $W$.

\punkt Let $W^*$ be the Hodge structure\index{terms}{Hodge structure@$\R$-Hodge
structure} of weight\index{terms}{weight!of a Hodge structure} $-1$
dual to $W$.  The sets of polarizations\index{terms}{polarization}
on $W$ and on $W^*$ are, of course, in a natural one-to-one
correspondence. It will be more convenient for us to identify the
set of Hermitian-Hodge metrics\index{terms}{metric!Hermitian-Hodge}
on $V$ with the set of polarizations on $W^*$ rather then on $W$.

Assume that the metric $h$ on the equivariant quaternionic
vector\index{terms}{quaternionic vector space!equivariant}
space $V$ is Hermitian-Hodge, and let $S \in \Lambda^2(W) \subset
\Lambda^2(V \otimes \C)$ be the corresponding
polarization\index{terms}{polarization}. Extend $h$ to an Hermitian
metric\index{terms}{metric!Hermitian} $h_I$ on the complex vector
space $V$ with the preferred\index{terms}{preferred complex
structure} complex structure $V_I$, and let
$$
\omega_I \in V_I \otimes \overline{V_I} \subset \Lambda^2(V
\otimes_\R \C) 
$$
be the imaginary part of the corresponding Hermitian
metric\index{terms}{metric!Hermitian} on the
dual space $V_I^*$. Let $i:V_J \to \overline{V_J}$ the action of the
element $I(\sqrt{-1}) \in \h$. By \eqref{i.conj} we have
$$
\omega_I(a,b) = h(a,i(b)) = h(a,C\overline{b}) = S(a,b) 
$$
for every $a,b \in V_J \subset V \otimes \C$. Since $\omega_I$ is
real, and $V \otimes \C = V_J \oplus \overline{V_J}$, the $2$-forms
$\omega_I$ and $S$ are related by
\begin{equation}\label{omega.and.Omega}
\omega_I = \half(S + \nu(S)), 
\end{equation}
where $\nu:V \otimes \C \to \overline{V \otimes \C}$ is the usual
complex conjugation. 

\section{Hodge bundles and quaternionic
manifolds}\label{hbqm.section}\index{terms}{Hodge bundle}\index{terms}{quaternionic manifold}

\subsection{Hodge bundles}\label{hb.sub}

\punkt Throughout the rest of the paper, our main tool in studying
hyperk\"ahler structures\index{terms}{hyperk\"ahler structures} on
smooth manifolds will be the equivalence between equivariant
quaternionic vector spaces\index{terms}{quaternionic vector
space!equivariant} and pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} of
weight\index{terms}{weight!of a Hodge structure} $1$, established in
Lemma~\ref{explicit.eqvs}.  In order to use it, we will need to
generalize this equivalence to the case of vector bundles over a
smooth manifold $M$, rather than just vector spaces. We will also
need to consider manifolds equipped with a smooth action of the
group $U(1)$, and we would like our generalization to take this
$U(1)$-action into account.

Such a generalization requires, among other things, an appropriate
notion of a vector bundle equipped with a pure $\R$-Hodge structure.
We introduce and study one version of such a notion in this section,
under the name of a Hodge bundle\index{terms}{Hodge bundle} (see
Definition~\ref{hodge.bundles}). 

\punkt Let $M$ be a smooth manifold equipped with a smooth
$U(1)$-action (or {\em a $U(1)$-manifold}, to simplify the
terminology), and let \index{terms}{involution $\iota$}$\iota:M \to
M$ be the action of the element $-1 \in U(1) \subset \C^*$.

\defn \label{hodge.bundles} 
An {\em Hodge bundle\index{terms}{Hodge bundle} of
weight\index{terms}{weight!of a Hodge bundle} $k$} on $M$ is a pair
$\langle\E,\conj\rangle$ of a $U(1)$-equi\-va\-ri\-ant complex
vector bundle $\E$ on $M$ and a $U(1)$-equivariant bundle map
\index{terms}{involution $\iota$}$\conj:\overline{\iota^*\E}(k) \to
\E$ satisfying $\conj \circ \iota^*\conj = \id$.  

Hodge bundles\index{terms}{Hodge bundle} of
weight\index{terms}{weight!of a Hodge bundle} $k$ over $M$ form a
tensor $\R$-linear additive category, denoted by $\prehodge_k(M)$.

\punkt\label{w.hodge} Let $\E$, $\F$ be two Hodge
bundles\index{terms}{Hodge bundle} on $M$ of weight\index{terms}{weight!of a Hodge bundle}s $m$ and
$n$. Say that a bundle map. or, more generally, a differential
operator $f:\E \to \F$ is {\em weakly Hodge\index{terms}{weakly
Hodge map}} if
\begin{enumerate}
\item \index{terms}{involution $\iota$}$f = \overline{\iota^*f}$, and 
\item there exists a decomposition $f = \sum_{0 \leq n-m} f_i$ with
$f_i$ being of degree $i$ with respect to the $U(1)$-equivariant
structure.  (In particular, $f=0$ unless $n \geq m$, and we always
have \index{terms}{involution $\iota$}$f_k = \overline{\iota^* f_{n-m-k}}$.)
\end{enumerate}

Denote by $\prehodge(M)$ the category of Hodge
bundles\index{terms}{Hodge bundle} of arbitrary weight\index{terms}{weight!of a Hodge bundle} on $M$, with
weakly Hodge bundle maps\index{terms}{weakly Hodge map} as
morphisms. For every $i$ the category $\prehodge_i(M)$ is a full
subcategory in $\prehodge(M)$.  Introduce also the category
$\prehodge^\D(M)$ with the same objects as $\prehodge(M)$ but with
weakly Hodge differential operators as morphisms. Both the
categories $\prehodge(M)$ and $\prehodge^\D(M)$ are additive
$\R$-linear tensor categories.

\punkt\label{H-type} For a weakly Hodge map\index{terms}{weakly
Hodge map} $f:\E \to \F$ call the canonical decomposition
$$
f = \sum_{0 \leq i \leq m-n} f_i 
$$
{\em the $H$-type\index{terms}{Hodge type1@$H$-type} decomposition}. 
\label{gamma.m}
For a Hodge bundle\index{terms}{Hodge bundle} $\E$ on $M$ of
non-negative weight\index{terms}{weight!of a Hodge bundle} $k$ let $\Gamma(\E) = \E \otimes \W_k^*$, where
$\W_k$ is the canonical pure $\R$-Hodges
structure\index{terms}{Hodge structure@$\R$-Hodge structure} introduced in
\ref{w.k}.  The universal properties of the Hodge structures $\W_k$
and Lemma~\ref{g.ex} generalize immediately to Hodge bundles. In
particular, $\Gamma$ extends to a functor $\Gamma:\prehodge_{\geq
0}(M) \to \prehodge_0(M)$ adjoint on the right to the canonical
embedding.

\punkt \label{u.trivial} If the $U(1)$-action on $M$ is trivial,
then Hodge bundles\index{terms}{Hodge bundle} of weight\index{terms}{weight!of a Hodge bundle} $i$ are the
same as real vector bundles $\E$ equipped with a Hodge type
bigrading\index{terms}{bigrading!Hodge type} $\E = \oplus_{p+q=i}
\E^{p,q}$ on the complexification $\E_\C = \E
\otimes_\R \C$. In particular, if $M=\PT$ is a single point, then
$\prehodge(M) \cong \prehodge^\D(M)$ is the category of pure
$\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge structure}. Weakly
Hodge bundle map\index{terms}{weakly Hodge map} are then the same as
weakly Hodge maps of $\R$-Hodge structures considered in
\ref{weakly.hodge}. (Thus the notion of a Hodge bundle is indeed a
generalization of the notion of a pure $\R$-Hodge structure.)

\punkt The categories of Hodge bundles\index{terms}{Hodge bundle}
are functorial in $M$, namely, for every smooth map $f:M_1 \to M_2$
of smooth $U(1)$-manifolds $M_1$, $M_2$ there exist pull-back
functors
\begin{align*}
f^*:\prehodge(M_1)    &\to \prehodge(M_1)   \\
f^*:\prehodge^\D(M_1) &\to \prehodge^\D(M_1).
\end{align*} 
In particular, let $M$ be a smooth $U(1)$-manifold and let $\pi:M
\to \PT$ be the canonical projection. Then every $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} $V$ of weight\index{terms}{weight!of a Hodge bundle} $i$
defines a constant Hodge bundle $\pi^*V$ on $M$, which we denote for
simplicity by the same letter $V$. Thus the trivial bundle $\R =
\Lambda^0(M) = \pi^*\R(0)$ has a natural structure of a Hodge bundle
of weight\index{terms}{weight!of a Hodge bundle} $0$.

\punkt \label{de.Rham} To give a first example of Hodge
bundles\index{terms}{Hodge bundle} and weakly Hodge
maps\index{terms}{weakly Hodge map}, consider a $U(1)$-manifold $M$
equipped with a $U(1)$-invariant almost complex structure $M_I$.
Let $\Lambda^i(M,\C) = \oplus \Lambda^{\cdot,i-\cdot}(M_I)$ be the
usual Hodge type\index{terms}{bigrading!Hodge type} decomposition of the
bundles $\Lambda^i(M,\C)$ of complex valued differential forms on
$M$. The complex vector bundles $\Lambda^{p,q}(M_I)$ are naturally
$U(1)$-equivariant. Let
$$
\conj:\Lambda^{p,q}(M_I) \to
\iota^*\overline{\Lambda^{q,p}(M_I)}
$$
\index{terms}{involution $\iota$}
be the usual complex conjugation, and introduce a $U(1)$-equivariant
structure on $\Lambda^\cdot(M,\C)$ by setting
$$
\Lambda^i(M,\C) = \bigoplus_{0 \leq j \leq i} \Lambda^{j,i-j}(M)(j). 
$$
The bundle $\Lambda^i(M,\C)$ with these $U(1)$-equivariant structure
and complex conjugation is a Hodge bundle of weight\index{terms}{weight!of a Hodge bundle} $i$ on $M$. The
de Rham differential $d_M$ is weakly Hodge, and the
$H$-type\index{terms}{Hodge type1@$H$-type} decomposition for $d_M$ is in this
case the usual Hodge type\index{terms}{bigrading!Hodge type} decomposition $d
= \6 + \bar\6$.

\punkt \rem Definition~\ref{hodge.bundles} is somewhat technical. It
can be heuristically rephrased as follows. For a complex vector
bundle $\E$ on $M$ the space of smooth global section
$C^\infty(M,\E)$ is a module over the algebra $C^\infty(M,\C)$ of
smooth $\C$-valued functions on $M$, and the bundle $\E$ is
completely defined by the module $C^\infty(M,\E)$. The $U(1)$-action
on $M$ induces a representation of $U(1)$ on the algebra
$C^\infty(M,\C)$. Let $\nu:C^\infty(M,\C) \to C^\infty(M,\C)$ be
composition of the complex conjugation and the map
$\iota\index{terms}{involution $\iota$}^*:C^\infty(M,\C) \to
C^\infty(M,\C)$. The map $\nu$ is an 
anti-complex involution; together with the $U(1)$-action it defines
a pure $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure} of
weight\index{terms}{weight!of a Hodge structure} $0$ on the algebra $C^\infty(M,\C)$. Giving a weight\index{terms}{weight!of a Hodge bundle} $i$
Hodge bundle\index{terms}{Hodge bundle} structure on $\E$ is then
equivalent to giving a weight\index{terms}{weight!of a Hodge structure} $i$ pure $\R$-Hodge structure on the
module $C^\infty(M,\E)$ such that the multiplication map
$$
C^\infty(M,\C) \otimes C^\infty(M,\E) \to C^\infty(M,\E)
$$
is a map of $\R$-Hodge structures.  

\subsection{Equivariant quaternionic
manifolds}\index{terms}{quaternionic manifold!equivariant} 

\punkt We now turn to our main subject, namely, various
dif\-feren\-ti\-al\--ge\-o\-met\-ric structures on smooth manifolds
associated to actions of the algebra $\h$ of
quaternions\index{terms}{quaternion algebra}. 

\punkt\defn A smooth manifold $M$ is called {\em
quaternionic\index{terms}{quaternionic manifold}} if it
is equipped with a smooth action of the algebra $\h$ on its
cotangent bundle $\Lambda^1(M)$. 

Let $M$ be a quaternionic manifold\index{terms}{quaternionic
manifold}.  Every algebra embedding $J:\C \to \h$ defines by
restriction an almost complex structure on the manifold $M$. Denote
this almost complex structure by $M_J$.

\punkt Assume that the manifold $M$ is equipped with a smooth action of the
group $U(1)$, and consider the standard action of $U(1)$ on the vector
space $\h$. Call the quaternionic
structure\index{terms}{quaternionic structure} and the $U(1)$-action on $M$
{\em compatible} if the action map 
$$
\h \otimes_\R \Lambda^1(M) \to \Lambda^1(M)
$$
is $U(1)$-equivariant. 

Equivalently, the quaternionic structure\index{terms}{quaternionic
structure} and the $U(1)$-action are compatible if the action
preserves the almost complex structure $M_I$, and the action map
$$
\h \otimes_\C \Lambda^{1,0}(M_I) \to \Lambda^{1,0}(M_I) 
$$
is $U(1)$-equivariant. 

\punkt \defn A quaternionic manifold\index{terms}{quaternionic
manifold} $M$ equipped with a compatible smooth 
$U(1)$-action is called {\em an equivariant quaternionic
manifold\index{terms}{quaternionic manifold!equivariant}}. 

For a $U(1)$-equivariant complex vector bundle $\E$ on $M$ denote by
$\E(k)$ the bundle $\E$ with $U(1)$-equivariant structure twisted by the
$1$-dimensional representation of weight\index{terms}{weight!of an $U(1)$-action} $k$.  Lemma~\ref{explicit.eqvs}
immediately gives the following.

\begin{lemma}\label{explicit.qm}
The category of quaternionic manifolds\index{terms}{quaternionic
manifold} is equivalent to the category of pairs $\langle M_I, j
\rangle$ of an almost complex manifold $M_I$ and a $\C$-linear
$U(1)$-equivariant smooth map $j:\Lambda^{0,1}(M_I)(1) \to
\Lambda^{0,1}(M_I)$ satisfying $j \circ \overline{j} = -\id$. 
\end{lemma}

\subsection{Quaternionic manifolds and Hodge
bundles}\index{terms}{quaternionic manifold}\index{terms}{Hodge bundle}

\punkt Let $M$ be a smooth $U(1)$-manifold.  Recall that we have
introduced in Subsection~\ref{hb.sub} a notion of a Hodge
bundle\index{terms}{Hodge bundle} on $M$. Hodge bundles arise
naturally in the study of quaternionic
structures\index{terms}{quaternionic structure} on $M$ in the
following way. Define a {\em quaternionic
bundle\index{terms}{quaternionic bundle}} on $M$ as a real
vector bundle $\E$ equipped with a left action of the algebra $\h$,
and let $\Bun(M,\h)$ be the category of smooth quaternionic vector
bundles on the manifold $M$. Let also $\Bun^{U(1)}(M,\h)$ be the
category of smooth quaternionic bundles $\E$ on $M$ equipped with a
$U(1)$-equivariant structure such that the $\h$-action map $\h \to
\Eend(\E)$ is $U(1)$-equivariant. Lemma~\ref{eqvs.hodge} immediately
generalizes to give the following.

\begin{lemma}\label{eqb.hodge}
The category $\Bun^{U(1)}(M,\h)$ is equivalent to the category of Hodge
bundles\index{terms}{Hodge bundle} of weight\index{terms}{weight!of a Hodge bundle} $1$ on $M$. 
\end{lemma}

\punkt Note that if the $U(1)$-manifold $M$ is equipped with an almost complex
structure, then the decomposition $\h = \overline{\C} \oplus I(\C)$ (see
\ref{h=c+c}) induces an isomorphism 
$$
\can:\h \otimes_{I(\C)} \Lambda^{0,1} \cong \Lambda^{1,0}(M) \oplus
\Lambda^{0,1}(M) \cong \Lambda^1(M,\C).
$$
The weight\index{terms}{weight!of a Hodge bundle} $1$ Hodge bundle\index{terms}{Hodge bundle} structure on
$\Lambda^1(M,\C)$ corresponding to the natural quaternionic
structure\index{terms}{quaternionic structure} on $\h
\otimes_{I(\C)} \Lambda^{0,1}(M)$ is the same as the one considered
in \ref{de.Rham}. 

\punkt Assume now that the smooth $U(1)$-manifold $M$ is equipped
with a compatible quaternionic structure, and let $M_I$ be the
preferred\index{terms}{preferred complex structure} almost complex structure on $M$.  Since $M_I$ is preserved
by the $U(1)$-action on $M$, the complex vector bundle
$\Lambda^{0,1}(M_I)$ of $(0,1)$-forms on $M_I$ is naturally
$U(1)$-equivariant.

The quaternionic structure\index{terms}{quaternionic structure} on
$\Lambda^1(M)$ induces by Lemma~\ref{eqb.hodge} a
weight\index{terms}{weight!of a Hodge bundle}-$1$ Hodge
bundle\index{terms}{Hodge bundle} structure on
$\Lambda^{0,1}(M_I)$. The corresponding $U(1)$-action on
$\Lambda^{0,1}(M_I)$ is induced by the action on $M_I$, and the real
structure map
$$
\conj:\Lambda^{1,0}(M_I)(1) \to \Lambda^{0,1}(M_I)
$$
is given by $\conj = \sqrt{-1} \left(
\iota\index{terms}{involution $\iota$}^* \circ j \right)$. (Here $j$
is induced by quaternionic structure\index{terms}{quaternionic
structure}, as in Lemma~\ref{explicit.qm}).

\punkt \label{lambda_j=lambda_i}
Let $M_J$ be the complementary\index{terms}{complementary complex
structure} almost complex structure on the 
equi\-va\-ri\-ant quaternionic manifold\index{terms}{quaternionic
manifold!equivariant} $M$. Recall that in 
\ref{complementary}\index{terms}{complementary complex structure} we
have defined for every equivariant quaternionic vector
space\index{terms}{quaternionic vector space!equivariant} $V$ the
standard isomorphism $H:V_J \to V_I$. This construction can be
immediately generalized to give a complex bundle isomorphism
$$
H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I). 
$$
Let $P:\Lambda^1(M,\C) \to \Lambda^{0,1}(M_J)$ be the natural projection,
and let $\Mult:\h \otimes_{I(\C)} \Lambda^{0,1}(M_I) \to
\Lambda^{0,1}(M_I)$ be the action map. By definition the diagram 
$$
\begin{CD}
\Lambda^1(M,\C)    @>{\can}>> \h \otimes_{I(\C)} \Lambda^{0,1}(M_I) \\
 @V{P}VV                       @V{\Mult}VV                          \\
\Lambda^{0,1}(M_J) @>{H}>>    \Lambda^{0,1}(M_I)
\end{CD}
$$
is commutative. Since the map $\Mult$ is compatible with the Hodge
bundle\index{terms}{Hodge bundle} structures, so is the projection
$P$.

\rem This may seems paradoxical, since the complex conjugation
$\conj$ on $\Lambda^1(M,\C)$ does not preserve $\Ker P =
\Lambda^{0,1}(M_J)$. However, under our definition of a Hodge
bundle\index{terms}{Hodge bundle} the conjugation on
$\Lambda^1(M,\C)$ is $\iota\index{terms}{involution $\iota$}^* \circ
\conj$ rather than $\conj$. Both 
$\conj$ and $\iota\index{terms}{involution $\iota$}^*$ interchange
$\Lambda^{1,0}(M_J)$ and 
$\Lambda^{0,1}(M_J)$.

\punkt The standard isomorphism $H:\Lambda^{0,1}(M_J) \to
\Lambda^{0,1}(M_I)$ does not commute with the Dolbeult
differentials. They are,
however, related by means of the Hodge bundle\index{terms}{Hodge bundle} structure on $\Lambda^{0,1}(M_I)$. Namely, we have the
following.

\begin{lemma}\label{d=d_0+d_0}
The Dolbeult differential
$D:\Lambda^0(M,\C) \to \Lambda^{0,1}(M_I)$ for the almost complex
structure $M_J$ is weakly Hodge{weakly Hodge map}. The
$U(1)$-invariant component $D_0$ in the
$H$-type\index{terms}{Hodge type1@$H$-type} decomposition $D = D_0 +
\overline{D_0}$ of the map $D$ coincides with the Dolbeult
differential for the almost
complex structure $M_I$.
\end{lemma}

\proof The differential $D$ is the composition $D = P \circ d_M$ of
the de Rham differential $d_M:\Lambda^0(M,\C) \to \Lambda^1(M,\C)$
with the canonical projection $P$. Since both $P$ and $d_M$ are
weakly Hodge\index{terms}{weakly Hodge map}, so is $D$. The rest
follows from the construction of the standard isomorphism $H$.
\endproof

\subsection{Holonomic derivations}

\punkt Let $M$ be a smooth $U(1)$-manifold. In order to give a
Hodge-theoretic description of the set of all equivariant
quaternionic structures\index{terms}{quaternionic structure} on
$M$, it is convenient to work not with various complex structures on
$M$, but with associated Dolbeult
differentials. To do this,
recall the following universal property of the cotangent bundle
$\Lambda^1(M)$.

\begin{lemma}\label{universal}
Let $M$ be a smooth manifold, and let $\E$ be a complex vector bundle on
$M$. Every differential operator $\6:\Lambda^0(M,\C) \to \E$ which is a
derivation with respect to the algebra structure on $\Lambda^0(M,\C)$
factors uniquely through the de Rham differential $d_M:\Lambda^0(M,\C) \to
\Lambda^1(M,\C)$ by means of a bundle map $P:\Lambda^1(M,\C) \to \E$. 
\end{lemma}

\punkt \label{holonomic}
We first use this universal property to describe almost complex structures.
Let $M$ be a smooth manifold equipped with a complex vector bundle $\E$.

\defn A derivation $D:\Lambda^0(M,\C) \to \E$ is called {\em
holonomic\index{terms}{holonomic derivation}} if 
the associated bundle map $P:\Lambda^1(M,\C) \to \E$ induces an isomorphism
of the subbundle $\Lambda^1(M,\R) \subset \Lambda^1(M,\C)$ of real
$1$-forms with the real vector bundle underlying $\E$. 

By Lemma~\ref{universal} the correspondence 
$$
M_I \mapsto \left\langle \Lambda^{0,1}(M_I), \bar\6 \right\rangle
$$
identifies the set of all almost complex structures $M_I$ on $M$ 
with the set of all pairs $\langle \E, D \rangle$ of
a complex vector bundle $\E$ and a holonomic\index{terms}{holonomic
derivation} derivation $D:\Lambda^0(M,\C) \to \E$. 

\punkt Assume now that the smooth manifold $M$ is equipped with smooth
action of the group $U(1)$. Then we have the following. 

\begin{lemma}\label{qm.hodge}
Let $\E$ be a weight\index{terms}{weight!of a Hodge bundle} $1$
Hodge bundle\index{terms}{Hodge bundle} on the smooth
$U(1)$-manifold $M$, and let 
$$
D:\Lambda^0(M,\C) \to \E
$$ 
be a weakly Hodge holonomic\index{terms}{holonomic derivation}
derivation. There exists a unique $U(1)$\--equi\-va\-ri\-ant quaternionic
structure\index{terms}{quaternionic structure} on $M$ such that
$\E \cong \Lambda^{0,1}(M_J)$ and $D$ is the Dolbeult
differential 
for the complementary\index{terms}{complementary complex structure} almost complex structures $M_J$ on $M$.
\end{lemma}

\proof Since the derivation $M$ is holonomic\index{terms}{holonomic
derivation}, it induces an almost complex structure $M_J$ on $M$. To
construct an almost complex structure $M_I$
complementary\index{terms}{complementary complex structure} to
$M_J$, consider the $H$-type\index{terms}{Hodge type1@$H$-type} decomposition $D
= D_0 + \overline{D_0}$ of the derivation $D:\Lambda^0(M,\C) \to \E$
(defined in \ref{H-type}).

The map $D_0$ is also a derivation. Moreover, it is
holonomic\index{terms}{holonomic derivation}. Indeed, by 
dimension count it is enough to prove that the associated bundle map
$P:\Lambda^1(M,\R) \to \E$ is injective. Since the bundle $\Lambda^1(M,\R)$
is generated by exact $1$-forms, it is enough to prove that any real valued
function $f$ on $M$ with $D_0f = 0$ is constant. However, since $D$ is
weakly Hodge\index{terms}{weakly Hodge map},
$$
Df = D_0 f + \overline{D_0} f = D_0 f + \overline{ D_0 \overline{f} } = D_0
f + \overline{ D_0 f}, 
$$
hence $D_0f = 0$ if and only if $Df=0$. Since $D$ is
holonomic\index{terms}{holonomic derivation}, $f$ is indeed 
constant. 

The derivation $D_0$, being holonomic\index{terms}{holonomic
derivation}, is the Dolbeult 
differential for an almost
complex structure $M_I$ on $M$.  Since $D_0$ is by definition
$U(1)$-equivariant, the almost complex structure $M_I$ is
$U(1)$-invariant. Moreover, $\E \cong \Lambda^{0,1}(M_I)$ as
$U(1)$-equivariant complex vector bundles.  By Lemma~\ref{eqb.hodge}
the weight\index{terms}{weight!of a Hodge bundle} $1$ Hodge
bundle\index{terms}{Hodge bundle} structure on 
$\E$ induces an equivariant quaternionic
bundle\index{terms}{quaternionic bundle} structure on $\E$
and, in turn, a structure of an equivariant quaternionic
manifold\index{terms}{quaternionic manifold!equivariant} on
$M$. The almost complex structure $M_I$ coincides by definition with
the preferred\index{terms}{preferred complex structure} almost complex structure.

It remains to notice that by Lemma~\ref{d=d_0+d_0} the Dolbeult
differential  
$\bar\6_J$ for the complementary\index{terms}{complementary complex structure} almost complex structure on $M$ indeed equals 
$D = D_0 + \overline{D_0}$. 
\endproof 

Together with Lemma~\ref{d=d_0+d_0}, this shows that the set of
equivariant quaternionic structures\index{terms}{quaternionic
structure} on the $U(1)$-manifold $M$ is naturally bijective to the
set of pairs $\langle \E, D \rangle$ of a weight\index{terms}{weight!of a Hodge bundle} $1$ Hodge bundle\index{terms}{Hodge bundle} $\E$
on $M$ and a weakly Hodge\index{terms}{weakly Hodge map}
holonomic\index{terms}{holonomic derivation} derivation
$D:\Lambda^0(M,\C) \to \E$.

\section{Hodge manifolds}\index{terms}{Hodge manifold}

\subsection{Integrability}

\punkt There exists a notion of integrability for quaternionic
manifolds\index{terms}{quaternionic manifold}
analogous to that for the almost complex ones.  

\defn A quaternionic manifold\index{terms}{quaternionic manifold}
$M$ is called {\em hypercomplex\index{terms}{hypercomplex manifold}}
if for two complementary\index{terms}{complementary complex structure} algebra embeddings $I,J:\C \to \h$ the
almost complex structures $M_I,M_J$ are integrable.

In fact, if $M$ is hypercomplex, then $M_I$ is integrable for any algebra
embedding $I:\C \to \h$. For a proof see, e.g., \cite{K}.

\punkt When a quaternionic manifold\index{terms}{quaternionic
manifold} $M$ is equipped with a compatible $U(1)$-action, there
exist a natural choice for a pair of almost complex structures on
$M$, namely, the preferred\index{terms}{preferred complex structure} and the complementary\index{terms}{complementary complex structure} one.

\defn An equivariant quaternionic manifold\index{terms}{quaternionic
manifold!equivariant} $M$ is called a {\em Hodge
manifold\index{terms}{Hodge manifold}} if both the
preferred\index{terms}{preferred complex structure} and the
equivariant almost complex structures $M_I$, $M_J$ are integrable.

Hodge manifolds\index{terms}{Hodge manifold} are the main object of
study in this paper.

\punkt \label{standard} There exists a simple Hodge-theoretic
description of Hodge manifolds\index{terms}{Hodge manifold} based on
Lemma~\ref{qm.hodge}. To give it (see
Proposition~\ref{explicit.hodge}), consider an equivariant
quaternionic manifold\index{terms}{quaternionic
manifold!equivariant} $M$, and let $M_J$ and $M_I$ be the
complementary\index{terms}{complementary complex structure} and the
preferred\index{terms}{preferred complex structure} complex
structures on $M$. The weight\index{terms}{weight!of a Hodge
bundle} $1$ Hodge bundle\index{terms}{Hodge bundle} structure on
$\Lambda^{0,1}(M_J)$ induces a weight\index{terms}{weight!of a Hodge
bundle} $i$ Hodge bundle structure on the bundle
$\Lambda^{0,i}(M_J)$ of $(0,i)$-forms on $M_J$. The standard
identification $H:\Lambda^{0,1}(M_J) \to
\Lambda^{0,1}(M_I)$ given in \ref{lambda_j=lambda_i} extends
uniquely to an algebra isomorphism $H:\Lambda^{0,i}(M_J) \to
\Lambda^{0,i}(M_I)$.

Let $D:\Lambda^{0\cdot}(M_J) \to \Lambda^{0,\cdot+1}(M_J)$ be the Dolbeault
differential for the almost complex manifold $M_J$.

\begin{lemma}\label{yet.another.lemma} 
The equivariant quaternionic manifold\index{terms}{quaternionic
manifold!equivariant} $M$ is 
Hodge\index{terms}{Hodge manifold} if and only if
the following holds. 
\begin{enumerate}
\item $M_J$ is integrable, that is, $D \circ D = 0$, and 
\item the differential $D:\Lambda^{0,i}(M_J) \to \Lambda^{0,i+1}(M_J)$
is weakly Hodge\index{terms}{weakly Hodge map} for every $i \geq 0$. 
\end{enumerate}
\end{lemma}

\proof Assume first that the conditions \thetag{i}, \thetag{ii}
hold. Condition \thetag{i} means that the complementary\index{terms}{complementary complex structure} almost complex
structure $M_J$ is integrable. By \thetag{ii} the map $D$ is weakly
Hodge\index{terms}{weakly Hodge map}.

Let $D = D_0 + \overline{D_0}$ be the
$H$-type\index{terms}{Hodge type1@$H$-type} decomposition. The map $D_0$ is an
algebra derivation of $\Lambda^{0,\cdot}(M_I)$. Moreover, by
Lemma~\ref{d=d_0+d_0} the map $D_0:\Lambda^0(M,\C) \to
\Lambda^{0,1}(M_J)$ is the Dolbeault differential $\bar\6_I$ for the
preferred\index{terms}{preferred complex structure} almost complex structure $M_I$ on $M$. (Or, more
precisely, is identified with $\bar\6_I$ under the standard
isomorphism $H$.) But the Dolbeult
differential admits at most one
extension to a derivation of the algebra
$\Lambda^{0,\cdot}(M_J)$. Therefore $D_0 = \bar\6_I$ everywhere.

The composition $D_0 \circ D_0$ is the $(2,0)$-component in the
$H$-type\index{terms}{Hodge type1@$H$-type} decomposition of the map $D \circ
D$. Since $D \circ D = 0$,
$$
D_0 \circ D_0 = \bar\6_I \circ \bar\6_I = 0.
$$
Therefore the preferred\index{terms}{preferred complex structure} complex structure $M_I$ is also integrable, and the
manifold $M$ is indeed Hodge\index{terms}{Hodge manifold}. 

Assume now that $M$ is Hodge. The canonical projection $P:\Lambda^1(M,\C)
\to \Lambda^{0,1}(M_J)$ extends then to a multiplicative projection 
$$
P:\Lambda^\cdot(M,\C) \to \Lambda^{0,\cdot}(M_J)
$$
from the de Rham complex\index{terms}{de Rham complex} of the
complex manifold $M_I$ to the Dolbeault complex of the complex
manifold $M_J$.  The map $P$ is surjective and weakly
Hodge\index{terms}{weakly Hodge map}, moreover, it commutes with the
differentials. Since the de Rham differential preserves the
pre-Hodge structures, so does the Dolbeault differential $D$.
\endproof

\punkt Lemma~\ref{yet.another.lemma} and Lemma~\ref{qm.hodge} together
immediately give the following. 

\begin{prop}\label{explicit.hodge} 
The category of Hodge manifolds\index{terms}{Hodge manifold} is
equivalent to the category of triples $\langle M, \E, D \rangle$ of
a smooth $U(1)$-manifold $M$, a weight\index{terms}{weight!of a Hodge bundle} $1$ Hodge
bundle\index{terms}{Hodge bundle} $\E$ on $M$, and a weakly
Hodge\index{terms}{weakly Hodge map} algebra derivation
$$
D = D^\cdot:\Lambda^\cdot(\E) \to \Lambda^{\cdot+1}(\E)
$$
such that $D \circ D = 0$, and the first component 
$$
D^0:\Lambda^0(M,\C) = \Lambda^0(\E) \to \E = \Lambda^1(\E)
$$
is holonomic\index{terms}{holonomic derivation} in the sense of
\ref{holonomic}. 
\end{prop}

\subsection{The de Rham complex of a Hodge
manifold}\index{terms}{Hodge manifold}\index{terms}{de Rham complex}

\punkt Let $M$ be a Hodge manifold. In this subsection we study in
some detail the de Rham complex\index{terms}{de Rham complex}
$\Lambda^\cdot(M,\C)$ of the manifold $M$, in order to obtain
information necessary for the discussion of metrics on $M$ given in
the Subsection~\ref{polarization}. The reader is advised to skip
this subsection until needed.

\punkt Let $\Lambda^{0,\cdot}(M_J)$ be the Dolbeault complex for the
complementary\index{terms}{complementary complex structure} complex structure $M_J$ on $M$. By
Proposition~\ref{explicit.hodge} the complex vector bundle
$\Lambda^{0,i}(M_J)$ is a Hodge bundle\index{terms}{Hodge bundle} of
weight\index{terms}{weight!of a Hodge bundle} $i$ on $M$, and the Dolbeult
differential
$D:\Lambda^{0,\cdot}(M_J) \to \Lambda^{0,\cdot+1}(M_J)$ is weakly
Hodge\index{terms}{weakly Hodge map}. Therefore $D$ admits an
$H$-type\index{terms}{Hodge type1@$H$-type} decomposition $D = D_0 + \overline{D_0}$.

\punkt Consider the de Rham complex\index{terms}{de Rham complex}
$\Lambda^i(M,\C)$ of the smooth manifold $M$.  Let $\Lambda^i(M,\C)
= \oplus_{p+q} \Lambda^{p,q}(M_J)$ be the Hodge
type\index{terms}{bigrading!Hodge type} decomposition
for the complementary\index{terms}{complementary complex structure} complex structure $M_J$ on $M$, and let
$\nu:\Lambda^{p,q}(M_J) \to \overline{\Lambda^{q,p}(M_J)}$ be the
usual complex conjugation. Denote also
$$
f^\nu = \nu \circ f \circ \nu^{-1}
$$
for any map $f:\Lambda^\cdot(M,\C) \to \Lambda^\cdot(M,\C)$. 

Let $d_M:\Lambda^\cdot(M,\C) \to \Lambda^{\cdot+1}(M,\C)$ be the de
Rham differential, and let $d_M = D + D^\nu$ be the Hodge
type\index{terms}{bigrading!Hodge type} decomposition for the complementary\index{terms}{complementary complex structure}
complex structure $M_J$ on $M$.  Since the Dolbeult
differential, in turn, equals
$D = D_0 +
\overline{D_0}$, we have 
$$
d_M = D_0 + \overline{D_0} + D_0^\nu + \overline{D_0}^\nu.
$$ 


\punkt \label{6_I}
Let $\bar\6_I:\Lambda^\cdot(M,\C) \to \Lambda^{\cdot+1}(M,\C)$ be the
Dolbeult differential for the preferred\index{terms}{preferred complex structure} complex structure $M_I$ on $M$. As
shown in the proof of Lemma~\ref{yet.another.lemma}, the $(0,1)$-component
of the differential $\bar\6_I$ with respect to the complex structure $M_J$
equals $D_0$. Therefore the $(1,0)$-component of the complex-conjugate map
$\6_I = \bar\6_I^\nu$ equals $D_0^\nu$. Since $d_M = \bar\6_I + \6_I$, we
have
\begin{align*}
\bar\6_I &= D_0 + \overline{D_0}^\nu\\
\6_I &= \overline{D_0} + D_0^\nu
\end{align*}
 
\punkt \label{6_I^H}
The standard isomorphism $H:\Lambda^{0,1}(M_J) \to \Lambda^{0,1}(M_I)$
introduced in \ref{standard} extends uniquely to a bigraded algebra
isomorphism $H:\Lambda^{\cdot,\cdot}(M_J) \to
\Lambda^{\cdot,\cdot}(M_I)$. By definition of the map $H$, on
$\Lambda^0(M,\C)$ we have
\begin{align}\label{identities}
\begin{split}
\bar\6_I  &= H \circ D_0 \circ H^{-1}\\
\6_I      &= H \circ D_0^\nu \circ H^{-1}\\
d_M       &= \6_I + \bar\6_I = H \circ (D_0 + D_0^\nu) H^{-1}. 
\end{split}
\end{align}
The right hand side of the last identity is the algebra derivation
of the de Rham complex\index{terms}{de Rham complex}
$\Lambda^\cdot(M,\C)$. Therefore, by Lemma~\ref{universal} it holds
not only on $\Lambda^0(M,\C)$, but on the whole
$\Lambda^\cdot(M,\C)$. The Hodge type\index{terms}{bigrading!Hodge type}
decomposition for the preferred\index{terms}{preferred complex structure} complex structure $M_I$ then shows
that the first two identities also hold on the whole de Rham complex
$\Lambda^\cdot(M,\C)$.

\punkt Let now $\xi = I(\sqrt{-1}):\Lambda^{0,1}(M_J) \to
\Lambda^{1,0}(M_J)$ be the operator corresponding to the preferred\index{terms}{preferred complex structure}
almost complex structure $M_I$ on $M$. Let also $\xi = 0$ on
$\Lambda^0(M,\C)$ and $\Lambda^{1,0}(M_J)$, and extend $\xi$ to a
derivation $\xi:\Lambda^{\cdot,\cdot}(M_J) \to
\Lambda^{\cdot-1,\cdot+1}(M_J)$ by the Leibnitz
rule\index{terms}{Leibnitz rule}. We finish this subsection with the
following simple fact.

\begin{lemma}\label{xi.lemma}
On $\Lambda^{\cdot,0}(M_J) \subset \Lambda^\cdot(M,\C)$ we have
\begin{align}\label{xi.eq}
\begin{split}
\xi \circ D_0 + D_0 \circ \xi &= \overline{D}_0^\nu\\
\xi \circ \overline{D}_0 + \overline{D}_0 \circ \xi &= -D_0^\nu. 
\end{split}
\end{align}
\end{lemma}

\proof It is easy to check that both identities hold on
$\Lambda^0(M,\C)$. But both sides of these identities are algebra
derivations of $\Lambda^{\cdot,0}(M_J)$, and the right hand sides
are holonomic\index{terms}{holonomic derivation} in the sense of
\ref{holonomic}. Therefore by Lemma~\ref{universal} both identities
hold on the whole $\Lambda^{\cdot,0}(M_J)$.
\endproof

\subsection{Polarized Hodge
manifolds}\label{polarization}\index{terms}{Hodge manifold}

\punkt Let $M$ be a quaternionic manifold\index{terms}{quaternionic
manifold!equivariant}. A Riemannian 
metric\index{terms}{metric!Riemannian} $h$ on $M$
is called {\em Qua\-ter\-ni\-onic-\-Her\-mi\-ti\-an} if for every point $m
\in M$ the induced metric $h_m$ on the tangent bundle $T_mM$ is
Qua\-ter\-ni\-onic\--Her\-mi\-ti\-an
\index{terms}{metric!Quaternionic-Hermitian} 
in the sense of \ref{hyperherm}.  

\defn A {\em hyperk\"ahler manifold\index{terms}{hyperk\"ahler
manifold}} is a hypercomplex manifold\index{terms}{hypercomplex
manifold} $M$ equipped with a Quaternionic-Hermitian
metric\index{terms}{metric!Quaternionic-Hermitian} $h$ which is
K\"ahler for both integrable almost complex structures $M_I$, $M_J$
on $M$.

\rem In the usual definition (see, e.g., \cite{Bes}) the integrability of
the almost complex structures $M_I$, $M_J$ is omitted, since it is
automatically implied by the K\"ahler condition.

\punkt Let $M$ be a Hodge manifold\index{terms}{Hodge manifold}
equipped with a Riemannian metric\index{terms}{metric!Riemannian}
$h$.  The metric $h$ is called {\em
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge}} if it is
Quaternionic-Hermitian\index{terms}{metric!Quaternionic-Hermitian}
and, in addition, invariant under the $U(1)$-action on $M$.

\defn Say that the manifold $M$ is {\em polarized} by the
Hermitian-Hodge metric\index{terms}{metric!Hermitian-Hodge} $h$ if
$h$ is not only
Quaternionic-Hermitian\index{terms}{metric!Quaternionic-Hermitian},
but also hyperk\"ahler\index{terms}{metric!hyperk\"ahler}.

\punkt \label{positive} Let $M$ be a Hodge
manifold\index{terms}{Hodge manifold}. By
Proposition~\ref{explicit.hodge} the holomorphic cotangent bundle
$\Lambda^{1,0}(M_J)$ for the
complementary\index{terms}{complementary complex structure} complex
structure $M_J$ on $M$ is a Hodge bundle\index{terms}{Hodge bundle}
of weight\index{terms}{weight!of a Hodge bundle} $1$. The
holomorphic tangent bundle $\Theta(M_J)$ is therefore a Hodge bundle
of weight\index{terms}{weight!of a Hodge bundle} $-1$.  By \ref{pol}
the set of all Hermitian-Hodge
metrics\index{terms}{metric!Hermitian-Hodge} $h$ on $M$ is in
natural one-to-one correspondence with the set of all
polarizations\index{terms}{polarization} on the Hodge bundle
$\Theta(M_J)$.

Since $\theta(M)$ is of odd weight\index{terms}{weight!of an
$U(1)$-action}, its polarizations\index{terms}{polarization} are
skew-symmetric as bilinear forms and correspond therefore to smooth
$(2,0)$-forms on the complex manifold $M_J$. A $(2,0)$-form $\Omega$
defines a polarization on $\Theta(M_J)$ if and only if $\Omega \in
C^\infty(M,\Lambda^{2,0}(M_J))$ considered as a map
$$
\Omega:\R(-1) \to \Lambda^{2,0}(M_J)
$$
is a map of weight\index{terms}{weight!of a Hodge bundle} $2$ Hodge bundles\index{terms}{Hodge bundle}, and
for an arbitrary smooth section $\chi \in C^\infty(M,\Theta(M_J))$
we have
\begin{equation}\label{P}
\Omega(\chi,\overline{\iota^*(\chi)}) > 0.
\end{equation}  
\index{terms}{involution $\iota$}

\punkt Assume that the Hodge manifold\index{terms}{Hodge manifold}
$M$ is equipped with an Hermitian-Hodge
metric\index{terms}{metric!Hermitian-Hodge} $h$. Let $\Omega \in
C^\infty(M,\Lambda^{2,0}(M_J))$ be the corresponding
polarization\index{terms}{polarization} ,
and let $\omega_I \in C^\infty(M,\Lambda^{1,1}(M_I))$ be the
$(1,1)$-form on the complex manifold $M_I$ associated to the
Hermitian metric\index{terms}{metric!Hermitian} $h$.  Either one of
the forms $\Omega$, $\omega_I$ completely defines the metric $h$,
and by \eqref{omega.and.Omega} we have
$$
\Omega + \nu(\Omega) = \omega_I, 
$$
where $\nu:\Lambda^\cdot(M,\C) \to \Lambda^\cdot(M,\C)$ is the complex
conjugation. 

\begin{lemma}\label{pol.hm}
The Hermitian-Hodge metric\index{terms}{metric!Hermitian-Hodge} $h$
polarizes $M$ if and only if the corresponding $(2,0)$-form $\Omega$
on $M_J$ is holomorphic, that is,
$$
D\Omega = 0,
$$
where $D$ is the Dolbeult differential for complementary\index{terms}{complementary complex structure} complex structure $M_J$.
\end{lemma}

\proof Let $\omega_I,\omega_J \in \Lambda^2(M,\C)$ be the K\"ahler forms
for the metric\index{terms}{metric!K\"ahler} $h$ and complex
structures $M_I$, $M_J$ on $M$. The metric $h$ is
hyperk\"ahler\index{terms}{metric!hyperk\"ahler}, hence polarizes
$M$, if and only if $d_M\omega_I = d_M\omega_J = 0$.

Let $D = D_0 + \overline{D_0}$ be the
$H$-type\index{terms}{Hodge type1@$H$-type} decomposition and let
$H:\Lambda^{\cdot,\cdot}(M_J) \to \Lambda^{\cdot,\cdot}(M_J)$ be the
standard algebra identification introduced in \ref{6_I^H}.  By
definition $H(\omega_I) = \omega_J$. Moreover, by
\eqref{identities} $H^{-1} \circ d_M \circ H = D_0 +
\overline{D_0}^\nu$, hence  
$$
H(d_M\omega_J) = D_0 \omega_I + \overline{D_0}^\nu\omega_I,
$$ 
and the metric $h$ is
hyperk\"ahler\index{terms}{metric!hyperk\"ahler} if and only if 
\begin{equation}\label{K}
d_M\omega_I = (D_0+\overline{D_0}^\nu)\omega_I = 0
\end{equation}
But $2\omega_I = \Omega + \nu(\Omega)$. Since $\Omega$ is of Hodge
type\index{terms}{bigrading!Hodge type} $(2,0)$ with respect to the
complementary\index{terms}{complementary complex structure} complex
structure $M_J$, \eqref{K} is equivalent to 
$$
\overline{D_0} \Omega = D_0 \Omega = \overline{D_0}^\nu \Omega =
D_0^\nu \Omega = 0.
$$
Moreover, let $\xi$ be as in Lemma~\ref{xi.lemma}. Then $\xi(\Omega)
= 0$, and by \eqref{xi.eq} $D_0\Omega = \overline{D}_0\Omega = 0$
implies that $D_0^\nu\Omega = \overline{D}_0^\nu\Omega = 0$ as well.
It remains to notice that since the metric $h$ is
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge}, $\Omega$ is of
$H$-type\index{terms}{Hodge type1@$H$-type} $(1,1)$ as a section of the
weight\index{terms}{weight!of a Hodge bundle} $2$ Hodge 
bundle\index{terms}{Hodge bundle} $\Lambda^{2,0}(M_J)$.  Therefore
$D\Omega = 0$ implies $\overline{D_0}\Omega = D_0\Omega = 0$, and
this proves the lemma.  \endproof

\rem This statement is wrong for general hyperk\"ahler
manifolds\index{terms}{hyperk\"ahler manifold}
(eve\-ry\-thing in the given proof carries over, except for the
implication $D\Omega=0 \Rightarrow
D_0\Omega=\overline{D_0}\Omega=\overline{D_0}^\nu \Omega = D_0^\nu
\Omega = 0$, which depends substantially on the $U(1)$-action). To
describe general hyperk\"ahler
metrics\index{terms}{metric!hyperk\"ahler} in holomorphic terms, one
has to introduce the so-called 
{\em twistor spaces\index{terms}{twistor space}} (see, e.g.,
\cite{HKLR}).

\section{Regular Hodge manifolds}\index{terms}{Hodge manifold!regular} 

\subsection{Regular stable points} 

\punkt Let $M$ be a smooth manifold equipped with a smooth
$U(1)$-action with differential $\phi_M$ (thus $\phi_M$ is a smooth
vector field on $M$).  Since the group $U(1)$ is compact, the subset
$M^{U(1)} \subset M$ of points fixed under $U(1)$ is a smooth
submanifold.

Let $m \in M^{U(1)} \subset M$ be a point fixed under
$U(1)$. Consider the representation of $U(1)$ on the tangent space
$T_m$ to $M$ at $m$.  Call the fixed point $m$ {\em
regular\index{terms}{regular fixed point}} if every irreducible
subrepresentation of $T_m$ is either trivial or isomorphic to the
representation on $\C$ given by embedding $U(1) \subset \C^*$. (Here
$\C$ is considered as a $2$-dimensional real vector space.)  Regular
fixed points form a union of connected component of the smooth
submanifold $M^{U(1)} \subset M$.

\punkt Assume that $M$ is equipped with a complex structure preserved by
the $U(1)$-action.  Call a point $m \in M$ {\em stable} if for any $t \in
\R, t \geq 0$ there exists $\exp(\sqrt{-1}t\phi_M)m$, and the limit 
$$
m_0 \in M, m_0 = \lim_{t \to +\infty} \exp(\sqrt{-1}t\phi_M)m
$$
also exists. 

\enlargethispage{10mm}

\punkt For every stable point $m \in M$ the limit $m_0$ is obviously fixed
under $U(1)$.  Call a point $m \in M$ {\em regular
stable\index{terms}{regular stable point}} if it is stable
and the limit $m_0 \in M^{U(1)}$ is a regular fixed
point\index{terms}{regular fixed point}.

Denote by $M^{reg} \subset M$ the subset of all regular stable points. The
subset $M^{reg}$ is open in $M$. 

\ex Let $Y$ be a complex manifold with a holomorphic bundle $\E$ and let
$E$ be the total space of $\E$. Let $\C^*$ act on $M$ by dilatation along the
fibers. Then every point $e \in E$ is regular
stable\index{terms}{regular stable point}.

\punkt Let $M$ be a Hodge manifold\index{terms}{Hodge
manifold}. Recall that the $U(1)$-action on $M$ preserves the
preferred\index{terms}{preferred complex structure} complex structure $M_I$.

\defn A Hodge manifold\index{terms}{Hodge manifold!regular} $M$ is
called {\em regular} if $M_I^{reg} = M_I$.

\subsection{Linearization of regular Hodge
manifolds}\index{terms}{Hodge
manifold!regular}\index{terms}{linearization}  

\punkt Consider a regular Hodge manifold\index{terms}{Hodge
manifold!regular} $M$. Let $\Delta \subset \C$ be the unit disk equipped
with the multiplicative semigroup structure. The group $U(1) \subset
\Delta$ is embedded into $\Delta$ as the boundary circle.

\begin{lemma}\label{regular}
The action $a:U(1) \times M \to M$ extends uniquely to a holomorphic
action $\tilde{a}:\Delta \times M_I \to M_I$. Moreover, for every $x
\in \Delta \setminus \{0\}$ the action map $\wt{a}(x):M_I \to M_I$
is an open embedding.
\end{lemma}

\proof Since $M$ is regular\index{terms}{Hodge manifold!regular}, the
exponential flow $\exp(it\phi_M)$ of the differential $\phi_M$ of
the action is defined for all positive $t \in \R$. Therefore $a:U(1)
\times M \to M$ extends uniquely to a holomorphic action
$$
\tilde{a}:\Delta^* \times M_I \to M_I, 
$$
where $\Delta^* = \Delta \backslash \{0\}$ is the punctured
disk. Moreover, the exponential flow converges as $t \to +\infty$,
therefore $\tilde{a}$ extends to $\Delta \times M_I$
continuously. Since this extension is holomorphic on a dense open
subset, it is holomorphic everywhere. This proves the first claim.

To prove the second claim, consider the subset $\wt{\Delta} \subset
\Delta^*$ of points $x \in \Delta$ such that $\wt{a}(x)$ is
injective and \'etale. The subset $\wt{\Delta}$ is closed under
multiplication and contains the unit circle $U(1) \subset
\Delta^*$. Therefore to prove that $\wt{\Delta} = \Delta^*$, it
suffices to prove that $\wt{\Delta}$ contains the interval $]0,1]
\subset \Delta^*$.

By definition we have $\wt{a}(h) = \exp(-\sqrt{-1}\log h \phi_M)$
for every $h \in ]0,1] \subset \Delta^*$. Thus we have to prove that
if for some $t \in \R, t \geq 0$ and for two points $m_1,m_2 \in M$
we have
$$
\exp(\sqrt{-1}t\phi_M)(m_1) = \exp(\sqrt{-1}t\phi_M)(m_2), 
$$
then $m_1 = m_2$. Let $m_1$, $m_2$ be such two points and let 
$$
t = \inf\{t \in \R, t \geq 0, \exp(\sqrt{-1}t\phi_M)(m_1) =
\exp(\sqrt{-1}t\phi_M)(m_2)\}. 
$$
If the point $m_0 = \exp(\sqrt{-1}t\phi_M)(m_1) =
\exp(\sqrt{-1}t\phi_M)(m_2) \in M$ is not $U(1)$-invariant, then it
is a regular point for the vector field $\sqrt{-1}\phi_M$, and by
the theory of ordinary differential equations we have $t = 0$ and
$m_1 = m_2 = m_0$.

Assume therefore that $m_0 \in M^{U(1)}$ is $U(1)$-invariant. Since
the group $U(1)$ is compact, the vector field $\sqrt{-1}\phi_M$ has
only a simple zero at $m_0 \subset M^{U(1)} \subset M$. Therefore
$m_0 = \exp(\sqrt{-1}t\phi_M)m_1$ implies that the point $m_1 \in M$
also is $U(1)$-invariant, and the same is true for the point $m_2
\in M$. But $\wt{a}(\exp t)$ acts by identity on $M^{U(1)} \subset
M$. Therefore in this case we also have $m_1=m_2=m_0$.
\endproof

\punkt Let $V = M_I^{U(1)} \subset M_I$ be the submanifold of fixed points
of the $U(1)$ action. Since the action preserves the complex structure on
$M_I$, the submanifold $V$ is complex.

\begin{lemma}
There exists a unique $U(1)$-invariant holomorphic map 
$$
\rho_M:M_I \to V
$$
such that $\rho_M|_{V} = \id$. 
\end{lemma}

\proof For every point $m \in M$ we must have $\rho_M(m) =
\displaystyle\lim_{t \to +\infty} \exp(i t \phi_M)$, which proves uniqueness.

To prove that $\rho_M$ thus defined is indeed holomorphic, notice that the
diagram 
$$
\begin{CD}
M_I           @>{0 \times \id}>>   \Delta \times M\\
 @V{\rho_M}VV                         @VV{\tilde{a}}V\\
V             @>>>                 M_I
\end{CD}
$$
is commutative. Since the action $\tilde{a}:\Delta \times M_I \to M_I$ is
holomorphic, so is the map $\rho_M$. 
\endproof 

\punkt Call the canonical map $\rho_M:M_I \to M_I^{U(1)}$ the {\em
canonical projection} of the regular Hodge
manifold\index{terms}{Hodge manifold!regular} $M$ onto the
submanifold $V \subset M$ of fixed points.

\begin{lemma}
The canonical projection $\rho_M:M \to M^{U(1)}$ is submersive, that
is, for every point $m \in M$ the differential $d\rho_M:T_mM \to
T_{\rho(m)}M^{U(1)}$ of the map $\rho_M$ at $m$ is surjective.
\end{lemma}

\proof Since $\rho_M|_{M^{U(1)}} = \id$, the differential $d\rho_M$
is surjective at points $m \in V \subset M$. Therefore it is
surjective on an open neighborhood $U \supset V$ of $V$ in $M$. For
any point $m \in M$ there exists a point $x \in \Delta$ such that $x
\cdot m \in U$. Since $\rho_M$ is $\Delta$-invariant, this implies that
$d\rho_M$ is surjective everywhere on $M$.  
\endproof

\punkt Let $\Theta(M/V)$ be the relative tangent bundle of the holomorphic
map $\rho:M \to V$. Let $\Theta(M)$ and $\Theta(V)$ be the tangent bundles
of $M$ and $V$ and consider the canonical exact sequence of complex bundles
$$
0 \Arrow \Theta(M/V) \Arrow \Theta(M) \overset{d\rho_M}{\Arrow}
\rho^*\Theta(V) \Arrow 0,
$$
where $d\rho_M$ is the differential of the projection $\rho_M:M \to
V$.

The quaternionic structure\index{terms}{quaternionic structure} on
$M$ defines a $\C$-linear map $j:\Theta(M) \to
\overline{\Theta}(M)$. Restricting to $\Theta(M/V)$ and composing
with $d\rho_M$, we obtain a $\C$-linear map $j:\Theta(M/V)
\to \rho^*\overline{\Theta}(V)$. 

\punkt \label{overline.T} 
Let $\tv$ be the total space of the bundle $\overline{\Theta}(V)$
complex-conjugate to the tangent bundle $\Theta(V)$, and let
$\rho:\tv \to V$ be the projection. Let the group $U(1)$ act on
$\tv$ by dilatation along the fibers of the projection $\rho$.

Since the canonical projection $\rho_M:M \to V$ is $U(1)$-invariant,
the differential $\phi_M$ of the $U(1)$-action defines a section
$$
\phi_M \in C^\infty(M,\Theta(V/M)).
$$
The section $j(\phi_M) \in C^\infty(M,\rho_M^*\overline{\Theta}(V))$
defines a map $\Lin_M:M \to \tv$ such that $\Lin_M \circ \rho =
\rho_M:M \to V$. Call the map $\Lin_M$ {\em the
linearization\index{terms}{linearization}} of the regular Hodge
manifold\index{terms}{Hodge manifold!regular} $M$.

\begin{prop}\label{linearization}
The linearization map $\Lin_M$ is a $U(1)$-equivariant open embedding.  
\end{prop}

\proof The map $j:\Theta(M/V) \to \rho^*\overline{\Theta}(V)$ is of
degree $1$ with respect to the $U(1)$-action, while the section
$\phi_M \in C^\infty(M,\Theta(M/V))$ is $U(1)$-invariant. Therefore
the map $\Lin_M$ is $U(1)$-equivariant.

Consider the differential $d\Lin_M:T_m(M) \to T_m(\tv)$ at a point $m \in V
\subset M$. We have
$$
d\Lin_M = d\rho_M \oplus d\rho_M \circ j:T_m(M) \to T_m(V) \oplus
\overline{T}_m(V)
$$
with respect to the decomposition $T_m(\tv) = T_m(V) \oplus
\overline{T}(V)$.  The tangent space $T_m$ is a regular equivariant
quaternionic vector space\index{terms}{quaternionic vector
space!regular equivariant}. Therefore the map $d\Lin_M$ is bijective
at $m$ by Lemma~\ref{regular.quaternionic}. Since $\Lin_M$ is
bijective on $V$, this implies that $\Lin_M$ is an open embedding on
an open neighborhood $U
\subset M$ of the submanifold $V \subset M$.

To finish the proof of proposition, it suffices prove that the
linearization\index{terms}{linearization} map $\Lin_M:M_I \to \tv$
is injective and \'etale on the whole $M_I$. To prove injectivity,
consider arbitrary two points $m_1,m_2 \in M_I$ such that
$\Lin_M(m_1)=\Lin_M(m_2)$.  There exists a point $x \in \Delta
\setminus \{0\}$ such that $x \cdot m_1, x
\cdot m_2 \in U$. The map $\Lin_M$ is $U(1)$-equivariant and
holomorphic, therefore it is $\Delta$-equivariant, and we have
$$
\Lin_M(x \cdot m_1) = x \cdot \Lin_M(m_1) = x \cdot \Lin_M(m_2) =
\Lin_M(x \cdot m_2).
$$ 
Since the map $\Lin_M:U \to \tv$ is injective, this implies that $x
\cdot m_1 = x \cdot m_2$. By Lemma~\ref{regular} the action map
$x:M_I \to M_I$ is injective. Therefore this is possible only if
$m_1 = m_2$, which proves injectivity.

To prove that the linearization\index{terms}{linearization} map is
\'etale, note that by Lemma~\ref{regular} the action map $x:M_I \to
M_I$ is not only injective, but also \'etale. Since $\Lin_M$ is
\"etale on $U$, so is the composition $\Lin_M \circ x:M_I \to U \to
\tv$ is \'etale. Since $\Lin_M \circ x = x \circ \Lin_M$, the map
$\Lin_M:M_I \to \tv$ is
\'etale at the point $m_1 \in M_I$.

Thus the linearization\index{terms}{linearization} map is also
injective and \'etale on the whole of $M_I$. Hence it is indeed an
open embedding, which proves the propostion.
\endproof

\subsection{Linear Hodge manifold structures}\index{terms}{Hodge
manifold!linear}

\punkt By Proposition~\ref{linearization} every regular Hodge
manifold\index{terms}{Hodge manifold!regular} $M$ admits a canonical
open embedding $\Lin_M:M \to \tv$ into the total space $\tv$ of the
(complex-conjugate) tangent bundle to its fixed points submanifold
$V \subset M$. This embedding induces a Hodge
manifold\index{terms}{Hodge manifold} structure on a neighborhood of
the zero section $V \subset \tv$.

In order to use the linearization\index{terms}{linearization}
construction, we will need a characterization of all Hodge manifold
structures on neighborhoods of $V
\subset \tv$ obtained in this way (see \ref{lin.def}). It is convenient to
begin with an invariant characterization of the linearization map $\Lin_M:M
\to \tv$.

\punkt \label{tau} 
Let $V$ be an arbitrary complex manifold, let $\tv$ be the total space
of the complex-conjugate to the tangent bundle $\Theta(V)$ to
$V$, and let $\rho:\tv \to V$ be the canonical projection. Contraction with
the tautological section of the bundle $\rho^*\overline{\Theta(V)}$ defines
for every $p$ a bundle map 
$$
\tau:\rho^*\Lambda^{p+1}(V,\C) \to \rho^*\Lambda^{p}(V,\C),  
$$
which we call {\em the tautological map}. In particular, the
induced map
$$
\tau:C^\infty(V,\Lambda^{0,1}(V)) \to C^\infty(\tv,\C)
$$
identifies the space $C^\infty(V,\Lambda^{0,1}(V))$ of smooth $(0,1)$-forms
on $V$ with the subspace in $C^\infty(\tv,\C)$ of function linear along the
fibers of the projection $\tv \to V$.

\punkt Let now $M$ be a Hodge manifold\index{terms}{Hodge
manifold}. Let $V \subset M_I$ be the complex submanifold of
$U(1)$-fixed points, and let $\rho_M:M \to V$ be the canonical
projection. Assume that $M$ is equipped with a smooth
$U(1)$-equivariant map $f:M \to \tv$ such that $\rho_M = \rho \circ
f$. Let $\bar\6_I$ be the Dolbeult
differential for the preferred\index{terms}{preferred complex structure}
complex structure $M_I$ on $M$, and let $\phi \in \Theta(M/V)$ be
the differential of the $U(1)$-action on $M$. Let also
$j:\Lambda^{0,1}(M_I) \to \Lambda^{1,0}(M_I)$ be the map induced by
the quaternionic structure\index{terms}{quaternionic structure} on $M$.

\begin{lemma}\label{lin.char}
The map $f:M \to \tv$ coincides with the
linearization\index{terms}{linearization} map if and only if for
every $(0,1)$-form $\alpha \in C^\infty(V,\Lambda^{0,1}(V))$ we have
\begin{equation}\label{L}
f^*\tau(\alpha) = \langle \phi, j(\rho_M^*\alpha) \rangle.
\end{equation}
Moreover, if $f = \Lin_M$, then we have 
\begin{equation}\label{LL}
f^*\tau(\beta) = \langle \phi, j(f^*\beta) \rangle
\end{equation}
for every smooth section $\beta \in C^\infty(\tv,\rho^*\Lambda^1(V,\C))$. 
\end{lemma}

\proof Since functions on $\tv$ linear along the fibers separate points,
the correspondence
$$
f^* \circ \tau:C^\infty(V,\Lambda^{0,1}(V)) \to C^\infty(M,\C)
$$
characterizes the map $f$ uniquely, which proves the ``only if''
part of the first claim. Since by assumption $\rho_M = \rho \circ
f$, the equality \eqref{L} is a particular case of \eqref{LL} with
$\beta = \rho^*\alpha$. Therefore the ``if'' part of the first claim
follows from the second claim, which is a rewriting of the
definition of the linearization\index{terms}{linearization} map
$\Lin_M:M \to \tv$ (see
\ref{overline.T}).  
\endproof

\punkt Let now $\Lin_M:M \to \tv$ be the
linearization\index{terms}{linearization} map for the 
regular Hodge manifold\index{terms}{Hodge manifold!regular}
$M$. Denote by $U \subset \tv$ the image of $\Lin_M$. The subset $U
\subset \tv$ is open and $U(1)$-invariant. In addition, the
isomorphism $\Lin_M:M \to U$ induces a regular Hodge
manifold\index{terms}{Hodge manifold!regular} structure on $U$.

Denote by $\Lin_U$ the linearization\index{terms}{linearization} map
for the regular Hodge manifold\index{terms}{Hodge manifold!regular}
$U$. Lemma~\ref{lin.char} implies the following.

\begin{corr}\label{lin.lin}
We have $\Lin_M \circ \Lin_U = \Lin_M$, thus the
linearization\index{terms}{linearization} map 
$$
\Lin_U:U \to \tv
$$
coincides with the given embedding $U \hookrightarrow \tv$.
\end{corr}

\proof Let $\alpha \in C^\infty(V,\Lambda^{0,1}(V)$ be a
$(0,1)$-form on $V$. By Lemma~\ref{lin.char} we have
$\Lin_U^*\tau(\alpha) = \langle \phi_U, j_U(\rho_U^*\alpha)\rangle$,
and it suffices to prove that
$$
\Lin_M^*(\Lin_U^*(\tau(\alpha))) = \langle \phi_M, j_M(\rho_M^*\alpha)
\rangle.
$$
By definition we have $\rho_M = \rho_U \circ \Lin_M$. Moreover, the
map $\Lin_M$ is $U(1)$-equivariant, therefore it sends $\phi_M$ to
$\phi_U$. Finally, by definition it commutes with the quaternionic
structure\index{terms}{quaternionic structure} map $j$. Therefore 
$$
\Lin_M^*(\Lin_U^*(\tau(\alpha))) = \Lin_M^*(\langle \phi_U,
j_U(\rho_U^*\alpha) \rangle)= \langle \phi_M, j_M(\rho_M^*\alpha)
\rangle,
$$
which proves the corollary. 
\endproof 

\punkt \defn \label{lin.def} Let $U \subset \tv$ be an open
$U(1)$-invariant neighborhood of the zero section $V \subset \tv$. A
Hodge manifold\index{terms}{Hodge manifold} structure on $\tv$ is
called {\em linear\index{terms}{Hodge manifold!linear}} if the
associated linearization map $\Lin_U:U \to \tv$ coincides with the
given embedding $U \hookrightarrow \tv$.

By Corollary~\ref{lin.lin} every Hodge manifold structure on a subset $U
\subset \tv$ obtained by the
linearization\index{terms}{linearization} construction is linear. 

\punkt We finish this section with the following simple observation, which
we will need in the next section.

\begin{lemma}\label{aux2}
Keep the notations of Lemma~\ref{lin.char}. Moreover, assume given a
subspace $\A \subset C^\infty(\tv,\rho^*\Lambda^1(V,\C))$ such that
the image of $\A$ under the restriction map
$$
\Res:C^\infty(\tv,\rho^*\Lambda^1(V,\C)) \to
C^\infty(V,\Lambda^1(V,\C)) 
$$
onto the zero section $V \subset \tv$ is the whole space
$C^\infty(V,\Lambda^1(V,\C))$. If \eqref{LL} holds for every section
$\beta \in \A$, then it holds for every smooh section 
$$
\beta \in C^\infty(\tv,\rho^*\Lambda^1(V,\C)).
$$
\end{lemma}

\proof By assumptions sections $\beta \in \A$ generate the
restriction of the bundle $\rho^*\Lambda^1(V,\C)$ onto the zero
section $V \subset \tv$. Therefore there exists an open neighborhood
$U \subset \tv$ of the zero section $V \subset \tv$ such that the
$C^\infty(U,\C)$-submodule 
$$
C^\infty(U,\C) \cdot \A \subset C^\infty(U,\rho^*\Lambda^1(V,\C)) 
$$
is dense in the space $C^\infty(U,\rho^*\Lambda^1(V,\C))$ of smooth
sections of the pull-back bundle $\rho^*\Lambda^1(V,\C)$. Since the
equality \eqref{LL} is continuous and linear with respect to
multiplication by smooth functions, it holds for all sections $\beta
\in C^\infty(U,\rho^*\Lambda^1(V,\C))$. Since it is also compatible
with the natural unit disc action on $M$ and $\tv$, it holds for all
sections $\beta \in C^\infty(\tv,\rho^*\Lambda^1(V,\C))$ as well.
\endproof

\section{Tangent bundles as Hodge
manifolds}\label{section.5}\index{terms}{Hodge manifold}

\subsection{Hodge connections}\index{terms}{Hodge connection}

\punkt The linearization\index{terms}{linearization} construction
reduces the study of arbitrary regular Hodge
manifolds\index{terms}{Hodge manifold!regular} to the study of
linear Hodge manifold\index{terms}{Hodge manifold!linear} structures
on a neighborhood $U \subset \tv$ of the zero section $V
\subset \tv$ of the total space of the complex conjugate to the
tangent bundle of a complex manifold $V$. In this section we use the
theory of Hodge bundles\index{terms}{Hodge bundle} developed in
Subsection~\ref{hb.sub} in order to describe Hodge
manifold\index{terms}{Hodge manifold} structures on $U$ in terms of
connections on the locally trivial fibration $U \to V$ of a certain
type, which we call {\em Hodge connections\index{terms}{Hodge
connection}} (see \ref{hodge.con}). It is this description, given in
Proposition~\ref{equiv}, which we will use in the latter part of the
paper to classify all such Hodge manifold structures.

\punkt \label{conn}
We begin with some preliminary facts about connections on locally
trivial fibrations. Let $f:X \to Y$ be an arbitrary smooth map of
smooth manifolds $X$ and $Y$. Assume that the map $f$ is submersive,
that is, the codifferential
$\del_f:f^*\Lambda^1(Y) \to \Lambda^1(X)$ is an injective bundle
map.  Recall that a {\em connection\index{terms}{connections!on a
fibration}} on $f$ is by definition a splitting $\Theta:\Lambda^1(X)
\to f^*\Lambda^1(Y)$ of the canonical embedding $\del_f$.

Let $d_X$ be the de Rham differential on the smooth manifold
$X$. Every connection $\Theta$ on $f:X \to Y$ defines an algebra
derivation
$$
D = \Theta \circ d_X:\Lambda^0(X) \to f^*\Lambda^1(Y), 
$$
satisfying 
\begin{equation}\label{conn.eq}
D \rho^* h = \rho^* d_Y h
\end{equation}
for every smooth function $h \in C^\infty(Y,\R)$. Vice versa, by the
universal property of the cotangent bundle (Lemma~\ref{universal})
every algebra derivation $D:\Lambda^0(X) \to \Lambda^1(Y)$
satisfying \eqref{conn.eq} comes from a unique
connection\index{terms}{connections!on a fibration} $\Theta$
on $f$.

\punkt Recall also that a connection $\Theta$ is called {\em flat}
if the associated derivation $D$ extends to an algebra
derivation
$$
D:f^*\Lambda^\cdot(Y) \to f^*\Lambda^{\cdot+1}(Y)
$$
so that $D \circ D = 0$. The splitting $\Theta:\Lambda^1(X) \to
f^*\Lambda^1(Y)$ extends in this case to an algebra map 
$$
\Theta:\Lambda^\cdot(X) \to f^*\Lambda^\cdot(Y)
$$
compatible with the de Rham differential $d_X:\Lambda^\cdot(X) \to
\Lambda^{\cdot+1}(X)$. 

\punkt We will need a slight generalization of the notion of connection. 

\defn Let $f:X \to Y$ be a smooth submersive morphism of complex
manifolds. A {\em $\C$-valued connection\index{terms}{connections!$\C$-valued}
} $\Theta$ on $f$ is a splitting $\Theta:\Lambda^1(Y,\C)
\to f^*\Lambda^1(X,\C)$ of the
codifferential map $\del
f:f^*\Lambda^1(Y,\C) \to \Lambda^1(X,\C)$ of complex vector
bundles. A $\C$-valued connection $\Theta$ is called {\em flat} if
the associated algebra derivation
$$
D = \Theta \circ d_X:\Lambda^0(X,\C) \to f^*\Lambda^1(Y,\C)
$$
extends to an algebra derivation 
$$
D:f^*\Lambda^\cdot(Y,\C) \to f^*\Lambda^{\cdot+1}(Y,\C)
$$
satisfying $D \circ D = 0$. 

As in \ref{conn}, every derivation $D:\Lambda^0(X,\C) \to
f^*\Lambda^1(Y,\C)$ satisfying \eqref{conn.eq} comes from a unique
$\C$-valued connection\index{terms}{connections!$\C$-valued} $\Theta$
on $f:X \to Y$.

\rem By definition for every flat connection on $f:X \to Y$ the
subbundle of horizontal vectors in the the tangent bundle
$\Theta(X)$ is an involutive distribution. By Frobenius
Theorem\index{terms}{Frobenius Theorem} this implies that the
connection\index{terms}{connections!on a fibration} defines locally a
trivialization of the fibration $f$.

This is no longer true for flat $\C$-valued
connections\index{terms}{connections!$\C$-valued}: the
subbundle of horizontal vectors in $\Theta(X) \otimes \C$ is only
defined over $\C$, and the Frobenius Theorem\index{terms}{Frobenius
Theorem} does not apply. One can try to correct this by replacing
the splitting $\Theta:\Lambda^1(X,\C) \to f^*\Lambda^1(Y,\C)$ with
its real part $\Re\Theta:\Lambda^1(X) \to \Lambda^1(Y)$, but this
real part is, in general, no longer flat.

\punkt For every $\C$-valued connection $\Theta:\Lambda^1(X,\C) \to
f^*\Lambda^1(Y,\C)$ on a fibration $f:X \to Y$ the kernel
$\Ker\Theta \subset \Lambda^1(X,\C)$ is canonically isomorphic to
the quotient $\Lambda^1(X,\C)/\del_f(f^*\Lambda^1(Y,\C))$, and the
composition
$$
R = \Theta \circ d_X:\Lambda^1(X,\C)/\del_f(f^*\Lambda^1(Y,\C) \cong
\Ker\Theta \to f^*\Lambda^2(Y,\C) 
$$
is in fact a bundle map. This map is called {\em the curvature} of
the $\C$-valued connection\index{terms}{connections!$\C$-valued}
$\Theta$. The connection $\Theta$ is flat if and only if its
curvature $R$ vanishes.

\punkt \label{conn.setup}
Let now $M$ be a complex manifold, and let $U \subset \tm$ be an
open neighborhood of the zero section $M \subset \tm$ in the total
space $\tm$ of the complex-conjugate to the tangent bundle to
$M$. Let $\rho:U \to M$ be the natural projection. Assume that $U$
is invariant with respect to the natural action of the unit disc
$\Delta \subset \C$ on $\tm$.

\punkt Since $M$ is complex, by \ref{de.Rham} the bundle
$\Lambda^1(M,\C)$ is equipped with a Hodge bundle\index{terms}{Hodge bundle} structure of weight\index{terms}{weight!of a Hodge bundle} $1$. The pullback bundle
$\rho^*\Lambda^1(M,\C)$ is then also equipped with a weight\index{terms}{weight!of a Hodge bundle} $1$
Hodge bundle structure.

Our description of the Hodge manifold\index{terms}{Hodge manifold}
structures on the subset $U \in \tm$ is based on the following
notion.

\defn \label{hodge.con} A {\em Hodge connection\index{terms}{Hodge
connection}} on the pair $\langle M, U\rangle$ is a $\C$-valued
connection\index{terms}{connections!$\C$-valued} on $\rho:U \to M$
such that the associated derivation
$$
D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)
$$
is weakly Hodge\index{terms}{weakly Hodge map} in the sense of
\ref{w.hodge}. A Hodge connection\index{terms}{Hodge connection} is
called {\em flat} if it extends to a weakly Hodge derivation
$$
D:\rho^*\Lambda^\cdot(M,\C) \to \rho^*\Lambda^{\cdot+1}(M,\C)
$$
satisfying $D \circ D = 0$. 

\punkt Assume given a flat Hodge connection\index{terms}{Hodge
connection} $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ on the
pair $\langle U,M \rangle$, and assume in addition that the
derivation $D$ is holonomic\index{terms}{holonomic derivation} in
the sense of \ref{holonomic}. Then 
the pair $\langle D, \rho^*\Lambda^1(M,\C)\rangle$ defines by
Proposition~\ref{explicit.hodge} a Hodge manifold\index{terms}{Hodge
manifold} structure on $U$.

It turns out that every Hodge manifold structure on $U$ can be
obtained in this way. Namely, we have the following. 

\begin{prop}\label{equiv}
There correspondence $D \mapsto \langle \rho^*\Lambda^1(M,\C), D
\rangle$ is a bijection between the set of all flat Hodge
connections\index{terms}{Hodge connection} $D$ on the pair $\langle
U, M\rangle$ such that $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$
is holonomic\index{terms}{holonomic derivation} in the sense of
\ref{holonomic}, and the set of all 
Hodge manifold\index{terms}{Hodge manifold} structures on the
$U(1)$-manifold $U$ such that the projection $\rho:U_I \to M$ is
holomorphic for the preferred\index{terms}{preferred complex structure} complex structure $U_I$ on $U$.
\end{prop}

\punkt The crucial part of the proof of Proposition~\ref{equiv} is
the following observation. 

\begin{lemma}\label{ident}
Assume given a Hodge manifold\index{terms}{Hodge manifold} structure
on the $U(1)$-manifold $U \subset \tm$. Let
$\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ be the
codifferential of the projection $\rho:U \to M$, and let
$P:\Lambda^1(U,\C) \to \Lambda^{0,1}(U_J)$ be the canonical
projection. The bundle map given by the composition
$$
P \circ \del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{0,1}(U_J)
$$
is an isomorphism of complex vector bundles. 
\end{lemma}

\proof Since the bundles $\rho^*\Lambda^1(M,\C)$ and
$\Lambda^{0,1}(U_J)$ are of the same rank, and the maps
$\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ and $P \circ
\del_\rho$ are equivariant with respect to the action of the unit
disc on $U$, it suffices to prove the claim on $M \subset U$. Let $m
\in M$ be an arbitrary point, and let $V = T^*_m\tm$ be the
cotangent bundle at $m$ to the Hodge manifold\index{terms}{Hodge
manifold} $U \subset \tm$. Let also $V^0 \subset V$ be the subspace
of $U(1)$-invariant vectors in $V$.

The space $V$ is an equivariant quaternionic vector
space\index{terms}{quaternionic vector space!equivariant}. Moreover,
the fibers of the bundles $\rho^*\Lambda^1(M,\C)$ and
$\Lambda^{0,1}(U_J)$ at the point $m$ are complex vector spaces, and
we have canonical identifications
\begin{align*}
\rho^*\Lambda^1(M,\C)|_m &\cong V^0_I \oplus \overline{V^0_I},\\
\Lambda^{0,1}(U_J)|_m &\cong V_J. 
\end{align*}
Under these identifications the map $P \circ \del_\rho$ at the point
$m$ coincides with the action map $V^0_I \oplus \overline{V^0_I} \to
V_J$, which is invertible by Lemma~\ref{regular.quaternionic}.
\endproof

\punkt By Proposition~\ref{explicit.hodge} every Hodge
manifold\index{terms}{Hodge manifold} structure on $U$ is given by a
pair $\langle \E, D\rangle$ of a Hodge bundle\index{terms}{Hodge bundle} $\E$ on $U$ of weight\index{terms}{weight!of a Hodge bundle}
$1$ and a holonomic\index{terms}{holonomic derivation} derivation
$D:\Lambda^0(U,\C) \to 
\E$. Lemma~\ref{ident} gives an isomorphism $\E \cong
\rho^*\Lambda^1(M,\C)$, so that $D$ becomes a flat $\C$-valued
connection\index{terms}{connections!$\C$-valued} on $U$ over $M$. To
prove Proposition~\ref{equiv} it suffices now to prove the
following.

\begin{lemma}
The complex vector bundle isomorphism 
$$
P \circ \del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{0,1}(U_J)
$$
associated to a Hodge manifold\index{terms}{Hodge manifold}
structure on $U$ is compatible with the Hodge
bundle\index{terms}{Hodge bundle} structures if
and only if the projection $\rho:U_I \to M$ is holomorphic for the
preferred\index{terms}{preferred complex structure} complex structure $U_I$ on $U$.
\end{lemma}

\proof The preferred\index{terms}{preferred complex structure} complex structure $U_I$ induces a Hodge
bundle\index{terms}{Hodge bundle} structure of weight\index{terms}{weight!of a Hodge bundle} $1$ on
$\Lambda^1(U,\C)$ by \ref{de.Rham}, and the canonical projection
$P:\Lambda^1(U,\C) \to \Lambda^{0,1}(U_J)$ is compatible with the
Hodge bundle structures by \ref{lambda_j=lambda_i}. If the
projection $\rho:U_I \to M$ is holomorphic, then the codifferential
$\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ sends the
subbundles $\rho^*\Lambda^{1,0}(M),\rho^*\Lambda^{0,1}(M) \subset
\rho^*\Lambda^1(M,\C)$ into, respectively, the subbundles
$\Lambda^{1,0}(U_I),\Lambda^{0,1}(U_I) \subset
\Lambda^1(U,\C)$. Therefore the map $\del_\rho:\rho^*\Lambda^1(M,\C)
\to \Lambda^1(U,\C)$ is compatible with the Hodge
bundle\index{terms}{Hodge bundle} structures,
which implies the ``if'' part of the lemma.

To prove the ``only if'' part, assume that $P \circ \del_\rho$ is a
Hodge bundle\index{terms}{Hodge bundle} isomorphism. Since the
complex conjugation $\nu:\overline{\Lambda^{0,1}(U_I)} \to
\Lambda^{1,0}(U_J)$ is compatible with the Hodge bundle structures,
the projection $\overline{P}:\Lambda^1(U,\C) \to \Lambda^{1,0}(U_J)$
and the composition $\overline{P} \circ
\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^{1,0}(U_J)$ are also
compatible with the Hodge bundle structures. Therefore the map
$$
P \oplus \overline{P}:\Lambda^1(U,\C) \to \Lambda^{1,0}(U_J) \oplus
\Lambda^{0,1}(U_J) 
$$
is a Hodge bundle isomorphism, and the composition 
$$
\del_\rho \circ (P \oplus \overline{P}):\rho^*\Lambda^1(M,\C) \to
\Lambda^1(U,\C)
$$ 
is a Hodge bundle map. Therefore the codifferential
$\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)$ is compatible
with the Hodge bundle structures. This means precisely that the
projection $\rho:U_I \to M$ is holomorphic, which finishes the proof
of the lemma and of Proposition~\ref{equiv}.  
\endproof

\subsection{The relative de Rham complex of $U$ over
$M$}\label{relative.de.rham.sub}\index{terms}{de Rham complex} 

\punkt Keep the notation of the last subsection. To use
Proposition~\ref{equiv} in the study of Hodge
manifold\index{terms}{Hodge manifold} structures on the open subset
$U \subset \tm$, we will need a way to check whether a given Hodge
connection\index{terms}{Hodge connection} on the pair $\langle U,M
\rangle$ is holonomic\index{terms}{holonomic derivation} in the
sense of \ref{holonomic}. We will also 
need to rewrite the linearity condition\index{terms}{linearity
condition} \ref{lin.def} for a Hodge manifold structure on $U$ in
terms of the associated Hodge connection $D$. To do this, we will
use the so-called {\em relative de Rham complex} of $U$ over
$M$\index{terms}{de Rham complex!relative}. For the convenience of
the reader, and to fix notation, we recall here its definition and
main properties.

\punkt Since the projection $\rho:U \to M$ is submersive, the
codifferential 
$$
\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)
$$ 
is injective. The {\em relative cotangent bundle} $\Lambda^1(U/M,C)$
is by definition the quotient bundle
$$
\Lambda^1(U/M,\C) = \Lambda^1(U,\C)/\del_\rho(\rho^*\Lambda^1(M,\C)). 
$$
Let $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ be the natural
projection. We have by definition the short exact sequence 
\begin{equation}\label{ex.seq}
\begin{CD}
0 @>>> \rho^*\Lambda^1(M,\C) @>\del_\rho>> \Lambda^1(U,\C) @>\pi>>
\Lambda^1(U/M,\C) @>>> 0
\end{CD}
\end{equation}
of complex vector bundles on $U$. 

\punkt The composition $d^r = \pi \circ d_U$ of the de Rham differential
$d_U$ with the projection $\pi$ is an algebra derivation 
$$
d^r:\Lambda^0(U,\C) \to \Lambda^1(U/M,\C),
$$
called {\em the relative de Rham differential}. It is a first order
differential operator, and $d^rf = 0$ if and only if the smooth
function $f:U \to \C$ factors through the projection $\rho:U \to
M$. 

Let $\Lambda^\cdot(U/M,\C)$ be the exterior algebra of the bundle
$\Lambda^1(U/M,\C)$. The projection $\pi$ extends to an algebra
map 
$$
\pi:\Lambda^\cdot(U,\C) \to \Lambda^\cdot(U/M,\C).
$$
The differential $d^r$ extends to an algebra derivation 
$$
d^r:\Lambda^\cdot(U/M,\C) \to \Lambda^{\cdot+1}(U/M,\C)
$$
satisfying $d^r \circ d^r = 0$, and we have $\pi \circ d_U = d^r
\circ \pi$. The differential graded algebra $\langle
\Lambda^\cdot(U/M,\C),d^r\rangle$ is called {\em the relative de
Rham complex\index{terms}{de Rham complex!relative}} of $U$ over $M$. 

\punkt Since the relative de Rham differential $d^r$ is linear with
respect to multiplication by functions of the form $\rho^*f$ with $f
\in C^\infty(M,\C)$, it extends canonically to an operator
$$
d^r:\rho^*\Lambda^i(M,\C) \otimes \Lambda^\cdot(U/M,\C) \to
\rho^*\Lambda^i(M,\C) \otimes \Lambda^{\cdot+1}(U/M,\C). 
$$
The two-step filtration $\rho^*\Lambda^1(M,\C) \subset
\Lambda^1(U,\C)$ induces a filtration on the de Rham
complex\index{terms}{de Rham complex} $\Lambda^\cdot(U,\C)$, and the
$i$-th associated graded quotient of this filtration is isomorphic
to the complex $\langle \rho^*\Lambda^i(M,\C) \otimes
\Lambda^\cdot(U/M,\C), d^r \rangle$.

\punkt Since $U \subset \tm$ lies in the total space of the
complex-conjugate to the tangent bundle to $M$, we have a canonical
algebra isomorphism
$$
\can:\overline{\rho^*\Lambda^\cdot(M,\C)} \to \Lambda^\cdot(U/M,\C).
$$
Let $\tau:C^\infty(M,\Lambda^1(M,\C)) \to C^\infty(U,\C)$ be
the tautological map sending a $1$-form to the corresponding linear
function on $\tm$, as in \ref{tau}. Then for every smooth
$1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$ we have
\begin{equation}\label{can.and.tau}
\can(\rho^*\alpha) = d^r\tau(\alpha). 
\end{equation}

\punkt The complex vector bundle $\Lambda^1(U/M,\C)$ has a natural
real structure, and it is naturally $U(1)$-equivariant. Moreover,
the decomposition $\Lambda^1(M,\C) = \Lambda^{1,0}(M) \oplus
\Lambda^{0,1}(M)$ induces a decomposition
$$
\Lambda^1(U/M,\C) = \can(\Lambda^{1,0}(M)) \oplus
\can(\Lambda^{0,1}(M)).
$$
This allows to define, as in \ref{de.Rham}, a canonical Hodge
bundle\index{terms}{Hodge bundle} structure of
weight\index{terms}{weight!of a Hodge bundle} $1$ on
$\Lambda^1(U/M,\C)$. It gives rise to a Hodge bundle structure on
$\Lambda^i(U/M,\C)$ of weight\index{terms}{weight!of a Hodge bundle}
$i$, and the relative de Rham differential
$$
d^r:\Lambda^\cdot(U/M,\C) \to \Lambda^{\cdot+1}(U/M,\C)
$$ 
is weakly Hodge\index{terms}{weakly Hodge map}.

\punkt The canonical isomorphism
$\can:\overline{\rho^*\Lambda^1(M,\C)} \to \Lambda^1(U/M,\C)$ is not
compatible with the Hodge bundle\index{terms}{Hodge bundle}
structures. The reason for this is that the real structure on the
Hodge bundles $\Lambda^\cdot(U/M,\C)$ is, by definition
\ref{de.Rham}, twisted by
$\iota\index{terms}{involution $\iota$}^*$, where
$\iota\index{terms}{involution $\iota$}:\tm \to \tm$ is 
the action of $-1 \in U(1) \subset \C$. Therefore, while $\can$ is
$U(1)$-equivariant, it is not real. To correct this, introduce an
involution $\zeta\index{terms}{involution $\zeta$}:\Lambda^1(M,\C)
\to \Lambda^1(M,\C)$ by 
\begin{equation}\label{zeta}
\zeta = \begin{cases} \id &\text{
on }\Lambda^{1,0}(M) \subset 
\Lambda^1(M,\C) \\
-\id &\text{ on }\Lambda^{0,1}(M) \subset \Lambda^1(M,\C)
\end{cases}
\end{equation}
\index{terms}{involution $\zeta$}
and set 
\begin{equation}\label{eta}
\eta = \can \circ
\rho^*\overline{\zeta}:\rho^*\overline{\Lambda^1(M,\C)} \to
\rho^*\overline{\Lambda^1(M,\C)} \to \Lambda^1(U/M,\C)
\end{equation}
\index{terms}{involution $\zeta$}
Unlike $\can$, the map $\eta$ preserves the Hodge bundle structures.

It will also be convenient to twist the tautological map
$\tau:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ by the involution
$\zeta\index{terms}{involution $\zeta$}$. Namely, define a map $\sigma:\rho^*\Lambda^1(M,\C) \to
\Lambda^0(U,\C)$ by
\begin{equation}\label{sigma1}
\sigma = \tau \circ \rho^*\overline{\zeta}:
\rho^*\overline{\Lambda^1(M,\C)} \to
\rho^*\overline{\Lambda^1(M,\C)} \to \Lambda^0(U/M,\C)  
\end{equation}
\index{terms}{involution $\zeta$}
By \eqref{can.and.tau} the twisted tautological map $\sigma$ and the
canonical map $\eta$ satisfy
\begin{equation}\label{eta.and.sigma}
\eta(\rho^*\alpha) = d^r\sigma(\alpha) 
\end{equation}
for every smooth $1$-form $\alpha \in C^\infty(M,\Lambda^1(M,\C))$. 

\punkt Let $\phi \in \Theta(U)$ be the differential of the canonical
$U(1)$-action on $U \subset \tm$. The vector field $\phi$ is real
and tangent to the fibers of the projection $\rho:U \to
M$. Therefore the contraction with $\phi$ defines an algebra
derivation 
\begin{align*}
\Lambda^{\cdot+1}(U/M,\C) &\to \Lambda^\cdot(U/M,\C)\\
\alpha &\mapsto \langle \phi, \alpha \rangle
\end{align*}
The following lemma gives a relation between this derivation, the
canonical weakly Hodge\index{terms}{weakly Hodge map} map
$\eta:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U/M,\C)$ given by
\eqref{eta}, and the tautological map $\tau:\rho^*\Lambda^1(M,\C)
\to \Lambda^0(U,\C)$.

\begin{lemma}\label{phi.and.tau}
For every smooth section $\alpha \in
C^\infty(U,\rho^*\Lambda^1(M,\C))$ we have 
$$
\sqrt{-1}\tau(\alpha) = \langle \phi, \eta(\alpha) \rangle \in
C^\infty(U,\C). 
$$
\end{lemma}

\proof Since the equality that we are to prove is linear with
respect to multiplication by smooth functions on $U$, we may assume
that the section $\alpha$ is the pull-back of a smooth $1$-form
$\alpha \in C^\infty(M,\Lambda^1(M,\C))$. The Lie derivative
$\LL_\phi:\Lambda^\cdot(U,\C) \to \Lambda^\cdot(U,\C)$ with respect
to the vector field $\phi$ is compatible with the projection
$\pi:\Lambda^\cdot(U,\C) \to \Lambda^\cdot(U/M,\C)$ and defines
therefore an algebra derivation $\LL_\phi:\Lambda^\cdot(U/M,\C) \to
\Lambda^\cdot(U/M,\C)$. The Cartan homotopy
formula\index{terms}{Cartan Homotopy Formula} gives
\begin{equation}\label{eqqq.1}
\LL_\phi \tau(\alpha) = \langle \phi, d^r\tau(\alpha) \rangle. 
\end{equation}
The function $\tau(\alpha)$ on $\tm$ is by definition $\R$-linear
along the fibers of the projection $\rho:\tm \to M$. The subspace
$\tau(C^\infty(M,\Lambda^1(M,\C))) \subset C^\infty(U,\C)$ of such
functions decomposes as 
$$
\tau(C^\infty(M,\Lambda^1(M,\C))) =
\tau(C^\infty(M,\Lambda^{1,0}(M))) \oplus
\tau(C^\infty(M,\Lambda^{0,1}(M,\C))), 
$$
and the group $U(1)$ acts on the components with weight\index{terms}{weight!of an $U(1)$-action} $1$ and
$-1$. Therefore the derivative $\LL_\phi$ of the $U(1)$-action acts
on the components by multiplication with $\sqrt{-1}$ and
$-\sqrt{-1}$. By definition of the involution
$\zeta\index{terms}{involution $\zeta$}$ (see 
\eqref{zeta}) this can be written as 
\begin{equation}\label{eqqq.2}
\LL_\phi \tau(\alpha) = \sqrt{-1}\tau(\zeta(\alpha)).
\end{equation}
\index{terms}{involution $\zeta$}
On the other hand, by \eqref{can.and.tau} and the definition of the
map $\eta$ we have 
\begin{equation}\label{eqqq.3}
d^r\tau(\alpha) = \can(\alpha) = \eta(\zeta(\alpha)). 
\end{equation}
\index{terms}{involution $\zeta$}
Substituting \eqref{eqqq.2} and \eqref{eqqq.3} into \eqref{eqqq.1}
gives  
$$
\sqrt{-1}\tau(\zeta(\alpha)) = \langle \phi, \eta(\zeta\index{terms}{involution $\zeta$}(\alpha)), 
$$
\index{terms}{involution $\zeta$}
which is equivalent to the claim of the lemma. 
\endproof 

\subsection{Holonomic Hodge connections}\index{terms}{Hodge
connection!holonomic}  

\punkt We will now describe a convenient way to check whether a
given Hodge connection\index{terms}{Hodge connection} $D$ on the
pair $\langle U,M \rangle$ is holonomic\index{terms}{holonomic
derivation} in the sense of \ref{holonomic}. To do this, we proceed
as follows. 

Consider the restriction $\Lambda^1(U,\C)|_M$ of the bundle
$\Lambda^1(U,\C)$ to the zero section $M \subset U \subset \tm$, and
let 
$$
\Res:\Lambda^1(U,\C)|_M \to \Lambda^1(M,\C)
$$
be the restriction map. The kernel of the map $\Res$ coincides with
the conormal bundle to the zero section $M \subset U$, which we
denote by\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$S^1(M,\C)$. The map
$\Res$ splits the 
restriction of exact sequence \eqref{ex.seq} onto the zero section
$M \subset U$, and we have the direct sum decomposition
\begin{equation}\label{hor.vert}
\Lambda^1(U,\C)|_M =
S^1(M,\C) \oplus
\Lambda^1(M,\C).   
\end{equation}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt\label{S1} The $U(1)$-action on $U \subset \tm$ leaves the
zero section $M \subset U$ invariant and defines therefore a
$U(1)$-action on the conormal bundle
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$S^1(M,\C)$. Together
with the usual real structure twisted by the action of the map
$\iota\index{terms}{involution $\iota$}:\tm \to 
\tm$, this defines a Hodge bundle\index{terms}{Hodge bundle}
structure of weight\index{terms}{weight!of a Hodge bundle} $0$ on
the bundle \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$S^1(M,\C)$. 

Note that the automorphism $\iota\index{terms}{involution
$\iota$}:\tm \to \tm$ acts as $-\id$ on the Hodge
bundle\index{terms}{Hodge bundle}\index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} $S^1(M,\C)$, so that the real structure
on $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)$ is
minus the usual one. Moreover, as a complex vector bundle the
conormal bundle $S^1(M,\C)$ to $M \subset
\tm$ is canonically isomorphic to the cotangent bundle
$\Lambda^1(M,\C)$. The Hodge type
bigrading\index{terms}{bigrading!Hodge type} on $S^1(M,\C)$ is given
in terms of this isomorphism by
$$
S^1(M,\C) = S^{1,-1}(M)
\oplus S^{-1,1}(M) \cong \Lambda^{1,0}(M)
\oplus \Lambda^{0,1}(M) = \Lambda^1(M,\C).
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt Let 
$$
C_{lin}^\infty(U,\C) = \tau(C^\infty(M,\Lambda^1(M,\C))) \subset
C^\infty(U,\C) 
$$
be the subspace of smooth functions linear along the fibers of the
canonical projection $\rho:U \subset \tm \to M$. The relative de
Rham differential defines an isomorphism
\begin{equation}\label{iso}
d^r:C_{lin}^\infty(U,\C) \to C^\infty(M,S^1(M,\C)). 
\end{equation}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
This isomorphism is compatible with the canonical Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} of
weight\index{terms}{weight!of a Hodge structure} $0$ on both 
spaces, and it is linear with respect to multiplication by smooth
functions $f \in C^\infty(M,\C)$.

\punkt \label{pr.part} Let now $D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ be a Hodge connection\index{terms}{Hodge
connection} on the pair $\langle U,M \rangle$, and let
$\Theta:\Lambda^1(U,\C) \to \rho^*\Lambda^(M,\C)$ be the
corresponding bundles map. Since $D$ is a $\C$-valued
connection\index{terms}{connections!$\C$-valued},
the restriction $\Theta|_M$ onto the zero section $M \subset M$
decomposes as
\begin{equation}\label{hor.vert.1}
\Theta = D_0 \oplus \id:S^1(M,\C) \oplus \Lambda^1(M,\C) \to 
\Lambda^1(M,\C) 
\end{equation}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
with respect to the direct sum decomposition \eqref{hor.vert} for a
certain bundle map $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$.   

\defn The bundle map \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ is called the {\em
principal part} of the Hodge connection\index{terms}{Hodge
connection!principal part of} $D$. 

\punkt Consider the map \index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} $D_0:C^\infty(M,S^1(M,\C)) \to 
C^\infty(M,\Lambda^1(M,\C))$ on the spaces of smooth sections
induced by the principal part $D_0$ of a Hodge
connection\index{terms}{Hodge connection} $D$.
Under the isomorphism \eqref{iso} this map coincides with the
restriction of the composition
$$
\Res \circ D:C^\infty(U,\C) \to C^\infty(U,\rho^*\Lambda^1(M,\C))
\to C^\infty(M,\Lambda^1(M,\C))
$$
onto the subspace $C^\infty_{lin}(U,\C) \subset
C^\infty(U,\C)$. Each of the maps $\Res$, $D$ is weakly
Hodge\index{terms}{weakly Hodge map}, so that this composition also
is weakly Hodge. Since the isomorphism \eqref{iso} is compatible
with the Hodge bundle\index{terms}{Hodge bundle} structures, this
implies that the principal part $D_0$ of the Hodge
connection\index{terms}{Hodge connection} $D$ is a weakly Hodge
bundle map\index{terms}{weakly Hodge map}. In particular, it is
purely imaginary with respect to the usual real structure on the
conormal bundle \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$S^1(M,\C)$.

\punkt We can now formulate the main result of this subsection. 

\begin{lemma}\label{aux1}
A Hodge connection\index{terms}{Hodge connection!holonomic}
$D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ on the pair $\langle
U,M \rangle$ is holonomic in the sense of \ref{holonomic} on an open
neighborhood $U_0 \subset U$ of the zero section $M \subset U$ if
and only if its principal part \index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} $D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ 
is a complex vector bundle isomorphism.
\end{lemma}

\proof By definition the derivation $D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is holonomic\index{terms}{holonomic
derivation} in the sense of \ref{holonomic} if and only if the
corresponding map
$$
\Theta:\Lambda^1(U,\R) \to \rho^*\Lambda^1(M,\C)
$$
is an isomorphism of real vector bundles. This is an open condition.
Therefore the derivation $D$ is holonomic\index{terms}{holonomic
derivation} on an open neighborhood $U_0 \supset M$ of the zero
section $M \subset U$ if and only if the map $\Theta$ is an
isomorphism on the zero section $M \subset U$ itself.

According to \eqref{hor.vert.1}, the restriction $\Theta|_M$
decomposes as $\Theta_M = D_0 + \id$, and the principal part
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$D_0:S^1(M,\C) \to
\Lambda^1(M,\C)$ of the Hodge connection\index{terms}{Hodge
connection} $D$ is purely imaginary with respect to the usual real
structure on $\Lambda^1(U,\C)|_M$, while the identity map
$\id:\rho^*\Lambda^1(M,\C) \to
\rho^*\Lambda^1(M,\C)$ is, of course, real. Therefore $\Theta_M$ is
an isomorphism if and only is $D_0$ is an isomorphism, which proves
the lemma.  \endproof

\subsection{Hodge connections and
linearity}\label{hodge.lin.subsec}\index{terms}{Hodge
connection}\index{terms}{linearity condition}

\punkt \label{aux} Assume now given a Hodge
manifold\index{terms}{Hodge manifold} structure on the subset $U
\subset \tm$, and let $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$
be the associated Hodge connection\index{terms}{Hodge connection} on
the pair $\langle U, M\rangle$ given by Proposition~\ref{equiv}. We
now proceed to rewrite the linearity
condition\index{terms}{linearity condition} \ref{lin.def} in terms
of the Hodge connection $D$.

Let $j:\Lambda^1(U,\C) \to \overline{\Lambda^1(U,\C)}$ be the
canonical map defined by the quaternionic
structure\index{terms}{quaternionic structure} on $U$, and let
$\iota\index{terms}{involution $\iota$}^*:\Lambda^1(U,\C) \to
\iota\index{terms}{involution $\iota$}^*\Lambda^1(U,\C)$ be the
action of the canonical involution $\iota\index{terms}{involution $\iota$}:U \to U$. Let also $D^\iota\index{terms}{involution $\iota$}:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be the operator
$\iota\index{terms}{involution $\iota$}^*$-conjugate to the Hodge
connection $D$.

We begin with the following identity.

\begin{lemma}\label{aux.lemma}
For every smooth function $f \in C^\infty(U,\C)$ we have 
$$
d^rf = \halfi\pi(j(\del_\rho(D-D^\iota)(f))),
$$
\index{terms}{involution $\iota$}
where $\pi:\Lambda^1(U,\C) \to \Lambda^1(U/M,\C)$ is the canonical
projection, and 
$$
\del_\rho:\rho^*\Lambda^1(M,\C) \to \Lambda^1(U,\C)
$$ 
is the codifferential of the projection
$\rho:U \to M$.
\end{lemma}

\proof By definition of the Hodge connection\index{terms}{Hodge
connection} $D$ the Dolbeault derivative $\bar\6_Jf$ coincides with
the $(0,1)$-component of the $1$-form $\del_\rho(Df) \in
\Lambda^1(U,\C)$ with respect to the
complementary\index{terms}{complementary complex structure} complex
structure $U_J$ on $U$. Therefore
$$
\bar\6_J f = \half \del_\rho(Df) + \halfi j(\del_\rho(Df)). 
$$
Applying the complex conjugation $\nu:\Lambda^\cdot(U,\C) \to
\overline{\Lambda^\cdot(U,\C)}$ to this equation, we get
\begin{align*}
\begin{split}
\6_J f &= \nu\left(\half \del_\rho(D\nu(f)) + \halfi
j(\del_\rho(D\nu(f))))\right) = \\
&= \half \nu(\del_\rho(D\nu(f))) - \halfi j(\nu(\del_\rho(D\nu(f)))). 
\end{split}
\end{align*}
Since the map $\del_\rho \circ D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is weakly Hodge\index{terms}{weakly Hodge
map}, we have
$$
\del_\rho(D(\iota^*\nu(f))) = \iota^*\nu(\del_\rho(Df)).
$$
\index{terms}{involution $\iota$}
Therefore $\nu(\del_\rho(D(f))) = \del_\rho(D^\iota(\nu(f)))$, and
we have 
$$
\6_J f = \half \del_\rho(D^\iota f) - \halfi j(\del_\rho(D^\iota f)). 
$$
\index{terms}{involution $\iota$}
Thus the de Rham derivative $d_Uf$ equals 
$$
d_Uf = \6_Jf + \bar\6_Jf = \half \del_\rho((D+D^\iota)f) + \halfi
j(\del_\rho((D-D^\iota)f)). 
$$
\index{terms}{involution $\iota$}
Now, by definition $\del_\rho \circ \pi = 0$. Therefore 
$$
d^rf = \pi(d_Uf) = \halfi\pi(j(\del_\rho((D-D^\iota)f))),
$$
\index{terms}{involution $\iota$}
which is the claim of the lemma. 
\endproof 

\punkt We will also need the following fact. It can be derived
directly from Lemma~\ref{aux.lemma}, but it is more convenient to
use Lemma~\ref{aux1} and the fact that the Hodge
connection\index{terms}{Hodge connection!holonomic}
$D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is holonomic.

\begin{lemma}\label{aux3}
In the notation of Lemma~\ref{aux.lemma}, let 
$$
\A = \del_\rho\left((D-D^\iota)\left(C_{lin}^\infty(U,\C)\right)\right)
\subset C^\infty(U, \rho^*\Lambda^1(M,\C))
$$ 
\index{terms}{involution $\iota$}
be the subspace of sections $\alpha \in
C^\infty(U,\rho^*\Lambda^1(M,\C))$ of the form $\alpha =
\del_\rho((D-D^\iota\index{terms}{involution $\iota$})f)$, where $f
\in C^\infty(U,\C)$ lies in the 
subspace $C_{lin}^\infty(U,\C) \subset C^\infty(U,\C)$ of smooth
functions on $U$ linear along the fibers of the projection $\rho:U
\to M$. The restriction $\Res(\A) \subset
C^\infty(M,\Lambda^1(M,\C))$ of the subspace $\A$ onto the zero
section $M \subset U$ is the whole space
$C^\infty(M,\Lambda^1(M,\C))$.
\end{lemma}

\proof Let $D_0 = \Res \circ D:C^\infty_{lin}(U,\C) \to
C^\infty(M,\Lambda^1(M,\C))$ be the principal part of the Hodge
connection\index{terms}{Hodge connection!principal part of} $D$ in
the sense of Definition~\ref{pr.part}. Since the canonical
automorphism $\iota\index{terms}{involution $\iota$}:\tm \to \tm$
acts as $-\id$ on $C_{lin}^\infty(U,\C)$, we have
$D_0^\iota\index{terms}{involution $\iota$} = - D_0$. Therefore
\begin{multline*}
\Res(\A) = \Res \circ (D - D^\iota)\left(C_{lin}^\infty(U,\C)\right) = \\ 
=(D_0-D^\iota_0)\left(C_{lin}^\infty(U,\C)\right) = 
D_0\left(C_{lin}^\infty(U,\C)\right). 
\end{multline*}
\index{terms}{involution $\iota$}
Since the Hodge connection\index{terms}{Hodge connection!holonomic}
$D$ is holonomic, this space coincides with the whole
$C^\infty(M,\Lambda^1(M,\C))$ by Lemma~\ref{aux1}.  \endproof

\punkt We now apply Lemma~\ref{aux.lemma} to prove the following
criterion for the linearity\index{terms}{linearity condition} of the
Hodge manifold\index{terms}{Hodge manifold} structure on $U$ defined
by the Hodge connection\index{terms}{Hodge connection}
$D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$.

\begin{lemma}\label{explicit.lin}
The Hodge manifold structure on $U \subset \tm$ corresponding to a
Hodge connection $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is
linear\index{terms}{linearity condition} in the sense of
\ref{lin.def} if and only if for every smooth function $f \in
C^\infty(U,\C)$ linear along the fibers of the projection $\rho:U
\subset \tm \to M$ we have
\begin{equation}\label{expl.lin}
f = \half\sigma\left((D-D^\iota)f\right), 
\end{equation}
\index{terms}{involution $\iota$}
where $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0(U,\C)$ is the
twisted tautological map introduced in \eqref{sigma1}, and
$D^\iota\index{terms}{involution $\iota$}:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is the operator 
$\iota\index{terms}{involution $\iota$}^*$-conjugate to $D$, as in \ref{aux}.
\end{lemma}

\proof By Lemma~\ref{lin.char} the Hodge manifold\index{terms}{Hodge
manifold} structure on $U$ is linear\index{terms}{Hodge
manifold!linear} if and only if for every $\alpha \in
C^\infty(U,\rho^*\Lambda^1(M,\C))$ we have
\begin{equation}\label{eee.to.prove}
\langle \phi, j(\alpha) \rangle = \tau(\alpha),  
\end{equation}
where $\phi$ is the differential of the $U(1)$-action on $U$,
$j:\Lambda^1(U,\C) \to \overline{\Lambda^1(U,\C)}$ is the operator
given by the quaternionic structure\index{terms}{quaternionic
structure} on $U$, and $\tau:\rho^*\Lambda^1(M,\C) \to
\Lambda^0(U,\C)$ is the tautological map sending a $1$-form on $M$
to the corresponding linear function on $\tm$, as in
\ref{tau}. Moreover, by Lemma~\ref{aux3} and Lemma~\ref{aux2} the
equality \eqref{eee.to.prove} holds for all smooth sections $\alpha
\in C^\infty(U,\rho^*\Lambda^1(M,\C))$ if and only if it holds for
sections of the form
\begin{equation}\label{special}
\alpha = \halfi\del_\rho((D-D^\iota)f), 
\end{equation}
\index{terms}{involution $\iota$}
where $f \in C_{lin}^\infty(U,\C) \subset C^\infty(U,\C)$ is linear
along the fibers of $\rho:U \to M$. 

Let now $f \in C^\infty(U,\C)$ be a smooth function on $U$ linear
along the fibers of $\rho:U \to M$, and let $\alpha$ be as in
\eqref{special}. Since $\phi$ is a vertical vector field on $U$ over
$M$, we have $\langle \phi, j(\alpha) \rangle = \langle \phi,
\pi(j(\alpha))\rangle$, where $\pi:\Lambda^1(U,\C) \to
\Lambda^1(U/M,\C)$ is the canonical projection. By
Lemma~\ref{aux.lemma}
\begin{equation}\label{thelema}
\langle \phi, j(\alpha) \rangle = \langle \phi, \pi(j(\alpha))
\rangle = \langle \phi, d^rf \rangle. 
\end{equation}
Since the function $f$ is linear along the fibers of $\rho:U \to M$,
we can assume that $f = \sigma(\beta)$ for a smooth $1$-form $\beta
\in C^\infty(M,\Lambda^1(M,\C)$. Then by \eqref{eta.and.sigma} and
by Lemma~\ref{phi.and.tau} the right hand side of \eqref{thelema} is
equal to
$$
\langle \phi, d^rf \rangle = \langle \phi, d^r(\sigma(\beta))
\rangle = \langle \phi, \eta(\beta) \rangle = \sqrt{-1}
\tau(\beta). 
$$
Therefore, \eqref{eee.to.prove} is equivalent to 
\begin{equation}\label{horus}
\sqrt{-1}\tau(\beta) =
\tau\left(\halfi\del_\rho((D-D^\iota)\sigma(\beta))\right).
\end{equation}
\index{terms}{involution $\iota$}
But we have $\tau = \sigma \circ \zeta\index{terms}{involution
$\zeta$}$, where $\zeta\index{terms}{involution
$\zeta$}:\rho^*\Lambda^1(M,\C) \to \rho^*\Lambda^1(M,\C)$ is the 
invulution introduced in \eqref{zeta}. In particular, the map
$\zeta\index{terms}{involution $\zeta$}$ is invertible, so that
\eqref{horus} is in turn equivalent 
to
$$
\sigma(\beta) = \half\sigma(\del_\rho((D-D^\iota)\sigma(\beta))),
$$
\index{terms}{involution $\iota$}
or, substituting back $f = \sigma(\beta)$, to 
$$
f = \half\sigma(\del_\rho((D-D^\iota)f)),
$$
\index{terms}{involution $\iota$}
which is exactly the condition \eqref{expl.lin}. 
\endproof 

\punkt \defn \label{hodge.conn.lin} 
A Hodge connection\index{terms}{Hodge connection!linear} $D$ on the
pair $\langle U, M \rangle$ is called {\em linear} if it satisfies
the condition~\ref{expl.lin}.

We can now formulate and prove the following more useful version of
Proposition~\ref{equiv}.

\begin{prop}\label{equiv.bis}
Every linear Hodge connection\index{terms}{Hodge connection!linear}
$$
D\!:\!\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)
$$ 
on the pair $\langle U, M\rangle$ defines a linear Hodge
manifold\index{terms}{Hodge manifold!linear} structure on an open
neighborhood $V \subset U$ of the zero section $M \subset U$, and
the canonical projection $\rho:V_I \to M$ is holomorphic for the
preferred\index{terms}{preferred complex structure} complex
structure $V_I$ on $V$. Vice versa, every such linear Hodge manifold
structure on $U$ comes from a unique linear Hodge connection $D$ on
the pair $\langle U,M \rangle$.
\end{prop}

\proof By Proposition~\ref{equiv} and Lemma~\ref{explicit.lin}, to
prove this proposition suffices to prove that if a Hodge
connection\index{terms}{Hodge connection} $D:\Lambda^0(U,\C) \to
\rho^*\Lambda^1(M,\C)$ is linear, then it is
holonomic\index{terms}{Hodge connection!holonomic} in the sense
of \ref{holonomic} on a open neighborhood $V \subset U$ of the zero
section $M \subset U$. Lemma~\ref{aux1} reduces this to proving that
the principal part \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$D_0:S^1(M,\C) \to \Lambda^1(M,\C)$ of a linear Hodge
connection\index{terms}{Hodge connection!linear} 
$D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ is a bundle
isomorphism.

Let $D:\Lambda^0(U,\C) \to \rho^*\Lambda^1(M,\C)$ be such
connection. By \eqref{expl.lin} we have
$$
\half \sigma \circ (D_0 -
D_0^\iota) =
\id:S^1(M,\C) \to 
\Lambda^1(M,\C) \to S^1(M,\C).
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\index{terms}{involution $\iota$}
Since $\sigma:\Lambda^1(M,\C) \to
S^1(M,\C)$\index{terms}{Hodge bundle!$S^1(M,\protect\C)$} is a
bundle isomorphism, so is the bundle map $D_0 - D_0^\iota
\index{terms}{involution $\iota$}:
S^1(M,\C) \to \Lambda^1(M,\C)$. As in the proof of Lemma~\ref{aux3},
we have $D_0 = - D_0^\iota\index{terms}{involution $\iota$}$. Thus
$D_0 = \half(D_0-D_0^\iota\index{terms}{involution $\iota$}):
S^1(M,\C) \to \Lambda^1(M,\C)$ also is a bundle isomorphism, which
proves the proposition.
\endproof

\section{Formal completions}\label{formal.section}

\subsection{Formal Hodge manifolds}\index{terms}{Hodge manifold!formal}

\punkt Proposition~\ref{equiv.bis} reduces the study of arbitrary
regular Hodge manifolds\index{terms}{Hodge manifold!regular} to the
study of connections of a certain type on a neighborhood $U \subset
\tm$ of the zero section $M \subset \tm$ in the total space $\tm$ of
the tangent bundle to a complex manifold $M$.  To obtain further
information we will now restrict our attention to the {\em formal}
neighborhood\index{terms}{formal neighborhood} of this zero
section. This section contains the appropriate definitions. We study
the convergence of our formal series in Section~\ref{convergence}.

\punkt Let $X$ be a smooth manifold and let $\Bun(X)$ be the category of
smooth real vector bundles over $X$. Let also $\Diff(X)$ be the
category with the same objects as $\Bun(X)$ but with differential operators
as morphisms.

Consider a closed submanifold $Z \subset X$. For every two real
vector bundles $\E$ and $\F$ on $X$ the vector space $\Hom(\E,\F)$
of bundle maps from $\E$ to $\F$ is naturally a module over the ring
$C^\infty(X)$ of smooth functions on $X$.  Let $\J_Z \subset
C^\infty(X)$ be the ideal of functions that vanish on $Z$ and let
$\Hom_Z(\E,\F)$ be the $\J_Z$-adic completion\index{terms}{formal
completion} of the $C^\infty(X)$-module $\Hom(\E,\F)$.

For any three bundles $\E,\F,\G$ the composition map 
$$
\Mult:\Hom(\E,\F) \otimes \Hom(\F,\G) \to \Hom(\E,\G)
$$ 
is $C^\infty(X)$-linear, hence extends to a map
$$
\Mult:\Hom_Z(\E,\F) \otimes \Hom_Z(\F,\G) \to \Hom_Z(\E,\G). 
$$
Let $\Bun_Z(X)$ be the category with the same objects as $\Bun(X)$
and for every two objects $\E$, $\F \in \Ob\Bun(X)$ with
$\Hom_Z(\E,\F)$ as the space of maps between $\F$ anf $\F$.  The
category $\Bun_Z(X)$, as well as $\Bun(X)$, is an additive tensor
category.

\punkt The space of differential operators $\Diff(\E,\F)$ is also a
$C^\infty(X)$ module, say, by left multiplication. Let $\Diff_Z(\E,\F)$ be
its $\J_Z$-completion. The composition maps in $\Diff(X)$ are no longer
$C^\infty(X)$-linear. However, they still are compatible with the
$\J_Z$-adic topology, hence extend to completions. Let $\Diff_Z(X)$ be the
category with the same objects as $\Bun(X)$ and with $\Diff_Z(\E,\F)$ as
the space of maps between two objects $\E,\F \in \Ob \Bun(X)$.

By construction we have canonical {\em $Z$-adic
completion\index{terms}{formal completion} functors}
$$
\Bun(X) \to \Bun_Z(X) \text{  and  } \Diff(X) \to \Diff_Z(X). 
$$
Call the categories $\Bun_Z(X)$ and $\Diff_Z(X)$ {\em the $Z$-adic
completions\index{terms}{formal completion}} of the categories
$\Bun(X)$ and $\Diff(X)$.

\punkt When the manifold $X$ is equipped with a smooth action of
compact Lie group $G$ fixing the submanifold $Z$, the completion
construction extends to the categories of $G$-equivariant bundles on
$M$.  When $G = U(1)$, the categories $\prehodge(X)$ and
$\prehodge^\D(X)$ defined in \ref{w.hodge} also admit canonical
completions, denoted by $\prehodge_Z(X)$ and $\prehodge_Z^\D(X)$.

\punkt Assume now that the manifold $X$ is equipped with a smooth
$U(1)$-action fixing the smooth submanifold $Z \subset X$. 

\defn A {\em formal quaternionic structure\index{terms}{quaternionic
structure!formal}} on $X$ along the submanifold $Z \subset X$ 
is given by an algebra map
$$
\Mult: \h \to \Eend_{\Bun_Z(X)} \left(\Lambda^1(X)\right)
$$
from the algebra $\h$ to the algebra $\Eend_{\Bun_Z(X)}
\left(\Lambda^1(X)\right)$ of endomorphisms of the cotangent bundle
$\Lambda^1(X)$ in the category $\Bun_Z(X)$.
A formal quaternionic structure\index{terms}{quaternionic
structure!formal equivariant} is called {\em equivariant} if the
map $\Mult$ is equivariant with respect to the natural $U(1)$-action
on both sides. 

\punkt Note that Lemma~\ref{universal} still holds in the situation
of formal completions\index{terms}{formal completion}. Consequently,
everything in Section~\ref{hbqm.section} carries over word-by-word
to the case of formal quaternionic
structures\index{terms}{quaternionic structure!formal}. In
particular, by Lemma~\ref{qm.hodge} 
giving a formal equivariant quaternionic
structure\index{terms}{quaternionic structure!formal equivariant} on
$X$ along $Z$ is equivalent to giving a pair $\langle
\E, D \rangle$ of a Hodge bundle\index{terms}{Hodge bundle} $\E$ on
$X$ and a holonomic\index{terms}{holonomic derivation} algebra
derivation $D:\Lambda^0(X) \to \E$ in $\prehodge_Z^\D(X)$.

\punkt The most convenient way to define Hodge
manifold\index{terms}{Hodge manifold} structures on $X$ in a formal
neighborhood\index{terms}{formal neighborhood} of $Z$ is by means of
the Dolbeault complex, as in Proposition~\ref{explicit.hodge}.

\defn A {\em formal Hodge manifold\index{terms}{Hodge
manifold!formal} structure} on $X$ along $Z$ is a pair of a Hodge
bundle\index{terms}{Hodge bundle} $\E \in \Ob \prehodge_Z(X)$ of
weight\index{terms}{weight!of a Hodge bundle} $1$ and an algebra
derivation $D^\cdot:\Lambda^\cdot\E \to
\Lambda^{\cdot+1}\E$ in $\prehodge^\D_Z(X)$ such that
$D^0:\Lambda^0(\E) \to \E$ is holonomic\index{terms}{holonomic
derivation} and $D^0 \circ D^1 = 0$. 

\punkt Let $U \subset X$ be an open subset containing $Z \subset
X$. For every Hodge manifold\index{terms}{Hodge manifold} structure
on $U$ the $Z$-adic completion\index{terms}{formal completion} functor
defines a formal Hodge manifold\index{terms}{Hodge manifold!formal}
structure on $X$ along $Z$. Call it {\em the $Z$-adic
completion\index{terms}{formal completion}} of the given structure on
$U$.

\rem Note that a Hodge manifold\index{terms}{Hodge manifold}
structure on $U$ is completely defined by the
preferred\index{terms}{preferred complex structure} and the
complementary\index{terms}{complementary complex structure} complex
structures $U_I$, $U_J$, hence always real-analytic by the
Newlander-Nirenberg Theorem\index{terms}{Newlander--Nirenberg
Theorem}.  Therefore, if two Hodge manifold structures on $U$ have
the same completion, they coincide on every connected component of
$U$ intersecting $Z$.

\subsection{Formal Hodge manifold structures on tangent
bundles}\index{terms}{Hodge manifold!formal}

\punkt Let now $M$ be a complex manifold, and let $\tm$ be the total
space of the complex-conjugate to the tangent bundle to $M$ equipped
with an action of $U(1)$by dilatation along the fibers of the
projection $\rho:\tm \to M$. All the discussion above applies to the
case $X = \tm$, $Z = M \subset \tm$. Moreover, the linearity
condition\index{terms}{linearity condition} in the form given in
Lemma~\ref{lin.char} generalizes immediately to the formal case.

\defn A formal Hodge manifold\index{terms}{Hodge manifold!formal}
structure on $\tm$ along $M$ is called {\em
linear\index{terms}{Hodge manifold!linear}} if for every
smooth $(0,1)$-form $\alpha \in C^\infty(M,\Lambda^{0,1}(M))$ we
have
$$
\tau(\alpha) = \langle \phi, j(\rho^*) \rangle \in
C^\infty_M(\tm,\C), 
$$
where $j$ is the map induced by the formal quaternionic
structure\index{terms}{quaternionic structure!formal} on $\tm$ and
$\phi$ and $\tau$ are as in Lemma~\ref{lin.char}.

\punkt As in the non-formal case, linear Hodge
manifold\index{terms}{Hodge manifold!linear} structures
on $\tm$ along $M \subset \tm$ can be described in terms of
differential operators of certain type. 

\defn \label{formal.hodge.con} A {\em formal Hodge
connection\index{terms}{Hodge connection!formal}} on $\tm$ along $M
\subset \tm$ is an algebra derivation
$$
D:\Lambda^0(\tm,\C) \to \rho^*\Lambda^1(M,\C)
$$
in $\prehodge_M^\D(\tm)$ such that for every smooth function $f \in
C^\infty(M,\C)$ we have $D\rho^*=\rho^*d_Mf$, as in
\eqref{conn.eq}. A formal Hodge connection\index{terms}{Hodge
connection!formal} is called {\em flat} if it extends to an algebra
derivation
$$
D:\rho^*\Lambda^\cdot(M,\C) \to \Lambda^{\cdot+1}(M,\C) 
$$
in $\prehodge_M^\D(\tm)$ such that $D \circ D = 0$. A formal Hodge
connection\index{terms}{Hodge connection!formal} is called {\em
linear\index{terms}{Hodge connection!linear}} if it satisfies the
condition \eqref{expl.lin} of Lemma~\ref{explicit.lin}, that is, for
every function $f \in C^\infty_{lin}(\tm,\C)$ linear along the
fibers of the projection $\rho:\tm \to M$ we have
$$
f = \half\sigma\left((D-D^\iota)f\right), 
$$
\index{terms}{involution $\iota$}
where $\sigma:\rho^*\Lambda^1(M,\C) \to \Lambda^0(\tm,\C)$ is the
twisted tautological map introduced in \eqref{sigma1}, the
automorphism $\iota\index{terms}{involution $\iota$}:\tm \to \tm$ is
the multiplication by $-1 \in
\C$ on every fiber of the projection $\rho:\tm \to M$, and
$D^\iota\index{terms}{involution $\iota$}:\Lambda^0(\tm,\C) \to
\rho^*\Lambda^1(M,\C)$ is the 
operator $\iota\index{terms}{involution $\iota$}^*$-conjugate to
$D$, as in \ref{aux}. 

The discussion in Section~\ref{section.5} generalizes immediately to
the formal case and gives the following. 

\begin{lemma}\label{form.hdg} 
Linear formal Hodge manifold\index{terms}{Hodge manifold!formal}
structures on $\tm$ along the zero section $M \subset \tm$ are in a
natural one-to-one correspondence with linear flat formal Hodge
connections\index{terms}{Hodge
connection!formal}\index{terms}{Hodge connection!linear} on $\tm$ along
$M$.
\end{lemma}


\subsection{The Weil algebra}\index{terms}{Weil algebra}

\punkt Let, as before, $M$ be a complex manifold and let $\tm$ be
the total space of the complex conjugate to its tangent bundle, as
in \ref{overline.T}. In the remaining part of this section we give a
description of the set of all formal Hodge
connections\index{terms}{Hodge connection!formal} on $\tm$ along $M$
in terms of certain differential operators on $M$ rather than on
$\tm$. We call such operators {\em extended\index{terms}{extended
connection} connections} on $M$ (see
\ref{ext.con} for the definition). Together with a complete
classification of extended connections given in the next Section,
this description provides a full classification of regular Hodge
manifolds\index{terms}{Hodge manifold!regular} ``in the formal
neighborhood\index{terms}{formal neighborhood} of the subset of
$U(1)$-fixed points''.

\punkt Before we define extended\index{terms}{extended connection}
connections in Subsection~\ref{ext.con.subsec}), we need to
introduce a certain algebra bundle in $\prehodge(M)$ which we call
{\em the Weil algebra\index{terms}{Weil algebra}}.  We begin with
some preliminary facts.

Recall (see, e.g., \cite{Del}) that every additive category $\A$
admits a canonical completion $\llim\A$ with respect to filtered
projective limits. The category $\llim\A$ is also additive, and it
is tensor if $\A$ was tensor.  Objects of the canonical completion
$\llim\A$ are called {\em pro-objects in $\A$}.

\punkt\label{rho_*} 
Let $\rho:\tm \to M$ be the canonical projection. Extend the
pullback functor $\rho^*:\Bun(M) \to \Bun(\tm)$ to a functor
$$
\rho^*:\Bun(M) \to \Bun_M(\tm) 
$$
to the $M$-adic completion\index{terms}{formal completion}
$\Bun_M(\tm)$. The functor $\rho^*$ admits a right adjoint direct
image functor
$$
\rho_*:\Bun_M(\tm) \to \llim\Bun(M).
$$
Moreover, the functor $\rho_*$ extends to a functor
$$
\rho_*:\Diff_M(\tm) \to \llim\Diff(M). 
$$
Denote by $\B^0(M,\C) = \rho_*\Lambda^0(\tm)$ the direct image under
the projection $\rho:\tm \to M$ of the trivial bundle
$\Lambda^0(\tm)$ on $\tm$.

The compact Lie group $U(1)$ acts on $\tm$ by dilatation along the
fibers, and the functor $\rho_*:\Diff_M(\tm) \to \llim\Diff(M)$
obviously extends to a functor $\rho_*:\prehodge^\D_M(\tm) \to
\llim\prehodge(M)$. The restriction of $\rho_*$ to the subcategory
$\prehodge_M(\tm) \subset \prehodge^\D_M(\tm)$ is adjoint on the
right to the pullback functor $\rho^*:\prehodge(M) \to
\prehodge_M(\tm)$.

\punkt The constant bundle $\Lambda^0(\tm)$ is canonically a Hodge
bundle\index{terms}{Hodge bundle} of weight\index{terms}{weight!of a
Hodge bundle} $0$. Therefore $\B^0(M,\C) = \rho_*\Lambda^0(M,\C)$ is
also a Hodge bundle of weight\index{terms}{weight!of a Hodge bundle}
$0$.  Moreover, it is a commutative algebra bundle in
$\llim\prehodge_0(M)$. Let $S^1(M,\C)$ be the conormal bundle to the zero
section $M \subset \tm$ equipped with a Hodge bundle structure of
weight\index{terms}{weight!of a Hodge bundle} $0$ as in \ref{S1},
and denote by $S^i(M,\C)$ the $i$-th symmetric power of the Hodge
bundle $S^1(M,\C)$. Then the algebra bundle $\B^0(M,\C)$ in
$\llim\prehodge_0(M)$ is canonically isomorphic
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
\B^0(M,\C) \cong \widehat{S}^\cdot(M,\C)
$$
to the completion $\widehat{S}^\cdot(M,\C)$ of the symmetric algebra
$S^\cdot(M,\C)$ of the Hodge bundle\index{terms}{Hodge bundle}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$S^1(M,\C)$ with
respect to the augmentation ideal $S^{>0}(M,\C)$. 

Since the $U(1)$-action on $M$ is trivial, the category
$\prehodge(M)$ of Hodge bundles\index{terms}{Hodge bundle} on $M$ is
equivalent to the category of pairs $\langle \E, \conj \rangle$ of a
complex bundle $\E$ equipped with a Hodge type
bigrading\index{terms}{bigrading!Hodge type}
$$
\E = \bigoplus_{p,q} \E^{p,q}
$$
and a real structure $\conj:\E^{p,q} \to \overline{\E^{q,p}}$. The
Hodge type bigrading on $\B^0(M,\C)$ is induced by the Hodge type
bograding \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$S^1(M,\protect\C)
= S^{1,-1}(M) \oplus S^{-1,1}(M)$ on the 
generators subbundle $S^1(M,\C) \subset \B^0(M,\C)$, which was 
described in \ref{S1}.

\rem The complex vector bundle $S^1(M,\C)$ is canonically isomorphic 
to the cotangent bundle $\Lambda^1(M,\C)$. However, the Hodge
bundle\index{terms}{Hodge bundle}\index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} 
structures on these two bundles are different (in fact, they have
different weight\index{terms}{weight!of a Hodge bundle}s).

\punkt \label{Weil.defn}
Consider the pro-bundles  
$$
\B^\cdot(M,\C) = \rho_*\rho^*\Lambda^\cdot(M,\C)
$$
on $M$. The direct sum $\oplus\B^\cdot(M,\C)$ is a graded algebra in
$\llim\Bun(M,\C)$. Moreover, since for every $i \geq 0$ the bundle
$\Lambda^i(M,\C)$ is a Hodge bundle\index{terms}{Hodge bundle} of
weight\index{terms}{weight!of a Hodge bundle} $i$ (see \ref{de.Rham}), $\B^i(M,\C)$ is also a Hodge bundle
of weight\index{terms}{weight!of a Hodge bundle} $i$. Denote by
$$
\B^i(M,\C) = \bigoplus_{p+q=i} \B^{p,q}(M,\C) 
$$
the Hodge type bigrading\index{terms}{bigrading!Hodge type} on $\B^i(M,\C)$. 

The Hodge bundle\index{terms}{Hodge bundle} structures on
$\B^\cdot(M,\C)$ are compatible with the multiplication. By the
projection formula
$$
\B^\cdot(M,\C) \cong \B^0(M,\C) \otimes \Lambda^\cdot(M,\C), 
$$
and this isomorphism is compatible with the Hodge bundle structures on
both sides. 

\defn Call the algebra $\B^\cdot(M,\C)$ in $\llim\prehodge(M)$ {\em
the Weil algebra\index{terms}{Weil algebra}} of the complex manifold $M$. 

\punkt \label{iota.Weil} 
The canonical involution $\iota\index{terms}{involution $\iota$}:\tm
\to \tm$ induces an algebra 
involution $\iota\index{terms}{involution $\iota$}^*:\B^\cdot(M,\C)
\to \B^\cdot(M,\C)$. It acts on generators as follows
$$
\iota^* = -\id:S^1(M,\C) \to S^1(M,\C) \qquad  
\iota^* =  \id:\Lambda^1(M,\C) \to \Lambda^1(M,\C).   
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\index{terms}{involution $\iota$}
For every operator $N:\B^p(M,\C) \to \B^q(M,\C)$, $p$ and $q$
arbitrary, we will denote by
$$
N^\iota = \iota^* \circ N \circ \iota^*:\B^p(M,\C) \to
\B^q(M,\C)
$$ 
\index{terms}{involution $\iota$}
the operator $\iota^*$-conjugate to $N$. 

\punkt \label{sigma}
The twisted tautological map $\sigma:\rho^*\Lambda^1(M,\C) \to
\Lambda^0(\tm,\C)$ introduced in \ref{sigma1} induces via the
functor $\rho_*$ a map $\sigma:\B^1(M,\C) \to \B^0(M,\C)$. Extend
this map to a derivation
$$
\sigma:\B^{\cdot+1}(M,\C) \to \B^\cdot(M,\C)
$$
by setting $\sigma = 0$ on $S^1(M,\C) \subset \B^0(M,\C)$.  The 
derivation $\B^{\cdot+1}(M,\C) \to \B^\cdot(M,\C)$ is not weakly
Hodge\index{terms}{weakly Hodge map}. However, it is real with
respect to the real structure on the Weil algebra\index{terms}{Weil
algebra} $\B^\cdot(M,\C)$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt \label{C.Weil} 
By definition of the twisted tautological map (\ref{sigma1},
\ref{tau}), the derivation $\sigma:\B^{\cdot+1}(M,\C) \to
\B^\cdot(M,\C)$ maps the subbundle $\Lambda^1(M,\C) \subset
\B^1(M,\C)$ to the subbundle $S^1(M,\C) \subset \B^0(M,\C)$ and 
defines a complex vector bundle isomorphism $\sigma:\Lambda^1(M,\C)
\to S^1(M,\C)$. To describe this isomorphism explicitly, recall that 
sections of the bundle $\B^0(M,\C)$ are the same as formal germs
along $M \subset \tm$ of smooth functions on the manifold $\tm$. The
sections of the subbundle $S^1(M,\C) \subset \B^0(M,\C)$ form the 
subspace of functions linear along the fibers of the canonical
projection $\rho:\tm \to M$. The isomorphism $\sigma:\Lambda^1(M,\C)
\to S^1(M,\C)$ induces an isomorphism between the space of smooth
$1$-forms on the manifold $M$ and the space of smooth functions on
$\tm$ linear long the fibers of $\rho:\tm \to M$. This isomorphism
coincides with the tautological one on the subbundle $\Lambda^{1,0}
\subset \Lambda^1(M,\C)$, and it is minus the tautological
isomorphism on the subbundle $\Lambda^{0,1} \subset
\Lambda^1(M,\C)$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

Denote by \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
C = \sigma^{-1}:S^1(M,\C) \to \Lambda^1(M,\C) 
$$ 
the bundle isomorphism inverse to $\sigma$. Note that the complex
vector bundle isomorphism $\sigma:\Lambda^1(M,\C) \to S^1(M,\C)$ is
real. Moreover, it sends the subbundle $\Lambda^{1,0}(M) \subset
\Lambda^1(M,\C)$ to $S^{1,-1}(M) \subset S^1(M,\C)$, and it sends 
$\Lambda^{0,1}(M)$ to $S^{-1,1}(M)$. Therefore the inverse
isomorphism $C:S^1(M,\C) \to \Lambda^1(M,\C)$ is weakly
Hodge\index{terms}{weakly Hodge map}. It coincides with the
tautological isomorphism on the subbundle $S^{1,-1} \subset
S^1(M,\C)$, and it equals minus the tautological isomorphism on the
subbundle $S^{-1,1} \subset S^1(M,\C)$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\subsection{Extended connections}\label{ext.con.subsec}

\punkt We are now ready to introduce the
extended\index{terms}{extended connection} connections. Keep the
notation of the last subsection.

\defn \label{ext.con} An {\em extended\index{terms}{extended
connection} connection} on a complex manifold $M$ is a differential
operator \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$D:S^1(M,\C) \to
\B^1(M,\C)$ which is weakly Hodge\index{terms}{weakly Hodge map} in
the sense of
\ref{w.hodge} and satisfies
\begin{equation}\label{e.c}
D(fa) = fDa + a \otimes df
\end{equation}
for any smooth function $f$ and a local section $a$ of the pro-bundle
$\B^0(M,\C)$. 

\punkt \label{red}
Let $D$ be an extended\index{terms}{extended connection} connection
on the manifold $M$. By \ref{Weil.defn} we have canonical bundle
isomorphisms
$$
\B^1(M,\C) \cong \B^0(M,\C) \otimes \Lambda^1(M,\C) \cong \bigoplus_{i
\geq 0} S^i(M,\C) \otimes \Lambda^1(M,\C). 
$$
Therefore the operator \index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} $D:S^1 \to \B^1$ decomposes
\begin{equation}\label{aug.con}
D = \sum_{p \geq 0}D_p, \quad D_p:S^1(M,\C) \to S^i(M,\C) \otimes
\Lambda^1(M,\C).  
\end{equation}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
By \eqref{e.c} all the components $D_p$ except for the $D_1$ are
weakly Hodge bundle maps\index{terms}{weakly Hodge map} on $M$,
while
$$
D_1:S^1(M,\C) \to
S^1(M,\C) \otimes
\Lambda^1(M,\C)  
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
is a connection\index{terms}{connections!on a vector bundle} in the
usual sense on the Hodge bundle\index{terms}{Hodge bundle}
$S^1(M,\C)$.

\defn The weakly Hodge bundle map\index{terms}{weakly Hodge map}
$D_0:S^1(M,\C) \to
\Lambda^1(M,\C)$ is called {\em the principal 
part} of the extended\index{terms}{extended
connection!principal part of} connection $D$ on $M$. The
connection\index{terms}{connections!on a vector bundle} $D_1$ is called
{\em the reduction} of the extended\index{terms}{extended
connection!reduction of!} connection $D$.  
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt Extended connection on $M$ are related to formal Hodge
connections\index{terms}{Hodge connection!formal} on the total space
$\tm$ of the complex-conjugate to the tangent bundle to $M$ by means
of the direct image functor
$$
\rho_*:\prehodge^\D_M(\tm) \to \llim\prehodge^\D(M).
$$ 
Namely, let $D:\Lambda^0(M,\C) \to \rho^*\Lambda^1(M,\C)$ be a
formal Hodge connection on $\tm$ along $M$ in the sense of
\ref{formal.hodge.con}. The restriction of the operator
$$
\rho_*D:\B^0(M,\C) \to \B^1(M,\C)
$$
to the generators subbundle $S^1(M,\C) \subset \B^0(M,\C)$ is then an
extended\index{terms}{extended connection} connection on $M$ in the
sense of \ref{ext.con}. The principal part
$D_0:S^1(M,\C)
\to \Lambda^1(M,\C)$ of the Hodge 
connection\index{terms}{Hodge connection} $D$ in the sense of
\ref{pr.part} coincides with the principal part of the extended
connection $\rho_*D$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt We now write down the counterparts of the flatness and
linearity conditions\index{terms}{linearity condition} on a Hodge
connection\index{terms}{Hodge connection} on $\tm$ for the
associated extended connection\index{terms}{extended connection} on
$M$. We begin with the linearity condition \ref{hodge.conn.lin}. Let
$D:S^1(M,\C) \to \B^1(M,\C)$ be an extended connection on $M$, let
$\sigma:\B^{\cdot+1}(M,\C) \to \B^\cdot(M,\C)$ be the algebra
derivation introduced in \ref{sigma}, and let
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
D^\iota:\B^\cdot(M,\C) \to
\B^{\cdot+1}(M,\C)  
$$
\index{terms}{involution $\iota$}
be the operator $\iota^*$-conjugate
to $D$ as in \ref{iota.Weil}.  

\defn \label{lin.ext.con}
An extended\index{terms}{extended connection!linear} connection $D$
is called {\em linear\index{terms}{extended connection!linear}} if
for every local section $f$ of the bundle $S^1(M,\C)$ we have 
$$
f = \half \sigma((D-D^\iota)f).
$$
\index{terms}{involution $\iota$}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
This is, of course, the literal rewriting of
Definition~\ref{hodge.conn.lin}. In particular, a formal Hodge
connection\index{terms}{Hodge connection!formal} $D$ on $\tm$ is
linear\index{terms}{Hodge connection!linear} if and only if so is
the extended\index{terms}{extended connection} connection $\rho_*D$
on $M$.

\punkt \label{deriv} 
Next we rewrite the flatness condition \ref{hodge.con}. Again, let
$$
D:S^1(M,\C) \to \B^1(M,\C)
$$ 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
be an extended\index{terms}{extended connection} connection on $M$.
Since the algebra pro-bundle $\B^0(M,\C)$ is freely generated by the
subbundle $S^1(M,\C) \subset \B^1(M,\C)$, by \eqref{e.c} the
operator $D:S^1(M,\C) \to \B^1(M,\C)$ extends uniquely to an 
algebra derivation
$$
D:\B^0(M,\C) \to \B^1(M,\C). 
$$
Moreover, we can extend this derivation even further to a
derivation of the Weil algebra\index{terms}{Weil algebra} 
$$
D:\B^\cdot(M,\C) \to \B^{\cdot+1}(M,\C) 
$$
by setting 
\begin{equation}\label{D=d}
D(f \otimes \alpha) = Df \otimes \alpha + f \otimes d\alpha
\end{equation} 
for any smooth section $f \in C^\infty(M,\B^0(M,\C))$ and any smooth
form $\alpha \in C^\infty(M,\Lambda^\cdot(M,\C))$. We will call this
extension {\em the derivation, associated to the extended
connection\index{terms}{extended connection!derivation associated
to} $D$}.

Vice versa, the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot(M,\C)$ is generated by the subbundles
$$
S^1(M,\C),\Lambda^1(M,\C)
\subset \B^\cdot(M,\C).  
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Moreover, for every algebra derivation $D:\B^\cdot(M,\C) \to
\B^{\cdot+1}(M,\C)$ the condition \eqref{D=d} completely defines the
restriction of $D$ to the generator subbundle $\Lambda^1(M,\C)
\subset \B^1(M,\C)$. Therefore an algebra derivation
$D:\B^\cdot(M,\C) \to \B^\cdot(M,\C)$ satisfying \eqref{D=d} is
completely determined by its restriction to the generators subbundle
$S^1(M,\C) \to \B^1(M,\C)$. If the derivation $D$ is weakly
Hodge\index{terms}{weakly Hodge map}, then this restriction is an
extended\index{terms}{extended connection} connection on $M$.
 
\punkt \defn \label{flat.ext.con}
The extended\index{terms}{extended connection!flat} connection $D$ is
called {\em flat} if the associated derivation satisfies $D \circ D = 0$.

If a formal Hodge connection\index{terms}{Hodge connection!formal}
$D$ on $\tm$ is flat in the sense of \ref{hodge.con}, then we have a
derivation $D:\rho^*\Lambda^\cdot(M,\C) \to
\rho^*\Lambda^{\cdot+1}(M,\C)$ such that $D \circ D = 0$. The
associated derivation $\rho_*D:\B^\cdot(M,\C) \to
\B^{\cdot+1}(M,\C)$ satisfies \eqref{D=d}. Therefore the
extended\index{terms}{extended connection} connection
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$\rho_*D:S^1(M,\C) \to
\B^1(M,\C)$ is also flat.

\punkt It turns out that one can completely recover a Hodge
connection\index{terms}{Hodge connection} $D$ on $\tm$ from the
corresponding extended\index{terms}{extended connection} connection
$\rho_*D$ on $M$. More precisely, we have the following.

\begin{lemma}
The correspondence $D \mapsto \rho_*D$ is a bijection between the
set of formal Hodge connections\index{terms}{Hodge
connection!formal} on $\tm$ along $M \subset \tm$ and the set of
extended\index{terms}{extended connection} connections on $M$. A
connection $D$ is flat, resp. linear if and only if $\rho_*D$ is
flat, resp. linear.
\end{lemma}

\proof To prove the first claim of the lemma, it suffices to prove
that every extended\index{terms}{extended connection} connection on
$M$ comes from a unique formal Hodge connection\index{terms}{Hodge
connection!formal} on the pair $\langle \tm,M\rangle$. In general, 
the functor $\rho_*$ is not fully faithful on the category
$\Diff(M)$, in other words, it does not induce an isomorphism on the
spaces of differential operators between vector bundles on
$\tm$. However, for every complex vector bundle $\E$ on $\tm$ the
functor $\rho_*$ does induce an isomorphism
$$
\rho_*:\Der_M(\Lambda^0(M,\C),\E) \cong
\Der_{\B^0(M,\C)}(\B^0(M,\C),\rho_*\E) 
$$
between the space of {\em derivations} from $\Lambda^0(M,\C)$ to
$\F$ completed along $M \subset \tm$ and the space of derivations
from the algebra $\B^0(M,\C) = \rho_*\Lambda^0(M,\C)$ to the
$\B^0(M,\C)$-module $\rho_*\E$. Therefore every derivation
$$
D':\B^0(M,\C) \to \B^1(M,\C) = \rho_*\rho^*\Lambda^1(M,\C)
$$ 
must be of the form $D'=\rho_*D$ for some derivation
$$
D:\Lambda^0(\tm,\C) \to \rho^*\Lambda^1(M,\C)
$$ 
It is easy to check that $D$ is a Hodge
connection\index{terms}{Hodge connection} if and only if
$D'=\rho_*D$ is weakly Hodge\index{terms}{weakly Hodge map} and
satisfies \eqref{e.c}. By \ref{deriv} the space of all such
derivations $D':\B^0(M,\C) \to \B^1(M,\C)$ coincides with the space
of all extended\index{terms}{extended connection} connections on $M$,
which proves the first claim of the lemma.

Analogously, for every extended\index{terms}{extended connection}
connection $D' = \rho_*D$ on $M$, the canonical extension of the
operator $D'$ to an algebra derivation $D':\B^\cdot(M,\C) \to
\B^{\cdot+1}(M,\C)$ constructed in
\ref{deriv} must be of the form $\rho_*D$ for a certain weakly
Hodge\index{terms}{weakly Hodge map} differential operator
$D:\rho^*\Lambda^\cdot(M,\C) \to \Lambda^{\cdot+1}(M,\C)$. If the
extended\index{terms}{extended connection} connection $D'$ is flat,
then $D' \circ D' = 0$. Therefore $D \circ D = 0$, which means that
the Hodge connection\index{terms}{Hodge connection} $D$ is
flat. Vice versa, if the Hodge connection $D$ is flat, then it
extends to a weakly Hodge derivation $D:\rho^*\Lambda^\cdot(M,\C)
\to \Lambda^{\cdot+1}(M,\C)$ so that $D \circ D = 0$. The equality
$D \circ D = 0$ implies, in particular, that the operator $\rho_*D$
vanishes on the sections of the form
$$
Df = df \in C^\infty(M,\Lambda^1(M,\C)) \subset
C^\infty(M,\B^1(M,\C)), 
$$
where $f \in C^\infty(M,\C)$ is a smooth function on $M$. Therefore
$\rho_*D$ coincides with the de Rham differential on the subbundle
$\Lambda^1(M,\C) \subset \B^1(M,\C)$. Hence by \ref{deriv} it is
equal to the canonical derivation $D':\B^\cdot(M,\C) \to
\B^{\cdot+1}(M,\C)$. Since $D \circ D = 0$, we have $D' \circ D' =
0$, which means that the extended\index{terms}{extended connection}
connection $D'$ is flat.

Finally, the equivalence of the linearity
conditions\index{terms}{linearity condition} on the Hodge
connection\index{terms}{Hodge connection} $D$ and on the
extended\index{terms}{extended connection} connection $D' = \rho_*D$
is trivial and has already been noted in
\ref{lin.ext.con}.  \endproof

This lemma together with Lemma~\ref{form.hdg} reduces the
classification of linear formal Hodge manifold\index{terms}{Hodge
manifold!formal}\index{terms}{Hodge manifold!linear} structures on 
$\tm$ along the zero section $M \subset \tm$ to the classification
of extended\index{terms}{extended connection} connections on the
manifold $M$ itself.

\section{Preliminaries on the Weil
algebra}\label{Weil.section}\index{terms}{Weil algebra} 

\subsection{The total de Rham
complex}\label{de.rham.sub}\index{terms}{de Rham complex!total|(}

\punkt Before we proceed further in the study of
extended\index{terms}{extended connection} 
connections on a complex manifold $M$, we need to establish some
linear-algebraic facts on the structure of the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ defined in
\ref{Weil.defn}. We also need to introduce an auxiliary Hodge
bundle\index{terms}{Hodge bundle} algebra on $M$ which we call {\em
the total Weil algebra\index{terms}{Weil algebra!total}}. This is
the subject of this section. Most of the facts here are of a
technical nature, and the reader is advised to skip this section
until needed.

\punkt We begin with introducing and studying a version of the de
Rham complex\index{terms}{de Rham complex} of a complex manifold $M$
which we call {\em the total de Rham complex}. Let $M$ be a smooth
complex $U(1)$-manifold. Recall 
that by \ref{de.Rham} the de Rham complex\index{terms}{de Rham
complex} $\Lambda^\cdot(M,\C)$ of the complex manifold $M$ is
canonically a Hodge bundle\index{terms}{Hodge bundle} algebra on
$M$.  Let $\Lambda_{tot}^\cdot(M) = \Gamma(\Lambda^\cdot(M,\C))$ be
the weight\index{terms}{weight!of a Hodge bundle} $0$ Hodge bundle obtained by applying the functor
$\Gamma$ defined in \ref{gamma.m} to the de Rham algebra
$\Lambda^\cdot(M,\C)$. By \ref{gamma.tensor} the bundle
$\Lambda_{tot}^\cdot(M)$ carries a canonical algebra structure. By
\ref{de.Rham} the de Rham differential $d_M$ is weakly
Hodge\index{terms}{weakly Hodge map}. Therefore it induces an
algebra derivation $d_M:\Lambda^\cdot_{tot}(M) \to
\Lambda^{\cdot+1}_{tot}(M)$ which is compatible with the Hodge
bundle structure and satisfies $d_M \circ d_M = 0$.

\defn The weight\index{terms}{weight!of a Hodge bundle} $0$ Hodge bundle\index{terms}{Hodge bundle} algebra
$\Lambda_{tot}^\cdot(M)$ is called {\em the total de Rham
complex} of the complex manifold $M$.

\punkt By definition
$$
\Lambda^i_{tot}(M) = \Gamma(\Lambda^i(M,\C)) = \Lambda^i(M,\C) \otimes \W^*_i,  
$$
where $\W^*_i = S^i\W^*_1$ is the symmetric power of the $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} $\W^*_1$, as in
\ref{w.k}. To describe the structure of the algebra
$\Lambda^\cdot_{tot}(M)$, we will use the following well-known
general fact. (For the sake of completeness, we have included a
sketch of its proof, see \ref{gen.symm.proof}.)

\begin{lemma}\label{symm}
Let $A$, $B$ be two objects in an arbitrary $\Q$-linear symmetric
tensor category $\A$, and let $\CC^\cdot = S^\cdot(A \otimes B)$ be
the sum of symmetric powers of the object $A \otimes B$. Note that
the object $\CC^\cdot$ is naturally a commutative algebra in $\A$ in
the obvious sense. Let also $\wt{\CC}^\cdot = \bigoplus_k S^kA
\otimes S^kB$ with the obvious commutative algebra structure. The
isomorphism $\CC^1 \cong \wt{\CC}^1 \cong A \otimes B$ extends to a
surjective algebra map $\CC^\cdot \to \wt{\CC}^\cdot$, and its
kernel $\J^\cdot \subset \CC^\cdot$ is the ideal generated by the
subobject $\J^2 = \Lambda^2(A) \otimes \Lambda^2(B) \subset S^2(A
\otimes B)$.
\end{lemma}

\punkt The category of complexes of Hodge bundles\index{terms}{Hodge bundle} on $M$ is obviously $\Q$-linear and tensor. Applying
Lemma~\ref{symm} to $A = \W^*_1$, $B = \Lambda^1(M,\C) [1]$
immediately gives the following.
 
\begin{lemma}\label{total.rel}
The total de Rham complex $\Lambda_{tot}^\cdot(M)$ of the complex
manifold $M$ is generated by its first component
$\Lambda^1_{tot}(M)$, and the kernel of the canonical surjective
algebra map
$$
\Lambda^\cdot(\Lambda_{tot}^1(M)) \to \Lambda_{tot}^\cdot(M) 
$$
from the exterior algebra of the bundle $\Lambda^1_{tot}(M)$ to
$\Lambda^\cdot_{tot}(M)$ is the ideal generated by the subbundle
$$
\Lambda^2\W_1 \otimes S^2(\Lambda^1(M,\C)) \subset S^2(\Lambda^1_{tot}(M)).
$$
\end{lemma}

\punkt \label{S} We can describe the Hodge bundle\index{terms}{Hodge bundle} $\Lambda^1_{tot}(M)$ more explicitly in the following
way. By definition, as a $U(1)$-equivariant complex vector bundle it
equals
$$
\Lambda^1_{tot}(M) = \Lambda^1(M,\C) \otimes \W_1^* = \left(
\Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(0)\right) \otimes \left(\C(0)
\oplus \C(-1)\right),
$$
where $\Lambda^{p,q}(M)(i)$ is the $U(1)$-equivariant bundle
$\Lambda^{p,q}(M)$ tensored with the $1$-dimensional representation
of weight\index{terms}{weight!of an $U(1)$-action} $i$, and $\C(i)$ is the constant $U(1)$-bundle
corresponding to the representation of weight\index{terms}{weight!of an $U(1)$-action} $i$. If we denote
\begin{align*}
S^1(M,\C) &=
\Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(-1) \subset 
\Lambda^1_{tot}(M),\\ 
\Lambda^1_{ll}(M) &= \Lambda^{1,0}(M) \subset \Lambda^1_{tot}(M),\\
\Lambda^1_{rr}(M) &= \Lambda^{0,1}(M) \subset \Lambda^1_{tot}(M),
\end{align*}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
then we have 
$$
\Lambda^1_{tot}(M) = S^1(M,\C) \oplus \Lambda^1_{ll}(M) \oplus 
\Lambda^1_{rr}(M).
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
The complex conjugation $\conj:\Lambda^1_{tot}(M) \to
\iota\index{terms}{involution
$\iota$}^*\overline{\Lambda^1_{tot}(M)}$ preserves the subbundle  
$$
S^1(M,\C) \subset
\Lambda^1_{tot}(M,\C) 
$$ 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
and interchanges $\Lambda^1_{ll}(M)$ and $\Lambda^1_{rr}(M)$.

\punkt\label{S.Hodge.type} If the $U(1)$-action on the manifold $M$
is trivial, then Hodge bundles\index{terms}{Hodge bundle} are the
same as bigraded complex vector bundles with a real structure. In
this case the Hodge bigrading\index{terms}{bigrading!Hodge type} on
the Hodge bundle $\Lambda^1_{tot}(M,\C)$ is given by
\begin{align*}
\left(\Lambda^1_{tot}(M)\right)^{1,-1} &= S^{1,-1}(M,\C) =
\Lambda^{1,0}(M)(1),\\ 
\left(\Lambda^1_{tot}(M)\right)^{-1,1} &= S^{-1,1}(M,\C) =
\Lambda^{0,1}(M)(-1),\\ 
\left(\Lambda^1_{tot}(M)\right)^{0,0} &= \Lambda^1_{ll}(M) \oplus
\Lambda^1_{rr}(M) = \Lambda^1(M,\C).
\end{align*}
Under these identifications, the real structure on
$\Lambda^1_{tot}(M,\C)$ is minus the one induced by the usual real
structure on the complex vector bundle $\Lambda^1(M,\C)$.

\rem The Hodge bundle\index{terms}{Hodge bundle}
$S^1(M,\C)$ is canonically isomorphic to the conormal bundle to the
zero section $M \subset \tm$, which we have described in \ref{S1}.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt \label{gamma.use}
Recall now that we have defined in \ref{l.r} canonical embeddings
$\gamma_l,\gamma_r:\W_p^* \to \W_k^*$ for every $0 \leq p \leq
k$. Since $\W_0^* = \C$, for every $p,q \geq 0$ we have by
\eqref{p+q} a short exact sequence
\begin{equation}\label{cap.cup}
\begin{CD}
0 @>>> \C @>>> \W_p^* \oplus \W_q^* @>{\gamma_l \oplus \gamma_r}>>
\W_{p+q}^* @>>> 0
\end{CD}
\end{equation}
of complex vector spaces. Recall also that the embeddings
$\gamma_l$, $\gamma_r$ are compatible with the natural maps
$\can:\W_p^* \otimes \W_q^* \to \W_{p+q}^*$. Therefore the
subbundles defined by 
\begin{align*}
\Lambda^k_l(M) &= \bigoplus_{0 \leq p \leq k} \gamma_l(\W_p^*) \otimes
\Lambda^{p,k-p}(M) \subset \Lambda^k_{tot}(M) = \bigoplus_{0 \leq p \leq k} \W_k^*
\otimes \Lambda^{p,k-p}(M)\\ 
\Lambda^k_r(M) &= \bigoplus_{0 \leq p \leq k} \gamma_r(\W_p^*) \otimes
\Lambda^{k-p,p}(M) \subset \Lambda^k_{tot}(M) = \bigoplus_{0 \leq p \leq k} \W_k^*
\otimes \Lambda^{k-p,p}(M)
\end{align*}
are actually subalgebras in the total de Rham
complex $\Lambda^\cdot_{tot}(M)$.

\punkt \label{l.r.rel} 
To describe the algebras $\Lambda^\cdot_l(M)$ and
$\Lambda^\cdot_r(M)$ explicitly, note that we obviously have
$\Lambda^\cdot_{tot}(M) = \Lambda^\cdot_l(M) +
\Lambda^\cdot_r(M)$. Moreover, in the notation of \ref{S} we have
\begin{align*}
\Lambda^1_l(M) &= S^1(M,\C) \oplus \Lambda^1_{ll}(M) \subset
\Lambda^1_{tot}(M),\\ 
\Lambda^1_r(M) &= S^1(M,\C) \oplus \Lambda^1_{rr}(M) \subset
\Lambda^1_{tot}(M). 
\end{align*}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
By Lemma~\ref{symm}, the algebra
$$
\Lambda^\cdot_l(M) = \left(\bigoplus_p \W_p^* \otimes
\Lambda^{p,0}(M)\right) \otimes \left( \bigoplus_q \Lambda^{0,q}(M) \right)
$$ 
is the subalgebra in the total de Rham complex
$\Lambda^\cdot_{tot}(M)$ generated by $\Lambda^1_l(M)$, and the
ideal of relations is generated by the subbundle
$$
S^2(\Lambda^{1,0}(M)) \otimes \Lambda^2(\W_1^*) \subset
\Lambda^2(\Lambda^1_l(M)). 
$$
Analogously, the subalgebra $\Lambda^\cdot_r(M) \subset
\Lambda^1_{tot}(M)$ is generated by $\Lambda^1_r(M)$, and the relations
are generated by
$$
S^2(\Lambda^{0,1}(M)) \otimes \Lambda^2(\W_1^*) \subset
\Lambda^2(\Lambda^1_r(M)). 
$$

\punkt We will also need to consider the ideals in these algebras
defined by
\begin{align*}
\Lambda^k_{ll}(M) &= \bigoplus_{1 \leq p \leq k} \gamma_l(\W_p^*) \otimes
\Lambda^{p,k-p}(M) \subset \Lambda^k_l(M)\\
\Lambda^k_{rr}(M) &= \bigoplus_{1 \leq p \leq k} \gamma_r(\W_p^*) \otimes
\Lambda^{k-p,p}(M) \subset \Lambda^k_r(M)
\end{align*}
The ideal $\Lambda^\cdot_{ll}(M) \subset \Lambda^1_l(M)$ is
generated by the subbundle $\Lambda^1_{ll}(M) \subset
\Lambda^1_l(M)$, and the ideal $\Lambda^\cdot_{rr}(M) \subset
\Lambda^1_r(M)$ is generated by the subbundle $\Lambda^1_{rr}(M)
\subset \Lambda^1_r(M)$.

\punkt \label{left.right} Denote by $\Lambda^\cdot_o(M) =
\Lambda^\cdot_l(M) \cap \Lambda^\cdot_r(M) \subset
\Lambda^\cdot_{tot}(M)$ the intersection of the subalgebras
$\Lambda_l^\cdot(M)$ and $\Lambda^\cdot_r(M)$.  Unlike either of
these subalgebras, the subalgebra $\Lambda^\cdot_o(M) \subset
\Lambda^\cdot_{tot}(M)$ is compatible with the weight\index{terms}{weight!of a Hodge bundle} $0$ Hodge
bundle\index{terms}{Hodge bundle} structure on the total de Rham
complex.  By \eqref{cap.cup} we have a short exact sequence
\begin{equation}\label{shrt}
\begin{CD}
0 @>>> \Lambda^\cdot(M,\C) @>>> \Lambda^\cdot_l(M) \oplus \Lambda^\cdot_r(M)
@>>> \Lambda^\cdot_{tot}(M) @>>> 0
\end{CD}
\end{equation}
of complex vector bundles on $M$. Therefore the algebra
$\Lambda^\cdot_o(M)$ is isomorphic, as a complex bundle algebra, to
the usual de Rham complex\index{terms}{de Rham complex}
$\Lambda^\cdot(M,\C)$. As a Hodge bundle\index{terms}{Hodge bundle}
algebra it is canonically isomorphic to the exterior algebra of the
Hodge bundle $S^1(M,\C)$ of weight\index{terms}{weight!of a Hodge
bundle} $0$ on the manifold $M$. 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

Finally, note that the short exact sequence \eqref{shrt} induces a
direct sum decomposition
$$
\Lambda^\cdot_{tot}(M) \cong \Lambda^\cdot_{ll}(M) \oplus
\Lambda^\cdot_o(M) \oplus \Lambda^\cdot_{rr}(M). 
$$
  
\punkt \rem The total de Rham complex $\Lambda^\cdot_{tot}(M)$ is related to
Simpson\index{names}{Simpson, C.}'s theory of Higgs
bundles\index{terms}{Higgs bundle} (see \cite{Shiggs}) in the
following way.  Recall that Simpson\index{names}{Simpson, C.} has
proved that every (sufficiently stable) complex bundle $\E$ on a
compact complex manifold $M$ equipped with a flat connection
$\nabla$ admits a unique Hermitian
metric\index{terms}{metric!Hermitian} $h$ such that $\nabla$ and the
$1$-form $\theta = \nabla - \nabla^h \in
C^\infty(M,\Lambda^1(\Eend\E))$ satisfy the so-called {\em
harmonicity condition}. He also has shown that this condition is
equivalent to the vanishing of a certain curvature-like tensor $R
\in \Lambda^2(M,\Eend\E)$ which he associated canonically to every
pair $\langle \nabla, \theta
\rangle$.

Recall that flat bundles $\langle \E, \nabla \rangle$ on the
manifold $M$ are in one-to-one correspondence with free differential
graded modules $\E \otimes \Lambda^\cdot(M,\C)$ over the de Rham
complex\index{terms}{de Rham complex} $\Lambda^\cdot(M,\C)$.  It
turns out that complex bundles $\E$ equipped with a flat connection
$\nabla$ and a $1$-form $\theta \in C^\infty(M, \Lambda^1(\Eend\E))$
such that Simpson\index{names}{Simpson, C.}'s tensor $R$ vanishes are
in natural one-to-one correspondence with free differential graded
modules $\E \otimes \Lambda^\cdot_{tot}(M)$ over the total de Rham
complex $\Lambda^\cdot_{tot}(M)$. Moreover, a pair
$\langle\theta,\nabla\rangle$ comes from a variation of pure
$\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure} on $\E$ if
and only if there exists a Hodge bundle\index{terms}{Hodge bundle}
structure on $\E$ such that the product Hodge bundle structure on
the free module $\E \otimes \Lambda^\cdot_{tot}(M)$ is compatible
with the differential.

\punkt\label{gen.symm.proof}
\proof[Proof of Lemma~\ref{symm}]
For every $k \geq 0$ let $G = \Sigma_k \times \Sigma_k$ be the
product of two copies of the symmetric group $\Sigma_k$ on $k$
letters. Let $\V_k$ be the $\Q$-representation of $G_k$ induced from
the trivial representation of the diagonal subgroup $\Sigma_k
\subset G_k$. The representation $\V_k$ decomposes as
$$
\V_k = \bigoplus_V V \boxtimes V, 
$$
where the sum is over the set of irreducible representations $V$ of
$\Sigma_k$. We obviously have 
$$
\CC^k = \Hom_{G_k}\left(\V_k, A^{\otimes k} \otimes B^{\otimes
k}\right) = \bigoplus_V \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right)
\otimes \Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) .
$$
Let $\J^\cdot \subset \CC^\cdot$ be the ideal generated by
$\Lambda^2A \otimes \Lambda^2B \subset S^2(A \otimes B)$. It is easy
to see that
$$
\J^k = \sum_{1 \leq l \leq k-1} \bigoplus_V
\Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \otimes
\Hom_{\Sigma_k}\left(V,A^{\otimes k}\right) \subset \CC^k,
$$
where the first sum is taken over the set of $k-1$ subgroups
$\Sigma_2 \subset \Sigma_k$, the $l$-th one transposing the $l$-th
and the $l+1$-th letter, while the second sum is taken over all
irreducible constituents $V$ of the representation of $\Sigma_k$
induced from the sign representation of the corresponding $\Sigma_2
\subset \Sigma_k$. Now, there is obviously only one irreducible
representation of $\Sigma_k$ which is not encountered as an index in
this double sum, namely, the trivial one. Hence $\CC^k / \J^k = S^kA
\otimes S^kB$, which proves the lemma.  
\endproof

\subsection{The total Weil
algebra}\label{t.W.sub}\index{terms}{Weil algebra!total}

\punkt \label{S.and.Lambda} Assume from now on that the
$U(1)$-action on the complex manifold $M$ is trivial. We now turn to
studying the Weil algebra\index{terms}{Weil algebra} of the manifold
$M$. Let $S^1(M,\C) = S^{1,-1}(M,\C) \oplus S^{-1,1}(M,\C)$ be the
weight\index{terms}{weight!of a Hodge bundle} $0$ Hodge
bundle\index{terms}{Hodge bundle} on $M$ introduced in \ref{S1}. To
simplify notation, denote
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\begin{align*}
S^\cdot &= \widehat{S}^\cdot(S^1(M,\C))\\ 
\Lambda^\cdot &= \Lambda^\cdot(M,\C), 
\end{align*} 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
where $\widehat{S}^\cdot$ is the completed symmetric power, and let 
$$
\B^\cdot = \B^\cdot(M,\C) = S^\cdot \otimes \Lambda^\cdot 
$$
be the Weil algebra\index{terms}{Weil algebra} of the complex
manifold $M$ introduced in \ref{Weil.defn}. Recall that the algebra
$\B^\cdot$ carries a natural Hodge bundle\index{terms}{Hodge bundle}
structure. In particular, it is equipped with a Hodge type
bigrading\index{terms}{bigrading!Hodge type}
$\B^i = \sum_{p+q=i} \B^{p,q}$.

\punkt \label{aug} We now introduce a different bigrading on the
Weil algebra\index{terms}{Weil algebra} $\B^\cdot$. The commutative
algebra $\B^\cdot$ is freely generated by the subbundles
$$
S^1 = S^{1,-1} \oplus S^{-1,1} \subset \B^0 \quad\text{ and }\quad
\Lambda^1 = \Lambda^{1,0} \oplus \Lambda^{0,1} \subset \B^1, 
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
therefore to define a multiplicative
bigrading\index{terms}{bigrading!augmentation} on the algebra
$\B^\cdot$ it suffices to assign degrees to these generator
subbundles $S^{1,-1},S^{-1,1},\Lambda^{1,0},\Lambda^{0,1} \subset
\B^\cdot$.

\defn The {\em augmentation
bigrading\index{terms}{bigrading!augmentation}} on $\B^\cdot$ is the
multiplicative bigrading defined by setting
\begin{align*}
\deg S^{1,-1} &= \deg \Lambda^{1,0} = ( 1, 0 )\\
\deg S^{-1,1} &= \deg \Lambda^{0,1} = ( 0, 1 )
\end{align*}
on generators $S^{1,-1},S^{-1,1},\Lambda^{1,0},\Lambda^{0,1} \subset
\B^{\cdot,\cdot}$. 

We will denote by $\B^{\cdot,\cdot}_{p,q}$ the component of the Weil
algebra\index{terms}{Weil algebra} of augmentation
bidegree\index{terms}{bidegree!augmentation|(} $(p,q)$. For any linear
map $a:\B^\cdot \to \B^\cdot$ we will denote by $a =
\sum_{p,q}a_{p,q}$ its decomposition with respect to the
augmentation bidegree.

It will also be useful to consider a coarser {\em augmentation
grading\index{terms}{grading!augmentation}} on $\B^\cdot$, defined
by $\deg\B^\cdot_{p,q} = p + q$. We will denote by $\B^\cdot_k =
\bigoplus_{p+q=k}\B^\cdot_{p,q}$ the component of $\B^\cdot$ of
augmentation degree\index{terms}{degree!augmentation} $k$.

\punkt Note that the Hodge bidegree\index{terms}{bidegree!Hodge} and
the augmentation bidegree are, in general, independent. Moreover,
the complex conjugation $\conj:\B^\cdot \to \overline{\B^\cdot}$
sends $\B^\cdot_{p,q}$ to $\overline{\B^\cdot_{q,p}}$. Therefore the
augmentation bidegree components $\B^\cdot_{p,q} \subset \B^\cdot$
are not Hodge subbundles. However, the coarser augmentation
grading\index{terms}{grading!augmentation} is compatible with the
Hodge structures, and the augmentation
degree\index{terms}{degree!augmentation} $k$-component $\B^i_k
\subset \B^i$ carries a natural Hodge bundle\index{terms}{Hodge bundle} structure of weight\index{terms}{weight!of a Hodge bundle}
$i$. Moreover, the sum $\B^\cdot_{p,q} + \B^\cdot_{q,p}
\subset \B^\cdot$ is also a Hodge subbundle.

\punkt \label{total.Weil} We now introduce an auxiliary
weight\index{terms}{weight!of a Hodge bundle} $0$ 
Hodge algebra\index{terms}{Hodge bundle} bundle on $M$, called the
total Weil algebra\index{terms}{Weil algebra!total}.  Recall that we
have defined in \ref{gamma.m} a functor $\Gamma:\prehodge_{\geq
0}(M) \to \prehodge_0(M)$ adjoint on the right to the canonical
embedding. Consider the Hodge bundle $\B_{tot}^\cdot =
\Gamma(\B^\cdot)$ of weight\index{terms}{weight!of a Hodge bundle}
$0$ on $M$.  By \ref{gamma.tensor} the 
multiplication on $\B^\cdot$ induces an algebra structure on
$\Gamma(\B^\cdot)$.

\defn The Hodge algebra bundle $\B_{tot}^\cdot$ of
weight\index{terms}{weight!of a Hodge bundle} $0$ is 
called {\em the total Weil algebra\index{terms}{Weil algebra!total}}
of the complex manifold $M$.

\rem For a more conceptual description of the functor $\Gamma$ and
the total Weil algebra\index{terms}{Weil algebra!total}, see
Appendix.

\punkt By definition of the functor $\Gamma$ we have $\B_{tot}^k =
\B^k \otimes \W_k^* = S^\cdot \otimes \Lambda^k \otimes \W_k^* =
S^\cdot \otimes \Lambda^k_{tot}$, where $\Lambda^k_{tot} = \Lambda^k
\otimes \W_\cdot^* = \Gamma(\Lambda^k)$ is the total de Rham
complex\index{terms}{de Rham complex!total|)} introduced in
Subsection~\ref{de.rham.sub}.  We have also introduced in
Subsection~\ref{de.rham.sub} Hodge bundle\index{terms}{Hodge bundle}
subalgebras $\Lambda^\cdot_o, \Lambda^\cdot_l, \Lambda^\cdot_r
\subset \Lambda^\cdot_{tot}$ in the total de Rham complex
$\Lambda^\cdot_{tot}$ and ideals $\Lambda^\cdot_{ll} \subset
\Lambda^\cdot_l$, $\Lambda^\cdot_{rr} \subset \Lambda^\cdot_r$ in
the algebras $\Lambda^\cdot_l$, $\Lambda^\cdot_r$. Let
\begin{align*}
\B_o^k &= S^\cdot \otimes \Lambda_o^k \subset \B_{tot}^k\\
\B_l^k &= S^\cdot \otimes \Lambda_l^k \subset \B_{tot}^k\\
\B_r^k &= S^\cdot \otimes \Lambda_r^k \subset \B_{tot}^k
\end{align*}
be the associated subalgebras in the total Weil
algebra\index{terms}{Weil algebra!total} $\B_{tot}^\cdot$ and let
\begin{align*}
\B_{ll}^k &= S^\cdot \otimes \Lambda_{ll}^k \subset \B_l^k\\
\B_{rr}^k &= S^\cdot \otimes \Lambda_{rr}^k \subset \B_r^k
\end{align*}
be the corresponding ideals in the Hodge bundle algebras
$\B^\cdot_l$, $\B^\cdot_r$.

By \ref{left.right} we have bundle isomorphisms $\Lambda^\cdot_{tot} =
\Lambda^\cdot_l + \Lambda^\cdot_r$ and $\Lambda_o^\cdot =
\Lambda^\cdot_l \cap \Lambda^\cdot_r$, and the direct sum
decomposition $\Lambda_{tot}^\cdot \cong \Lambda_{ll}^\cdot \oplus
\Lambda_o^\cdot \oplus \Lambda_{rr}^\cdot$. Therefore we also have
\begin{align}\label{drct}
\begin{split}
\B^\cdot_{tot} &= \B_l^\cdot + \B_r^\cdot = \B^\cdot_{ll} \oplus
\B^\cdot_o \oplus \B^\cdot_{rr}\\
\B_o^\cdot &= \B_l^\cdot \bigcap \B_r^\cdot \subset \B^\cdot_{tot}
\end{split}
\end{align}
Moreover, the algebra $\Lambda^\cdot_o$ is isomorphic to the usual
de Rham complex\index{terms}{de Rham complex} $\Lambda^\cdot$,
therefore the subalgebra $\B_o^\cdot \subset \B^\cdot_{tot}$ is
isomorphic to the usual Weil algebra\index{terms}{Weil algebra}
$\B^\cdot$. These isomorphisms are {\em not} weakly
Hodge\index{terms}{weakly Hodge map}.

\punkt The total Weil algebra\index{terms}{Weil algebra!total}
carries a canonical weight\index{terms}{weight!of a Hodge bundle}
$0$ Hodge bundle\index{terms}{Hodge bundle} structure, and we will
denote the corresponding Hodge type 
grading\index{terms}{bigrading!Hodge type} by upper indices:
$\B_{tot}^\cdot = \oplus_p\left(\B_{tot}^\cdot\right)^{p,-p}$. The
augmentation bigrading\index{terms}{bigrading!augmentation} on the
Weil algebra\index{terms}{Weil algebra} introduced in \ref{aug}
extends to a bigrading of the total Weil algebra\index{terms}{Weil
algebra!total}, which we will denote by lower indices. In general, 
both these grading and the direct sum decomposition \eqref{drct} are
independent, so that, in general, for every $i \geq 0$ we have a
decomposition
$$
\B_{tot}^i = \bigoplus_{n,p,q} \left(\B_{ll}^i\right)_{p,q}^{n,-n}
\oplus \left(\B_o^i\right)_{p,q}^{n,-n} \oplus
\left(\B_{rr}^i\right)_{p,q}^{n,-n}. 
$$ 
We would like to note, however, that some terms in this
decomposition vanish when $i = 0,1$. Namely, we have the following
fact.

\begin{lemma}\label{total.aug}
Let $n,k$ be arbitrary integers such that $k \geq 0$.  
\begin{enumerate}
\item If $n+k$ is odd, then $\left(\B^0_{tot}\right)_k^{n,-n} = 0$. 
\item If $n+k$ is even, then $\left(\B^1_{ll}\right)_k^{n,-n} =
\left(\B^1_{rr}\right)_k^{n,-n} = 0$, while if $n+k$ is odd, then
$\left(\B^1_o\right)_k^{n,-n} = 0$.
\end{enumerate}
\end{lemma}

\proof
\begin{enumerate}
\item The bundle $\B^0_{tot}$ by definition coincides with $\B^0$, and
it is generated by the subbundles $S^{1,-1},S^{-1,1} \subset
\B^0$. Both these subbundles have augmentation
degree\index{terms}{degree!augmentation} $1$ and Hodge
degree\index{terms}{degree!Hodge} $\pm 1$, so that the sum $n+k$ of
the Hodge degree with the augmentation degree is even. Since both
gradings are multiplicative, for all non-zero components
$\B^{n,-n}_k \subset \B^0$ the sum $n+k$ must also be even.

\item By definition we have $\B^1_{tot} = \B^0 \otimes \Lambda^1_{tot}$. The
subbundle $\Lambda^1_{tot} \subset \B^1_{tot}$ has augmentation
degree\index{terms}{degree!augmentation} $1$,
and it decomposes 
$$
\Lambda^1_{tot} = \Lambda^1_o \oplus \Lambda^1_{ll} \oplus
\Lambda^1_{rr}.
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
By \ref{S} we have $\Lambda^1_o \cong S^1 = S^{1,-1} \oplus
S^{-1,1}$ as Hodge bundles\index{terms}{Hodge bundle}, so that the
Hodge degrees\index{terms}{degree!Hodge} on $\Lambda^1_o \subset
\Lambda^1_{tot}$ are odd. On the other hand, the subbundles
$\Lambda^1_{ll},\Lambda^1_{rr} \subset \Lambda^1_{tot}$ are by
\ref{S.Hodge.type} of Hodge bidegree\index{terms}{bidegree!Hodge}
$(0,0)$. Therefore the sum $n+k$ of the Hodge and the augmentation
degrees is even for $\Lambda^1_o$ and odd for $\Lambda^1_{ll}$ and
$\Lambda^1_{rr}$. Together with \thetag{i} this proves the claim.
\endproof
\end{enumerate}

\subsection{Derivations of the Weil algebra}\index{terms}{Weil algebra}

\punkt We will now introduce certain canonical derivations of the
Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ which will
play an important part in the rest of the paper. First of all, to
simplify notation, for any two linear maps $a,b$ let
$$
\{ a,b \} = a \circ b + b \circ a 
$$
be their anticommutator, and for any linear map $a:\B^\cdot \to
\B^{\cdot+i}$ let $a = \sum_{p+q=i} a^{p,q}$ be the Hodge
type\index{terms}{bigrading!Hodge type} decomposition. The following fact is
well-known, but we have included a proof for the sake of
completeness.

\begin{lemma}
For every two odd derivations $P,Q$ of a
graded-commutative algebra $\A$, their anticommutator $\{P,Q\}$ is
an even derivation of the algebra $\A$. 
\end{lemma}

\proof Indeed, for every $a,b \in \A$ we have
\begin{align*}
\{P,Q\}(ab) &= P(Q(ab)) + Q(P(ab)) \\
& = P(Q(a)b + (-1)^{\deg a}aQ(b)) +
Q(P(a)b + (-1)^{\deg a}aP(b)) \\
& = P(Q(a))b + (-1)^{\deg Q(a)}Q(a)P(b)
+ (-1)^{\deg a}P(a)Q(b) \\
&\quad + aP(Q(b)) + Q(P(a))b + (-1)^{\deg P(a)}P(a)Q(b) \\
&\quad + (-1)^{\deg a}Q(a)P(b) + aQ(P(b)) \\
& = P(Q(a))b - (-1)^{\deg a}Q(a)P(b)
+ (-1)^{\deg a}P(a)Q(b) \\
&\quad + aP(Q(b)) + Q(P(a))b - (-1)^{\deg a}P(a)Q(b) \\
&\quad + (-1)^{\deg a}Q(a)P(b) + aQ(P(b)) \\
& = P(Q(a))b + aP(Q(b)) + Q(P(a))b + a Q(P(b)) \\
& = \{P,Q\}(a)b + a\{P,Q\}(b).
\end{align*} 
\endproof 

\punkt \label{C.and.sigma} \index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} Let $C:S^1 \to \Lambda^1$ be the
canonical weakly Hodge map\index{terms}{weakly Hodge map} introduced
in \ref{C.Weil}. Extend $C$ to an algebra derivation $C:\B^\cdot \to
\B^{\cdot+1}$ by setting $C = 0$ on $\Lambda^1 \subset \B^1$. By
\ref{C.Weil} the derivation $C$ is weakly Hodge.  The composition
$$
C \circ C = \half\{C,C\}:\B^\cdot \to \B^{\cdot+2} 
$$
is also an algebra derivation, and it obviously vanishes on the
generator subbundles $S^1,\Lambda^1 \subset \B^\cdot$. Therefore $C
\circ C = 0$ everywhere.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
 
Let also $\sigma:\B^{\cdot+1} \to \B^\cdot$ be the derivation
introduced in \ref{sigma}.  The derivation $\sigma$ is not weakly
Hodge\index{terms}{weakly Hodge map}; however, it is real and admits
a decomposition $\sigma = \sigma^{-1,0} + \sigma^{0,-1}$ into
components of Hodge types\index{terms}{bigrading!Hodge type} $(-1,0)$ and
$(0,-1)$. Both these components are algebra derivations of the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot$. We obviously have
$\sigma \circ \sigma = \sigma^{-1,0} \circ \sigma^{-1,0} =
\sigma^{0,-1} \circ \sigma^{0,-1} = 0$ on generators
$S^1,\Lambda^1 \subset \B^\cdot$, and, therefore, on the whole Weil
algebra\index{terms}{Weil algebra}.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\rem Up to a sign the derivations $C,\sigma$ and their Hodge
bidegree\index{terms}{bidegree!Hodge} components coincide with the
so-called {\em Koszul differentials\index{terms}{Koszul
differential}} on the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot = S^\cdot \otimes
\Lambda^\cdot$.

\punkt \label{total.C} The derivation $C:\B^\cdot \to \B^{\cdot+1}$
is by definition weakly Hodge\index{terms}{weakly Hodge
map}. Applying the functor $\Gamma$ to it, we obtain a derivation
$C:\B_{tot}^\cdot \to \B_{tot}^{\cdot+1}$ of the total Weil
algebra\index{terms}{Weil algebra!total} $\B_{tot}^\cdot$ preserving
the weight\index{terms}{weight!of a Hodge bundle} $0$ Hodge
bundle\index{terms}{Hodge bundle} structure on 
$\B_{tot}^\cdot$. The canonical identification $\B^\cdot \cong
\B^\cdot_o \subset \B_{tot}^\cdot$ is compatible with the derivation
$C:\B^\cdot_{tot} \to \B^{\cdot+1}_{tot}$. Moreover, by
\ref{C.and.sigma} this derivation satisfies $C \circ C =
0:\B^\cdot_{tot} \to \B^{\cdot+2}_{tot}$. Therefore the total Weil
algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}$ equipped
with the derivation $C$ is a complex of Hodge bundles of
weight\index{terms}{weight!of a Hodge bundle} $0$. 

The crucial linear algebraic property of the total Weil
algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}$ of the
manifold $M$ which will allow us to classify flat
extended\index{terms}{extended connection!flat} connections on $M$ is
the following.

\begin{prop}\label{ac}
Consider the subbundle  
\begin{equation}\label{sbcmp}
\bigoplus_{p,q \geq 1}\left(\B^\cdot_{tot}\right)_{p,q} \subset
\B^\cdot_{tot}
\end{equation}
of the total Weil algebra\index{terms}{Weil algebra!total}
$\B^\cdot_{tot}$ consisting of the components of augmentation
bidegrees $(p,q)$ with $p,q \geq 1$. This subbundle equipped with
the differential $C:\left(\B^\cdot_{tot}\right)_{\cdot,\cdot} \to
\left(\B^{\cdot+1}_{tot}\right)_{\cdot,\cdot}$ is an acyclic complex
of Hodge bundles\index{terms}{Hodge bundle} of
weight\index{terms}{weight!of a Hodge bundle} $0$ on $M$. 
\end{prop}

\punkt We sketch a more or less simple and conceptual proof of
Proposition~\ref{ac} in the Appendix. However, in order to be able
to study in Section~\ref{convergence} the analytic properties of our
formal constructions, we will need an explicit contracting homotopy
for the complex \eqref{sbcmp}, which we now introduce.

The restriction of the derivation $C:\B^0_{tot} \to \B^1_{tot}$ to
the subbundle $S^1 \subset \B_0 \cong \B^0_{tot}$ induces a Hodge
bundle\index{terms}{Hodge bundle} isomorphism
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
C:S^1 \to \Lambda^1_o \subset \Lambda^1_{tot} \subset \B^1_{tot}.
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Define a map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ by 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\begin{equation}\label{sigma.c.eq}
\sigma_{tot} = 
\begin{cases} 
0 \quad &\text{ on }\quad\Lambda^1_{ll},\Lambda^1_{rr} \subset
\Lambda^1_{tot},\\ 
C^{-1} \quad &\text{ on }\quad\Lambda^1_o \subset \Lambda^1_{tot}.
\end{cases}
\end{equation}
The map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ preserves the Hodge 
bundle\index{terms}{Hodge bundle} structures of
weight\index{terms}{weight!of a Hodge bundle} $0$ on both 
sides. Moreover, its restriction to the subbundle $\Lambda^1 \cong
\Lambda^1_o \subset \Lambda^1_{tot}$ coincides with the canonical
map $\sigma:\Lambda^1 \to S^1$ introduced in \ref{C.and.sigma}. 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt \label{sigma.l} \index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} Unfortunately, unlike $\sigma:\Lambda^1 \to
S^1$, the map $\sigma_{tot}:\Lambda^1_{tot} \to S^1$ does {\em not}
admit an extension to a derivation $\B^{\cdot+1}_{tot} \to
\B_{tot}^\cdot$ of the total Weil algebra\index{terms}{Weil
algebra!total} $\B^\cdot_{tot}$. We will extend it to a {\em bundle map}
$\sigma_{tot}:\B^{\cdot+1}_{tot} \to \B^\cdot_{tot}$ in a somewhat
roundabout way. To do this, define a map $\sigma_l:\Lambda^1_l \to
S^0 \subset \B^0$ by
$$
\sigma_l =  
\begin{cases}
0 \quad &\text{ on } \quad\Lambda_{ll}^1 \subset \Lambda_l^1 \quad
\text{ and on }\quad \left(\Lambda^1_o\right)^{-1,1} \subset
\Lambda^1_{tot}, \\ 
C^{-1} \quad &\text{ on } \quad\left(\Lambda^1_o\right)^{1,-1} \subset
\Lambda^1_{tot}.
\end{cases}
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
and set $\sigma_l = 0$ on $S^1$. By \ref{gamma.use} we have
$$
\Lambda^1_l = \Lambda^{1,0} \oplus \left( \Lambda^{0,1} \otimes
\W_1^*\right). 
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
The map $\sigma_l:\Lambda^1_l \to S^1$ vanishes on the second
summand in this direct sum, and it equals $C^{-1}:\Lambda^{1,0} \to
S^{1,-1} \subset S^1$ on the first summand. The restriction of the map
$\sigma_l$ to the subbundle 
$$
\Lambda^{1,0} \oplus \Lambda^{0,1} = \Lambda^1 \cong \Lambda^1_o \subset
\Lambda^1_l
$$ 
vanishes on $\Lambda^{0,1}$ and equals $C^{-1}$ on
$\Lambda^{1,0}$. Thus it is equal to the Hodge
type\index{terms}{bigrading!Hodge type}-$(0,-1)$ component
$\sigma^{0,-1}:\Lambda^1 \to S^1$ of the canonical map
$\sigma:\Lambda^1 \to S^1$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\punkt By \ref{l.r.rel} the algebra $\B^\cdot_l$ is generated by the
bundles $S^1$ and $\Lambda^1_l$, and the ideal of relations is
generated by the subbundle
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\begin{equation}\label{rell}
S^2\left(\Lambda^{0,1}\right) \otimes \Lambda^2\left(\W_1^*\right) 
\subset \Lambda^2\left(\Lambda^1_l\right).
\end{equation}
Since the map $\sigma_l:\Lambda^1_l \to S^1$ vanishes on
$\Lambda^{0,1} \otimes \W_1^* \subset \Lambda^1_l$, it extends to an
algebra derivation $\sigma_l:\B_l^{\cdot+1} \to \B^\cdot_l$. The
restriction of the derivation $\sigma_l$ to the subalgebra $\B^\cdot
\cong \B^\cdot_0 \subset \B_{tot}^\cdot$ coincides with the
$(0,-1)$-component $\sigma^{0,-1}$ of the derivation
$\sigma:\B^{\cdot+1} \to \B^\cdot$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

Analogously, the $(-1,0)$-component $\sigma^{-1,0}$ of the
derivation $\sigma:\B^{\cdot+1} \to \B^\cdot$ extends to an algebra
derivation $\sigma_r:\B_r^{\cdot+1} \to \B_r^\cdot$ of the
subalgebra $\B^\cdot_r \subset \B_{tot}^\cdot$. By definition, the
derivation $\sigma_l$ preserves the decomposition $\B_l^\cdot =
\B_{ll} \oplus \B_o^\cdot$, while the derivation $\sigma_r$
preserves the decomposition $\B_r^\cdot = \B_{rr}^\cdot \oplus
\B_o^\cdot$. Both these derivations vanish on $\B_{tot}^0$, therefore
both are maps of $\B_{tot}^0$-modules. In addition, the compositions
$\sigma_l \circ \sigma_l$ and $\sigma_r \circ \sigma_r$ vanish on
generator and, therefore, vanish identically.

\punkt \label{sigma.c}
Extend both $\sigma_l$ and $\sigma_r$ to the whole $\B_{tot}^\cdot$ by
setting
\begin{equation}\label{nol}
\sigma_l = 0 \text{ on }\B^\cdot_r \qquad \sigma_r = 0 \text{ on
}\B^\cdot_l, 
\end{equation}
and let
$$
\sigma_{tot} = \sigma_l + \sigma_r:\B^{\cdot+1}_{tot} \to \B_{tot}^\cdot.
$$
On $\Lambda^1_{tot} \subset \B^1_{tot}$ this is the same map as in
\eqref{sigma.c.eq}. The bundle map $\sigma_{tot}:\B_{tot}^{\cdot+1}
\to \B_{tot}^\cdot$ preserves the direct sum decomposition
\eqref{drct}, and its restriction to $\B_o^\cdot \subset
\B^\cdot_{tot}$ coincides with the derivation $\sigma$.  Note that
neither of the maps $\sigma_l$, $\sigma_r$, $\sigma_{tot}$ is a
derivation of the total Weil algebra\index{terms}{Weil
algebra!total} $\B_{tot}^\cdot$. However, all these maps are linear with
respect to the $\B_{tot}^0$-module structure on $\B_{tot}^\cdot$ and
preserve the decomposition \eqref{drct}. The map
$\sigma_{tot}:\B^{\cdot+1}_{tot} \to \B^\cdot_{tot}$ is equal to
$\sigma_l$ on $\B^\cdot_{ll} \subset \B^\cdot_{tot}$, to $\sigma_r$
on $\B^\cdot_{rr}$ and to $\sigma:\B^{\cdot+1} \to \B^\cdot$ on
$\B^\cdot \cong \B^\cdot_o \subset \B^\cdot_{tot}$. Since $\sigma_l
\circ \sigma_l = \sigma_r \circ \sigma_r = \sigma \circ \sigma = 0$,
we have $\sigma_{tot} \circ \sigma_{tot} = 0$.

\punkt The commutator 
$$
h = \{C, \sigma_{tot}\}:\B_{tot}^\cdot \to \B_{tot}^\cdot
$$
of the maps $C$ and $\sigma_{tot}$ also preserves the decomposition
\eqref{drct}, and we have the following. 

\begin{lemma}\label{h.acts}
The map $h$ acts as multiplication by $p$ on
$\left(\B_{ll}^\cdot\right)_{p,q}$, as multiplication by $q$ on
$\left(\B_{rr}^\cdot\right)_{p,q}$ and as multiplication by $(p+q)$
on $\left(\B_o^\cdot\right)_{p,q}$.
\end{lemma}

\proof It suffices to prove the claim separately on each term in the
decomposition \eqref{drct}. By definition $\sigma_{tot} = \sigma_l +
\sigma_r$, and $h = h_l + h_r$, where $h_l = \{\sigma_l,C\}$ and
$h_r = \{\sigma_r,C\}$. Moreover, $h_l$ vanishes on $\B_{rr}^\cdot$
and $h_r$ vanishes on $\B_{ll}^\cdot$. Therefore it suffices to
prove that $h_l = p\id$ on $\left(\B^\cdot_l\right)_{p,q}$ and that
$h_r = q\id$ on $\left(\B^\cdot_r\right)_{p,q}$. The proofs of these
two identities are completely symmetrical, and we will only give a
proof for $h_l$.

The algebra $\B^\cdot_l$ is generated by the subbundles
$S^1 \subset \B^0_l$ and $\Lambda^1_l \subset \B^1_l$. The augmentation
bidegree decomposition of $S^1$ is by definition given by
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
S^1_{1,0} = S^{1,-1} \qquad\qquad S^1_{0,1} = S^{-1,1},
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
while the augmentation bidegree decomposition of $\Lambda^1_l$ is
given by
$$
\left(\Lambda^1_l\right)_{1,0} = \Lambda^{1,0} \qquad
\left(\Lambda^1_l\right)_{0,1} = \Lambda^{0,1} \otimes \W_1^*. 
$$
By the definition of the map $\sigma_l:\Lambda^1_l \to S^1$ (see
\ref{sigma.l}) we have $h_l = \{C,\sigma_l\} = \id$ on
$\Lambda^{1,0}$ and $S^{1,-1}$, and $h_l = 0$ on $\Lambda^{0,1}
\otimes \W_1^*$ and on $S^{-1,1}$. Therefore for every $p,q \geq 0$
we have $h_l=p\id$ on the generator subbundles $S^1_{p,q}$ and on
$\left(\Lambda^1_l\right)_{p,q}$. Since the map $h_l$ is a
derivation and the augmentation bidegree is multiplicative, the same
holds on the whole algebra $\B^\cdot_l =
\oplus\left(\B^\cdot_l\right)_{p,q}$. 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\endproof

Lemma~\ref{h.acts} shows that the map $\sigma_{tot}$ is a homotopy,
contracting the subcomplex \eqref{sbcmp} in the total Weil
algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}$, which
immediately implies Proposition~\ref{ac}.

\rem In fact, in our classification of flat
extended\index{terms}{extended connection!flat} connections given in
Section~\ref{main.section} it will be more convenient for us to use
Lemma~\ref{h.acts} directly rather than refer to
Proposition~\ref{ac}.

\punkt We finish this section with the following corollary of
Lemma~\ref{total.aug} and Lemma~\ref{h.acts}, which we will need in
Section~\ref{convergence}.

\begin{lemma}\label{h.on.b1}
Let $n=\pm 1$. If the integer $k \geq 1$ is odd, then the map
$h:\B^\cdot_{tot} \to \B^\cdot_{tot}$ acts on
$\left(\B_{tot}^1\right)^{n,-n}_k$ 
by multiplication by $k$. If $k = 2m \geq 1$ is even, then the
endomorphism $h:\left(\B_{tot}^1\right)^{n,-n}_k \to
\left(\B_{tot}^1\right)^{n,-n}_k$ is diagonalizable, and its only
eigenvalues are $m$ and $m-1$.
\end{lemma}

\proof If $k$ is odd, then  by Lemma~\ref{total.aug} we have
$\left(\B_{tot}^1\right)^{n,-n}_k = \left(\B_o^1\right)^{n,-n}_k$,
and Lemma~\ref{h.acts} immediately implies the claim.  Assume that
the integer $k = 2m$ is even. By Lemma~\ref{total.aug} we have
$$
\left(\B_{tot}^\cdot\right)^{n,-n}_k =
\left(\B_{ll}^\cdot\right)^{n,-n}_k \oplus
\left(\B_{rr}^\cdot\right)^{n,-n}_k = \B^{n,-n}_{k-1} \otimes
\left(\Lambda^1_{ll} \oplus \Lambda^1_{rr}\right).
$$
The bundle $\B^0$ is generated by subbundles $S^{1,-1}$ and
$S^{-1,1}$. The first of these subbundles has augmentation bidegree
$(1,0)$, while the second one has augmentation bidegree
$(0,1)$. Therefore for every augmentation bidegree component
$\B^{n,-n}_{p,q} \subset \B^{n,-n}_{k-1}$ we have $p - q = n$ and
$p+q=k-1$. This implies that $\B^{n,-n}_{k-1} = \B^{n,-n}_{p,q}$
with $p=m-(1-n)/2$ and $q=m-(1+n)/2$.
  
By definition the augmentation bidegrees of the bundles
$\Lambda^1_{ll}$ and $\Lambda^1_{rr}$ are, respectively, $(0,1)$ and
$(1,0)$. Lemma~\ref{h.acts} shows that the only eigenvalue of the
map $h$ on $\left(\B^1_{ll}\right)_{k+1}$ is $p=(m-(1-n)/2))$, while
its only eigenvalue on $\left(\B^1_{rr}\right)_{k+1}$ is
$q=(m-(1+n)/2)$. Since $n = \pm 1$, one of these numbers equals $m$
and the other one equals $m-1$.  
\endproof

\section{Classification of flat extended
connections}\label{main.section}\index{terms}{extended connection!flat}

\subsection{K\"ah\-le\-ri\-an connections}\index{terms}{connections!K\"ahlerian} 

\punkt Let $M$ be a complex manifold. In
Section~\ref{formal.section} we have shown that formal Hodge
manifold\index{terms}{Hodge manifold!formal} structures on the
tangent bundle $\tm$ are in one-to-one correspondence with linear
flat extended connections\index{terms}{extended
connection!linear}\index{terms}{extended connection!flat} on the manifold
$M$ (see \ref{ext.con}--\ref{flat.ext.con} for the definitions). It
turns out that flat linear extended connections on $M$ are, in turn,
in natural one-to-one correspondence with differential operators of
a much simpler type, namely, connections on the cotangent bundle
$\Lambda^{1,0}(M)$ satisfying certain vanishing conditions
(Theorem~\ref{kal=ext}). We call such
connections\index{terms}{connections!K\"ahlerian} {\em
K\"ah\-le\-ri\-an}. In this section we use the results of
Section~\ref{Weil.section} establish the correspondence between
extended connections\index{terms}{extended connection} on $M$ and
K\"ah\-le\-ri\-an connections on $\Lambda^{1,0}(M)$.

\punkt We first give the definition of K\"ah\-le\-ri\-an
connections. Assume that the manifold $M$ is equipped with a
connection
$$
\nabla:\Lambda^1(M) \to \Lambda^1(M) \otimes \Lambda^1(M) 
$$
on its cotangent bundle $\Lambda^1(M)$. Let 
\begin{align*}
T &= \Alt \circ \nabla - d_M:\Lambda^1(M) \to \Lambda^2(M) \\
R &= \Alt \nabla \circ \nabla:\Lambda^1(M) \to \Lambda^1(M) \otimes
\Lambda^2(M) 
\end{align*}
be its torsion and curvature, and let $R = R^{2,0} + R^{1,1} +
R^{0,2}$ be the decomposition of the curvature according to the
Hodge type\index{terms}{bigrading!Hodge type}.  

\defn The connection\index{terms}{connections!K\"ahlerian} $\nabla$
is called {\em K\"ah\-le\-ri\-an} if  
\begin{align*}
T &= 0 \tag{i}\\
R^{2,0} &= R^{0,2} = 0 \tag{ii}
\end{align*}

\ex The Levi-Civita connection\index{terms}{connections!Levi-Civita}
on a K\"ahler manifold is K\"ah\-le\-ri\-an.  

\rem The condition $T=0$ implies, in particular, that the component 
$$
\nabla^{0,1}:\Lambda^{1,0}(M) \to \Lambda^{1,1}(M)
$$
of the connection\index{terms}{connections!on a vector bundle}
$\nabla$ coincides with the Dolbeault differential. Therefore a
K\"ah\-le\-ri\-an connection is always
holomorphic\index{terms}{connections!holomorphic}
\index{terms}{connections!K\"ahlerian}. 

\punkt Recall that in \ref{red} we have associated to any
extended\index{terms}{extended connection} connection $D$ on $M$ a
connection $\nabla$ on the cotangent bundle $\Lambda^1(M,\C)$ called
the reduction\index{terms}{extended connection!reduction of} of
$D$. We can now formulate the main result of this section.

\begin{theorem}\label{kal=ext}
\begin{enumerate}
\item If an extended connection $D$ on $M$ is flat and
linear\index{terms}{extended
connection!linear}\index{terms}{extended connection!flat}, then  
its reduction $\nabla$ is K\"ah\-le\-ri\-an. 
\item Every K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian} $\nabla$ on the
cotangent bundle $\Lambda^1(M,\C)$ is the reduction of a unique
linear flat extended connection $D$ on $M$.
\end{enumerate}
\end{theorem} 

The rest of this section is taken up with the proof of
Theorem~\ref{kal=ext}. To make it more accessible, we first give an
informal outline. The actual proof starts with
Subsection~\ref{pf.first.sub}, and it is independent from the rest
of this subsection.

\punkt Assume given a K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian} $\nabla$ on the
manifold $M$. To prove Theorem~\ref{kal=ext}, we have to construct a
flat linear extended connection\index{terms}{extended
connection!linear}\index{terms}{extended connection!flat} $D$ on $M$
with reduction $\nabla$.  Every extended connection decomposes into
a series $D = \sum_{k \geq 0}D_k$ as in
\eqref{aug.con}, and, since $\nabla$ is the reduction of $D$, we
must have $D_1 = \nabla$. We begin by checking in
Lemma~\ref{lin.aug} that if $D$ is linear, then $D_0 = C$, where
$C:S^1(M,\C) \to \Lambda^1(M,\C)$ is as in \ref{C.Weil}. The sum $C
+ \nabla$ is already a linear extended connection on $M$. By
\ref{deriv} it extends to a derivation $D_{\leq 1}$ of the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ of the manifold
$M$, but this derivation does not necessarily satisfy $D_{\leq 1}
\circ D_{\leq 1} = 0$, thus the extended
connection\index{terms}{extended connection} $D_{\leq 1}$ is 
not necessarily flat.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

We have to show that one can add the ``correction terms'' $D_k, k
\geq 2$ to $D_{\leq 1}$ so that $D = \sum_k D_k$ satisfies all the
conditions of Theorem~\ref{kal=ext}. To do this, we introduce in
\ref{red.Weil} a certain quotient $\wB^\cdot(M,\C)$ of the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$, called the
reduced Weil algebra\index{terms}{Weil algebra!reduced}. The reduced
Weil algebra\index{terms}{Weil algebra!reduced} is defined in such a
way that for every extended\index{terms}{extended connection}
connection $D$ the associated derivation $D:\B^\cdot(M,\C) \to
\B^{\cdot+1}(M,\C)$ preserves the kernel of the surjection $\B^\cdot
\to \wB^\cdot$, thus inducing a derivation $\wD:\wB^\cdot(M,\C) \to
\wB^{\cdot+1}(M,\C)$. Moreover, the algebra $\wB^\cdot(M,\C)$ has
the following two properties:
\begin{enumerate}
\item \label{first.prop} The derivation $\wD:\wB^\cdot(M,\C) \to
\wB^{\cdot+1}(M,\C)$ satisfies $\wD \circ \wD = 0$ if and only if
the connection\index{terms}{connections!K\"ahlerian} $D_1$ is
K\"ah\-le\-ri\-an.
\item \label{second.prop} Let $\wD$ be the weakly
Hodge\index{terms}{weakly Hodge map} derivation of the quotient
algebra $\wB^\cdot(M,\C)$ induced by an arbitrary linear extended
connection\index{terms}{extended connection!linear} $D_{\leq 1}$ and
such that $\wD \circ \wD = 0$. Then the derivation $\wD$ lifts
uniquely to a weakly Hodge derivation $D$ of the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ such that $D
\circ D = 0$, and the derivation also $D$ comes from a linear
extended connection on $M$ (see Proposition~\ref{main} for a precise
formulation of this statement).
\end{enumerate}

\punkt The property \ref{first.prop} is relatively easy to check,
and we do it in the end of the proof, in
Subsection~\ref{pf.last.sub}. The rest is taken up with establishing
the property \ref{second.prop}. The actual proof of this statement
is contained in Proposition~\ref{main}, and
Subsection~\ref{pf.first.sub} contains the necessary preliminaries.

Recall that we have introduced in \ref{aug} a new grading on the
Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$, called the
augmentation grading\index{terms}{grading!augmentation}, so that the
component $D_k$ in the decomposition $D = \sum_kD_k$ is of
augmentation degree\index{terms}{degree!augmentation} $k$. In order
to lift $\wD$ to a derivation $D$ so that $D \circ D = 0$, we begin
with the given lifting $D_{\leq 1}$ and then add components $D_k, k
\geq 2$, one by one, so that on each step for $D_{\leq k} = D_{\leq
1} + \sum_{2 \leq p \leq k}$ the composition $D_{\leq k}
\circ D_{\leq k}$ is zero in augmentation degrees from $0$ to
$k$. In order to do it, we must find for each $k$ a solution to the
equation
\begin{equation}\label{tslv}
D_0 \circ D_k  = - R_k,
\end{equation}
where $R_k$ is the component of augmentation
degree\index{terms}{degree!augmentation} $k$ in the composition
$D_{\leq k-1} \circ D_{\leq k-1}$. This solution must be weakly
Hodge\index{terms}{weakly Hodge map}, and the
extended\index{terms}{extended connection} connection $D_{\leq k} =
D_{\leq k-1} + D_k$ must be linear\index{terms}{extended
connection!linear}.

We prove in Lemma~\ref{lin.aug} that since $D_{\leq 0}$ is linear,
we may assume that $D_0 = C$. In addition, since $\wD \circ \wD =
0$, we may assume by induction that the image of $R_k$ lies in the
kernel $\I^\cdot$ of the quotient map $\B^\cdot(M,\C) \to
\wB^\cdot(M,\C)$.

\punkt \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
In order to analyze weakly Hodge maps\index{terms}{weakly Hodge map}
from $S^1(M,\C)$ to the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot(M,\C)$, we apply the functor $\Gamma:\prehodge_{\geq 0}(M)
\to \prehodge_0(M)$ constructed in \ref{gamma.m} to the bundle
$\B^\cdot(M,\C)$ to obtain the total Weil algebra\index{terms}{Weil
algebra!total} $\B_{tot}^\cdot(M,\C) = \Gamma(\B^\cdot(M,\C))$ of
weight\index{terms}{weight!of a Hodge bundle} $0$, which we studied
in Subsection~\ref{t.W.sub}. The Hodge bundle\index{terms}{Hodge
bundle} $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)$
on the manifold $M$ is of weight\index{terms}{weight!of a Hodge
bundle} $0$, and, by the universal property of the functor $\Gamma$,
weakly Hodge maps from $S^1(M,\C)$ to $\B^\cdot(M,\C)$ are in
one-to-one correspondence with Hodge bundle maps from
$S^1(M,\C)$ to the total Weil algebra\index{terms}{Weil algebra!total}
$\B^\cdot_{tot}(M,\C)$. The canonical map $C:S^1(M,\C) \to
\B^1(M,\C)$ extends to a derivation $C:\B^\cdot_{tot}(M,\C) \to
\B^{\cdot+1}_{tot}(M,\C)$. Moreover, the weakly Hodge map
$R_k:S^1(M,\C) \to \B^2(M,\C)$ defines a Hodge bundle map
$R_k^{tot}:S^1(M,\C) \to \B^2_{tot}(M,\C)$, and solving \eqref{tslv}
is equivalent to finding a Hodge bundle map $D_k:S^1(M,\C) \to
\B^1_{tot}(M,\C)$ such that
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\begin{equation}\label{tslvc}
C \circ D_k = - R_k.
\end{equation}

\punkt Recall that by \ref{C.and.sigma} the derivation
$$
C:\B^\cdot_{tot}(M,\C) \to \B^{\cdot+1}_{tot}(M,\C)
$$ 
satisfies $C \circ C = 0$, so that the total Weil
algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}(M,\C)$
becomes a complex with differential $C$. The crucial part of the
proof of Theorem~\ref{kal=ext} consists in noticing that the
subcomplex $\I_{tot}^\cdot(M,\C) = \Gamma(\I^\cdot(M,\C)) \subset
\B^\cdot_{tot}(M,\C)$ of the total Weil algebra\index{terms}{Weil
algebra!total} $\B^\cdot_{tot}(M,\C)$ corresponding to the kernel 
$\I^\cdot(M,\C) \subset \B^\cdot(M,\C)$ of the quotient map
$\B^\cdot(M,\C) \to \wB^\cdot(M,\C)$ is canonically contractible.
This statement is analogous to Proposition~\ref{ac}, and we prove it
in the same way. Namely, we check that the subcomplex
$\I^\cdot_{tot} \subset \B^\cdot_{tot}$ is preserved by the bundle
map $\sigma_{tot}:\B^{\cdot+1}_{tot} \to \B^\cdot_{tot}$ constructed
in \ref{sigma.c}, and that the anticommutator $h =
\{\sigma_{tot},C\}:\B^\cdot_{tot}(M,\C) \to \B^\cdot_{tot}(M,\C)$ is
invertible on the subcomplex $\I_{tot}^\cdot(M,\C) \subset
\B^\cdot_{tot}(M,\C)$ (Corollary~\ref{h.inv} of
Lemma~\ref{h.acts}). We also check that $C \circ R_k^{tot} = 0$,
which implies that the Hodge bundle\index{terms}{Hodge bundle} map
\begin{equation}\label{sltn}
D_k = -h^{-1} \circ \sigma_{tot} \circ R_k^{tot}:S^1(M,\C) \to \B^1_{tot}(M,\C)
\end{equation}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
provides a solution to the equation \eqref{tslvc}. 

\punkt To establish the property \ref{second.prop}, we have to
insure additionally that the extended connection $D = D_{\leq k}$ is
linear\index{terms}{extended connection!linear}, and and we have to
show that the solution $D_k$ of \eqref{tslvc} with this property is
unique. This turns out to be pretty straightforward. We show in
Lemma~\ref{lin.aug} that $D_{\leq k}$ is linear if and only if
\begin{equation}\label{lnr}
\sigma_{tot} \circ D_k = 0.
\end{equation}
Moreover, we show that the homotopy $\sigma_{tot}:\B^{\cdot+1}_{tot}(M,\C)
\to \B^\cdot_{tot}(M,\C)$ satisfies $\sigma_{tot} \circ \sigma_{tot} =
0$. Therefore the solution $D_k$ to \eqref{tslvc} given by
\eqref{sltn} satisfies \eqref{lnr} automatically.

The uniqueness of such a solution $D_k$ follows from the
invertibility of $h = C \circ \sigma_{tot} + \sigma_{tot} \circ C$. Indeed,
for every two solutions $D_k,D_k'$ to \eqref{tslv}, both satisfying
\eqref{lnr}, their difference $P = D_k - D'_k$ satisfies $C \circ P
= 0$. If, in addition, both $D_k$ and $D'_k$ were to satisfy
\eqref{lnr}, we would have had $\sigma_{tot} \circ P = 0$. Therefore $h
\circ P = 0$, and $P$ has to vanish.

\punkt These are the main ideas of the proof of
Theorem~\ref{kal=ext}. The proof itself begins in the next
subsection, and it is organized as follows. In
Subsection~\ref{pf.first.sub} we express the linearity
condition\index{terms}{linearity condition} on an
extended\index{terms}{extended connection} connection $D$ in terms of
the associated derivation $D_{tot}:\B^\cdot_C \to
\B^{\cdot+1}_{tot}$ of the total Weil algebra\index{terms}{Weil
algebra!total} $\B^\cdot_{tot}$ of the manifold $M$. After that, we 
introduce in Subsection~\ref{pf.third.sub} the reduced Weil
algebra\index{terms}{Weil algebra!reduced} $\wB^\cdot(M,\C)$ and
prove Proposition~\ref{main}, thus reducing Theorem~\ref{kal=ext} to
a statement about derivations of the reduced Weil
algebra\index{terms}{Weil algebra!reduced}. Finally, in
Subsection~\ref{pf.last.sub} we prove this statement.

\rem In the Appendix we give, following
Deligne\index{names}{Deligne, P.} and Simpson\index{names}{Simpson, C.}, a
more geometric description of the functor $\Gamma:\prehodge_{\geq 0}
\to \prehodge$ and of the total Weil algebra\index{terms}{Weil
algebra!total} $\B^\cdot_{tot}(M,\C)$, which allows to give a simpler and
more conceptual proof for the key parts of Theorem~\ref{kal=ext}.

\subsection{Linearity and the total Weil
algebra}\label{pf.first.sub}\index{terms}{Weil algebra!total}

\punkt \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Assume given an extended\index{terms}{extended connection}
connection $D:S^1 \to \B^1$ on the manifold $M$, and extend it to a
derivation $D:\B^\cdot \to \B^{\cdot+1}$ of the Weil
algebra\index{terms}{Weil algebra} as in
\ref{deriv}.  Let $D = \sum_{k\geq 0} D_k$ be the augmentation
degree\index{terms}{degree!augmentation} decomposition.  The
derivation $D$ is weakly Hodge\index{terms}{weakly Hodge map} and
defines therefore a derivation $D = \sum_{k\geq 0}
D_k:\B_{tot}^\cdot \to \B_{tot}^{\cdot+1}$ of the total Weil
algebra\index{terms}{Weil algebra!total} $\B_{tot}^\cdot$.

Before we begin the proof of Theorem~\ref{kal=ext}, we give the
following rewriting of the linearity
condition\index{terms}{linearity condition} \ref{lin.ext.con} on
the extended\index{terms}{extended connection} connection $D$ in
terms of the total Weil algebra\index{terms}{Weil algebra!total}.

\begin{lemma}\label{lin.aug}
The extended\index{terms}{extended connection} connection $D$ is
linear\index{terms}{extended connection!linear} if and only if
$D_0=C$ and $\sigma_{tot} \circ D_k = 0$ on $S^1 \subset \B_{tot}^0$
for every $k \geq 0$. 
\end{lemma}

\proof \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Indeed, by Lemma~\ref{total.aug} for odd integers $k$ and $n
= \pm 1$ the subbundle $\left(\B_o^1\right)^{n,-n}_{k+1} \subset
\B_{tot}^1$ vanishes. Therefore the map $D_k:S^1 \to \B_{tot}^1$ factors 
through $\B_{ll}^1 \oplus \B_{rr}^1$.  Since by definition
(\ref{sigma.c}) we have $\sigma_{tot}=0$ on both $\B_{ll}^1$ and
$\B_{rr}^1$, for odd $k$ we have $\sigma_{tot} \circ D_k = 0$ on
$S^1$ regardless of the extended\index{terms}{extended connection}
connection $D$. On the other hand, for even $k$ we have
$\left(\B_{tot}^1\right)^{n,-n}_{k+1} =
\left(\B_o^1\right)^{n,-n}_{k+1}$. Therefore on
$S^1$ we have $\sigma_{tot}
\circ D_k = \sigma \circ D_k$ (where $\sigma:\B^{\cdot+1}
\to \B^\cdot$ is as in \ref{C.and.sigma}). Moreover, since
$\sigma:\Lambda^1 \to S^1$ is an isomorphism, $D_0 = C$ is
equivalent to $\sigma \circ D_0 = \id:S^1 \to \Lambda^1 \to
S^1$. Therefore the condition of the lemma is equivalent to the
following
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\begin{equation}\label{tprv}
\sigma \circ D_k = 
\begin{cases} 
\id, \quad &\text{ for } k=0\\
0,   \quad &\text{ for even integers } k > 0.
\end{cases}
\end{equation}
Let now $\iota\index{terms}{involution $\iota$}^*:\B^\cdot \to
\B^\cdot$ be the operator given by the 
action of the canonical involution $\iota\index{terms}{involution
$\iota$}:\tm \to \tm$, as in 
\ref{iota.Weil}, and let $D^\iota\index{terms}{involution $\iota$} =
\sum_{k \geq 0} D_k^\iota\index{terms}{involution $\iota$} = 
\iota\index{terms}{involution $\iota$}^* \circ D
\circ(\iota\index{terms}{involution $\iota$}^*)^{-1}$ be the
operator $\iota\index{terms}{involution $\iota$}^*$-conjugate to the
derivation $D$. The operator $\iota\index{terms}{involution
$\iota$}^*$ acts as $-\id$ on $S^1\index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} \subset \B^0$ and as $\id$ on $\Lambda^1
\subset \B^1$. Since it is an algebra automorphism, it acts as
$(-1)^{i+k}$ on $\B_k^i \subset \B^\cdot$. Therefore
$D_k^\iota\index{terms}{involution $\iota$} = (-1)^{k+1}D_k$, and
\eqref{tprv} is equivalent to
$$
\sigma \circ \half(D - D^\iota) =
\id:S^1 \to \B^1 \to S^1,   
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\index{terms}{involution $\iota$}
which is precisely the definition of a linear extended
connection\index{terms}{extended connection!linear}. 
\endproof 

\subsection{The reduced Weil
algebra}\label{pf.third.sub}\index{terms}{Weil algebra!reduced}

\punkt \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
We now begin the proof of Theorem~\ref{kal=ext}. Our first step is
to reduce the classification of linear\index{terms}{extended
connection!linear} flat extended\index{terms}{extended
connection!flat} connections $D:S^1 \to \B^1$ on the manifold $M$ to
the study of derivations of a certain quotient $\wB^\cdot$ of the
Weil algebra\index{terms}{Weil algebra} $\B^\cdot$. We introduce
this quotient in this subsection under the name of reduced Weil
algebra\index{terms}{Weil algebra!reduced}. We then show that every
extended\index{terms}{extended connection} connection $D$ on $M$
induces a derivation $\wD:\wB^\cdot \to \wB^{\cdot+1}$ of the
reduced Weil algebra\index{terms}{Weil algebra!reduced}, and that a
linear flat extended connection $D$ on $M$ is completely defined by
the derivation $\wD$.

\punkt By Lemma~\ref{h.acts} the anticommutator $h =
\{C,\sigma_{tot}\}:\B^\cdot_{tot} \to \B^\cdot_{tot}$ of the
canonical bundle endomorphisms $C,\sigma_{tot}$ of the total Weil
algebra\index{terms}{Weil algebra!total} $\B_{tot}^\cdot$ is
invertible on every component $\left(\B^\cdot_{tot}\right)_{p,q}$ of
augmentation bidegree $(p,q)$ with $p,q \geq 1$.

The direct sum $\oplus_{p,q \geq
1}\left(\B_{tot}^\cdot\right)_{p,q}$ is an ideal in the total Weil
algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}$, and it is
obtained by applying the functor $\Gamma$ to the ideal $\oplus_{p,q
\geq 1}\B^\cdot_{p,q}$ in the Weil algebra\index{terms}{Weil
algebra} $\B^\cdot$. For technical reasons, it will be more
convenient for us to consider the smaller subbundle
$$
\I^\cdot = \bigoplus_{p \geq 2,q \geq 1}\B^\cdot_{p,q} +
\bigoplus_{p \geq 1, q \geq 2} \B^\cdot_{p,q} \subset \B^\cdot 
$$
of the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$. The
subbundle $\I^\cdot$ is a Hodge subbundle in $\B^\cdot$, and it is
an ideal with respect to the multiplication in $\B^\cdot$.

\punkt \defn \label{red.Weil} The reduced Weil
algebra\index{terms}{Weil algebra!reduced} $\wB^\cdot =
\wB^\cdot(M,\C)$ of the manifold $M$ is the quotient
$$
\wB^\cdot = \B^\cdot / \I^\cdot
$$
of the full Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ by
the ideal $\I^\cdot$.

The reduced Weil algebra\index{terms}{Weil algebra!reduced}
decomposes as
$$
\wB^\cdot = \B^\cdot_{1,1} \oplus \bigoplus_{p \geq 0}\B^\cdot_{p,0}
\oplus \bigoplus_{q \geq 0}\B^\cdot_{0,q} 
$$
with respect to the augmentation
bigrading\index{terms}{bigrading!augmentation} on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot$. The two summands on
the right are equal to
\begin{align*}
\bigoplus_{p \geq 0}\B_{p,0}^\cdot &= \bigoplus_{p geq
0}S^p\left(S^{1,-1}\right) \otimes \Lambda^{\cdot,0}, \\ 
\bigoplus_{q \geq 0}\B_{0,q}^\cdot &= \bigoplus_{q geq
0}S^q\left(S^{-1,1}\right) \otimes \Lambda^{0,\cdot},
\end{align*} 

\punkt Since $\I^\cdot$ is a Hodge subbundle, the reduced Weil
algebra\index{terms}{Weil algebra!reduced} carries a canonical Hodge
bundle\index{terms}{Hodge bundle} structure compatible with the
multiplication. It also obviously inherits the augmentation
bigrading\index{terms}{bigrading!augmentation}, and defines an ideal
$\I_{tot}^\cdot = \Gamma(\I^\cdot)
\subset \B_{tot}^\cdot$ in the total Weil algebra\index{terms}{Weil
algebra!total} $\B_{tot}^\cdot$. Lemma~\ref{h.acts} immediately 
implies the following fact.

\begin{corr}\label{h.inv}
The map $h=\{C,\sigma_{tot}\}:\B_{tot}^\cdot \to \B_{tot}^\cdot$ is invertible
on $\I_{tot}^\cdot \subset \B_{tot}^\cdot$. 
\end{corr} 

\punkt \label{wD} Let now $D:\B^\cdot \to \B^\cdot$ be the
derivation associated to the extended\index{terms}{extended
connection!derivation associated to} connection $D$ as in 
\ref{deriv}. The derivation $D$ does not increase the augmentation
bidegree, it preserves the ideal
$\I^\cdot \subset \B^\cdot$ and defines therefore a weakly
Hodge\index{terms}{weakly Hodge map} derivation of the reduced Weil
algebra\index{terms}{Weil algebra!reduced} $\wB^\cdot$, which we
denote by $\wD$. If the extended\index{terms}{extended connection}
connection $D$ is flat, then the derivation $\wD$ satisfies $\wD
\circ \wD = 0$.

We now prove that every derivation $\wD:\wB^\cdot \to \wB^{\cdot+1}$
of this type comes from a linear\index{terms}{extended
connection!linear} flat extended\index{terms}{extended connection!flat}
connection $D$, and that the connection 
$D$ is completely defined by $\wD$.  More precisely, we have the
following.

\begin{prop}\label{main}\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Let $D:S^1 \to \B^1$ be a linear\index{terms}{extended
connection!linear} but not necessarily flat
extended\index{terms}{extended connection!flat} connection on $M$,
and let $\wD:\wB^\cdot \to
\wB^{\cdot+1}$ be the associated weakly Hodge\index{terms}{weakly
Hodge map} derivation of the reduced Weil
algebra\index{terms}{Weil algebra!reduced} $\wB^\cdot$. Assume that
$\wD \circ \wD = 0$.

There exists a unique weakly Hodge\index{terms}{weakly Hodge map}
bundle map \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$P:S^1 \to
\I^1$ such that the extended\index{terms}{extended connection}
connection $D' = D + P:S^1 \to \B^1$ is linear and flat.
\end{prop}

\punkt \proof Assume given a linear\index{terms}{extended
connection!linear} extended connection $D$ satisfying the condition
of Proposition~\ref{main}. To prove the proposition, we have to
construct a weakly Hodge\index{terms}{weakly Hodge map} map
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$P:S^1 \to
\I^1$ such that the extended\index{terms}{extended connection}
connection $D+P$ is linear and flat. We 
do it by induction on the augmentation
degree\index{terms}{degree!augmentation}, that is, we construct
one-by-one the terms $P_k$ in the augmentation degree decomposition
$P = \sum_k P_k$. The identity $\wD \circ \wD = 0$ is the base of
the induction, and the induction step is given by applying the
following lemma to $D + \sum_{i=2}^k P_k$, for each $k \geq 1$ in
turn.

\begin{lemma}\label{main.ind}
Assume given a linear extended connection\index{terms}{extended
connection!linear} $D:S^1\index{terms}{Hodge 
bundle!$S^1(M,\protect\C)$} \to \B^{\cdot+1}$ 
on $M$ and let $D:\B^\cdot \to \B^{\cdot+1}$ also denote the
associated derivation.  Assume also that the composition $D \circ
D:\B^\cdot \to \B^{\cdot+2}$ maps $S^1$ into $\I^2_{>k} = \oplus_{p>k}\I^2_p$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
There exists a unique weakly Hodge\index{terms}{weakly Hodge map}
bundle map $P_k:S^1 \to \I^1_{k+1}$ such that the
extended\index{terms}{extended connection} connection $D' = D +
P_k:S^1 \to \B^1$ is linear\index{terms}{extended
connection!linear}, and for the associated derivation $D':\B^\cdot
\to \B^{\cdot+1}$ the composition $D' \circ D'$ maps
$S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$ into
$\I^2_{>k+1} = \oplus_{p>k+1}\I^2_p$.
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\end{lemma}

\proof Let $D:\B_{tot}^\cdot \to \B_{tot}^\cdot$ be the derivation
of the total Weil algebra\index{terms}{Weil algebra!total}
associated to the extended\index{terms}{extended
connection!derivation associated to} connection $D$, and let  
$$
R:\left(\B_{tot}^\cdot\right)_\cdot \to
\left(\B_{tot}^{\cdot+2}\right)_{\cdot+k}
$$ 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
be the component of augmentation
degree\index{terms}{degree!augmentation} $k$ of the composition $D
\circ D:S^1 \to \B^2_{tot}$. Note that by
\ref{deriv} the map $R$ vanishes on the subbundle $\Lambda^1_{tot}
\subset
\left(\B^1_{tot}\right)_1$. Moreover, the composition $C \circ
R:\B^\cdot_{tot} \to \B^{\cdot+3}$ of the map $R$ with the canonical
derivation $C:\B^\cdot_{tot} \to \B^{\cdot+1}_{tot}$ vanishes on the
subbundle $S^1 \subset
\B^0_{tot}$. Indeed, since $C$ maps $S^1$ into $\Lambda^1_{tot}$,
the composition $C \circ R$ is 
equal to the commutator $[C,R]:S^1 \to \B^3_{tot}$.  Since the
extended connection $D$ is by assumption
linear\index{terms}{extended connection!linear}, we have $D_0=C$, and
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\begin{align*}
C \circ R = [C,R] &= \sum_{0 \leq p \leq k}[C,D_p \circ D_{k-p}] =\\
&= [C,\{C,D_k\}] + \sum_{1 \leq p \leq k-1}[C,D_p \circ D_{k-p}].
\end{align*}
Since $C \circ C = 0$, the first term in the right hand side
vanishes. Let $\Theta = \sum_{1 \leq p \leq k-1}D_p:\B^\cdot_{tot} \to
\B^{\cdot+1}_{tot}$. Then the second term is the component of
augmentation degree\index{terms}{degree!augmentation} $k$ in the
commutator $[C,\Theta \circ \Theta]:S^1 \to \B^3_{tot}$. By
assumption $\{D,D\}=0$ in augmentation degrees $<k$. Therefore we
have $\{C,\Theta\} = - \{\Theta,\Theta\}$ in augmentation degrees $<
k$. Since $\Theta$ increases the augmentation degree, this implies
that in augmentation degree $k$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
[C,\Theta \circ \Theta] = \{C,\Theta\} \circ \Theta - \Theta \circ
\{C,\Theta\} = [\Theta,\{\Theta,\Theta\}],
$$
which vanishes tautologically. 

\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
The set of all weakly Hodge\index{terms}{weakly Hodge map} maps
$P:S^1 \to \I_k^1$ coincides with the set of all maps $P:S^1 \to
\left(\I_{tot}^1\right)_k$ preserving the Hodge
bundle\index{terms}{Hodge bundle} structures. Let $P$ be such a map,
and let $D':\B_{tot}^\cdot \to \B_{tot}^{\cdot+1}$ be the derivation
associated to the extended\index{terms}{extended
connection!derivation associated to} connection $D' = D + P$.  
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}

Since the extended connection $D$ is by assumption
linear\index{terms}{extended connection!linear}, by
Lemma~\ref{lin.aug} the extended\index{terms}{extended connection}
connection $D'$ is linear if and only if $\sigma_{tot} \circ P = 0$.
Moreover, since the augmentation
degree\index{terms}{degree!augmentation}-$0$ component of the
derivation $D$ equals $C$, the augmentation degree-$k$ component
$Q:S^1 \to \B^2_{tot}$ in the composition $D' \circ D'$ is equal to
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
Q = R + \{C,(D'-D)\}.
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
By definition $D'-D:\B_{tot}^\cdot \to \B_{tot}^{\cdot+1}$ equals $P$ on
$S^1 \subset \B_{tot}^0$ and vanishes on $\Lambda^1_{tot} \subset
\B_{tot}^1$. Since $C$ maps $S^1$ into $\Lambda^1_{tot} \subset \B_{tot}^1$, we
have $Q = R + C \circ P$. Thus, a map $P$ satisfies the condition of
the lemma if and only if
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
\begin{cases}
C \circ P = -R\\
\sigma_{tot} \circ P = 0
\end{cases}
$$
To prove that such a map $P$ is unique, note that these equations imply 
$$
h \circ P = (\sigma_{tot} \circ C + C \circ \sigma_{tot}) \circ P =
\sigma_{tot} \circ R, 
$$
and $h$ is invertible by Corollary~\ref{h.inv}. To prove that such a
map $P$ exists, define $P$ by 
$$
P = -h^{-1} \circ \sigma_{tot} \circ R:S^1 \to \I^1_{k+1}.
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
The map $h = \{C,\sigma_{tot}\}$ and its inverse $h^{-1}$ commute with
$C$ and with $\sigma_{tot}$.  Since $\sigma_{tot} \circ \sigma_{tot} = C \circ
C0$, we have $\sigma_{tot} \circ P = h^{-1} \circ \sigma_{tot} \circ
\sigma_{tot} \circ R = 0$. On the other hand, $C \circ R = 0$. Therefore
\begin{align*}
C \circ P &= - C \circ h^{-1} \circ \sigma_C \circ R = - h^{-1} \circ
C \circ \sigma_{tot} \circ R = \\
&= h^{-1} \circ \sigma_{tot} \circ C \circ R - h^{-1} \circ h \circ R = -R.
\end{align*}
This finishes the proof of the lemma and of Proposition~\ref{main}.
\endproof

\subsection{Reduction of extended
connections}\label{pf.last.sub}\index{terms}{extended
connection!reduction of}

\punkt We now complete the proof of Theorem~\ref{kal=ext}.  First we
will need to identify explicitly the low Hodge
bidegree\index{terms}{bidegree!Hodge} components
of the reduced Weil algebra\index{terms}{Weil algebra!reduced}
$\wB^\cdot$. The following is easily checked by direct inspection.

\begin{lemma}
We have 
\begin{align*}
&\wB^{2,-1} \oplus \wB^{1,0} \oplus \wB^{0,1} \oplus \wB^{-1,2} =
\Lambda^1 \oplus \left(S^1 \otimes \Lambda^1\right) \subset \wB^1\\ 
&\wB^{3,-2} \oplus \wB^{2,-1} \oplus \wB^{1,1} \oplus \wB^{-1,2}
\oplus \wB^{-2,3} = \\
&\qquad\qquad\qquad\qquad\qquad\qquad = \left(S^{1,-1} \otimes
\Lambda^{2,0}\right) \oplus \left(S^{-1,1} \otimes
\Lambda^{0,2}\right) \oplus \Lambda^2 \subset \wB^2   
\end{align*}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\end{lemma}

\punkt \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Let now $\nabla:S^1\to S^1 \otimes \Lambda^1$ be
an arbitrary real connection\index{terms}{connections!on a vector
bundle} on the bundle $S^1$. The operator
$$
D = C + \nabla:S^1 \to
\Lambda^1 \oplus \left(\Lambda^1 \otimes S^1\right)  \subset \B^1 
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
is then automatically weakly Hodge\index{terms}{weakly Hodge map}
and defines therefore an extended\index{terms}{extended connection}
connection on $M$. This connection 
is linear\index{terms}{extended connection!linear} by
Lemma~\ref{lin.aug}. Extend $D$ to a derivation $D:\B^\cdot \to
\B^{\cdot+1}$ as in \ref{deriv}, and let $\wD:\wB^\cdot \to
\wB^{\cdot+1}$ be the associated derivation of the reduced Weil
algebra\index{terms}{Weil algebra!reduced}.

\begin{lemma}\label{kal=red}
The derivation $\wD$ satisfies $\wD \circ \wD = 0$ if and only if
the connection\index{terms}{connections!K\"ahlerian} $\nabla$ is
K\"ah\-le\-ri\-an.  
\end{lemma}

\proof Indeed, the operator $\wD \circ \wD$ is weakly
Hodge\index{terms}{weakly Hodge map}, hence factors through a bundle
map
\begin{multline*}
\wD \circ \wD: S^1 \to \wB^{3,-1} \oplus \wB^{2,0} \oplus \wB^{1,1}
\oplus \wB^{0,2} \oplus \wB^{-1,3} = \\
= \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus \left(S^{-1,1}
\otimes \Lambda^{0,2}\right) \oplus \Lambda^2 \subset \B^2.  
\end{multline*}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
By definition we have 
$$
\wD \circ \wD = (C + \nabla) \circ (C + \nabla) = \{C, \nabla\} + \{
\nabla, \nabla \}.
$$
An easy inspection shows that the sum is direct, and the first
summand equals 
$$
\{C, \nabla\} = T \circ C:S^1 \to \Lambda^2,  
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
where $T$ is the torsion of the
connection\index{terms}{connections!on a vector bundle} $\nabla$,
while the second summand equals
$$
\{\nabla, \nabla\} = R^{2,0} \oplus R^{0,2}: S^{1,-1} \oplus
S^{-1,1} \to \left(S^{1,-1} \otimes \Lambda^{2,0}\right) \oplus
\left(S^{-1,1} \otimes \Lambda^{0,2}\right), 
$$
where $R^{2,0}$, $R^{0,2}$ are the Hodge type\index{terms}{bigrading!Hodge
type} components of the curvature of the connection $\nabla$. Hence
$\wD \circ \wD = 0$ if and only if $R^{2,0} = R^{0,2} = T = 0$,
which proves the lemma and finishes the proof of
Theorem~\ref{kal=ext}.
\endproof

\punkt We finish this section with the following corollary of
Theorem~\ref{kal=ext} which gives an explicit expression for the
augmentation degree\index{terms}{degree!augmentation}-$2$ component
$D_2$ of a flat linear\index{terms}{extended connection!linear}
extended\index{terms}{extended connection!flat} connection $D$ on the
manifold $M$. We will need this expression in Section~\ref{metrics.section}.

\begin{corr}\label{D.2}
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Let $D = \sum_{k \geq 0}D_k:S^1 \to \B^1$ be a flat
linear\index{terms}{extended connection!linear}
extended\index{terms}{extended connection!flat} 
connection on $M$, so that $D_0 = C$ and $D_1$ 
is a K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian} on $M$. We have   
$$
D_2 = \frac{1}{3}\sigma \circ R, 
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
where $\sigma:\B^{\cdot+1} \to \B^\cdot$ is the canonical derivation
introduced in \ref{sigma}, and $R = D_1 \circ
D_1:S^1 \to S^1\otimes \Lambda^{1,1} \subset \B^2$ is the curvature of the
K\"ah\-le\-ri\-an connection $D_1$.
\end{corr}

\proof \index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
Extend the connection $D$ to a derivation $D_C = \sum_{k \geq
0}D^{tot}_k:\B^\cdot_{tot} \to \B^{\cdot+1}_{tot}$ of the total Weil
algebra\index{terms}{Weil algebra!total}. By the construction used
in the proof of Lemma~\ref{main.ind} we have $D^{tot}_2 = h^{-1}
\circ \sigma_{tot} \circ R_{tot}:S^1 \to \left(\B^1_{tot}\right)_3$,
where $h:\B^\cdot_{tot} \to \B^\cdot_{tot}$ is as in
Lemma~\ref{h.acts}, the map $\sigma_{tot}:\B^{\cdot+1}_{tot} \to
\B^\cdot_{tot}$ is the canonical map constructed in \ref{sigma.c},
and $R^{tot}:S^1 \to \left(\B^2_{tot}\right)_3$ is the square
$R_{tot} = D^{tot}_1 \circ D^{tot}_1$ of the derivation
$D^{tot}_1:\B^\cdot_{tot} \to \B^{\cdot+1}_{tot}$. By
Lemma~\ref{h.on.b1} the map $h$ acts on $\left(\B_{tot}^1\right)_3$
by multiplication by $3$. Moreover, it is easy to check that
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
$$
\left(B^2_{tot}\right)_3 = \left(S^1 \otimes \Lambda^2\right) \oplus
\left(S^{-1,1} \otimes \Lambda^{2,0}\right)\oplus
\left(S^{1,-1} \otimes \Lambda^{0,2}\right), 
$$
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
and the map $R^{tot}:S^1 \to \left(B^2_{tot}\right)_3$ sends $S^1$
into the first summand in this decomposition and coincides with the
curvature $R:S^1 \to S^1\index{terms}{Hodge
bundle!$S^1(M,\protect\C)$} \otimes  \Lambda^2 \subset
\left(B^2_{tot}\right)_3$ of the K\"ah\-le\-ri\-an 
connection\index{terms}{connections!K\"ahlerian} $D_1$. Therefore
$\sigma_{tot} \circ R^{tot} = \sigma \circ R$, which proves the claim. 
\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}
\endproof

\section{Metrics}\label{metrics.section}

\subsection{Hyperk\"ahler metrics on Hodge
manifolds}\index{terms}{Hodge
manifold}\index{terms}{metric!hyperk\"ahler} 

\punkt Let $M$ be a complex manifold equipped with a
K\"ah\-le\-ri\-an connection\index{terms}{connections!K\"ahlerian}
$\nabla$, and consider the associated 
linear formal Hodge manifold\index{terms}{Hodge
manifold!formal}\index{terms}{Hodge manifold!linear} structure on the
tangent bundle $\tm$. In this section we construct a natural
bijection between the set of all
polarizations\index{terms}{polarization} on the Hodge 
manifold\index{terms}{Hodge manifold} $\tm$ in the sense of
Subsection~\ref{polarization} and the set of all K\"ahler
metrics\index{terms}{metric!K\"ahler} on $M$ compatible with the 
given connection $\nabla$.

\punkt \label{restr} Let $h$ be a hyperk\"ahler
metric\index{terms}{metric!hyperk\"ahler} on $\tm$, or, more 
generally, a formal germ of such a metric in the neighborhood of the
zero section $M \subset \tm$. Assume that the metric $h$ is
compatible with the given hypercomplex
structure\index{terms}{hypercomplex manifold} and
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge} in the sense of
\ref{hermhodge}, and let $\omega_I$ be the K\"ahler form associated
to $h$ in the preferred\index{terms}{preferred complex structure} complex structure $\tm_I$ on $\tm$.

Let $h_M$ be the restriction of the metric $h$ to the zero section
$M \subset \tm$, and let $\omega \in C^\infty(M,\Lambda^{1,1}(M))$
be the associated real $(1,1)$-form on the complex manifold
$M$. Since the embedding $M \subset \tm_I$ is holomorphic, the form
$\omega$ is the restriction onto $M$ of the form $\omega_I$. In
particular, it is closed, and the
metric\index{terms}{metric!K\"ahler} $h_M$ is therefore K\"ahler. 

\punkt The main result of this section is the following. 

\begin{theorem}\label{metrics}
Restriction onto the zero section $M \subset \tm$ defines a
one-to-one correspondence between 
\begin{enumerate}
\item K\"ahler metrics\index{terms}{metric!K\"ahler} on $M$ 
compatible with the K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian} $\nabla$, and
\item formal germs in the neighborhood on $M \subset \tm$ of
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge} hyperk\"ahler 
metrics\index{terms}{metric!hyperk\"ahler} on $\tm$ compatible with 
the given formal Hodge manifold\index{terms}{Hodge manifold!formal}
structure.
\end{enumerate}
\end{theorem}

\noindent 
The rest of this section is devoted to the proof of this theorem. 

\punkt In order to prove Theorem~\ref{metrics}, we reformulate it in
terms of polarizations\index{terms}{polarization} rather than
metrics. Recall (see \ref{positive}) that a {\em polarization} of
the formal Hodge manifold\index{terms}{Hodge manifold!formal} $\tm$
is by definition a $(2,0)$-form $\Omega \in
C^\infty_M(\tm,\Lambda^{2,0}(\tm_J))$ for the
complementary\index{terms}{complementary complex structure} complex
structure $\tm_J$ which is holomorphic, real and of
$H$-type\index{terms}{Hodge type1@$H$-type} $(1,1)$ with respect to the
canonical Hodge bundle\index{terms}{Hodge bundle} structure on
$\Lambda^{2,0}(\tm_J)$, and satisfies a certain positivity condition
\eqref{P}.

\punkt By Lemma~\ref{pol.hm}
Hermitian-Hodge\index{terms}{metric!Hermitian-Hodge} hyperk\"ahler
metrics\index{terms}{metric!hyperk\"ahler} on $\tm$ are in
one-to-one correspondence with
polarizations\index{terms}{polarization}. Let $h$ be a K\"ahler
metric\index{terms}{metric!K\"ahler} on $M$, and let $\omega_I$ and 
$\omega$ be the K\"ahler forms for $h$ on $\tm_I$ and on $M \subset
\tm_I$.  The corresponding polarization $\Omega
\in C^\infty(\tm,\Lambda^{2,0}(\tm_J))$ satisfies by
\eqref{omega.and.Omega} 
\begin{equation}\label{pol2kal}
\omega_I = \half\left(\Omega + \nu(\Omega)\right) \in \Lambda^2(\tm,\C),
\end{equation}
where $\nu:\Lambda^2(\tm,\C) \to \overline{\Lambda^2(\tm,\C)}$ is
the usual complex conjugation. 

\punkt Let $\rho:\tm \to M$ be the natural projection, and let 
$$
\Res:\rho_*\Lambda^{\cdot,0}(\tm_J) \to \Lambda^\cdot(M,\C)
$$
be the map given by the restriction onto the zero section $M \subset
\tm$. Both bundles are naturally Hodge bundles\index{terms}{Hodge
bundle} of the same weight\index{terms}{weight!of a Hodge bundle} on
$M$ in the sense of \ref{hodge.bundles}, and the bundle map $\Res$
preserves the Hodge bundle structures. Since $\Omega$ is of
$H$-type\index{terms}{Hodge type1@$H$-type} $(1,1)$, the form $\Res\Omega \in
C^\infty(M,\Lambda^2(M,\C))$ is real and of Hodge
type\index{terms}{bigrading!Hodge type} $(1,1)$. By
\eqref{pol2kal}
\begin{multline*}
\Res\Omega = \half\left(\Res\Omega + \overline{\Res\Omega}\right) =
\half(\Omega+\nu(\Omega))|_{M \subset \tm} = \\
= \omega_I|_{M \subset
\tm} = \omega \in \Lambda^{1,1}(M).
\end{multline*}
Therefore to prove Theorem~\ref{metrics}, it suffices to prove the
following.  
\begin{itemize}
\item \label{to.prove} For every
polarization\index{terms}{polarization} $\Omega$ of the formal Hodge
manifold\index{terms}{Hodge manifold!formal} $\tm$ the restriction
$\omega = \Res\Omega \in C^\infty(\Lambda^{1,1}(M))$ is compatible
with the connection\index{terms}{connections!on a vector bundle}
$\nabla$, that is, $\nabla\omega=0$. Vice versa, every real positive
$(1,1)$-form $\omega \in C^\infty(\Lambda^{1,1}(M))$ satisfying
$\nabla\omega=0$ extends to a polarization $\Omega$ of $\tm$, and
such an extension is unique.
\end{itemize}
This is what we will actually prove. 

\subsection{Preliminaries}\label{drm.mod}

\punkt\label{descr} We begin with introducing a convenient model for
the holomorphic de Rham algebra\index{terms}{de Rham
complex!holomorphic} $\Lambda^{\cdot,0}(\tm_J)$ of the complex
manifold $\tm_J$, which would be independent of the Hodge
manifold\index{terms}{Hodge manifold} structure on $\tm$.  To
construct such a model, consider the relative de Rham
complex\index{terms}{de Rham complex!relative}
$\Lambda^\cdot(\tm/M,\C)$ of $\tm$ over $M$ (see
\ref{relative.de.rham.sub} for a reminder of its definition and main
properties). Let $\pi:\Lambda^\cdot(\tm,\C) \to
\Lambda^\cdot(\tm/M,\C)$ be the canonical projection. Recall that
the bundle $\Lambda^i(\tm/M,\C)$ of relative $i$-forms on $\tm$ over
$M$ carries a natural structure of a Hodge bundle\index{terms}{Hodge
bundle} of weight\index{terms}{weight!of a Hodge bundle}
$i$. Moreover, we have introduced in \eqref{eta} a 
Hodge bundle isomorphism
$$
\eta:\rho^*\Lambda^\cdot(M,\C) \to \Lambda^\cdot(\tm/M,\C) 
$$
between $\Lambda^i(\tm/M,\C)$ and the pullback
$\rho^*\Lambda^\cdot(M,\C)$ of the bundle of $\C$-valued $i$-forms
on $M$.

\begin{lemma}\label{ident.bis}
\begin{enumerate}
\item The projection $\pi$ induces an algebra isomorphism 
$$
\pi:\Lambda^{\cdot,0}(\tm_J) \to \Lambda^\cdot(\tm/M,\C) 
$$
compatible with the natural Hodge bundle\index{terms}{Hodge bundle}
structures.
\item Let $\alpha \in C^\infty_M(\tm,\Lambda^{i,0}(\tm_J))$ be a
smooth $(i,0)$-form on $\tm_J$, and consider the smooth $i$-form 
$$
\beta = \eta^{-1}\pi(\alpha) \in C^\infty_M(\tm,\Lambda^i(\tm,\C)) 
$$
on $\tm$. The forms $\alpha$ and $\beta$ have the same restriction
to the zero section $M \subset \tm$. 
\end{enumerate}
\end{lemma}

\proof Since $\eta$, $\pi$ and the restriction map are compatible
with the algebra structure on $\Lambda^\cdot(M,\C)$, it suffices to
prove both claims for $\Lambda^1(M,\C)$. For every bundle $\E$ on
$\tm$ denote by $\E|_{M \subset \tm}$ the restriction of $\E$ to the
zero section $M \subset \tm$. Consider the bundle map
$$
\chi = \eta \circ \Res:\Lambda^{1,0}(\tm_J)|_{M \subset \tm} \to
\Lambda^1(M,\C) \to \Lambda^1(\tm/M,\C)|_{M \subset \tm}. 
$$
The second claim of the lemma is then equivalent to the identity
$\chi = \pi$.\! Moreover, note that the contraction with
the canonical vector field $\phi$ on $\tm$ defines an injective map
$i_\phi:C^\infty(M,\Lambda^1(\tm/M,\C)|_M) \to
C^\infty(\tm,\C)$. Therefore it suffices to prove that $i_\phi \circ
\chi = i_\phi \circ \pi$.

Every smooth section $s$ of the bundle $\Lambda^{1,0}(\tm_J)|_{M
\subset \tm}$ is of the form
$$
s = (\rho^*\alpha + \sqrt{-1}j\rho^*\alpha)|_{M \subset \tm}, 
$$
where $\alpha \in C^\infty(M,\Lambda^1(M,\C))$ is a smooth $1$-form
on $M$, and 
$$
j:\Lambda^1(\tm,\C) \to \overline{\Lambda^1(\tm,\C)}
$$
is the map induced by the quaternionic
structure\index{terms}{quaternionic structure} on $\tm$. For such a
section $s$ we have $\Res(s) = \alpha$, and by \eqref{phi.and.tau}
$i_\phi(\chi(s)) = \sqrt{-1}\tau(\alpha)$, where
$\tau:C^\infty(M,\Lambda^1(M,\C)) \to C^\infty(\tm,\C)$ is the
tautological map introduced in \ref{tau}. On the other hand, since
$\pi
\circ \rho^* = 0$, we have 
$$
i_\phi(\pi(s)) = i_\phi(\pi(\sqrt{-1}j\rho^*\alpha)) =
i_\phi(\sqrt{-1}j\rho^*\alpha).
$$ 
Since the Hodge manifold\index{terms}{Hodge manifold} structure on
$\tm$ is linear\index{terms}{Hodge manifold!linear}, this equals
$$
i_\phi(\pi(s)) = \sqrt{-1}i_\phi(j(\rho^*\alpha)) =
\sqrt{-1}\tau(\alpha) = \chi(s), 
$$
which proves the second claim of the lemma. Moreover, it shows that
the restriction of the map $\pi$ to the zero section $M \subset \tm$
is an isomorphism. As in the proof of Lemma~\ref{ident}, this
implies that the map $\pi$ is an isomorphism on the whole $\tm$,
which proves the first claim and finishes the proof of the lemma. 
\endproof 

\punkt Lemma~\ref{ident.bis}~\thetag{i} allows to define the bundle
isomorphism
$$
\pi^{-1} \circ \eta:\rho^*\Lambda^\cdot(M,\C) \to
\Lambda^\cdot(\tm/M,\C) \to \Lambda^{\cdot,0}(\tm_J), 
$$
between $\rho^*\Lambda^\cdot(M,\C)$ and $\Lambda^{\cdot,0}(\tm_J)$,
and it induces an isomorphism
$$
\rho_*(\eta \circ \pi^{-1}):\rho_*\rho^*\Lambda^\cdot(M,\C) \cong
\rho_*\Lambda^{\cdot,0}(\tm_J)
$$
between the direct images of these bundles under the canonical
projection $\rho:\tm \to M$. 

On the other hand, by adjunction we have the canonical embedding
$$
\Lambda^\cdot(M,\C) \hookrightarrow \rho_*\rho^*\Lambda^\cdot(M,\C),
$$
and by the projection formula it extends to an isomorphism
$$
\rho_*\rho^*\Lambda^\cdot(M,\C) \cong \Lambda^\cdot(M,\C) \otimes
\B^0, 
$$
where $\B^0 = \rho_*\Lambda^0(\tm,\C)$ is the $0$-th component of
the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ of $M$. All
these isomorphisms are compatible with the Hodge
bundle\index{terms}{Hodge bundle} structures and with the
multiplication.

\punkt \label{ident.punkt}
It will be convenient to denote the image $\rho_*(\eta \circ
\pi^{-1})\left(\Lambda^\cdot(M,\C)\right) \subset
\rho_*\Lambda^{\cdot,0}(\tm_J)$ by $L^\cdot(M,\C)$ or, to simplify
the notation, by $L^\cdot$. (The algebra $L^\cdot(M,\C)$ is, of
course, canonically isomorphic to $\Lambda^\cdot(M,\C)$.) We then
have the identification
\begin{equation}\label{ident.zero}
L^\cdot \otimes \B^0 \cong \rho_*\rho^*\Lambda^\cdot(M,\C) \cong 
\rho_*\Lambda^{\cdot,0}(\tm_J). 
\end{equation}
This identification is independent of the Hodge
manifold\index{terms}{Hodge manifold} structure on $\tm$. Moreover,
by Lemma~\ref{ident.bis}~\thetag{ii} the restriction map
$\Res:\rho_*\Lambda^{\cdot,0}(\tm_J) \to \Lambda^\cdot(M,\C)$ is
identified under \eqref{ident.zero} with the canonical projection
$L^\cdot \otimes \B^0 \to L^\cdot \otimes \B^0_0 \cong L^\cdot$.

By Lemma~\ref{ident} we also have the identification
$\rho_*\Lambda^{0,\cdot}(\tm_J) \cong \B^\cdot$. Therefore
\eqref{ident.zero} extends to an algebra isomorphism
\begin{equation}\label{ident.formula}
\rho_*\Lambda^{\cdot,\cdot}(\tm_J) \cong
\rho_*\Lambda^\cdot(\tm/M,\C) \otimes \Lambda^\cdot(M,\C) \cong
L^\cdot \otimes \B^\cdot. 
\end{equation}
This isomorphism is also compatible with the Hodge
bundle\index{terms}{Hodge bundle} structures on both sides.

\subsection{The Dolbeult differential on $\protect\tm_J$}

\punkt Our next goal is to express the Dolbeult
differential $\bar\6_J$ of the
complex manifold $\tm_J$ in terms of the model for the de Rham
complex\index{terms}{de Rham complex} $\Lambda^{\cdot,\cdot}(\tm_J)$
given by \eqref{ident.formula}. For every $k \geq 0$ denote by
$$
D:L^k \otimes \B^\cdot \to L^k \otimes \B^{\cdot+1}. 
$$
the differential operator induced by
$\bar\6_J:\Lambda^{\cdot,\cdot}(\tm_J) \to
\Lambda^{\cdot,\cdot+1}(\tm_J)$ under \eqref{ident.formula}. The
operator $D$ is weakly Hodge\index{terms}{weakly Hodge map}. It
satisfies the Leibnitz rule\index{terms}{Leibnitz rule} with respect
to the algebra structure on $L^\cdot \otimes \B^\cdot$, and we have
$D \circ D = 0$. By definition for $k=0$ it coincides with the
derivation $D:\B^\cdot \to \B^{\cdot+1}$ defined by the Hodge
manifold\index{terms}{Hodge manifold} structure on $\tm$. For $k >
0$ the complex $\langle L^k \otimes \B^\cdot, D \rangle$ is a free
differential graded module over the Weil algebra\index{terms}{Weil
algebra} $\langle\B^\cdot,D\rangle$.

\punkt \label{dr.L}
The relative de Rham differential $d^r$ (see
Subsection~\ref{relative.de.rham.sub}) induces under the isomorphism
\eqref{ident.formula} an algebra derivation
$$
d^r:L^\cdot \otimes \B^\cdot \to L^{\cdot+1} \otimes \B^\cdot. 
$$
The derivation $d^r$ also is weakly Hodge\index{terms}{weakly Hodge
map}, and we have the following.

\begin{lemma}\label{D.dr}
The derivations $D$ and $d^r$ commute, that is,  
$$
\{D,d^r\} = 0:L^\cdot \otimes \B^\cdot \to L^{\cdot+1} \otimes
\B^{\cdot+1}. 
$$
\end{lemma}

\proof The operator $\{D,d^r\}$ satisfies the Leibnitz
rule\index{terms}{Leibnitz rule}, so it suffices to prove that it
vanishes on $\B^0$, $\B^1$ and $L^1 \otimes \B^0$.  Moreover, the
$\B^0$-modules $\B^1$ and $L^1 \otimes \B^0$ are generated,
respectively, by local sections of the form $Df$ and $d^rf$, $f \in
\B^0$. Since $\{D,d^r\}$ commutes with $D$ and $d^r$, it suffices to
prove that it vanishes on $\B^0$. Finally, $\{D,d^r\}$ is continuous
in the adic topology on $\B^0$. Since the subspace
$$
\{fg|f,g \in \B^0, Df = d^rg = 0\} \subset \B^0
$$
is dense in this topology, it suffices to prove that for a local
section $f \in \B^0$ we have $\{D,d^r\}f=0$ if either $d^rf=0$ of
$Df=0$.

It is easy to see that for every local section $\alpha \in \B^\cdot$
we have $d^r\alpha = 0$ if and only if $\alpha \in \B^\cdot_0$ is of
augmentation degree\index{terms}{degree!augmentation} $0$. By
definition the derivation $D$ preserves the component $\B^\cdot_0
\subset \B^\cdot$ of augmentation degree $0$ in $\B^0$. Therefore
$d^rf=0$ implies $d^rDf=0$ and consequently $\{D,d^r\}f=0$. This
handles the case $d^rf=0$. To finish the proof, assume given a local
section $f \in \B^0$ such that $Df=0$. Such a section by definition
comes from a germ at $M \subset \tm$ of a holomorphic function
$\wh{f}$ on $\tm_J$. Since $\wh{f}$ is holomorphic, we have
$\bar\6_J\wh{f}=0$ and $d\wh{f} =
\6_J\wh{f}$. Therefore $d^rf = \pi(d\wh{f}) = \pi(\6_Jf)$, and
$$
Dd^rf = \pi(\bar\6_J\6_J\wh{f}) = -\pi(\6_J\bar\6_J\wh{f}) = 0, 
$$
which, again, implies $\{D,d^r\}f = 0$. 
\endproof

\punkt Let now $\nabla=D_1:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$} \to S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$} \otimes 
\Lambda^1$ be the reduction of the extended\index{terms}{extended
connection!reduction of} connection $D$. It induces a
connection\index{terms}{connections!on a vector bundle} on the bundle
$L^1 \cong S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$, and this
connection extends by the Leibnitz rule\index{terms}{Leibnitz rule}
to a connection on the exterior algebra $L^\cdot$ of the bundle
$L^1$, which we will also denote by $\nabla$.

Denote by $R = \nabla \circ \nabla:L^\cdot \to L^\cdot \otimes
\Lambda^2$ the curvature of the connection $\nabla$. Since $\nabla
\circ \nabla = \half\{\nabla,\nabla\}$, the operator $R$ also
satisfies the Leibnitz rule\index{terms}{Leibnitz rule} with respect
to the multiplication in $L^\cdot$.

\punkt Introduce the augmentation grading\index{terms}{grading!augmentation} on the bundle $L^\cdot \otimes \B^\cdot$ by setting $\deg
L^\cdot = 0$. The derivation $D:L^\cdot \otimes \B^\cdot \to L^\cdot
\otimes \B^{\cdot+1}$ obviously does not increase the augmentation
degree\index{terms}{degree!augmentation}, and we have the
decomposition $D = \sum_{k \geq 0}D_k$. On the other hand, the
derivation $d^r$ preserves the augmentation degree. Therefore
Lemma~\ref{D.dr} implies that for every $k \geq 0$ we have
$\{D_k,d^r\} = 0$. This in turn implies that $D_0 = 0$ on $L^p$ for
$p > 0$, and therefore $D_0 = \id \otimes C:L^p \otimes
\B^\cdot \to L^p \otimes \B^{\cdot+1}$. Moreover, this allows to
identify explicitly the components $D_1$ and $D_2$ of the derivation
$D:L^\cdot \to L^\cdot \otimes \B^1$. Namely, we have the following.

\begin{lemma}\label{D.1.2}
We have 
\begin{align*}
D_1 &= \nabla:L^\cdot \to L^\cdot \otimes \B^1_1 = L^\cdot \otimes
\Lambda^1\\ 
D_2 &= \frac{1}{3}\sigma \circ R:L^\cdot \to L^\cdot \otimes \B^1_2,
\end{align*}
where $\sigma = \id \otimes \sigma:L^\cdot \otimes \B^{\cdot+1} \to
L^\cdot \otimes \B^\cdot$ is as in \ref{C.and.sigma}. 
\end{lemma}

\proof Since both sides of these identities satisfy the Leibnitz
rule\index{terms}{Leibnitz rule} with respect to the multiplication
in $L^\cdot$, it suffices to prove them for $L^1$. But $d^r:\B^0 \to
L^1 \otimes \B^0$ restricted to $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$} \subset \B^0$ becomes an
isomorphism $d^r:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$} \to L^1$. Since $\{D_1,d^r\} = \{D_2,d^r\} =
0$, it suffices to prove the identities with $L^1$ replaced with
$S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}$. The first one then becomes the definition of $\nabla$, and
the second one is Corollary~\ref{D.2}.  \endproof

\subsection{The proof of Theorem~\ref{metrics}}

\punkt We can now prove Theorem~\ref{metrics} in the form
\ref{to.prove}. We begin with the following corollary of
Lemma~\ref{D.1.2}.

\begin{corr}\label{red.corr}
Let $\I^\cdot \subset \B^\cdot$ be the ideal introduced in
\ref{red.Weil}. An arbitrary smooth section $\alpha \in
C^\infty(M,L^\cdot)$ satisfies
\begin{equation}\label{rrd}
D\alpha \in C^\infty(M, L^\cdot \otimes \I^1) \subset C^\infty(M,
L^\cdot \otimes \B^1)
\end{equation}
if and only if $\nabla \alpha = 0$. 
\end{corr}

\proof Again, both the identity \eqref{rrd} and the equality
$\nabla\alpha$ are compatible with the Leibnitz
rule\index{terms}{Leibnitz rule} with respect to the multiplication
in $\alpha$. Therefore it suffices to prove that they are equivalent
for every $\alpha \in L^1$. By definition of the ideal $\I^\cdot$
the equality \eqref{rrd} holds if and only if $D_1\alpha = D_2\alpha
= 0$. By Lemma~\ref{D.1.2} this is equivalent to $\nabla\alpha =
\sigma \circ R(\alpha) = 0$. But since $R = \nabla \circ \nabla$,
$\nabla\alpha = 0$ implies $\sigma \circ R(\alpha) = 0$, which
proves the claim.  \endproof

\punkt Let now $\Omega \in C^\infty(M, \rho_*\Lambda^{2,0}(\tm_J)
\cong L^2 \otimes \B^0$ be a polarization\index{terms}{polarization}
of the Hodge manifold\index{terms}{Hodge manifold} $\tm_J$, so that
$\Omega$ is of Hodge type\index{terms}{bigrading!Hodge type} $(1,1)$ and
$D\Omega=0$. Let $\omega = \Res\Omega \in
C^\infty(M,\Lambda^{1,1}(M,\C))$ be its restriction, and let $\Omega
= \sum_{k \geq 0}\Omega_k$ be its augmentation
degree\index{terms}{degree!augmentation} decomposition.

As noted in \ref{ident.punkt}, the restriction map
$\Res:\rho_*\Lambda^{\cdot,0}(\tm_J) \to \Lambda^\cdot(M,\C)$ is
identified under the isomorphism \eqref{ident.formula} with the
projection $\LL^\cdot \otimes \B^0 \to L^\cdot$ onto the component
of augmentation degree\index{terms}{degree!augmentation}
$0$. Therefore $\omega = \Omega_0$. Since the augmentation
degree-$1$ component $\left(L^2 \otimes B^0_{tot}\right)^{1,1}_1=0$
and $D\Omega = 0$, we have $\nabla\omega = D_1\Omega_0 = 0$, which
proves the first claim of Theorem~\ref{metrics}.

\punkt To prove the second claim of the theorem, let $\omega$ be a
K\"ahler form on $M$ compatible with the
connection\index{terms}{connections!on a vector bundle} $\nabla$, so
that $\nabla\omega = 0$. We have to show that there exists a unique
section $\Omega = \sum_{k \geq 0}\Omega_k \in C^\infty(M, L^2
\otimes \B^0)$ which is of Hodge type\index{terms}{bigrading!Hodge type}
$(1,1)$ and satisfies $D\Omega=0$ and $\Omega_0 = \omega$. As in the
proof of Theorem~\ref{kal=ext}, we will use induction on $k$. Since
$\Omega$ is of Hodge type\index{terms}{bigrading!Hodge type} $(1,1)$, we must
have $\Omega_1 = 0$, and by Corollary~\ref{red.corr} we have
$D(\Omega_0 + \Omega_1) \in C^\infty(M, L^2 \otimes \I^1)$, which
gives the base of our induction. The induction step is given by
applying the following proposition to $\sum_{0 \geq p \geq
k}\Omega_k$ for each $k \geq 1$ in turn.

\begin{prop}\label{metrics.ind}
Assume given integers $p,q,k; p,q \geq 0, k \geq 1$ and assume given
a section $\alpha \in C^\infty(M, L^{p+q} \otimes \B^0)$ of Hodge
type\index{terms}{bigrading!Hodge type} $(p,q)$ such that
$$
D\alpha \in C^\infty(M, L^{p+q} \otimes \I^1_{\geq k}), 
$$
where $\I^\cdot_{\geq k} = \oplus_{m \geq k}\I^\cdot_m$. Then there
exists a unique section 
$$
\beta \in C^\infty(M, L^{p+q} \otimes \B_k)
$$ 
of the same Hodge type\index{terms}{bigrading!Hodge type} $(p,q)$ and
such that 
$$
D(\alpha + \beta) \in C^\infty(M, L^{p+q} \otimes \I^1_{\geq k+1}).
$$
\end{prop}

\proof Let $\B_{tot}^\cdot$ be the total Weil
algebra\index{terms}{Weil algebra!total} introduced in
\ref{total.Weil}, and consider the free module $L^{p+q} \otimes
\B_{tot}^\cdot$ over $\B_{tot}^\cdot$ generated by the Hodge
bundle\index{terms}{Hodge bundle} $L^{p+q}$. This module carries a
canonical Hodge bundle structure of weight\index{terms}{weight!of a
Hodge bundle} $p+q$.  Consider the maps 
$C:\B_{tot}^\cdot \to \B_{tot}^{\cdot+1}$,
$\sigma_{tot}:\B^{\cdot+1} \to \B_{tot}^\cdot$ introduced in
\ref{total.C} and \ref{sigma.c}, and let $C = \id \otimes
C,\sigma_{tot} = \id \otimes \sigma_{tot}:L^{p+q} \otimes \B_{tot}
\to L^{p+q} \otimes \B_{tot}$ be the associated endomorphisms of the
free module $L^{p+q} \otimes \B_{tot}^\cdot$.

The maps $C$ and $\sigma_{tot}$ preserve the Hodge
bundle\index{terms}{Hodge bundle} structure.  The commutator $h =
\{C, \sigma_{tot}\}: \B^\cdot_{tot} \to \B^\cdot_{tot}$ is
invertible on $\I_{tot}^\cdot \subset \B_{tot}^\cdot$ by
Corollary~\ref{h.inv} and acts as $k\id$ on $\B^0_k \subset
\B^\cdot_{tot}$. Therefore the endomorphism
$$
h = \id \otimes h = \{C,\sigma_{tot}\}:L^{p+q} \otimes \B^\cdot_{tot} \to
L^{p+q} \otimes \B^\cdot_{tot}
$$
is invertible on $L^{p+q} \otimes \I^\cdot_{tot}$ and acts as $k\id$ on
$L^{p+q} \otimes \B^0_k$.

Since the derivation $D:L^{p+q} \otimes \B^\cdot \to L^{p+q} \otimes
\B^{\cdot+1}$ is weakly Hodge\index{terms}{weakly Hodge map}, it
induces a map $D^{tot}:L^{p+q} \otimes \B_{tot}^\cdot \to L^{p+q}
\otimes \B_{tot}^{\cdot+1}$, and $D\alpha \in L^{p+q} \otimes
\I^1_{\geq k}$ if and only if the same holds for $D^{tot}\alpha$. To
prove uniqueness, note that $D_0 = C$ is injective on $L^{p+q}
\otimes \B^0_k$. If there are two sections $\beta,\beta'$ satisfying
the conditions of the proposition, then $D_0(\beta-\beta')=0$, hence
$\beta = \beta'$.

To prove existence, let $\gamma = (D^{tot}\alpha)_k$ be the component of
the section $D^{tot}\alpha$ of augmentation
degree\index{terms}{degree!augmentation} $k$. Since $D^{tot} \circ
D^{tot}=0$, we have $C\gamma = D^{tot}_0\gamma = 0$ and $C \circ \sigma_{tot}
\gamma = h \gamma$.  Let $\beta = - \frac{1}{k} \circ
\sigma_{tot}(\gamma)$. The section $\beta$ is of Hodge
type\index{terms}{bigrading!Hodge type} $(p,q)$ and of augmentation degree
$k$. Moreover, it satisfies
$$
D^{tot}_0\beta = C\beta = -Ch^{-1}\sigma_{tot}(\gamma) = -h^{-1} \circ C
\circ \sigma_{tot} \gamma = -h^{-1} \circ h\gamma = -\gamma.
$$
Therefore $D_{tot}(\alpha+\beta)$ is indeed a section of $L^{p+q}
\otimes (I^1_{tot})_{\geq k+1}$, which proves the proposition and
finishes the proof of Theorem~\ref{metrics}. 
\endproof 

\subsection{The cotangent bundle}

\punkt For every K\"ahler manifold $M$ Theorem~\ref{metrics}
provides a canonical formal hyperk\"ahler
structure\index{terms}{hyperk\"ahler structures!formal} on the total
space $\tm$ of the complex-conjugate to the tangent bundle to
$M$. In particular, we have a closed holomorphic $2$-form $\Omega_I$
for the preferred\index{terms}{preferred complex structure} complex structure $\tm_I$ on $M$. 

Let $T^*M$ be the total space to the cotangent bundle to $M$
equipped with the canonical holomorphic symplectic form $\Omega$. To
obtain a hyperk\"ahler metric\index{terms}{metric!hyperk\"ahler} of
the formal neighborhood\index{terms}{formal neighborhood} of the
zero section $M \subset T^*M$, one can apply an appropriate version
of the Darboux-Weinstein\index{names}{Weinstein, A.}
Theorem\index{terms}{Darboux--Weinstein Theorem}, which gives a local
symplectic isomorphism $\kappa:\tm \to T^*M$ in a neighborhood of
the zero section. However, this theorem is not quite standard in the
holomorphic and formal situations. For the sake of completeness, we
finish this section with a sketch of a construction of such an
isomorphism $\kappa:\tm \to T^*M$ which can be used to obtain a
hyperk\"ahler metric\index{terms}{metric!hyperk\"ahler} on $T^*M$
rather than on $\tm$.

\punkt We begin with the following preliminary fact on the
holomorphic de Rham complex\index{terms}{de Rham
complex!holomorphic} of the manifold $\tm_I$.

\begin{lemma}\label{de.rham.exact}
Assume given either a formal Hodge manifold\index{terms}{Hodge
manifold!formal} structure on the $U(1)$-manifold $\tm$ along the
zero section $M \subset \tm$, or an actual Hodge
manifold\index{terms}{Hodge manifold} structure on an open
neighborhood $U \subset \tm$ of the zero section.
\begin{enumerate}
\item For every point $m \in M$ there exists an open neighborhood $U
\subset \tm_I$ such that the spaces $\Omega^\cdot(U)$ of holomorphic
forms on the complex manifold $\tm_I$ (formally completed along $M
\subset \tm$ if necessary) equipped with the holomorphic de Rham
differential $\6_I:\Omega^\cdot(U) \to \Omega^{\cdot+1}(U)$ form an
exact complex. 
\item If the subset $U \subset \tm$ is invariant under the
$U(1)$-action on the manifold $\tm$, then the same is true for the
subspaces $\Omega^\cdot_k(U) \subset \Omega^\cdot(U)$ of forms of
weight\index{terms}{weight!of an $U(1)$-action} $k$ with respect to
the $U(1)$-action.
\item Assume further that the canonical projection $\rho:\tm_I \to
M$ is holomorphic for the preferred\index{terms}{preferred complex
structure} complex structure $\tm_I$ on $\tm$. Then both these
claims also hold for the spaces $\Omega^\cdot(U/M)$ of relative
holomorphic forms on $U$ over $M$.
\end{enumerate}
\end{lemma}

\proof The claim \thetag{i} is standard. To prove \thetag{ii}, note
that, both in the formal and in the analytic situation, the spaces
$\Omega^\cdot(U)$ are equipped with a natural topology. Both this
topology and the $U(1)$-action are preserved by the holomorphic
Dolbeult differential $\6_I$.

The subspaces $\Omega^\cdot_{fin}(U) \subset \Omega^\cdot(U)$ of
$U(1)$-finite vectors are dense in the natural topology. Therefore
the complex $\langle \Omega^\cdot_{fin}(U),\6_I\rangle$ is also
exact. Since the group $U(1)$ is compact, we have
$$
\Omega^\cdot_{fin}(U) = \bigoplus_k\Omega^\cdot_k(U),
$$
which proves \thetag{ii}. The claim \thetag{iii} is, again,
standard. 
\endproof 

\punkt We can now formulate and prove the main result of this
subsection.

\begin{prop}
Assume given a formal polarized Hodge manifold\index{terms}{Hodge
manifold!formal} structure on the manifold $\tm$ along the zero
section $M \subset \tm$ such that the canonical projection
$\rho:\tm_I \to M$ is holomorphic for the preferred\index{terms}{preferred complex structure} complex
structure $\tm_I$ on $\tm$. Let $\Omega_I$ be the associated formal
holomorphic $2$-form on $\tm_I$. Let $T^*M$ be the total space of
the cotangent bundle to the manifold $M$ equipped with a canonical
holomorphic symplectic form $\Omega$. There exists a unique
$U(1)$-equivariant holomorphic map $\kappa:\tm_I \to T^*M$, defined
in a formal neighborhood\index{terms}{formal neighborhood} of the
zero section, which commutes with the canonical projections onto $M$
and satisfies $\Omega_I = \kappa^*\Omega$. Moreover, if the
polarized Hodge manifold\index{terms}{Hodge manifold} structure on
$\tm_I$ is defined in an open neighborhood $U \subset \tm$ of the
zero section $M \subset \tm$, then the map $\kappa$ is also defined
in a (possibly smaller) open neighborhood of the zero section.
\end{prop}

\proof By virtue of the uniqueness, the claim is local on $M$, so
that we can assume that the whole $M$ is contained in a
$U(1)$-invariant neighborhood $U \subset \tm_I$ satisfying the
conditions of Lemma~\ref{de.rham.exact}. Holomorphic maps $\kappa:U
\to T^*M$ which commute with the canonical projections onto $M$ are
in a natural one-to-one correspondence with holomorphic sections
$\alpha$ of the bundle $\rho^*\Lambda^{1,0}(M)$ on $\tm_I$. Such a
map $\kappa$ is $U(1)$-equivariant if and only if the corresponding
$1$-form $\alpha \in \Omega^1(U)$ is of
weight\index{terms}{weight!of an $U(1)$-action} $1$ with respect to 
the $U(1)$-action. Moreover, it satisfies $\kappa^*\Omega=\Omega_I$
if and only if $\6_I\alpha = \Omega_I$. Therefore to prove the
formal resp. analytic parts of the proposition it suffices to prove
that there exists a unique holomorphic formal resp. analytic section
$\alpha \in C^\infty(U,\rho^*\Lambda^{1,0}(M)$ which is of
weight\index{terms}{weight!of an $U(1)$-action} 
$1$ with respect to the $U(1)$-action and satisfies
$\6_I\alpha=\Omega_I$.

The proof of this fact is the same in the formal and in the analytic
situations. By definition of the polarized Hodge
manifold\index{terms}{Hodge manifold} the $2$-form $\Omega_I \in
\Omega^2(U)$ is of weight\index{terms}{weight!of an $U(1)$-action}
$1$ with respect to the 
$U(1)$-action. Therefore by Lemma~\ref{de.rham.exact}~\thetag{ii}
there exists a holomorphic $1$-form $\alpha \in \Omega^1(U)$ of
weight\index{terms}{weight!of an $U(1)$-action} $1$ with respect to
the $U(1)$-action and such that 
$\6_I\alpha = \Omega_I$. Moreover, the image of the form $\Omega_I$
under the canonical projection $\Omega^2(U) \to \Omega^2(U/M)$ is
zero. Therefore by Lemma~\ref{de.rham.exact}~\thetag{iii} we can
arrange so that the image of the form $\alpha$ under the projection
$\Omega^1(U) \to \Omega^1(U/M)$ is also zero, so that $\alpha$ is in
fact a section of the bundle $\rho^*\Lambda^{1,0}(M)$. This proves
the existence part. To prove uniqueness, note that every two such
$1$-forms must differ by a form of the type $\6_if$ for a certain
holomorphic function $f \in \Omega^0(U)$. Moreover, by
Lemma~\ref{de.rham.exact}~\thetag{ii} we can assume that the
function $f$ is of weight\index{terms}{weight!of an $U(1)$-action}
$1$ with respect to the $U(1)$-action.  On 
the other hand, by Lemma~\ref{de.rham.exact}~\thetag{iii} we can
assume that the function $f$ is constant along the fibers of the
canonical projection $\rho:\tm_I \to M$. Therefore we have $f=0$
identically on the whole $U$.  \endproof


\section{Convergence}\label{convergence}

\subsection{Preliminaries}

\punkt Let $M$ be a complex manifold. By Theorem~\ref{kal=ext} every
K\"ah\-le\-ri\-an connection\index{terms}{connections!K\"ahlerian}
$\nabla:\Lambda^1(M,\C) \to \Lambda^1(M,\C) \otimes \Lambda^1(M,\C)$
on the cotangent bundle $\Lambda^1(M,\C)$ to the manifold $M$
defines a flat linear extended\index{terms}{extended connection!flat}
connection\index{terms}{extended connection!linear}
$D:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to \B^1(M,\C)$
on $M$ and therefore a formal Hodge connection\index{terms}{Hodge
connection!formal} $D$ on the total space $\tm$ of the 
complex-conjugate to the tangent bundle to $M$. By
Proposition~\ref{equiv.bis}, this formal Hodge
connection\index{terms}{Hodge connection!formal} defines, in turn, a
formal Hodge manifold structure\index{terms}{Hodge manifold!formal}
on $\tm$ in the formal neighborhood\index{terms}{formal
neighborhood} of the zero section $M \subset \tm$.

In this section we show that if the K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian}
$\nabla$ is real-analytic, then the corresponding formal Hodge
manifold structure\index{terms}{Hodge manifold!formal} on $\tm$ is
the completion of an actual Hodge manifold\index{terms}{Hodge
manifold} structure on an open neighborhood $U \subset \tm$ of the
zero section $M \subset \tm$. We also show that if the connection
$\nabla$ comes from a K\"ahler metric\index{terms}{metric!K\"ahler} 
$\omega$ on $M$, then the corresponding
polarization\index{terms}{polarization} $\Omega$ of the 
formal Hodge manifold $\tm$ defined in Theorem~\ref{metrics}
converges in a neighborhood $U' \subset U \subset \tm$ of the zero
section $M \subset \tm$ to a polarization of the Hodge manifold
structure on $U'$. Here is the precise formulation of these results.

\begin{theorem}\label{converge}
Let $M$ be a complex manifold equipped with a real-analytic
K\"ah\-le\-ri\-an connection\index{terms}{connections!K\"ahlerian}
$\nabla:\Lambda^1(M,\C) \to 
\Lambda^1(M,\C) \otimes \Lambda^1(M,\C)$ on its cotangent bundle
$\Lambda^1(M,\C)$. There exists an open neighborhood $U \subset \tm$
of the zero section $M \subset \tm$ in the total space $\tm$ of the
complex-conjugate to the tangent bundle to $M$ and a Hodge
manifold\index{terms}{Hodge manifold} structure on $U \subset \tm$
such that its completion along the zero section $M \subset \tm$
defines a linear\index{terms}{extended connection!linear} flat
extended\index{terms}{extended connection!flat} connection $D$ on $M$
with reduction $\nabla$. 

Moreover, assume that $M$ is equipped with a K\"ahler
metric\index{terms}{metric!K\"ahler} 
$\omega$ such that $\nabla\omega=0$, and let $\Omega \in
C^\infty_M(\tm,\Lambda^2(\tm,\C))$ be the formal
polarization\index{terms}{polarization!formal} of the Hodge
manifold\index{terms}{Hodge manifold} structure on $U \subset
\tm$ along $M \subset \tm$. Then there exists an open neighborhood
$U' \subset U$ of $M \subset U$ such that $\Omega \in
C^\infty(U',\Lambda^2(\tm,\C)) \subset
C^\infty_M(\tm,\Lambda^2(\tm,\C))$.
\end{theorem}

\punkt \label{formal.Weil}
We begin with some preliminary observations. First of all, the
question is local on $M$, therefore we may assume that $M$ is an
open neighborhood of $0$ in the complex vector space $V = \C^n$. Fix
once and for all a real structure and an Hermitian
metrics\index{terms}{metric!Hermitian} on the 
vector space $V$, so that it is isomorphic to its dual $V \cong
V^*$.

The subspace $\J \subset C^\infty(M,\C)$ of functions vanishing at
$0 \in M$ is an ideal in the algebra $C^\infty(M,\C)$, and $\J$-adic
topology on $C^\infty(M,\C)$ extends canonically to the de Rham
algebra\index{terms}{de Rham complex} $\Lambda^\cdot(M,\C)$ of the
manifold $M$ and, further, to the Weil algebra\index{terms}{Weil
algebra} $\B^\cdot(M,\C)$ of $M$ introduced in
\ref{Weil.defn}. Instead of working with bundle algebra
$\B^\cdot(M,\C)$ on $M$, it will be convenient for us to consider
the vector space
$$
\B^\cdot = C^\infty_\J(M,B^\cdot(M,\C)),
$$ 
which is by definition the $\J$-adic completion\index{terms}{formal
completion} of the space $C^\infty(M,\B^\cdot(M,\C))$ of global
sections of the Weil algebra\index{terms}{Weil algebra}. This vector
space is canonically a (pro-)algebra over $\C$. Moreover, the Hodge
bundle\index{terms}{Hodge bundle} structure on $\B^\cdot(M,\C)$
induces an $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure}
on the algebra $\B^\cdot$.

\punkt The $\J$-adic completion\index{terms}{formal completion}
$C^\infty_\J(M,\C)$ of the space of smooth functions on $M$ is
canonically isomorphic to the completion $\wh{S}^\cdot(V)$ of the
symmetric algebra of the vector space $V \cong V^*$. The cotangent
bundle $\Lambda^1(M,\C)$ is isomorphic to the trivial bundle $\V$
with fiber $V$ over $M$, and the completed de Rham
algebra\index{terms}{de Rham complex}
$C^\infty_\J(M,\Lambda^\cdot(M,\C))$ is isomorphic to the product
$$
C^\infty_\J(M,\Lambda^\cdot(M,\C)) \cong S^\cdot(V) \otimes
\Lambda^\cdot(V). 
$$
This is a free graded-commutative algebra generated by two copies of
the vector space $V$, which we denote by $V_1 = V \subset
\Lambda^1(V)$ and by $V_2 = V \subset S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(V)$. It is convenient to
choose the trivialization $\Lambda^1(M,\C) \cong \V$ in such a way
that the de Rham differential $d_M:\Lambda^\cdot(M,\C) \to
\Lambda^{\cdot+1}(M,\C)$ induces an identity map $d_M:V_2 \to V_1
\subset C^\infty_\J(M,\Lambda^1(M,\C))$.

\punkt The complex vector bundle $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)$ on $M$ is also
isomorphic to the trivial bundle $\V$. Choose a trivialization
$S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \cong \V$ in such a way that the canonical map
$C:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to \Lambda^1(M,\C)$ is the identity map. Denote by
$$
V_3 = V \subset C^\infty(M,S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)) \subset \B^0 
$$
the subset of constant sections in $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \cong \V$. Then the 
Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ becomes
isomorphic to the product
$$
\B^i \cong \wh{S}^\cdot(V_2 \oplus V_3) \otimes \Lambda^i(V_1)
$$
of the completed symmetric algebra $\wh{S}^\cdot(V_2 \oplus V_3)$ of
the sum $V_2 \oplus V_3$ of two copies of the vector space $V$ and
the exterior algebra $\Lambda^\cdot(V_1)$ of the third copy of the
vector space $V$.

\punkt Recall that we have introduced in \ref{aug} a grading on the
Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ which we
call {\em the augmentation
grading\index{terms}{grading!augmentation}}. It induces a grading on
the the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$. The augmentation
grading on $\B^\cdot$ is multiplicative, and it is obtained by
assigning degree $1$ to the generator subspaces $V_1,V_3 \subset
\B^\cdot$ and degree $0$ to the generator subspace $V_2 \subset
\B^\cdot$. As in \ref{aug}, we will denote the augmentation grading
on $\B^\cdot$ by lower indices.

We will now introduce yet another grading on the algebra $\B^\cdot$
which we will call {\em the total grading\index{terms}{grading!total}}. It is by definition the multiplicative grading obtained
by assigning degree $1$ to {\em all} the generators $V_1,V_2,V_3
\subset \B^\cdot$ of the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot$ We will denote by $\B^\cdot_{k,n} \subset \B^\cdot$ the
component of augmentation degree\index{terms}{degree!augmentation}
$k$ and total degree\index{terms}{degree!total} $n$. Note that
by definition $n,k \geq 0$ and, moreover, $n \geq k$.

\rem In \ref{aug} we have also defined a finer {\em augmentation
bigrading\index{terms}{bigrading!augmentation}} on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ \, and it this
bigrading that was denoted by double lower indices throughout
Section~\ref{main.section}. We will now longer need the augmentation
bigrading, so there is no danger of confusion.

\punkt The trivialization of the cotangent bundle to $M$ defines an
isomorphism $\tm \cong M \times V$ and a constant Hodge
connection\index{terms}{Hodge connection} on the pair $\langle \tm,M
\rangle$. The corresponding extended\index{terms}{extended
connection} connection $D^{const}:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to \B^1(M,\C)$ is the sum of the trivial
connection 
$$
\nabla^{const}_1:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)
\to S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \otimes 
\Lambda^1(M,\C) \subset \B^1(M,\C)
$$ 
on $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \cong \V$ and
the canonical isomorphism  
$$
C = \id:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to
\Lambda^1(M,\C) \subset \B^1(M,\C). 
$$ 
The derivation $D^{const}:\B^\cdot \to \B^{\cdot+1}$ of the Weil
algebra\index{terms}{Weil algebra} associated to the
extended\index{terms}{extended connection!derivation associated to}
connection $D^{const}$ by \ref{deriv} is equal to
\begin{align*}
D^{const} &= C = \id:V_3 \to V_1\\
D^{const} &= d_M = \id:V_2 \to V_1\\
D^{const} &= d_M = 0 \text{ on }V_1
\end{align*}
on the generator spaces $V_1,V_2,V_3 \subset \B^\cdot$. In
particular, the derivation $D^{const}$ preserves the total
degree\index{terms}{degree!total}.

\punkt Let now $D:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)
\to \B^1(M,\C)$ be the an arbitrary linear\index{terms}{extended
connection!linear} extended connection on the manifold $M$, and let 
$$
D = \sum_{k \geq 0}D_k:\B^\cdot \to \B^{\cdot+1}
$$
be the derivation of the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot$ associated to the extended\index{terms}{extended
connection!derivation associated to} connection $D$ by 
\ref{deriv}. The derivation $D$ admits a finer decomposition
$$
D = \sum_{k,n \geq 0} D_{k,n}:\B^\cdot \to \B^{\cdot+1}
$$
according to both the augmentation and the total
degree\index{terms}{degree!total}\index{terms}{degree!augmentation}
on $\B^\cdot$. The summand $D_{k,n}$ by definition raises the
augmentation degree by $k$ and the total degree by $n$.

\punkt Since the extended\index{terms}{extended connection}
connection $D:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to
\B^1(M,\C)$ is linear\index{terms}{extended connection!linear}, its
component $D_0:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to
\Lambda^1(M,\C)$ of augmentation
degree\index{terms}{degree!augmentation} $0$ coincides with the
canonical isomorphism 
$C:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to
\Lambda^1(M,\C)$. Therefore the restriction of the derivation 
$D:\B^\cdot \to \B^{\cdot+1}$ to the generator subspace $V_3 \subset
S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \subset \B^0$ satisfies
$$
D_0 = C = D^{const}_0 = D^{const}_{0,0}:V_3 \to V_1 \subset \B^1.
$$
In particular, all the components $D_{0,n}$ except for $D_{0,0}$
vanish on the subspace $V_3 \subset \B^\cdot$. 

The restriction of the derivation $D$ to the subspace
$\Lambda^\cdot(M,\C) \subset \B^\cdot(M,\C)$ by definition coincides
with the de Rham differential $d_M:\Lambda^\cdot(M,\C) \to
\Lambda^{\cdot+1}(M,\C)$. Therefore on the generator subspaces
$V_1,V_2 \subset \B^\cdot$ we have $D = d_M = D^{const}$. In
particular, all the components $D_{k,n}$ except for $D_{1,0}$ vanish
on the subspaces $V_1,V_2 \subset \B^\cdot$.

\punkt The fixed Hermitian metric\index{terms}{metric!Hermitian} on 
the generator spaces $V_1 = V_2 = V_3 = V$ extends uniquely to a
metric\index{terms}{Weil algebra!standard metric on} on the whole
Weil algebra\index{terms}{Weil algebra} such that the multiplication
map $\B^\cdot \otimes \B^\cdot \to \B^\cdot$ is an isometry. We call
this metric {\em the standard metric} on $\B^\cdot$. We finish our 
preliminary observations with the following fact which we will use
to deduce Theorem~\ref{converge} from estimates on the components
$D_{n,k}$ of the derivation $D:\B^\cdot \to \B^{\cdot+1}$.

\begin{lemma}\label{est=conv}
Let $D = \sum_{n,k}D_{n,k}:\B^\cdot \to \B^{\cdot+1}$ be a
derivation associated to an extended\index{terms}{extended
connection!derivation associated to} connection $D$ on the manifold 
$M$.  Consider the norms $\|D_{k,n}\|$ of the restrictions
$D_{k,n}:V_3 \to \B^1_{n+1,k+1}$ of the derivations
$D_{k,n}:\B^\cdot \to \B^{\cdot+1}$ to the generator subspace $V_3
\subset \B^0$ taken with respect to the standard
metric\index{terms}{Weil algebra!standard metric on} on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot$. If for certain
constants $C,\eps > 0$ and for every natural $n \geq k \geq 0$ we
have
\begin{equation}\label{est}
\|D_{k,n}\| < C\eps^n,
\end{equation}
then the formal Hodge connection\index{terms}{Hodge
connection!formal} on $\tm$ along $M$ associated to $D$ converges to an
actual real-analytic Hodge connection\index{terms}{Hodge connection}
on the open ball of radius $\eps$ in $\tm$ with center at $0 \in M
\subset \tm$. Conversely, if the extended\index{terms}{extended
connection} connection $D$ comes from a real-analytic Hodge
connection on an open neighborhood $U \subset
\tm$, and if the Taylor series for this Hodge connection converge in
the closed ball of radius $\eps$ with center at $0 \in M \subset
\tm$, then there exists a constant $C>0$ such that \eqref{est} holds
for every $n,k \geq 0$.
\end{lemma}

\proof The constant Hodge connection\index{terms}{Hodge connection}
$D^{const}$ is obviously defined on the whole $\tm$, and every other
formal Hodge connection\index{terms}{Hodge connection!formal} on
$\tm$ is of the form
$$
D = D^{const} + d^r \circ \Theta:\Lambda^0(\tm,\C) \to
\rho^*\Lambda^1(M,\C), 
$$
where $d^r:\Lambda^0(\tm,\C) \to \Lambda^1(\tm/M,\C)$ is the
relative de Rham differential, and $\Theta \in
C^\infty_M(\tm,\Lambda^1(\tm/M,\C) \otimes \rho^*\Lambda^1(M,\C))$
is a certain relative $1$-form on the formal
neighborhood\index{terms}{formal neighborhood} of $M
\subset \tm$ with values in the bundle $\rho^*\Lambda^1(M,\C)$. Both
bundles $\Lambda^1(\tm/M,\C)$ and $\rho^*\Lambda^1(M,\C)$ are
canonically isomorphic to the trivial bundle $\V$ with fiber $V$ on
$\tm$. Therefore we can treat the $1$-form $\Theta$ as a formal germ
of a $\End(V)$-valued function on $\tm$ along $M$. The Hodge
connection\index{terms}{Hodge connection} $D$ converges on a subset
$U \subset \tm$ if and only if this formal germ comes from a
real-analytic $\End(V)$-valued function $\Theta$ on $U$.

The space of all formal Taylor series for $\End(V)$-valued functions
on $\tm$ at $0 \in M \subset \tm$ is by definition equal to $\End(V)
\otimes \B^0$. Moreover, for every $n \geq 0$ the component
$\Theta_n \in \B^0_n \otimes \End(V) = \Hom(V,\B^0_n \otimes V)$ of
total degree\index{terms}{degree!total} $n$ of the formal power
series for the function $\Theta$ at $0 \in \tm$ is equal to the
derivation
$$
\sum_{0 \leq k \leq n}D_{k,n}:V=V_3 \to V=V_1 \otimes \bigoplus_{0 \leq k
\leq n}\B^0_{k,n}.
$$
Every point $x \in \tm$ defines the ``evaluation at $x$'' map
$$
\ev_x:C^\infty(\tm,\End(V)) \to \C, 
$$
and the formal Taylor series for $\Theta \in \End(V) \otimes \B^0$
converges at the point $x \in \tm$ if and only if the series
$$
\Theta(x) = \sum_{n \geq 0}\ev_x(\Theta_n) \in \End(V)
$$
converges. But we have
$$
\|\ev_x(\Theta_n)\| = \left\|\sum_{0 \leq k \leq n}D_{k,n}\right\||x|^n,
$$
where $|x|$ is the distance from the point $x$ to $0 \in \tm$. Now
the application of standard criteria of convergence finishes the
proof of the lemma.  
\endproof

\subsection{Combinatorics}

\punkt We now derive some purely combinatorial facts needed to
obtain estimates for the components $D_{k,n}$ of the
extended\index{terms}{extended connection} connection $D$. First, let
$a_n$ be the Catalan numbers, that is, the numbers defined by the
recurrence relation
$$
a_n = \sum_{1 \leq k \leq n-1} a_k a_{n-k}
$$
and the initial conditions $a_1 = 1$, $a_n = 0$ for $n \leq 0$. As
is well-known, the generating function $f(z) = \sum_{k \geq
0}a_kz^k$ for the Catalan numbers satisfies the equation $f(z) =
f(z)^2 + z$ and equals therefore
$$
f(z) = \half - \sqrt{\frac{1}{4} - z}.
$$
The Taylor series for this function at $z=0$ converges for $4|z|
< 1$, which implies that 
$$
a_k < C(4+\eps)^k
$$
for some positive constant $C > 0$ and every $\eps > 0$. 

\punkt \label{b.n.k}
We will need a more complicated sequence of integers, numbered by
two natural indices, which we denote by $b_{k,n}$. The sequence
$b_{k,n}$ is defined by the recurrence relation
$$
b_{k,n} = \sum_{p,q;1 \leq p \leq k-1} \frac{q+1}{k} b_{p,q}b_{k-p,n-q}
$$
and the initial conditions 
$$
\begin{cases}
b_{k,n} &= 0 \quad \text{ for } \quad k \leq 0,\\
b_{k,n} &= 0 \quad \text{ for } \quad k = 1, n < 0,\\
b_{k,n} &= 1 \quad \text{ for } \quad k = 1, n \geq 0,
\end{cases}
$$
which imply, in particular, that if $n < 0$, then $b_{k,n} = 0$ for
every $k$. For every $k \geq 1$ let $g_k(z) = \sum_{n \geq
0}b_{k,n}z^n$ be the generating function for the numbers
$b_{k,n}$. The recurrence relations on $b_{k,n}$ give 
\begin{align*}
g_k(z) &= \frac{1}{k}\sum_{1 \leq p \leq
k-1}g_{k-p}(z)\left(1+z\frac{\6}{\6 z}\right)(g_p(z)) \\ 
&=\frac{1}{2k}\sum_{1 \leq p \leq k-1}\left(2+z\frac{\6}{\6
z}\right)(g_p(z)g_{k-p}(z)), 
\end{align*}
and the initial conditions give 
$$
g_1(z) = \frac{1}{1-z}.
$$

\punkt Say that a formal series $f(z)$ in the variable $z$ is {\em
non-negative} if all the terms in the series are non-negative real
numbers. The sum and product of two non-negative series and the
derivative of a non-negative series is also obviously
non-negative. For two formal series $s(z)$, $t(z)$ write $s(z) \ll
t(z)$ if the difference $t(z) - s(z)$ is a non-negative power
series.

Our main estimate for the generating functions $g_k(z)$ is the
following. 

\begin{lemma}
For every $k \geq 1$ we have 
$$
g_k(z) \ll a_k\frac{1}{(1-z)^{2k-1}}, 
$$
where $a_k$ are the Catalan numbers. 
\end{lemma}

\proof Use induction on $k$. For $k=1$ we have $g_1(z) =
\frac{1}{1-z}$ and $a_1 = 1$, which gives the base for
induction. Assume that the claim is proved for all $p < k$. Since
all the $g_n(z)$ are non-negative power series, this implies that for
every $p$, $1 \leq p \leq k-1$ we have
$$
g_p(z)g_{k-p}(z) \ll
a_pa_{k-p}\frac{1}{(1-z)^{2p-1}}\frac{1}{(1-z)^{2k-2p-1}} =
a_pa_{k-p}\frac{1}{(1-z)^{2k-2}}. 
$$
Therefore 
\begin{align*}
\left(2+z\frac{\6}{\6 z}\right)(g_p(z)g_{k-p}(z)) &\ll
a_pa_{k-p}\left(2+z\frac{\6}{\6 z}\right)\frac{1}{(1-z)^{2k-2}} \\
&= a_pa_{k-p}\left(\frac{2}{(1-z)^{2k-2}} +
\frac{(2k-2)z}{(1-z)^{2k-1}} \right) \\  
&= a_pa_{k-p}\left(\frac{2k-2}{(1-z)^{2k-1}} -
\frac{2k-4}{(1-z)^{2k-2}}\right) \\ 
&\ll (2k-2)a_pa_{k-p}\frac{1}{(1-z)^{2k-1}}.
\end{align*}
Hence
\begin{align*}
g_k(z) &= \frac{1}{2k}\sum_{1 \leq p \leq k-1}\left(2+z\frac{\6}{\6
z}\right)(g_p(z)g_{k-p}(z)) \\ 
&\ll \frac{2k-2}{2k}\sum_{1 \leq p \leq
k-1}a_pa_{k-p}\frac{1}{(1-z)^{2k-1}} \\ 
&\ll \frac{1}{(1-z)^{2k-1}}\sum_{1 \leq p \leq k-1}a_pa_{k-p} =
a_k\frac{1}{(1-z)^{2k-1}}, 
\end{align*}
which proves the lemma. 
\endproof 

\punkt This estimate yields the following estimate for the numbers
$b_{k,n}$. 

\begin{corr}\label{combin}
The power series 
$$
g(z) = \sum_{k,n} b_{k,n}z^{n+k} = \sum_{k \geq 1}g_k(z)z^k
$$
converges for $z < 3-\sqrt{8}$. Consequently, for every $C_2$ such
that $(3-\sqrt{8})C_2 > 1$ there exists a positive constant $C > 0$
such that
$$
b_{n,k} < CC_2^{n+k}
$$
for every $n$ and $k$. (One can take, for example, $C_2 = 6$.) 
\end{corr}

\proof Indeed, we have 
\begin{equation}\label{g(z)}
g(z) \ll \sum_{k \geq 1}a_kz^k\frac{1}{(1-z)^{2k-1}} =
(1-z)f\left(\frac{z}{(1-z)^2}\right), 
\end{equation}
where $f(z) = \frac{1}{2} - \sqrt{\frac{1}{4}-z}$ is the generating
function for the Catalan numbers. Therefore
$$
g(z) \ll (1-z)\left(\frac{1}{2} - \sqrt{\frac{1}{4} -
\frac{z}{(1-z)^2}}\right), 
$$
and the right hand side converges absolutely when 
$$
\frac{|z|}{(1-z)^2} < \frac{1}{4}. 
$$
Since $3 - \sqrt{8}$ is the root of the quadratic equation $(1-z)^2
= 4z$, this inequality holds for every $z$ such that $|z| < 3 -
\sqrt{8}$.  
\endproof

\punkt \label{b.n.k.m} To study
polarizations\index{terms}{polarization} of Hodge 
manifold\index{terms}{Hodge manifold} structures on $\tm$, we will
need yet another recursive sequence of integers, which we denote by
$b_{k,n}^m$. This sequence is defined by the recurrence relation
$$
b^m_{k,n} = \sum_{p,q;1 \leq p \leq k-1} \frac{q+m(k-p)}{k}
b^m_{p,q}b_{k-p,n-q}
$$
and the initial conditions 
$$
\begin{cases}
b^m_{k,n} &= 0 \quad \text{ unless } \quad k,m \leq 0,\\
b^m_{k,n} &= 0 \quad \text{ for } \quad k = 1, n < 0,\\
b^m_{k,n} &= 1 \quad \text{ for } \quad k = 1, n \geq 0. 
\end{cases}
$$

\punkt \label{c.k.n}
To estimate the numbers $b^m_{k,n}$, consider the auxiliary sequence
$c_{k,n}$ defined by setting
$$
c_{k,n} = \sum_{p,q;1 \leq p \leq k-1}c_{p.q}b_{k-p,n-q} \qquad k
\geq 2, 
$$
and $c_{k,n} = b_{k,n}$ for $k \leq 1$. The generating series $c(z)
= \sum_{k,n \geq 0}c_{k,n}z^{n+k}$ satisfies
$$
c(z) = c(z)g(z) + \frac{z}{1-z},
$$
so that we have $c(z) = \frac{z}{(1-z)(1-g(z))}$, which is
non-singular when $|z|, |g(z)| < 1$ and $g(z)$ is non-singular. 
By \eqref{g(z)} the latter inequality holds if 
$$
\left|(1-z)\left(\frac{1}{2} - \sqrt{\frac{1}{4} -
\frac{z}{(1-z)^2}}\right)\right| < 1,
$$
which holds in the whole disc where $g(z)$ converges, that is, for
$|z| < 3-\sqrt{8}$. Therefore, as in Corollary~\ref{combin}, we have
\begin{equation}\label{cnk}
c_{k,n} < C6^{n+k}
\end{equation}
for some positive constant $C$. 

\punkt We can now estimate the numbers $b^m_{k,n}$. 

\begin{lemma}\label{combin.2}
For every $m,k,n$ we have 
\begin{equation}\label{bnkm.indu}
b^m_{k,n} \leq (2m)^{k-1}c_{k,n}b_{k,n},
\end{equation}
where $c_{k,n}$ are as in \ref{c.k.n} and $b_{k,n}$ are the
numbers introduced in \ref{b.n.k}. Consequently, we have 
$$
b^m_{k,n} < C(72m)^{n+k+m}
$$
for some positive constant $C > 0$. 
\end{lemma}

\proof Use induction on $k$. The case $k=1$ follows from the initial
conditions. Assume the estimate \eqref{bnkm.indu} proved for all
$b^m_{p,q}$ with $p < k$. Note that by the recurrence relations we
have $b_{p,q} \leq b_{k,n}$ and $c_{p,q} \leq c_{k,n}$ whenever $p <
k$. Therefore
\begin{align*}
b^m_{k,n} &= \sum_{p,q;1 \leq p \leq k-1} \frac{q+m(k-p)}{k}
b^m_{p,q}b_{k-p,n-q} \\
&\leq \sum_{p,q;1 \leq p \leq k-1} \frac{q}{k}b^m_{p,q}b_{k-p,n-q} +
\sum_{p,q;1 \leq p \leq k-1} b^m_{p,q}b_{k-p,n-q} \\
&\leq \sum_{p,q;1 \leq p \leq k-1}2^{p-1} c_{p,q} \frac{q}{k}
b_{p,q}b_{k-p,n-q} + \sum_{p,q;1 \leq p \leq k-1} 2^{p-1} b_{p,q}
c_{p,q} b_{k-p,n-q} \\
&\leq 2^{k-2} c_{k,n}\sum_{p,q;1 \leq p \leq k-1} \frac{q+1}{k}
b_{p,q} b_{k-p,n-q} \\
&\quad + 2^{k-2}b_{k,n}\sum_{p,q;1 \leq p \leq k-1} 
c_{p,q} b_{k-p,n-q} \\
&= 2^{k-2}c_{k,n}b_{k,n} + 2^{k-2}c_{k,n}b_{k,n} = 2^{k-1}c_{k,n}b_{k,n},
\end{align*}
which proves \eqref{bnkm.indu} for $b^m_{k,n}$. The second estimate of
the lemma now follows from \eqref{cnk} and Corollary~\ref{combin}. 
\endproof 

\subsection{The main estimate}

\punkt \label{norms} Let now $D = \sum_{k,n}D_{n,k}:\B^\cdot \to
\B^{\cdot+1}$ be a derivation of the Weil algebra\index{terms}{Weil
algebra} $\B^\cdot$ associated to a flat
linear\index{terms}{extended connection!linear}
extended\index{terms}{extended connection!flat} connection 
on $M$.  Consider the restriction $D_{k,n}:\B^\cdot_{p,q} \to
\B^\cdot{p+k,q+n}$ of the derivation $D_{k,n}$ to the component
$\B^\cdot_{p,q}\subset\B^\cdot$ of augmentation
degree\index{terms}{degree!augmentation} $p$ and total
degree\index{terms}{degree!total} $q$. Since both $\B^\cdot_{p,q}$
and $\B^\cdot{p+k,q+n}$ are finite-dimensional vector spaces, the
norm of this restriction with respect to the standard
metric\index{terms}{Weil algebra!standard metric on} on
$\B^\cdot$ is well-defined. Denote this norm by $\|D_{k,n}\|_{p,q}$.

By Lemma~\ref{est=conv} the convergence of the Hodge
manifold\index{terms}{Hodge manifold} structure on $\tm$
corresponding to the extended\index{terms}{extended connection}
connection $D$ is related to the 
growth of the norms $\|D_{k,n}\|_{1,1}$.  Our main estimate on the
norms $\|D_{k,n}\|_{1,1}$ is the following.

\begin{prop}\label{estimate}
Assume that there exist a positive constant $C_0$ such that for
every $n$ the norms $\|D_{1,n}\|_{1,1}$ and $\|D_{1,n}\|_{0,1}$ 
satisfy 
$$
\|D_{1,n}\|_{1,1}, \|D_{1,n}\|_{1,0} < C_0^n.
$$
Then there exists a positive constant $C_1$ such that for every
$n,k$ the norm $\|D_{k,n}\|_{1,1}$ satisfies
$$
\|D_{k,n}\|_{1,1} < C_1^n.
$$
\end{prop}

\punkt \label{metr} In order to prove Proposition~\ref{estimate}, we
need some preliminary facts. Recall that we have introduced in
\ref{total.Weil} the total Weil algebra\index{terms}{Weil
algebra!total} $\B_{tot}^\cdot(M,\C)$ of the manifold $M$, and let
$$
\B^\cdot_{tot} = C^\infty_\J(M,\B_{tot}^\cdot(M,\C))
$$
be the algebra of its smooth sections completed at $0 \subset M$. By
definition for every $k \geq 0$ we have $\B_{tot}^k = \B^k \otimes
\W_k^*$, where $\W_k$ is the $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} of
weight\index{terms}{weight!of a Hodge structure} $k$ universal 
for weakly Hodge maps\index{terms}{weakly Hodge map}, as in
\ref{w.k.uni}. There exists a unique Hermitian
metric\index{terms}{metric!Hermitian} on $\W_k$ such 
that all the Hodge components $\W^{p,q} \subset \W_k$ are orthogonal
and all the Hodge degree\index{terms}{degree!Hodge} components
$w_k^{p,q}$ of the universal weakly Hodge map\index{terms}{weakly
Hodge map!universal} $w_k:\R(0) \to \W_k$ are isometries. This
metric defines a canonical Hermitian metric on $\W_k^*$.

\defn The {\em standard metric\index{terms}{Weil algebra!standard
metric on}} on the total Weil algebra\index{terms}{Weil 
algebra!total} $\B_{tot}^\cdot$ is the product of the canonical metric and
the standard metric on $\B^\cdot$.

\punkt By Lemma~\ref{total.rel} the total Weil
algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}$ is
generated by the subspaces $V_2,V_3 \subset \B^0 = \B^0_{tot}$ and
the subspace $V_1 \otimes \W_1^* \subset \B^1_{tot}$, which we
denote by $V_1^{tot}$. The ideal of relations for the algebra
$\B^\cdot_{tot}$ is the ideal in $S^\cdot(V_2 \oplus
V_3)\otimes\Lambda^\cdot(V_1^{tot})$ generated by
$S^2(V_1)\otimes\Lambda^2(\W_1^*) \subset \Lambda^2(V_1^{tot})$.

The direct sum decomposition \eqref{drct} induces a direct sum
decomposition 
$$
V^{tot}_1 = V^{ll}_1 \oplus \V^o_1 \oplus V^{rr}_1 
$$
of the generator subspace $V^{tot}_1 \subset \B^1_{tot}$. The
subspaces $V^o_1 \subset V^{tot}_1$ and $V^{ll}_1 \oplus V^{rr}_1
\subset V^{tot}_1$ are both isomorphic to the vector space
$V_1$. More precisely, the universal weakly Hodge
map\index{terms}{weakly Hodge map} $w_1:\R(0) \to \W_1$ defines a
projection
$$
P:V^{tot}_1 = V_1 \otimes \W_1^* \to V_1,
$$
and the restriction of the projection $P$ to either of the subspaces
$V^o_1,V^{ll}_1 \oplus V^{rr}_1 \subset V^{tot}_1$ is an
isomorphism. Moreover, either of these restrictions is an isometry
with respect to the standard metrics\index{terms}{Weil
algebra!standard metric on}. 

\punkt \label{P2}
The multiplication in $\B_{tot}^\cdot$ is not an isometry with
respect to this metric\index{terms}{Weil algebra!standard metric
on}. However, for every $b_1,b_2 \subset \B_{tot}^\cdot$ we 
have the inequality
$$
\|b_1b_2\| \leq \|b_1\|\cdot\|b_2\|. 
$$
Moreover, this inequality becomes an equality when $b_1 \subset
\B^0_{tot}$. In particular, if we extend the map $P:V^{tot}_1 \to V_1$ to a
$\B^0$-module map 
$$
P:\B^1_{tot} \to \B^1,
$$
then the restriction of the map $P$ to either of the subspaces
$\B^1_o, \B^1_{ll}\oplus\B^1_{rr} \subset \B^1_{tot}$ is an isometry
with respect to the standard metric\index{terms}{Weil
algebra!standard metric on}. Therefore the norm of the projection 
$P:\B^1_{tot} \to \B^1$ is at most $2$.

\punkt The total and augmentation
gradings\index{terms}{grading!total}\index{terms}{grading!augmentation}
on the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ extend to
gradings on the total Weil algebra\index{terms}{Weil algebra!total}
$\B_{tot}^\cdot$, also denoted by lower indices. The extended
connection $D$ on $M$ induces a derivation\index{terms}{extended
connection!derivation associated to} $D^{tot} = 
\sum_{n,k}D_{k,n}^{tot}:\B^\cdot_{tot} \to \B^{\cdot+1}_{tot}$. As
in \ref{norms}, denote by $\|D^{tot}_{k,n}\|_{p,q}$ the norm of the
map $D^{tot}_{k,n}:\left(\B^\cdot_{tot}\right)_{p,q} \to
\left(\B^\cdot_{tot}\right)_{p+k,q+n}$ with respect to the standard
metric\index{terms}{Weil algebra!standard metric on} on
$\B^\cdot_{tot}$. The derivations $D^{tot}_{k,n}$ are related to
$D_{k,n}$ by
$$
D_{k,n} = P \circ D^{tot}_{k,n}:\B^0 \to \B^1, 
$$
and we have the following.

\begin{lemma}\label{total.ne.total}
For every $k$, $n$ and $p=0,1$ we have 
$$
\|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1}.
$$
\end{lemma}

\proof By definition we have $\B^\cdot_{0,1} \oplus \B^\cdot_{1,1} =
V_1 \oplus V_2 \oplus V_3$. Moreover, the derivation $D_{k,n}$
vanishes on $V_1$, hence $D^{tot}_{k,n}$ vanishes on $V_1^{tot}$. Therefore
it suffices to compare their norms on $V_2 \oplus V_3 \subset \B^0 =
\B^0_{tot}$.

Since on $S^\cdot(V_2) \subset \B^0$ the derivation $D_{k,n}$
coincides with the de Rham differential, the derivation $D^{tot}_{k,n}$
maps the subspace $V_2$ into $\B^1_{ll}\oplus\B^1_{rr}$. Moreover,
by Lemma~\ref{total.aug} the derivation $D_{k,n}^{tot}$ maps $V_3$
either into $\B_o^1$ or into $\B^1_{ll} \oplus \B^1_{rr}$, depending
on the parity of the number $k$. Since $D_{k,n} = P \circ D^{tot}_{k,n}$
and the map $P:\B^1_{tot} \to \B^1$ is an isometry on both $\B^1_o
\subset \B^1_{tot}$ and $\B^1_{ll}\oplus\B^1_{rr} \subset \B^1_{tot}$, we
have $\|D_{k,n}\| = \|D^{tot}_{k,n}\|$ on both $V_2 \subset \B^0$ and
$V_3 \subset \B^0$, which proves the lemma.
\endproof

This lemma allows to replace the derivations $D_{k,n}$ in
Proposition~\ref{estimate} with associated derivations
$D_{k,n}^{tot}$ of the total Weil algebra\index{terms}{Weil
algebra!total} $\B_{tot}^\cdot$.

\punkt Since the extended\index{terms}{extended connection!flat}
connection $D$ is linear\index{terms}{extended connection!linear}
and flat, the construction used in the proof of Lemma~\ref{main.ind}
shows that
\begin{equation}\label{indu}
D^{tot}_k = h^{-1} \circ \sigma_{tot} \circ \sum_{1 \leq p \leq k-1}D_p^{tot}
\circ D^{tot}_{k-p}:V_3 \to \left(\B^1_{tot}\right)_{k+1},
\end{equation}
where $\sigma_{tot}:\B^{\cdot+1}_{tot} \to \B^\cdot_{tot}$ is the
canonical map constructed in \ref{sigma.c}, and $h:\B^\cdot_{tot}
\to \B^\cdot_{tot}$ is as in Lemma~\ref{h.acts}. Both $\sigma_{tot}$
and $h$ preserve the augmentation
degree\index{terms}{degree!total}. In order to obtain estimates on
$\|D_{k,n}\|_{1,1} =
\|D_{k,n}^{tot}\|_{1,1}$, we have to estimate the norms $\|h^{-1}\|$
and $\|\sigma_{tot}\|$ of the restrictions of maps $h^{-1}$ and
$\sigma_{tot}$ on the subspace $\left(\B_{tot}^\cdot\right)_{k+1}
\subset \B^\cdot_{tot}$.

By Lemma~\ref{h.on.b1} the map $h:\left(\B^\cdot\right)_{k+1} \to
\left(\B^\cdot\right)_{k+1}$ is diagonalizable, with eigenvalues
$k+1$ if $k$ is even and $(k+1)/2,(k-1)/2$ if $k$ is odd. Since $m
\geq 2$, in any case on $\left(\B^\cdot\right)_{k+1}$ we have
\begin{equation}\label{h.est}
\|h^{-1}\| < \frac{3}{k}.
\end{equation}

\punkt \label{sigma.est.punkt}
To estimate $\sigma_{tot}:\B^2_{tot} \to \B^2_{tot}$, recall that, as noted in
\ref{sigma.c}, the map $\sigma_{tot}$ is a map of $\B^0$-modules. The
space $\B^2_{tot} = \B^0 \otimes \left(\B^2_{tot}\right)_2$ is a
free $\B^0$-module generated by a finite-dimensional vector space
$\left(\B^2_{tot}\right)_2$. The map $\sigma_{tot}$ preserves the
augmentation degree\index{terms}{degree!augmentation}, hence it maps
$\left(\B^2_{tot}\right)_2$ into the finite-dimensional vector space
$\left(\B^1_{tot}\right)_2$. Therefore there exists a constant $K$
such that
\begin{equation}\label{sigma.est}
\|\sigma_{tot}\| \leq K
\end{equation}
on $\left(\B^2_{tot}\right)_2$. Since we have
$\|b_1b_2\|=\|b_1\|\cdot\|b_2\|$ for every $b_1 \in \B^0$, $b_2 \in
\B^2_{tot}$, and the map $\sigma_{tot}$ is $\B^0$-linear, the estimate
\eqref{sigma.est} holds on the whole $\B^2_{tot} = \B^0 \otimes
\left(\B^2_{tot}\right)_2$. We can assume, in addition, that $K \geq 1$. 

\rem In fact $K = 2$, but we will not need this. 

\punkt Let now $b_{k,n}$ be the numbers defined recursively in
\ref{b.n.k}. Our estimate for $\|D^{tot}_{k,n}\|_{p,q}$ is the
following. 

\begin{lemma}\label{est.indu}
In the assumptions and notations of Proposition~\ref{estimate}, we
have 
$$
\|D^{tot}_{k,n}\|_{p,q} < q(3K)^{k-1} C_0^nb_{k,n}
$$
for every $k,n$. 
\end{lemma}

\proof Use induction on $k$. The base of induction is the case
$k=1$, when the inequality holds by assumption. Assume that for some
$k$ we have proved the inequality for all 
$\|D^{tot}_{m,n}\|_{p,q}$ with $m < k$, and fix a number $n \geq k$.  

Consider first the restriction of $D^{tot}_{k,n}$ onto the generator
subspace $V_3 \subset \B^0$.  Taking into account the total
degree\index{terms}{degree!total}, we can rewrite \eqref{indu} as
$$
D^{tot}_{k,n} = h^{-1} \circ \sigma_{tot} \circ \sum_{1 \leq p \leq k-1}\sum_q
D^{tot}_{k-p,n-q} \circ D^{tot}_{p,q}:V_3 \to \left(\B^1_{tot}\right)_{k+1}. 
$$
Therefore the norm of the map $D^{tot}_{k,n}:V_3 \to \B_{tot}^1$ satisfies 
$$
\|D^{tot}_{k,n}\} \leq \|h^{-1}\| \cdot \|\sigma_{tot}\| \cdot \sum_{1 \leq p
\leq k-1}\sum_q\|D^{tot}_{k-p,n-q}\|_{p+1,q+1} \cdot \|D^{tot}_{p,q}\|_{1,1}. 
$$
Substituting into this the estimates \eqref{h.est},
\eqref{sigma.est} and the inductive assumption, we get 
$$
\|D^{tot}_{k,n}\| < \frac{3}{k}K\sum_{1 \leq p \leq
k-1}\sum_q(q+1)(3K)^{k-2}C_0^nb_{k-p,n-q}b_{p,q} = (3K)^{k-1}C_0^nb_{k,n}.
$$
Since by definition $D^{tot}_{k,n}$ vanishes on $V_2 \subset \B^0$ and
on $V^{tot}_1 \subset \B^1_{tot}$, this proves that 
$$
\|D^{tot}_{k,n}\|_{p,1} < (3K)^{k-1}C_0^nb_{k,n}
$$
when $p=0,1$. Since the map $D^{tot}_{k,n}:\B^\cdot_{tot} \to
\B^{\cdot+1}_{tot}$ is a derivation, the Leibnitz
rule\index{terms}{Leibnitz rule} and the triangle inequality show
that for every $p$, $q$
$$
\|D^{tot}_{k,n}\|_{p,q} < q(3K)^{k-1}C_0^nb_{k,n}, 
$$
which proves the lemma.
\endproof 

\punkt \proof[Proof of Proposition~\ref{estimate}]
By Lemma~\ref{total.ne.total} we have $\|D_{k,n}\|_{p,1} =
\|D^{tot}_{k,n}\|_{p,1}$, and by Lemma~\ref{est.indu} we have 
$$
\|D_{k,n}\|_{p,1} = \|D^{tot}_{k,n}\|_{p,1} < (3K)^{k-1}C_0^nb_{n,k}.
$$
Since $k \leq n$ and $K \geq 1$, this estimate together with
Corollary~\ref{combin} implies that
$$
\|D_{k,n}\|_{p,1} < C(3K)^{k-1}C_0^n6^{2n} < C(108KC_0)^n 
$$
for some positive constant $C > 0$, which proves the proposition. 
\endproof 

\punkt Proposition~\ref{estimate} gives estimates for the derivation
$D:\B^\cdot \to \B^{\cdot+1}$ or, equivalently, for the Dolbeault
differential 
$$
\bar\6_J:\Lambda^{0,\cdot}(\tm_J) \to \Lambda^{0,\cdot+1}(\tm_J)
$$ 
for the complementary\index{terms}{complementary complex structure}
complex structure $\tm_J$ on $\tm$ associated to the
extended\index{terms}{extended connection} connection $D$ on $M$. To
prove the second part of Theorem~\ref{converge}, we will need to
obtain estimates on the Dolbeult differential
$\bar\6_J:\Lambda^{p,0}(\tm_J) \to \Lambda^{p,1}(\tm_J)$ with $p >
0$. To do this, we use the model for the de Rham
complex\index{terms}{de Rham complex} $\Lambda^{\cdot,\cdot}(\tm_J)$
constructed in Subsection~\ref{drm.mod}.

Recall that in \ref{ident.punkt} we have identified the direct image
$\rho_*\Lambda^{\cdot,\cdot}(\tm_J)$ of the de Rham algebra of the
manifold $\tm$ with the free module over the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ generated by a
graded algebra bundle $L^\cdot(M,\C)$ on $M$. The Dolbeult
differential $\bar\6_J$ for the
complementary\index{terms}{complementary complex structure} complex
structure $\tm_J$ induces an algebra 
derivation $D:L^\cdot(M,\C) \otimes \B^\cdot(M,\C) \to L^\cdot(M,\C)
\otimes \B^{\cdot+1}(M,\C)$, so that the free module
$\rho_*\Lambda^{\cdot,\cdot}(\tm_J) \cong L^\cdot(M,\C) \otimes
\B^\cdot(M,\C)$ becomes a differential graded module over the Weil
algebra\index{terms}{Weil algebra}.

\punkt The algebra bundle $L^\cdot(M,\C)$ is isomorphic to the de
Rham algebra\index{terms}{de Rham complex} $\Lambda^\cdot(M,\C)$. In
particular, the bundle $L^1(M,\C)$ is isomorphic to the trivial
bundle $\V$ with fiber $V$ over $M$.  By \ref{dr.L} the relative de
Rham differential
$$
d^r:\Lambda^\cdot(\tm/M,\C) \to \Lambda^{\cdot+1}(\tm/M,\C)
$$
induces a derivation
$$
d^r:L^\cdot(M,\C) \otimes \B^\cdot(M,\C) \to
L^{\cdot+1}(M,\C) \otimes \B^\cdot(M,\C), 
$$
and we can choose the trivialization $L^1(M,\C) \cong \V$ in such a
way that $d^r$ identifies the generator subspace $V_3 \subset \B^0$
with the subspace of constant sections of $\V \cong L^1(M,\C)
\subset L^1(M,\C) \otimes \B^0(M,\C)$. 

\punkt \label{LLL}
Denote by
$$
\LL^{\cdot,\cdot} = C^\infty_\J(L^\cdot(M,\C) \otimes
\B_{tot}^\cdot(M,\C))
$$
the $\J$-adic completion\index{terms}{formal completion} of the space
of smooth sections of the algebra bundle $L^\cdot(M,\C) \otimes
\B_{tot}^\cdot(M,\C)$ on $M$.  The space $\LL^{\cdot,\cdot}$ is a
bigraded algebra equipped with the derivations
$d^r:\LL^{\cdot,\cdot} \to \LL^{\cdot+1,\cdot}$,
$D^{tot}:\LL^{\cdot,\cdot} \to \LL^{\cdot,\cdot+1}$, which commute
by Lemma~\ref{D.dr}. The algebra $\LL^{\cdot,\cdot}$ is the free
graded-commutative algebra generated by the subspaces $V_1,V_2,V_3
\subset \LL^{0,\cdot} = \B^\cdot$ and the subspace $V = d^r(V_3)
\subset \LL^{1,0}$, which we denote by $V_4$.

\punkt As in \ref{metr}, the given metric on the generator subspaces
$V_1=V_2=V_3=V_4=V$ extends uniquely to a multiplicative metric on
the algebra $\LL^{\cdot,\cdot}$, which we call {\em the standard
metric\index{terms}{Weil algebra!standard metric on}}. For every
$k > 0$, introduce the total and augmentation
gradings\index{terms}{grading!total}\index{terms}{grading!augmentation} on the free $\B^\cdot$-module $\LL^{k,\cdot} =
\Lambda^k(V_4) \otimes \B^\cdot$ by setting $\deg \Lambda^k(V_4) =
(0,0)$.  Let $D = D_{k,n}$ be the decomposition of the derivation
$D:\LL^{\cdot,\cdot} \to \LL^{\cdot,\cdot+1}$ with respect to the
total and the augmentation
degrees\index{terms}{degree!augmentation}\index{terms}{degree!total}.
Denote by $\|D_{k,n}\|^p_q$ the 
norm with respect to the standard metric of the restriction of the
derivation $D_{k,n}:\LL^{p,0} \to \LL^{p,1}$ to the component in
$\LL^{p,0}$ of total degree $q$.

\punkt Let now $\LL_{tot}^{\cdot,\cdot} = \Lambda^\cdot(V_4) \otimes
\B^\cdot_{tot}$ be the product of the exterior algebra
$\Lambda^\cdot(V_4)$ with the total Weil algebra\index{terms}{Weil
algebra!total} $\B^\cdot_{tot}$. We have the canonical
identification $\LL^{p,q}_{tot} = \LL^{p,q} \otimes \W_q^*$, and the canonical
projection $P:\LL^{\cdot,\cdot}_{tot} \to \LL^{\cdot,\cdot}$,
identical on $\LL^{\cdot,0}_{tot} = \LL^{\cdot,0}$.  As in \ref{P2},
the norm of the projection $P$ on $\LL^{\cdot,1}_{tot}$ is at most
$2$.

The derivation $D:\LL^{\cdot,0} \to \LL^{\cdot,1}$ induces a
derivation $D^{tot}:\LL_{tot}^{\cdot,0} \to \LL_{tot}^{\cdot,1}$,
related to $D$ by $D = P \circ D^{tot}$. The
gradings\index{terms}{grading!total}\index{terms}{grading!augmentation}
and the metric\index{terms}{Weil algebra!standard metric on} on
$\LL^{\cdot,\cdot}$ extend to $\LL^{\cdot,\cdot}_{tot}$, 
in particular, we have the decomposition $D^{tot} =
\sum_{k,n}D^{tot}_{k,n}$ with respect to the total and the
augmentation
degrees\index{terms}{degree!total}\index{terms}{degree!augmentation}.
Denote by $\|D_{k,n}^{tot}\|^p_q$ the norm with respect to the standard metric
of the restriction of the derivation $D^{tot}_{k,n}:\LL_{tot}^{p,0}
\to
\LL_C^{p,1}$ to the component in $\LL^{p,0}$ of total degree
$q$. Since $\|P\|\leq 2$ on $\LL^{\cdot,1}_{tot}$, we have
\begin{equation}\label{DcL}
\|D_{k,n}^{tot}\|^p_q \leq 2\cdot\|D_{k,n}\|^p_q.
\end{equation}

\punkt The estimate on the norms $\|D^{tot}_{k,n}\|^\LL_{p,q}$ that we
will need is the following.

\begin{lemma}\label{D2est}
In the notation of Lemma~\ref{est.indu}, we have 
$$
\|D^{tot}_{k,n}\|^p_q < 2(q+pk)(3K)^{k-1} C_0^nb_{k,n},
$$
for every $p,q,k,n$. 
\end{lemma}

\proof Since $D_{k,n}$ satisfies the Leibnitz
rule\index{terms}{Leibnitz rule}, it suffices to prove the estimate
for the restriction of the derivation $D^{tot}_{k,n}$ to the
generator subspaces $V_2,V_3,V_4 \subset \LL^{\cdot,0}$. By
\eqref{DcL} it suffices to prove that on $V_2,V_3,V_4$ we have
$$
\|D_{k,n}\|^p_q < (q+pk)(3K)^{k-1} C_0^nb_{k,n}. 
$$
On the generator subspaces $V_2,V_3 \subset \B^0$ we have $p=0$, and
this equality is the claim of Lemma~\ref{est.indu}. Therefore it
suffices to consider the restriction of the derivation $D_{k,n}$ to
the subspace $V_4 = d^r(V_3) \subset \LL^{1,0}$. We have $d^r \circ
D_{k,n} = D_{k,n} \circ d^r:V_3 \to \LL^{1,1}$. Moreover, $d^r =
\id:V_3 \to V_4$ is an isometry. Since the operator $d^r:\B^\cdot
\to \LL^{1,\cdot}$ satisfies the Leibnitz rule\index{terms}{Leibnitz
rule} and vanishes on the generators $V_1,V_2 \subset \B^\cdot$, the
norm $\|d^r\|_k$ of its restriction to the subspace $\B^\cdot_k$ of
augmentation degree\index{terms}{degree!augmentation} $k$ does not
exceed $k$. Therefore 
\begin{align*}
\|D_{k,n}\|^1_0 &= \left\|D_{k,n}|_{V_4}\right\| =
\left\|D_{k,n}\circ d^r|_{V_3}\right\| = \left\|d^r \circ
D_{k,n}|_{V_3}\right\| \\
&\leq \left\|D_{k,n}|_{V_3}\right|\cdot\left\|d^r_{\B^1_k}\right\| < k
(3K)^{k-1} C_0^nb_{k,n},
\end{align*}
which proves the lemma.
\endproof 


\subsection{The proof of Theorem~\ref{converge}} 

\punkt We can now prove Theorem~\ref{converge}. Let
$D:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \to \B^1(M,\C)$
be a flat linear\index{terms}{extended connection!linear} extended
connection on $M$. Assume that its reduction\index{terms}{extended
connection!reduction of}
$D_1=\nabla:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)
\to S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \otimes
\Lambda^1(M,\C)$ is a real-analytic
connection\index{terms}{connections!on a vector bundle} 
on the bundle $S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)$.

The operator $D_1:S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C)
\to S^1\index{terms}{Hodge bundle!$S^1(M,\protect\C)$}(M,\C) \otimes
\Lambda^1(M,\C) \subset \B^1(M,\C)$ considered as an
extended\index{terms}{extended connection} connection on $M$ defines
a Hodge connection\index{terms}{Hodge connection}
$D_1:\Lambda^0(\tm,\C) \to \rho^*\Lambda^1(M,\C)$ on the pair
$\langle \tm,M \rangle$, and this Hodge
connection\index{terms}{Hodge connection} is also
real-analytic. Assume further that the Taylor series at $0 \subset M
\subset \tm$ for the Hodge connection $D_1$ converge in the closed
ball of radius $\eps>0$.

\punkt Let $D = \sum_{n,k}D_{k,n}:\B^\cdot \to \B^{\cdot+1}$ be the
derivation of the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$
associated to the extended\index{terms}{extended
connection!derivation associated to} connection $D$.  Applying
Lemma~\ref{est=conv} to the Hodge connection\index{terms}{Hodge
connection} $D_1$ proves that there exists a constant $C>0$ such
that for every $n \geq 0$ the norm $\|D_{1,n}\|_{1,1}$ of the
restriction of the derivation $D_{1,n}$ to the generator subspace
$V_3 = \B^0_{1,1} \subset \B^0$ satisfies
$$
\|D_{1,n}\|_{1,1} < CC_0^n,
$$
where $C_0 = 1/\eps$. 

By definition the derivation $D_{1,n}$ vanishes on the generator
subspace $V_1 \subset \B^1$. If $n > 0$, then it also vanishes on
the generator subspace $V_2 = \B^0_{0,1} \subset \B^0$. If $n=0$,
then its restriction to $V_2 = \B^0_{0,1} \subset \B^0$ is the
identity isomorphism $D_{1,0}=\id:V_2 \to V_1$. In any case, we have
$\|D_{1,n}\|_{0,1} \leq 1$. Increasing if necessary the constant
$C_0$, we can assume that for any $n$ and for $p=0,1$ we have
$$
\|D_{1,n}\|_{p,1} < C_0^n.
$$

\punkt We can now apply our main estimate,
Proposition~\ref{estimate}. It shows that there exists a constant
$C_1 > 0$ such that for every $k,n$ we have
$$
\|D_{k,n}\|_{1,1} < C_1^n.
$$
Together with Lemma~\ref{est=conv} this estimate implies that the
formal Hodge connection\index{terms}{Hodge connection!formal} $D$ on
$\tm$ along $M \subset \tm$ corresponding to the
extended\index{terms}{extended connection} connection $D$ converges
to a real-analytic Hodge connection\index{terms}{Hodge connection}
on an open neighborhood $U \subset \tm$ of the zero section $M
\subset \tm$. This in turn implies the first claim of
Theorem~\ref{converge}.

\punkt To prove the second claim of Theorem~\ref{converge}, assume
that the manifold $M$ is equipped with a K\"ahler form $\omega$
compatible with the K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian} $\nabla$, so that
$\nabla\omega=0$. The differential operator $\nabla:\Lambda^{1,1}(M)
\to \Lambda^{1,1}(M) \otimes \Lambda^1(M,\C)$ is elliptic and
real-analytic. Since $\nabla\omega=0$, the K\"ahler form $\omega$ is
also real-analytic.

For every $p,q \geq 0$ we have introduced in \ref{LLL} the space
$\LL^{p,q}$, which coincides with the space of formal germs at $0
\in M \subset \tm$ of smooth forms on $\tm$ of
type\index{terms}{bigrading!Hodge type} $(p,q)$ with respect to the
complementary\index{terms}{complementary complex structure} complex
structure $\tm_J$. The spaces 
$\LL^{\cdot,\cdot}$ carry the total and the augmentation
gradings\index{terms}{grading!total}\index{terms}{grading!augmentation}.
Consider $\omega$ as an 
element of the vector space 
$\LL^{2,0}_0$, and let $\omega = \sum_n\omega_n$ be the total
degree\index{terms}{degree!total} decomposition.  The decomposition
$\omega = \sum_n \omega_n$ is the Taylor series decomposition for
the form $\omega$ at $0 \in M$. Since the form $\omega$ is
real-analytic, there exists a constant $C_2$ such that
\begin{equation}\label{omega.est}
\|\omega_n\| < C_2^n
\end{equation}
for every $n$. 

\punkt Let $\Omega = \sum_k \Omega_k = \sum_{k,n} \Omega_{k,n}
\subset \LL^{2,0}$ be the formal
polarization\index{terms}{polarization!formal} of the Hodge
manifold\index{terms}{Hodge manifold} 
$\tm$ at $M \subset \tm$ corresponding to the K\"ahler form $\omega$
by Theorem~\ref{metrics}. By definition we have $\omega =
\Omega_0$. Moreover, by construction used in the proof of
Proposition~\ref{metrics.ind} we have
\begin{equation}\label{Omega.rec}
\Omega_k = -\frac{1}{k}\sum_{1 \leq p \leq
k-1}\sigma_{tot}(D^{tot}_{k-p}\Omega_p), 
\end{equation}
where $D^{tot}:\LL^{2,0} \to \LL^{2,1}$ is the derivation associated
to the extended\index{terms}{extended connection!derivation
associated to} connection $D$ on $M$ and $\sigma_{tot}:\LL^{2,\cdot+1}
\to \LL^{2,\cdot}$ is the extension to $\LL^{2,\cdot} = L^2_0
\otimes \B^\cdot_{tot}$ of the canonical endomorphism of the total
Weil algebra\index{terms}{Weil algebra!total} $\B^\cdot_{tot}$
constructed in the proof of Proposition~\ref{metrics.ind}.

\punkt The map $\sigma_{tot}:\LL^{2,1} \to \LL^{2,0}$ is a map of
$\B^0_{tot}$-modules. Therefore, as in \ref{sigma.est.punkt}, there
exists a constant $K_1 > 0$ such that 
\begin{equation}\label{K1}
\|\sigma_{tot}\| < K_1
\end{equation}
on $\LL^{2,1}$. We can assume that $K_1 > 3K$, where $K$ is as in
\eqref{sigma.est}.  Together with the recursive formula
\eqref{Omega.rec}, this estimate implies the following estimate on
the norms $\|\Omega_{k,n}\|$ of the components $\Omega_{k,n}$ of the
formal polarization\index{terms}{polarization!formal} $\Omega$ taken
with respect to the standard metric\index{terms}{Weil
algebra!standard metric on}. 

\begin{lemma}
For every $k,n$ we have 
$$
\|\Omega_{k,n}\| < (2K_1)^{k-1}C^nb^2_{k,n},
$$
where $C = \max(C_0,C_2)$ is the bigger of the constants $C_0$,
$C_2$, and $b^2_{k,n}$ are the numbers defined recursively in
\ref{b.n.k.m}. 
\end{lemma}

\proof Use induction on $k$. The base of the induction is the case
$k=1$, where the estimate holds by \eqref{omega.est}. Assume the
estimate proved for all $\Omega_{p,n}$ with $p < k$, and fix a
number $n$. By \eqref{Omega.rec} we have
$$
\Omega_{k,n} = -\frac{1}{k}\sum_{1 \leq p \leq
k-1}\sum_q\sigma_{tot}(D^{tot}_{k-p,n-q}\Omega_{p,q}).
$$
Substituting the estimate \eqref{K1} together with the inductive
assumption and the estimate on $\|D^{tot}_{k-p,n-q}\|^2_q$ obtained in
Lemma~\ref{D2est}, we get
\begin{align*}
\|\Omega_{k,n}\| &< \frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_q
\|\sigma_{tot}\|\cdot\|D^{tot}_{k-p,n-q}\|^2_q\cdot\|\Omega_{p,q}\| \\
&< \frac{1}{k}\sum_{1 \leq p \leq k-1}\sum_qK_1 \cdot
2(q+2(k-p))(3K)^{k-p-1}C_0^{n-q}b_{k-p,n-q} \\
&\qquad\qquad\qquad\qquad\cdot (2K_1)^{p-1}C^qb_{p,q} \\
&< \frac{1}{k}(2K_1)^{k-1}C^n\sum_{1 \leq p \leq
k-1}\sum_q(q+2(k-p))b_{k-p,n-q}b^2_{p,q} \\
&= (2K_1)^{k-1}C^nb^2_{k,n},
\end{align*}
which proves the lemma. 
\endproof 

\punkt This estimate immediately implies the last claim of
Theorem~\ref{converge}. Indeed, together with Lemma~\ref{combin.2}
it implies that for every $k,n$ 
$$
\|\Omega_{k,n}\| < (C_3)^n
$$
for some constant $C_3>0$. But $\Omega = \sum_n\sum_{0 \leq k \leq
n}\Omega_{k,n}$ is the Taylor series decomposition for the formal
polarization\index{terms}{polarization!formal} $\Omega$ at $0
\subset M$. The standard convergence criterion shows that this
series converges in an open ball of radius $1/C_3 > 0$. Therefore
the polarization\index{terms}{polarization} $\Omega$ is indeed
real-analytic in a neighborhood of $0 \subset M \subset \tm$.
\endproof 



\section*{Appendix}
\refstepcounter{section}
\refstepcounter{subsection}
\renewcommand{\thesection}{A}
\addcontentsline{toc}{section}{Appendix}

\punkt In this appendix we describe a well-known Borel-Weyl type
localization construction for quaternionic vector
spaces\index{terms}{quaternionic vector space} (see,
e.g. \cite{HKLR}) which provides a different and somewhat more
geometric approach to many facts in the theory of Hodge
manifolds\index{terms}{Hodge manifold}. In particular, we establish,
following Deligne\index{names}{Deligne, P.} and
Simpson\index{names}{Simpson, C.} (\cite{De2}, \cite{Simpson}), a
relation between Hodge manifolds and the theory of mixed $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure}. For the sake of
simplicity, we consider only Hodge manifold structures on the formal
neighborhood\index{terms}{formal neighborhood} of $0 \in \R^{4n}$
instead of actual Hodge manifolds, as in
Section~\ref{convergence}. To save the space all proofs are either
omitted or only sketched.

\punkt Let $\SB$ be the Severi-Brauer variety associated to the
algebra $\h$, that is, the real algebraic variety of minimal right 
ideals in $\h$. The variety $\SB$ is a twisted $\R$-form of the
complex projective line $\cp^1$. 

For every algebra map $I:\C \to \h$ let the algebra $\h \otimes_\R
\C$ act on the $2$-dimensional complex vector space $\h_I$ by left
multiplication, and let $\widehat{I} \subset \h \otimes_\R \C$ be
the annihilator of the subspace $I(\C) \subset \h_I$ with respect to
this action. The subspace $\widehat{I}$ is a minimal right ideal in
$\h \otimes_\R \C$. Therefore it defines a $\C$-valued point
$\widehat{I} \subset \SB(\C)$ of the real algebraic variety
$\SB$. This establishes a bijection between the set $\SB(\C)$ and
the set of algebra maps from $\C$ to $\h$.

\punkt Let $\Shv(\SB)$ be the category of flat coherent sheaves on
$\SB$. Say that a sheaf $\E \in \Ob\Shv(\SB)$ is {\em of weight
\index{terms}{weight!of a sheaf on $\protect\C P^1$} $p$} 
if the sheaf $\E \otimes \C$ on $\cp^1 = \SB \otimes \C$ is a sum of
several copies of the sheaf $\calo(p)$. 

Consider a quaternionic vector space\index{terms}{quaternionic
vector space} $V$.  Let $\I \in \h
\otimes \calo_\SB$ be the tautological minimal left ideal in the
algebra sheaf $\h \otimes \calo_\SB$, and let $\loc{V} \in
\Ob\Shv(\SB)$ be the sheaf defined by
$$
\loc{V} = V \otimes \calo_\SB / \I \cdot V \otimes \calo_\SB. 
$$
The correspondence $V \mapsto \loc{V}$ defines a functor from
quaternionic vector spaces\index{terms}{quaternionic vector space}
to $\Shv(\SB)$. It is easy to check that this functor is a full
embedding, and its essential image is the subcategory of sheaves of weight
\index{terms}{weight!of a sheaf on $\protect\C P^1$} $1$. Call $\loc{V}$
{\em the localization} of the quaternionic vector
space\index{terms}{quaternionic vector space} $V$. For every algebra
map $\I:\C \to \h$ the fiber $\loc{V}|_{\widehat{I}}$ of the
localization $\loc{V}$ over the point $\widehat{I} \subset \SB(\C)$
corresponding to the map $i:\C \to \h$ is canonically isomorphic to
the real vector space $V$ with the complex structure $V_I$.

\punkt The compact Lie group $U(1)$ carries a canonical structure of
a real algebraic group. Fix an algebra embedding $I:\C \to \h$ and
let the group $U(1)$ act on the algebra $\h$ as in
\ref{u.acts.on.h}. This action is algebraic and induces therefore an
algebraic action of the group $U(1)$ on the Severi-Brauer variety
$\SB$. The point $\widehat{I}:\Spec\C \subset \SB$ is preserved by
the $U(1)$-action.  The action of the group $U(1)$ on the complement
$\SB \setminus \widehat{I}(\Spec\C) \subset \SB$ is free, so that
the variety $\SB$ consists of two $U(1)$-orbits. The corresponding
orbits of the complexified group $\C^* = U(1) \times \Spec\C$ on the
complexification $\SB \times \Spec\C \cong \cp$ are the pair of
points $0,\infty \subset \cp$ and the open complement $\cp \setminus
\{0,\infty\} \cong \C^* \subset \cp$.

Let $\Shv^{U(1)}(\SB)$ be the category of $U(1)$-equivariant flat
coherent sheaves on the variety $\SB$. The localization construction
immediately extends to give the equivalence $V \mapsto \loc{V}$
between the category of equivariant quaternionic vector
spaces\index{terms}{quaternionic vector space!equivariant} and the 
full subcategory in $\Shv^{U(1)}(\SB)$ consisting of sheaves of weight
\index{terms}{weight!of a sheaf on $\protect\C P^1$} $1$. For an equivariant
quaternionic vector space $V$, the 
fibers of the sheaf $\loc{V}$ over the point $\widehat{I} \subset
\SB(\C)$ and over the complement $\SB \setminus
\widehat{I}(\Spec\C)$ are isomorphic to the space $V$ equipped,
respectively, with the preferred\index{terms}{preferred complex
structure} and the complementary\index{terms}{complementary complex
structure} complex 
structures $V_I$ and $V_J$.

\punkt The category of $U(1)$-equivariant flat coherent sheaves on
the variety $\SB$ admits the following beautiful description, due to
Deligne\index{names}{Deligne, P.}. 

\begin{lemma}[ (\cite{De2},\cite{Simpson})]\label{DS}
\begin{enumerate}
\item For every integer $n$ the full subcategory
$$
\Shv^{U(1)}_n(\SB) \subset \Shv^{U(1)}(\SB)
$$ 
of sheaves of weight\index{terms}{weight!of a sheaf on $\protect\C
P^1$} $n$ is equivalent to the category of pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} of
weight\index{terms}{weight!of a Hodge structure} $n$.
\item The category of pairs $\langle \E, W_\cdot\rangle$ of a flat
$U(1)$-equivariant sheaf 
$$
\E \in \Ob\Shv^{U(1)}(\SB)
$$ 
and an increasing filtration $W_\cdot$ on $\E$ such that for every
integer $n$
\begin{equation}\label{hg}
W_n\E/W_{n-1}\E \quad \text{ is a sheaf of weight } \quad n \quad 
\text{ on } \quad \SB
\end{equation}
\index{terms}{weight!of a sheaf on $\protect\C P^1$}
is equivalent to the category of mixed $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure!mixed}. (In particular,
it is abelian.)
\end{enumerate}
\end{lemma}

\punkt For every pure $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge
structure} $V$ call the corresponding $U(1)$-e\-qui\-va\-ri\-ant
flat coherent sheaf on the variety $\SB$ {\em the localization} of
$V$ and denote it by $\loc{V}$. For the trivial $\R$-Hodge structure
$\R(0)$ of weight\index{terms}{weight!of a Hodge structure} $0$ the sheaf $\loc{\R(0)}$ coincides with the
structure sheaf $\calo$ on $\SB$. If $V$, $W$ are two pure
$\R$-Hodge structures, then the space $\Hom(\loc{V},\loc{W})$ of
$U(1)$-equivariant maps between the corresponding sheaves coincides
with the space of weakly Hodge maps\index{terms}{weakly Hodge map}
from $V$ to $W$ in the sense of Subsection~\ref{w.H.sub}.

For every pure $\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge
structure} $V$ the space $\Gamma(\SB,\loc{V})$ of the global
sections of the sheaf $\loc{V}$ is equipped with an action of the
group $U(1)$ and carries therefore a canonical $\R$-Hodge structure
of weight\index{terms}{weight!of a Hodge structure} $0$. This $\R$-Hodge structure is the same as the
universal $\R$-Hodge structure $\Gamma(V)$ of weight\index{terms}{weight!of a Hodge structure} $0$ constructed
in Lemma~\ref{g.ex}. This explains our notation for the functor
$\Gamma:\prehodge_{\geq 0} \to \prehodge_0$.

\punkt Assume given a complex vector space $V$ and let $M$ be the
formal neighborhood\index{terms}{formal neighborhood} of $0 \in
V$. Let $\B^\cdot$ be the Weil algebra\index{terms}{Weil algebra} of
the manifold $M$, as in \ref{formal.Weil}. For every $n \geq 0$ the
vector space $\B^n$ is equipped with an $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} of weight\index{terms}{weight!of a Hodge structure} $n$, so that
we can consider the localization $\loc{\B^n}$. The sheaf
$\oplus\loc{\B^\cdot}$ is a commutative algebra in the tensor
category $\Shv^{U(1)}(\SB)$. We will call {\em the localized Weil
algebra\index{terms}{Weil algebra!localized}}.

The augmentation grading\index{terms}{grading!augmentation} on
$\B^\cdot$ defined in \ref{aug} is compatible with the $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure}.  Therefore it defines
an augmentation grading on the localized Weil
algebra\index{terms}{Weil algebra!localized} $\loc{\B^\cdot}$. The
finer augmentation bigrading\index{terms}{bigrading!augmentation} on
$\B^\cdot$ does not define a bigrading on $\loc{\B^\cdot}$. However,
it does define a bigrading on the complexified algebra
$\loc{\B^\cdot}
\otimes \C$ of $\C^*$-equivariant sheaves on the manifold $\SB
\otimes \C \cong \cp$.

\punkt Assume now given a flat extended\index{terms}{extended
connection!flat} connection $D:\B^\cdot \to
\B^{\cdot+1}$ on $M$. Since the derivation $D$ is weakly
Hodge\index{terms}{weakly Hodge map}, it corresponds to a derivation
$D:\loc{\B^\cdot} \to \loc{\B^{\cdot+1}}$ of the localized Weil
algebra\index{terms}{Weil algebra!localized} $\loc{\B^\cdot}$.  It
is easy to check that the complex $\langle \loc{\B^\cdot}, D
\rangle$ is acyclic in all degrees but $0$. Denote by $\HH^0$ the
$0$-th cohomology sheaf $H^0(\loc{\B^\cdot})$. The sheaf $\HH^0$
carries a canonical algebra structure. Moreover, while the
derivation $D$ does not preserve the augmentation
grading\index{terms}{grading!augmentation} on
$\loc{\B^\cdot}$, it preserves the decreasing {\em augmentation
filtration} $\left(\loc{\B^\cdot}\right)_{\geq \cdot}$.  Therefore
we have a canonical decreasing filtration on the algebra $\HH^0$,
which we also call the augmentation filtration.

\punkt It turns out that the associated graded quotient algebra $\gr
\HH^0$ with respect to the augmentation filtration does not depend
on the extended\index{terms}{extended connection} connection $D$. To
describe it, introduce the $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure} $W$ of
weight\index{terms}{weight!of a Hodge structure} $-1$ by setting $W
= V$ as a real vector space and
\begin{align}\label{W.dfn}
\begin{split}
W^{-1,0} &= V \subset V \otimes_\R \C,\\ 
W^{0,-1} &= \overline{V} \subset V \otimes_\R \C.
\end{split}
\end{align} 
The $k$-th graded piece $\gr_k \HH^0$ with respect to the
augmentation filtration is then isomorphic to the symmetric power
$S^k(\loc{W})$ of the localization $\loc{W}$ of the $\R$-Hodge
structure $W$. In particular, it is a sheaf of
weight\index{terms}{weight!of a sheaf on $\protect\C P^1$} $-n$, so
that up to a change of numbering 
the augmentation filtration on $\HH^0$ satisfies the condition
\eqref{hg}. The extension data between these graded pieces depend on
the extended\index{terms}{extended connection} connection $D$. The
whole associated graded algebra $\gr \HH^0$ is isomorphic to the
completed symmetric algebra $\widehat{S}^\cdot(\loc{W})$.

\punkt Using standard deformation theory, one can show that the
algebra map $\HH^0 \to \loc{\B^0}$ is the universal map from the
algebra $\HH^0$ to a complete commutative pro-algebra in the tensor
category of $U(1)$-equivariant flat coherent sheaves of weight
\index{terms}{weight!of a sheaf on $\protect\C P^1$} $0$ on $\SB$. Moreover,
the localized Weil algebra\index{terms}{Weil algebra!localized}
$\loc{\B^\cdot}$ coincides with the relative de Rham
complex\index{terms}{de Rham complex!relative} of $\loc{\B^0}$ over
$\HH^0$. Therefore one can recover, up to an isomorphism, the whole
algebra $\loc{\B^\cdot}$ and, consequently, the
extended\index{terms}{extended connection} connection $D$, solely
from the algebra $\HH^0$ in $\Shv^{U(1)}(\SB)$. Together with
Lemma~\ref{DS}~\thetag{ii} this gives the following, due also to
Deligne\index{names}{Deligne, P.} (in a different form).

\begin{prop}\label{DS.prop}
The correspondence $\langle \B^\cdot, D \rangle \mapsto \HH^0$ is a
bijection between the set of all isomorphism classes of flat
extended\index{terms}{extended connection!flat} connections on $M$
and the set of all algebras $\HH^0$ in the tensor category of mixed
$\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge structure!mixed}
equipped with an isomorphism $\gr^W_{-1}\HH^0 \cong W$ between the
$-1$-th associated graded piece of the weight
filtration\index{terms}{weight filtration} on $\HH^0$ and the pure
$\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure} $W$ defined
in
\eqref{W.dfn} which induces for every $n \geq 0$ an isomorphism
$\gr^W_{-n}\HH^0 \cong S^nW$.
\end{prop}

\rem The scheme $\Spec\HH^0$ over $\SB$ coincides with the so-called
twistor space\index{terms}{twistor space} of the manifold $\tm$ with
the hypercomplex structure\index{terms}{hypercomplex structure}
given by the extended\index{terms}{extended connection} connection
$D$ (see \cite{HKLR} for the
definition). Deligne\index{names}{Deligne, P.}'s and
Simpson\index{names}{Simpson, C.}'s (\cite{De2}, \cite{Simpson})
approach differs from ours in that they use the language of twistor
spaces to describe the relation between $U(1)$-equivariant
hypercomplex manifolds\index{terms}{hypercomplex manifold} and mixed
$\R$-Hodge structures\index{terms}{Hodge structure@$\R$-Hodge
structure!mixed}. Since this requires some additional machinery, we have
avoided introducing twistor spaces in this paper.

\punkt We will now try to use the localization construction to
eludicate some of the complicated linear algebra used in
Section~\ref{main.section} to prove our main theorem. As we have
already noted, the category $\prehodge$ of pure $\R$-Hodge
structures\index{terms}{Hodge structure@$\R$-Hodge structure} with weakly Hodge
maps\index{terms}{weakly Hodge map} as morphisms is identified by
localization with the category $\Shv^{U(1)}(\SB)$ of
$U(1)$-equivariant flat coherent sheaves on $\SB$. Moreover, the
functor $\Gamma:\prehodge_{\geq 0} \to \prehodge_0$ introduced in
Lemma~\ref{g.ex} is simply the functor of global sections
$\Gamma(\SB,\cdot)$.

\punkt Consider the localized Weil algebra\index{terms}{Weil
algebra!localized} $\loc{\B^\cdot}$ with the derivation
$C:\loc{\B^\cdot} \to \loc{\B^{\cdot+1}}$ associated to the canonical weakly
Hodge\index{terms}{weakly Hodge map} derivation introduced in
\ref{C.and.sigma}. The differential graded algebra $\langle
\loc{\B^\cdot},C \rangle$ is canonically an algebra over the
completed symmetric algebra $\widehat{S}^\cdot(V)$ generated by the
constant sheaf on $\SB$ with the fiber $V$. Moreover, it is a free
commutative algebra generated by the complex
\begin{equation}\label{cmplx}
V \longrightarrow V(1)
\end{equation}
placed in degrees $0$ and $1$, where $V(1)$ is the
$U(1)$-equivariant sheaf of weight\index{terms}{weight!of a sheaf on
$\protect\C P^1$} $1$ on $\SB$ corresponding to the 
$\R$-Hodge structure\index{terms}{Hodge structure@$\R$-Hodge structure} given by
$V(1)^{1,0} = V$ and $V(1)^{0,1} = \overline{V}$.

\punkt The homology sheaves of the complex \eqref{cmplx} are
non-trivial only in degree $1$. This non-trivial homology sheaf is a
skyscraper sheaf concentrated in the point $\widehat{I}(\Spec\C)
\subset \SB$ with fiber $V$. The associated sheaf on the
complexification $\SB \otimes \C \cong \cp$ splits into the sum of
skyscraper sheaf with fiber $V$ concentrated at $0 \in \cp$ and the
skyscraper sheaf with fiber $\overline{V}$ concentrated at $\infty
\in \cp$. This splitting corresponds to the splitting of the complex
\eqref{cmplx} itself into the components of augmentation
bidegrees $(1,0)$ and $(0,1)$. 

\punkt Let now $\I^\cdot = \B^\cdot_{\geq 0,\geq 0}$ be the sum of
the components in the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot$ of augmentation
bidegree\index{terms}{bidegree!augmentation|)} greater or equal than
$(1,1)$. The subspace $\I^\cdot 
\subset \B^\cdot$ is compatible with the $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure}. The
crucial point in the proof of Theorem~\ref{kal=ext} is
Proposition~\ref{ac}, which claim the acyclycity of the complex
$\langle \Gamma(\I^\cdot), C
\rangle$. It is this fact that becomes almost obvious from the point
of view of the localization construction. To show it, we first prove
the following.

\begin{lemma}
The complex 
$$
\left\langle\loc{\I^\cdot},C\right\rangle
$$ 
of $U(1)$-equivariant sheaves on $\SB$ is acyclic.
\end{lemma}

\proof It suffices to prove that the complex $\loc{I^\cdot} \otimes
\C$ of sheaves on $\SB \otimes \C \cong \cp$ is acyclic. To prove
it, let $p,q \geq 1$ be arbitrary integer, and consider the
component $\left(\loc{B^\cdot}\right)_{p,q}$ of augmentation
bidegree\index{terms}{bidegree!augmentation} $(p,q)$ in the
localized Weil algebra\index{terms}{Weil algebra!localized}
$\loc{\B^\cdot}$. By definition we have
$$
\left(\loc{B^\cdot}\right)_{p,q} = \left(\loc{B^\cdot}\right)_{p,0}
\otimes \left(\loc{B^\cdot}\right)_{0,q} 
$$
Since the complex $\left(\loc{B^\cdot}\right)_{p,0} =
S^p\left(\loc{B^\cdot}\right)_{1,0}$ has homology concentrated at $0
\in \cp$, while the complex $\left(\loc{B^\cdot}\right)_{0,q} =
S^q\left(\loc{B^\cdot}\right)_{0,1}$ has homology concentrated at
$\infty \in \cp$, their product is indeed acyclic.
\endproof 

Now, we have $\Gamma(\I^\cdot) = \Gamma(\SB,\loc{\I^\cdot})$, and
the functor $\Gamma(\SB,\cdot)$ is exact on the full subcategory in
$\Shv^{U(1)}(\SB)$ consisting of sheaves of positive weight
\index{terms}{weight!of a sheaf on $\protect\C P^1$}. Therefore the
complex $\langle \Gamma(\I^\cdot),C \rangle$ 
is also acyclic, which gives an alternative proof of
Proposition~\ref{ac}.

\punkt We would like to finish the paper with the following
observation. Proposition~\ref{DS.prop} can be extended to the
following claim.

\begin{prop}
Let $M$ be a complex manifold.  There exists a natural bijection
between the set of isomorphism classes of germs of Hodge
manifold\index{terms}{Hodge manifold} structures on $\tm$ in the
neighborhood of the zero section $M \subset \tm$ and the set of
multiplicative filtrations $F^\cdot$ on the sheaf $\calo_\R(M)
\otimes \C$ of $\C$-valued real-analytic functions on $M$ satisfying
the following condition:
\begin{itemize}
\item For every point $m \in M$ let $\widehat{\calo}_m$ be the
formal completion\index{terms}{formal completion} of the local ring
$\calo_m$ of germs of real-analytic functions on $M$ in a
neighborhood of $m$ with respect to the maximal ideal. Consider the
filtration $F^\cdot$ on $\widehat{\calo}_m \otimes \C$ induced by
the filtration $f^\cdot$ on the sheaf $\calo_\R(M) \otimes \C$, and
for every $k \geq 0$ let $W_{-k}\widehat{\calo}_m \subset
\widehat{\calo}_m$ be the $k$-th power of the maximal ideal in
$\widehat{\calo}_m$. Then the triple $\langle \widehat{\calo}_m,
F^\cdot, W^\cdot \rangle$ is a mixed $\R$-Hodge
structure\index{terms}{Hodge structure@$\R$-Hodge structure!mixed}. (In particular,
$F^k = 0$ for $k > 0$.)
\end{itemize}
\end{prop}

If the Hodge manifold\index{terms}{Hodge manifold} structure on
$\tm$ is such that the projection $\rho:\tm_I \to M$ is holomorphic
for the preferred\index{terms}{preferred complex structure} complex
structure $\tm_I$ on $\tm$, then it is 
easy to see that the first non-trivial piece $F^0\calo_\R(M) \otimes
\C$ of the filtration $F^\cdot$ on the sheaf $\calo_\R(M) \otimes
\C$ coincides with the subsheaf $\calo(M) \subset \calo_\R(M)
\otimes \C$ of holomorphic functions on $M$. Moreover, since the
filtration $F^\cdot$ is multiplicative, it is completely defined by
the subsheaf $F^{-1}\calo_\R(M) \otimes \C \subset \calo_\R(M)
\otimes \C$. It would be very interesting to find an explicit
description of this subsheaf in terms of the K\"ah\-le\-ri\-an
connection\index{terms}{connections!K\"ahlerian} $\nabla$ on $M$
which corresponds to the Hodge manifold\index{terms}{Hodge manifold}
structure on $\tm$. 

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\medskip

\noindent
{\sc Independent University of Moscow\\
B. Vlassievsky, 11\\
Moscow, Russia, 121002\\
{\em E-mail:}
kaledin@@balthi.dnttm.rssi.ru}
