\index{terms}{hyperk\"ahler manifold}
\index{terms}{Levi-Civita connection}.  

\index{terms}{hyperk\"ahler quaternion
\index{terms}{quaternion algebra} 

\index{terms}{Levi-Civita connection}. 
\index{terms}{hyperk\"ahler manifold} 

\index{terms}{hyperk\"ahler manifold} $M$ is

\index{terms}{Hodge type} $(2,0)$

\index{terms}{hyperk\"ahler structure} on $M$, and if so,

\index{terms}{K\"ahler metric} $h$ on $M$. Indeed,

\index{terms}{hyperk\"ahler metric}. This

\index{people}{Yau} 

\index{terms}{Calabi-Yau

\index{terms}{hyperk\"ahler manifold} are essentially the same.

\index{terms}{formal neighborhood} of the origin $0 \in
Theorem\index{terms}{Darboux-Weinstein Theorem}. On the other hand,
hyperk\"ahler structures\index{terms}{hyperk\"ahler structure} on
this formal neighborhood\index{terms}{formal neighborhood} form an
metrics\index{terms}{hyperk\"ahler metric} defined on the
hyperk\"ahler manifolds\index{terms}{hyperk\"ahler manifold} given by
structure\index{terms}{hyperk\"ahler structure}. The goal of this
metric\index{terms}{K\"ahler metric}. The metric on $M$ extends to a
hyperk\"ahler metric\index{terms}{hyperk\"ahler metric} $h$ defined
in the formal neighborhood\index{terms}{formal neighborhood} of the
metric\index{terms}{formal hyperk\"ahler metric} $h$ converges to a
real-analytic metric\index{terms}{real-analytic metric} in an open
metrics\index{terms}{hyperk\"ahler metric} obtained by
structure\index{terms}{hyperk\"ahler structure} on a smooth manifold
manifold\index{terms}{hyperk\"ahler manifold}. On the other hand, it
metrics''. The K\"ahler metric\index{terms}{K\"ahler metric} on $M$
connection\index{terms}{holomorphic connection} $\nabla$ on the
call such connections\index{terms}{K\"ahlerian connection} {\em
hyperk\"ahler manifold\index{terms}{hyperk\"ahler manifold}
manifold\index{terms}{hypercomplex manifold} (see, e.g.,
condition\index{terms}{Hitchin's compatibility condition} on the
set of all K\"ah\-le\-ri\-an connections\index{terms}{K\"ahlerian
structures\index{terms}{hypercomplex structure} on the formal
neighborhood\index{terms}{formal neighborhood} of the zero section
structure\index{terms}{hypercomplex structure} is defined in an open
manifolds\index{terms}{$U(1)$-equivariant hyperk\"ahler manifold} and
the theory of $\R$-Hodge structures\index{terms}{$\R$-Hodge
manifolds\index{terms}{Hodge manifold} for the hypercomplex
manifolds\index{terms}{hypercomplex manifold} equipped with a
manifolds\index{terms}{hyperk\"ahler manifold}
so-called hyperk\"ahler reduction\index{terms}{hyperk\"ahler
manifolds\index{terms}{Hodge manifold}, as
algebra about quaternionic\index{terms}{quaternionic vector space}
vector spaces and $\R$-Hodge structures\index{terms}{$\R$-Hodge
the notion of {\em weakly Hodge map\index{terms}{weakly Hodge map}},
{\em Hodge bundle\index{terms}{Hodge bundle}} on a smooth manifold
almost quaternionic manifolds\index{terms}{quaternionic manifold}
structure\index{terms}{quaternionic structure} and a $U(1)$-action
an almost quaternionic structure\index{terms}{quaternionic
hypercomplex manifold\index{terms}{hypercomplex manifold} and
of {\em Hodge manifolds\index{terms}{Hodge manifold}}. We then
language of Hodge bundles\index{terms}{Hodge bundle}, to be used
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}{polarized Hodge
manifold\index{terms}{$U(1)$-equivariant hyperk\"ahler} equipped with
sense of Hitchin\index{people}{Hitchin}\index{terms}{Hitchin's
with arbitrary Hodge manifolds\index{terms}{Hodge manifold} and
$U(1)$-fixed points\item{terms}{regular fixed point} every such
with weight\index{terms}{weight of an $U(1)$-action}s $0$ and
linearization\index{terms}{linearization}} of the regular Hodge
manifold\index{terms}{regular Hodge manifold}.
Theorem\index{terms}{Darboux-Weinstein Theorem} in the symplectic
space\index{terms}Hitchin-Simpson moduli space of flat connections}
linearization\index{terms}{linearization} construction provides a
metric\index{terms}{hyperk\"ahler metric} on $\M^{reg}$ provided by
$M$. In Section 5 we describe Hodge manifolds\index{terms}{Hodge
Hodge connection\index{terms}{Hodge connection}}.
neighborhood\index{terms}{formal neighborhood} of the zero section
formal Hodge connections\index{terms}{formal Hodge connection} on
connections\item{terms}{extended connection}}. We also
algebra\index{terms}{Weil algebra}} of the manifold $M$. Extended
connections\item{terms}{extended connection} on the manifold $M$ and
manifolds\index{terms}{regular Hodge manifold}, we need to derive
some linear-algebraic facts on the Weil algebra\index{terms}{Weil
complex\index{terms}{de Rham complex} of a smooth complex manifold,
which we call {\em the total de Rham complex}\index{terms}{total de
define the so-called {\em total Weil algebra\index{terms}{total Weil
extended\item{terms}{extended connection} connections on $M$ are in
connections\index{terms}{K\"ahlerian connection} on the cotangent
polarizations\index{terms}{polarization}. After some preliminaries,
map\index{terms}{weakly Hodge map}. This approach also allows us to
\subsection{Quaternionic vector spaces}\index{terms}{quaternionic
\defn A {\em quaternionic vector space\index{terms}{quaternionic
over the algebra\index{terms}{quaternion algebra} $\h$. 
complex structure $V_I$ {\em the preferred\index{terms}{preferred complex structure} complex structure} on
space\index{terms}{equivariant quaternionic vector space} is a quaternionic
weight\index{terms}{weight of an $U(1)$-action} $i$} if $W = W^i$, that is, if an element $z \in \C^*$ acts
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
space\index{terms}{equivariant quaternionic vector space}. The action map 
quaternionic vector spaces\index{terms}{equivariant quaternionic
vector spaces which we will call regular\item{terms}{regular
\defn An equivariant quaternionic vector space\index{terms}{regular
of weight\index{terms}{weight of an $U(1)$-action} $1$. 
\proof Let $V^1_I \subset V_I$ be the weight\index{terms}{weight of
the $ gm$-action.  Since $V$ is regular\item{terms}{regular
space\index{terms}{regular equivariant quaternionic vector space} is a sum of
structure\index{terms}{$\R$-Hodge structure} has Hodge numbers
structure}\label{complementary}\index{terms}{complementary complex
$J$ are {\em complementary\index{terms}{complementary complex structure}} if
space\index{terms}{equivariant quaternionic vector space}.  The
embeddings complementary\index{terms}{complementary complex
complementary\index{terms}{complementary complex structure} 
complementary\index{terms}{complementary complex structure} to $I$
and speak of {\em the complementary\index{terms}{complementary
complementary\index{terms}{complementary complex structure} embedding $J:\C \to \h$ induces an isomorphism
isomorphism} between the complementary\index{terms}{complementary
complex structure} and the preferred\index{terms}{preferred complex
vector\index{terms}{regular equivariant quaternionic vector space}
\subsection{$\protect\R$-Hodge structures}\index{terms}{$\R$-Hodge structure} 
\punkt Recall that {\em a pure $\R$-Hodge structure $W$ of weight\index{terms}{weight of a Hodge structure}
$W^{p,q}$ is called {\em the Hodge type bigrading\index{terms}{Hodge
structure $\R(k)$ of weight\index{terms}{weight of a Hodge structure} $-2k$ is by definition the
bigrading\index{terms}{Hodge type bigrading}
For a pure $\R$-Hodge structure\index{terms}{$\R$-Hodge structure}
common $\R$-Hodge structure\index{terms}{$\R$-Hodge structure},
canonical $\R$-Hodge structure of weight\index{terms}{weight of a Hodge structure} $1$ with Hodge
bigrading\index{terms}{Hodge type bigrading}
structure\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $1$ with
structure\index{terms}{$\R$-Hodge structure}, and denote by $W_\R
$\conj$. Define the {\em Weil operator\index{terms}{Weil operator}}
structures of weight\index{terms}{weight of a Hodge structure} $0$ the Weil operator\index{terms}{Weil
structure\index{terms}{$\R$-Hodge structure} $W$ of weight\index{terms}{weight of a Hodge structure} $i$ let
spaces\index{terms}{regular equivariant quaternionic vector space}
category of pure $\R$-Hodge structures of weight\index{terms}{weight
structure\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $1$
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 $V_J$
is then a map of $\R$-Hodge structures\index{terms}{$\R$-Hodge
\subsection{Weakly Hodge maps}\label{w.H.sub}\index{terms}{weakly
structures\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $i$ is
Hodge type bigrading\index{terms}{Hodge type bigrading} and the Hodge
structures\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure}s
required to preserve the weight\index{terms}{weight of a Hodge structure}s, so that $\Hom(V,W)=0$ unless
Hodge\index{terms}{weakly Hodge map}} if it preserves the Hodge
type\index{terms}{Hodge type} $(p,m-n-p)$. Note that this condition
structures\index{terms}{$\R$-Hodge structure} of arbitrary weight\index{terms}{weight of a Hodge structure}
with weakly Hodge maps\index{terms}{weakly Hodge map} as
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
weight\index{terms}{weight of a Hodge structure} $n$.
structure\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $1$ with
\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
spaces. Moreover, for every pure $\R$-Hodge structure $V$ of weight\index{terms}{weight of a Hodge structure}
$\W_k$ is a pure $\R$-Hodge structure of weight\index{terms}{weight of a Hodge structure} $k$, with Hodge
\W_1$. For every pure $\R$-Hodge structure $V$ of weight\index{terms}{weight of a Hodge structure} $n$ the map
map\index{terms}{universal weakly Hodge map} from a pure $\R$-Hodge
structures\index{terms}{$\R$-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
type\index{terms}{Hodge type} decomposition
therefore the dual $\R$-Hodge structures\index{terms}{$\R$-Hodge
\W_{n+k}^*$. For every pure $\R$-Hodge structure $V$ of weight\index{terms}{weight of a Hodge structure} $k
\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}{$\R$-Hodge structure}
$V_{n+k}$ of weight\index{terms}{weight of a Hodge structure} $n+k$. By the universal property the map $f$
Hodge\index{terms}{weakly Hodge map}, but it decomposes $\gamma =
bigrading\index{terms}{Hodge type bigrading}. Denote $\gamma_l =
\punkt The map $\gamma_l$ if of Hodge type\index{terms}{Hodge type}
structures\index{terms}{$\R$-Hodge structure} $V_1$, $V_2$ of
non-negative weight\index{terms}{weight of a Hodge structure} a surjective map
Consider a quaternionic vector space $V$\index{terms}{quaternionic
metric\index{terms}{Euclidean metric}
\index{terms}{Quaternionic-Hermitian metric} if it is invariant
$V$ is equivariant\index{terms}{equivariant quaternionic vector
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric}} if it is
Let $V_I$ be the vector space $V$ with the preferred\index{terms}{preferred complex structure} complex structure $I$,
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric} if and only if
metric\index{terms}{Hermitian metric} on $V_I$, 
\punkt Recall that a {\em polarization\index{terms}{polarization}}
$S$ on a pure $\R$-Hodge structure\index{terms}{$\R$-Hodge
structure} $W$ of weight\index{terms}{weight of a Hodge structure}
operator\index{terms}{Weil operator}.)
space\index{terms}{equivariant quaternionic vector space} equipped
with an Euclidean metric\index{terms}{Euclidean metric} $h$, and let
structure\index{terms}{$\R$-Hodge structure} of
weight\index{terms}{weight of a Hodge structure} $1$ 
metric\index{terms}{Hermitian metric} $h_J$ on $V_J$, and let $S:W
The form $S$ is a polarization\index{terms}{polarization} if and
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric}. This gives a
\punkt Let $W^*$ be the Hodge structure\index{terms}{$\R$-Hodge
structure} of weight\index{terms}{weight of a Hodge structure} $-1$
dual to $W$.  The sets of polarizations\index{terms}{polarization}
set of Hermitian-Hodge metrics\index{terms}{Hermitian-Hodge metric}
vector\index{terms}{equivariant quaternionic vector space}
polarization\index{terms}{polarization}. Extend $h$ to an Hermitian
metric\index{terms}{Hermitian metric} $h_I$ on the complex vector
space $V$ with the preferred\index{terms}{preferred complex
metric\index{terms}{Hermitian metric} on the
manifolds}\label{hbqm.section}\index{terms}{Hodge
bundle}\index{terms}{quaternionic manifold}
hyperk\"ahler structures\index{terms}{hyperk\"ahler structure} on
quaternionic vector spaces\index{terms}{equivariant quaternionic
vector space} and pure $\R$-Hodge structures\index{terms}{$\R$-Hodge
structure} of weight\index{terms}{weight of a Hodge structure} $1$,
under the name of a Hodge bundle\index{terms}{Hodge bundle} (see
terminology), and let $\iota\index{terms}{involution $\iota$}:M \to M$ be the action of the element
\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
$U(1)$-equivariant bundle map $\conj:\overline{\iota\index{terms}{involution $\iota$}^*\E}(k) \to \E$
satisfying $\conj \circ \iota\index{terms}{involution $\iota$}^*\conj = \id$.
Hodge bundles\index{terms}{Hodge bundle} of weight\index{terms}{weight of a Hodge bundle} $k$ over $M$ form
bundles\index{terms}{Hodge bundle} on $M$ of weight\index{terms}{weight of a Hodge bundle}s $m$ and
operator $f:\E \to \F$ is {\em weakly Hodge\index{terms}{weakly
\item $f = \overline{\iota\index{terms}{involution $\iota$}^*f}$, and 
have $f_k = \overline{\iota\index{terms}{involution $\iota$}^* f_{n-m-k}}$.)
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
\punkt\label{H-type} For a weakly Hodge map\index{terms}{weakly
{\em the $H$-type\index{terms}{$H$-type} decomposition}. 
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
structure\index{terms}{$\R$-Hodge structure} introduced in
then Hodge bundles\index{terms}{Hodge bundle} of weight\index{terms}{weight of a Hodge bundle} $i$ are the
bigrading\index{terms}{Hodge type bigrading} $\E = \oplus_{p+q=i}
$\R$-Hodge structures\index{terms}{$\R$-Hodge structure}. Weakly
Hodge bundle map\index{terms}{weakly Hodge map} are then the same as
\punkt The categories of Hodge bundles\index{terms}{Hodge bundle}
structure\index{terms}{$\R$-Hodge structure} $V$ of weight\index{terms}{weight of a Hodge bundle} $i$
of weight\index{terms}{weight of a Hodge bundle} $0$.
bundles\index{terms}{Hodge bundle} and weakly Hodge
maps\index{terms}{weakly Hodge map}, consider a $U(1)$-manifold $M$
usual Hodge type\index{terms}{Hodge type} decomposition of the
\conj:\Lambda^{p,q}(M_I) \to \iota\index{terms}{involution $\iota$}^*\overline{\Lambda^{q,p}(M_I)} 
and complex conjugation is a Hodge bundle of weight\index{terms}{weight of a Hodge bundle} $i$ on $M$. The
$H$-type\index{terms}{$H$-type} decomposition for $d_M$ is in this
case the usual Hodge type\index{terms}{Hodge type} decomposition $d
$\iota\index{terms}{involution $\iota$}^*:C^\infty(M,\C) \to C^\infty(M,\C)$. The map $\nu$ is an
a pure $\R$-Hodge structure\index{terms}{$\R$-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
manifolds}\index{terms}{equivariant quaternionic manifold} 
quaternions\index{terms}{quaternion algebra}. 
quaternionic\index{terms}{quaternionic manifold}} if it
Let $M$ be a quaternionic manifold\index{terms}{quaternionic
structure\index{terms}{quaternionic structure} and the $U(1)$-action on $M$
Equivalently, the quaternionic structure\index{terms}{quaternionic
\punkt \defn A quaternionic manifold\index{terms}{quaternionic
manifold\index{terms}{equivariant quaternionic manifold}}. 
$1$-dimensional representation of weight\index{terms}{weight of an $U(1)$-action} $k$.  Lemma~\ref{explicit.eqvs}
The category of quaternionic manifolds\index{terms}{quaternionic
bundles}\index{terms}{quaternionic manifold}\index{terms}{Hodge bundle}
bundle\index{terms}{Hodge bundle} on $M$. Hodge bundles arise
structures\index{terms}{quaternionic structure} on $M$ in the
bundle\index{terms}{quaternionic bundle}} on $M$ as a real
bundles\index{terms}{Hodge bundle} of weight\index{terms}{weight of a Hodge bundle} $1$ on $M$. 
The weight\index{terms}{weight of a Hodge bundle} $1$ Hodge bundle\index{terms}{Hodge bundle} structure on
structure\index{terms}{quaternionic structure} on $\h
preferred\index{terms}{preferred complex structure} almost complex structure on $M$.  Since $M_I$ is preserved
The quaternionic structure\index{terms}{quaternionic structure} on
weight\index{terms}{weight of a Hodge bundle}-$1$ Hodge
bundle\index{terms}{Hodge bundle} structure on
is given by $\conj = \sqrt{-1} \left( \iota\index{terms}{involution
structure\index{terms}{quaternionic structure}, as in
Let $M_J$ be the complementary\index{terms}{complementary complex structure} almost complex structure on the
equi\-va\-ri\-ant quaternionic manifold\index{terms}{equivariant
\ref{complementary\index{terms}{complementary complex structure}} we have defined for every equivariant
quaternionic vector space\index{terms}{equivariant quaternionic
bundle\index{terms}{Hodge bundle} structures, so is the projection
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
differential\index{terms}{Dolbeault differential}s. They are,
however, related by means of the Hodge bundle\index{terms}{Hodge
The Dolbeult differential\index{terms}{Dolbeault differential}
$H$-type\index{terms}{$H$-type} decomposition $D = D_0 +
differential\index{terms}{Dolbeault differential} for the almost
weakly Hodge\index{terms}{weakly Hodge map}, so is $D$. The rest
quaternionic structures\index{terms}{quaternionic structures} on
differential\index{terms}{Dolbeault differential}s. To do this,
Let $\E$ be a weight\index{terms}{weight of a Hodge bundle} $1$ Hodge bundle\index{terms}{Hodge bundle} on
structure\index{terms}{quaternionic structure} on $M$ such that
differential\index{terms}{Dolbeault differential} 
for the complementary\index{terms}{complementary complex structure} almost complex structures $M_J$ on $M$.
structure $M_I$ complementary\index{terms}{complementary complex structure} to $M_J$, consider the
$H$-type\index{terms}{$H$-type} decomposition $D = D_0 +
weakly Hodge\index{terms}{weakly Hodge map},
differential\index{terms}{Dolbeault differential} for an almost
the weight\index{terms}{weight of a Hodge bundle} $1$ Hodge
bundle\index{terms}{Hodge bundle} structure on 
bundle\index{terms}{quaternionic bundle} structure on $\E$
manifold\index{terms}{equivariant quaternionic manifold} on
the preferred\index{terms}{preferred complex structure} almost complex structure.
differential\index{terms}{Dolbeault differential}  
$\bar\6_J$ for the complementary\index{terms}{complementary complex structure} almost complex structure on $M$ indeed equals 
equivariant quaternionic structures\index{terms}{quaternionic
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 derivation
\section{Hodge manifolds}\index{terms}{Hodge manifold}
manifolds\index{terms}{quaternionic manifold}
\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
\punkt When a quaternionic manifold\index{terms}{quaternionic
$M$, namely, the preferred\index{terms}{preferred complex structure} and the complementary\index{terms}{complementary complex structure} one.
\defn An equivariant quaternionic manifold\index{terms}{equivariant
Hodge manifold\index{terms}{Hodge manifold}} if both the preferred\index{terms}{preferred complex structure}
Hodge manifolds\index{terms}{Hodge manifold} are the main object of
description of Hodge manifolds\index{terms}{Hodge manifold} based on
quaternionic manifold\index{terms}{equivariant quaternionic
manifold} $M$, 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}
weight\index{terms}{weight of a Hodge bundle} $i$ Hodge bundle
The equivariant quaternionic manifold\index{terms}{equivariant
Hodge\index{terms}{Hodge manifold} if and only if
is weakly Hodge\index{terms}{weakly Hodge map} for every $i \geq 0$. 
hold. Condition \thetag{i} means that the complementary\index{terms}{complementary complex structure} almost complex
Hodge\index{terms}{weakly Hodge map}.
$H$-type\index{terms}{$H$-type} decomposition. The map $D_0$ is an
preferred\index{terms}{preferred complex structure} almost complex structure $M_I$ on $M$. (Or, more
differential\index{terms}{Dolbeault differential} admits at most one
$H$-type\index{terms}{$H$-type} decomposition of the map $D \circ
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}. 
from the de Rham complex\index{terms}{de Rham complex} of the
Hodge\index{terms}{weakly Hodge map}, moreover, it commutes with the
The category of Hodge manifolds\index{terms}{Hodge manifold} is
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
manifold}\index{terms}{Hodge manifold}\index{terms}{de Rham complex}
some detail the de Rham complex\index{terms}{de Rham complex}
complementary\index{terms}{complementary complex structure} complex structure $M_J$ on $M$. By
$\Lambda^{0,i}(M_J)$ is a Hodge bundle\index{terms}{Hodge bundle} of
weight\index{terms}{weight of a Hodge bundle} $i$ on $M$, and the Dolbeult
differential\index{terms}{Dolbeault differential}
Hodge\index{terms}{weakly Hodge map}. Therefore $D$ admits an
$H$-type\index{terms}{$H$-type} decomposition $D = D_0 + \overline{D_0}$.
\punkt Consider the de Rham complex\index{terms}{de Rham complex}
type\index{terms}{Hodge type} decomposition
for the complementary\index{terms}{complementary complex structure} complex structure $M_J$ on $M$, and let
type\index{terms}{Hodge type} decomposition for the complementary\index{terms}{complementary complex structure}
differential\index{terms}{Dolbeault differential}, in turn, equals
Dolbeult differential\index{terms}{Dolbeault differential} for the preferred\index{terms}{preferred complex structure} complex structure $M_I$ on $M$. As
of the de Rham complex\index{terms}{de Rham complex}
$\Lambda^\cdot(M,\C)$. The Hodge type\index{terms}{Hodge type}
decomposition for the preferred\index{terms}{preferred complex structure} complex structure $M_I$ then shows
\Lambda^{1,0}(M_J)$ be the operator corresponding to the preferred\index{terms}{preferred complex structure}
rule\index{terms}{Leibnitz rule}. We finish this subsection with the
manifolds}\label{polarization}\index{terms}{Hodge manifold}
\punkt Let $M$ be a quaternionic manifold\index{terms}{equivariant
metric\index{terms}{Riemannian metric} $h$ on $M$
Qua\-ter\-ni\-onic\--Her\-mi\-ti\-an\index{terms}{Quaternionic-Hermitian
\defn A {\em hyperk\"ahler manifold\index{terms}{hyperk\"ahler
manifold}} is a hypercomplex manifold\index{terms}{hypercomplex
metric\index{terms}{Quaternionic-Hermitian metric} $h$ which is
\punkt Let $M$ be a Hodge manifold\index{terms}{Hodge manifold}
equipped with a Riemannian metric\index{terms}{Riemannian metric}
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric}} if it is
Quaternionic-Hermitian\index{terms}{Quaternionic-Hermitian metric}
Hermitian-Hodge metric\index{terms}{Hermitian-Hodge metric} $h$ if
Quaternionic-Hermitian\index{terms}{Quaternionic-Hermitian metric},
but also hyperk\"ahler\index{terms}{hyperk\"ahler metric}.
manifold\index{terms}{Hodge manifold}. By
$\Lambda^{1,0}(M_J)$ 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}
Hodge bundle of weight\index{terms}{weight of a Hodge bundle} $-1$.  By \ref{pol} the set of all
Hermitian-Hodge metrics\index{terms}{Hermitian-Hodge metric} $h$ on
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
is a map of weight\index{terms}{weight of a Hodge bundle} $2$ Hodge bundles\index{terms}{Hodge bundle}, and
\Omega(\chi,\overline{\iota\index{terms}{involution $\iota$}^*(\chi)}) > 0.
\punkt Assume that the Hodge manifold\index{terms}{Hodge manifold}
metric\index{terms}{Hermitian-Hodge metric} $h$. Let $\Omega \in
polarization\index{terms}{polarization} ,
Hermitian metric\index{terms}{Hermitian metric} $h$.  Either one of
The Hermitian-Hodge metric\index{terms}{Hermitian-Hodge metric} $h$
where $D$ is the Dolbeult differential\index{terms}{Dolbeault
differential} for complementary\index{terms}{complementary complex structure} complex structure $M_J$.
for the metric\index{terms}{K\"ahler metric} $h$ and complex
hyperk\"ahler\index{terms}{hyperk\"ahler metric}, hence polarizes
$H$-type\index{terms}{$H$-type} decomposition and let
and the metric $h$ is hyperk\"ahler\index{terms}{hyperk\"ahler
type\index{terms}{Hodge type} $(2,0)$ with respect to the
complementary\index{terms}{complementary complex structure} complex structure $M_J$, \eqref{K} is equivalent to
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric}, $\Omega$ is of
$H$-type\index{terms}{$H$-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
manifolds\index{terms}{hyperk\"ahler manifold}
describe general hyperk\"ahler metrics\index{terms}{hyperk\"ahler
{\em twistor spaces\index{terms}{twistor space}} (see, e.g.,
\section{Regular Hodge manifolds}\index{terms}{regular Hodge manifold} 
regular\item{terms}{regular fixed point}} if every irreducible
stable\item{terms}{regular stable point}} if it is stable
point\item{terms}{regular fixed point}.
stable\item{terms}{regular stabel point}.
\punkt Let $M$ be a Hodge manifold\index{terms}{Hodge
preferred\index{terms}{preferred complex structure} complex structure $M_I$.
\defn A Hodge manifold\index{terms}{regular Hodge manifold} $M$ is
manifolds}\item{terms}{regular Hodge
manifold}\index{terms}{linearization}  
\punkt Consider a regular Hodge manifold\index{terms}{regular Hodge
\proof Since $M$ is regular\item{terms}{regular Hodge manifold}, the
manifold\index{terms}{regular Hodge manifold} $M$ onto the
The quaternionic structure\index{terms}{quaternionic structure} on
linearization\index{terms}{linearization}} of the regular Hodge
manifold\index{terms}{regular Hodge manifold} $M$.
quaternionic vector space\index{terms}{regular equivariant
linearization\index{terms}{linearization} map $\Lin_M:M_I \to \tv$
To prove that the linearization\index{terms}{linearization} map is
Thus the linearization\index{terms}{linearization} map is also
\subsection{Linear Hodge manifold structures}\index{terms}{linear
manifold\index{terms}{regular Hodge manifold} $M$ admits a canonical
manifold\index{terms}{Hodge manifold} structure on a neighborhood of
In order to use the linearization\index{terms}{linearization}
\punkt Let now $M$ be a Hodge manifold\index{terms}{Hodge
differential\index{terms}{Dolbeault differential} for the preferred\index{terms}{preferred complex structure}
the quaternionic structure\index{terms}{quaternionic structure} on $M$.
linearization\index{terms}{linearization} map if and only if for
definition of the linearization\index{terms}{linearization} map
linearization\index{terms}{linearization} map for the 
regular Hodge manifold\index{terms}{regular Hodge manifold}
manifold\index{terms}{regular Hodge manifold} structure on $U$.
Donote by $\Lin_U$ the linearization\index{terms}{linearization} map
for the regular Hodge manifold\index{terms}{regular Hodge manifold}
linearization\index{terms}{linearization} map $\Lin_U:U \to \tv$
structure\index{terms}{quaternion structure} map $j$. Therefore 
Hodge manifold\index{terms}{Hodge manifold} structure on $\tv$ is
called {\em linear\index{terms}{linear Hodge manifold}} if the
linearization\index{terms}{linearization} construction is linear. 
manifolds}\label{section.5}\index{terms}{Hodge manifold}
\subsection{Hodge connections}\index{terms}{Hodge connection}
\punkt The linearization\index{terms}{linearization} construction
manifolds\index{terms}{regular Hodge manifold} to the study of
linear Hodge manifold\index{terms}{linear Hodge manifold} structures
theory of Hodge bundles\index{terms}{Hodge bundle} developed in
manifold\index{terms}{Hodge manifold} structures on $U$ in terms of
type, which we call {\em Hodge connections\index{terms}{Hodge
that is, the codifferential\index{terms}{codifferential}
map.  Recall that a {\em connection\index{terms}{connection on a
connection\index{terms}{connection on a fibration} $\Theta$
manifolds. A {\em $\C$-valued connection\index{terms}{$\C$-valued
codifferential\index{terms}{codifferential} map $\del
$\C$-valued connection\index{terms}{$\C$-valued connection} $\Theta$
Theorem\index{terms}{Forbenius Theorem} this implies that the
connection\index{terms}{connection on a fibration} defines locally a
connections\index{terms}{$\C$-valued connection}: the
defined over $\C$, and the Frobenius Theorem\index{terms}{Frobenius
the $\C$-valued connection\index{terms}{$\C$-valued connection}
$\Lambda^1(M,\C)$ 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} $1$
Our description of the Hodge manifold\index{terms}{Hodge manifold}
\defn \label{hodge.con} A {\em Hodge connection\index{terms}{Hodge
connection\index{terms}{$\C$-valued connection} on $\rho:U \to M$
is weakly Hodge\index{terms}{weakly Hodge map} in the sense of
\ref{w.hodge}. A Hodge connection\index{terms}{Hodge connection} is
\punkt Assume given a flat Hodge connection\index{terms}{Hodge
Proposition~\ref{explicit.hodge} a Hodge manifold\index{terms}{Hodge
connections\index{terms}{Hodge connection} $D$ on the pair $\langle
Hodge manifold\index{terms}{Hodge manifold} structures on the
holomorphic for the preferred\index{terms}{preferred complex structure} complex structure $U_I$ on $U$.
Assume given a Hodge manifold\index{terms}{Hodge manifold} structure
codifferential\index{terms}{codifferential} of the projection $\rho:U \to M$, and let
cotangent bundle at $m$ to the Hodge manifold\index{terms}{Hodge
space\index{terms}{equivariant quaternionic vector space}. Moreover,
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}
connection\index{terms}{$\C$-valued connection} on $U$ over $M$. To
associated to a Hodge manifold\index{terms}{Hodge manifold}
bundle\index{terms}{Hodge bundle} structures if
preferred\index{terms}{preferred complex structure} complex structure $U_I$ on $U$.
\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
projection $\rho:U_I \to M$ is holomorphic, then the codifferential\index{terms}{codifferential}
bundle\index{terms}{Hodge bundle} structures,
Hodge bundle\index{terms}{Hodge bundle} isomorphism. Since the
is a Hodge bundle map. Therefore the codifferential\index{terms}{codifferential}
$M$}\label{relative.de.rham.sub}\index{terms}{de Rham complex} 
manifold\index{terms}{Hodge manifold} structures on the open subset
connection\index{terms}{Hodge connection} on the pair $\langle U,M
need to rewrite the linearity condition\index{terms}{linearity
$M$\index{terms}{relative de Rham complex}. For the convenience of
codifferential\index{terms}{codifferential} 
Rham complex\index{terms}{relative de Rham complex}} of $U$ over $M$. 
complex\index{terms}{de Rham complex} $\Lambda^\cdot(U,\C)$, and the
bundle\index{terms}{Hodge bundle} structure of weight\index{terms}{weight of a Hodge bundle} $1$ on
$\Lambda^i(U/M,\C)$ of weight\index{terms}{weight of a Hodge bundle} $i$, and the relative de Rham
\Lambda^{\cdot+1}(U/M,\C)$ is weakly Hodge\index{terms}{weakly Hodge
compatible with the Hodge bundle\index{terms}{Hodge bundle}
\ref{de.Rham}, twisted by $\iota\index{terms}{involution $\iota$}^*$, where $\iota\index{terms}{involution $\iota$}:\tm \to \tm$ is
involution $\zeta\index{terms}{involution $\zeta$}:\Lambda^1(M,\C) \to \Lambda^1(M,\C)$ by
\zeta\index{terms}{involution $\zeta$} = \begin{cases} \id &\text{ on }\Lambda^{1,0}(M) \subset
\rho^*\overline{\zeta\index{terms}{involution $\zeta$}}:\rho^*\overline{\Lambda^1(M,\C)} \to
$\zeta\index{terms}{involution $\zeta$}$. Namely, define a map $\sigma:\rho^*\Lambda^1(M,\C) \to
\sigma = \tau \circ \rho^*\overline{\zeta\index{terms}{involution $\zeta$}}:
canonical weakly Hodge\index{terms}{weakly Hodge map} map
formula\index{terms}{Cartan Homotopy Formula} gives
and the group $U(1)$ acts on the components with weight\index{terms}{weight of an $U(1)$-action} $1$ and
$-\sqrt{-1}$. By definition of the involution $\zeta\index{terms}{involution $\zeta$}$ (see
\LL_\phi \tau(\alpha) = \sqrt{-1}\tau(\zeta\index{terms}{involution $\zeta$}(\alpha)).
d^r\tau(\alpha) = \can(\alpha) = \eta(\zeta\index{terms}{involution $\zeta$}(\alpha)). 
\sqrt{-1}\tau(\zeta\index{terms}{involution $\zeta$}(\alpha)) = \langle \phi, \eta(\zeta\index{terms}{involution $\zeta$}(\alpha)), 
\subsection{Holonomic Hodge connections}\index{terms}{holonomic
given Hodge connection\index{terms}{Hodge connection} $D$ on the
denote by $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$. The map $\Res$ splits the restriction of
\Lambda^1(U,\C)|_M = S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \oplus \Lambda^1(M,\C).
$U(1)$-action on the conormal bundle $S^1\index{terms}{Hodge bundle
$\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 $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(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} $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$, so that the
real structure on $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ is minus the usual one. Moreover, as a
complex vector bundle the conormal bundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ to $M \subset
$\Lambda^1(M,\C)$. The Hodge type bigrading\index{terms}{Hodge type
bigrading} on $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ is given in terms of this isomorphism by
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) = S^{1,-1}(M) \oplus S^{-1,1}(M) \cong \Lambda^{1,0}(M)
d^r:C_{lin}^\infty(U,\C) \to C^\infty(M,S^1\index{terms}{Hodge
structures\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $0$ on both
\rho^*\Lambda^1(M,\C)$ be a Hodge connection\index{terms}{Hodge
connection\index{terms}{$\C$-valued connection},
\Theta = D_0 \oplus \id:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \oplus \Lambda^1(M,\C) \to
certain bundle map $D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \Lambda^1(M,\C)$.  
\defn The bundle map $D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \Lambda^1(M,\C)$ is called
connection\index{terms}{principal part of Hodge connection} $D$.
\punkt Consider the map $D_0:C^\infty(M,S^1\index{terms}{Hodge
connection\index{terms}{Hodge connection} $D$.
Hodge\index{terms}{weakly Hodge map}, so that this composition also
with the Hodge bundle\index{terms}{Hodge bundle} structures, this
connection\index{terms}{Hodge connection} $D$ is a weakly Hodge
bundle map\index{terms}{weakly Hodge map}. In particular, it is
conormal bundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$.
A Hodge connection\index{terms}{holonomic Hodge connection}
and only if its principal part $D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \Lambda^1(M,\C)$
$D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
\Lambda^1(M,\C)$ of the Hodge connection\index{terms}{Hodge
linearity}\label{hodge.lin.subsec}\index{terms}{Hodge
connection}\index{terms}{linearity condition}
manifold\index{terms}{Hodge manifold} structure on the subset $U
be the associated Hodge connection\index{terms}{Hodge connection} on
condition\index{terms}{linearity condition} \ref{lin.def} in terms
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\index{terms}{involution $\iota$}^*$-conjugate to the Hodge
d^rf = \halfi\pi(j(\del_\rho(D-D^\iota\index{terms}{involution $\iota$})(f))),
is the codifferential\index{terms}{codifferential} of the projection
\proof By definition of the Hodge connection\index{terms}{Hodge
complementary\index{terms}{complementary complex structure} complex
\rho^*\Lambda^1(M,\C)$ is weakly Hodge\index{terms}{weakly Hodge
\del_\rho(D(\iota\index{terms}{involution $\iota$}^*\nu(f))) = \iota\index{terms}{involution $\iota$}^*\nu(\del_\rho(Df)).
Therefore $\nu(\del_\rho(D(f))) = \del_\rho(D^\iota\index{terms}{involution $\iota$}(\nu(f)))$, and
\6_J f = \half \del_\rho(D^\iota\index{terms}{involution $\iota$} f) - \halfi j(\del_\rho(D^\iota\index{terms}{involution $\iota$}
d_Uf = \6_Jf + \bar\6_Jf = \half \del_\rho((D+D^\iota\index{terms}{involution $\iota$})f) + \halfi
j(\del_\rho((D-D^\iota\index{terms}{involution $\iota$})f)). 
d^rf = \pi(d_Uf) = \halfi\pi(j(\del_\rho((D-D^\iota\index{terms}{involution $\iota$})f))),
connection\index{terms}{holonomic Hodge connection}
\A = \del_\rho\left((D-D^\iota\index{terms}{involution $\iota$})\left(C_{lin}^\infty(U,\C)\right)\right)
\del_\rho((D-D^\iota\index{terms}{involution $\iota$})f)$, where $f \in C^\infty(U,\C)$ lies in the
connection\index{terms}{principal part of a Hodge connection} $D$ in
automorphism $\iota\index{terms}{involution $\iota$}:\tm \to \tm$
$D_0^\iota\index{terms}{involution $\iota$} = - D_0$. Therefore
\Res(\A) = \Res \circ (D - D^\iota\index{terms}{involution
=(D_0-D^\iota\index{terms}{involution
Since the Hodge connection\index{terms}{holonomic Hodge connection}
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}
linear\index{terms}{linearity condition} in the sense of
f = \half\sigma\left((D-D^\iota\index{terms}{involution $\iota$})f\right), 
$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}.
\proof By Lemma~\ref{lin.char} the Hodge manifold\index{terms}{Hodge
manifold} structure on $U$ is linear\index{terms}{linear Hodge
given by the quaternionic structure\index{terms}{quaternionic
\alpha = \halfi\del_\rho((D-D^\iota\index{terms}{involution $\iota$})f), 
\tau\left(\halfi\del_\rho((D-D^\iota\index{terms}{involution $\iota$})\sigma(\beta))\right).
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
$\zeta\index{terms}{involution $\zeta$}$ is invertible, so that
\sigma(\beta) = \half\sigma(\del_\rho((D-D^\iota\index{terms}{involution $\iota$})\sigma(\beta))),
f = \half\sigma(\del_\rho((D-D^\iota\index{terms}{involution $\iota$})f)),
A Hodge connection\index{terms}{linear Hodge connection} $D$ on the
Every linear Hodge connection\index{terms}{linear Hodge connection}
manifold\index{terms}{linear Hodge manifold} structure on
the preferred\index{terms}{preferred complex structure} complex structure $V_I$ on $V$. Vice versa, every such
connection\index{terms}{Hodge connection} $D:\Lambda^0(U,\C) \to
the principal part $D_0:S^1\index{terms}{Hodge bundle
connection\index{terms}{linear Hodge connection} 
\half \sigma \circ (D_0 - D_0^\iota\index{terms}{involution
$\iota$}) = \id:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
\Lambda^1(M,\C) \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C).
Since $\sigma:\Lambda^1(M,\C) \to S^1\index{terms}{Hodge bundle
$D_0 - D_0^\iota\index{terms}{involution
$\iota$}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
= - D_0^\iota\index{terms}{involution $\iota$}$. Thus $D_0 = \half(D_0-D_0^\iota\index{terms}{involution $\iota$}):S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
\subsection{Formal Hodge manifolds}\index{terms}{formal Hodge manifold}
regular Hodge manifolds\index{terms}{regular Hodge manifold} to the
neighborhood\index{terms}{formal neighborhood} of this zero
$\Hom_Z(\E,\F)$ be the $\J_Z$-adic completion\index{terms}{adic
completion\index{terms}{adic completion} functors}
completions\index{terms}{adic completion}} of the categories
\defn A {\em formal quaternionic structure\index{terms}{formal
A formal quaternionic structure\index{terms}{formal equivariant
of formal completions\index{terms}{formal completion}. Consequently,
to the case of formal quaternionic structures\index{terms}{formal
structure\index{terms}{formal equivariant quaternionic structure} on
\E, D \rangle$ of a Hodge bundle\index{terms}{Hodge bundle} $\E$ on
manifold\index{terms}{Hodge manifold} structures on $X$ in a formal
neighborhood\index{terms}{formal neighborhood} of $Z$ is by means of
\defn A {\em formal Hodge manifold\index{terms}{formal 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 $D^\cdot:\Lambda^\cdot\E \to
X$. For every Hodge manifold\index{terms}{Hodge manifold} structure
on $U$ the $Z$-adic completion\index{terms}{adic completion} functor
defines a formal Hodge manifold\index{terms}{formal Hodge manifold}
completion\index{terms}{adic completion}} of the given structure on
\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
Theorem\intex{terms}{Newlander-Nirenberg Theorem}.  Therefore, if
bundles}\index{terms}{formal Hodge manifold}
condition\index{terms}{linearity condition} in the form given in
\defn A formal Hodge manifold\index{terms}{formal Hodge manifold}
linear\index{terms}{linear Hodge manifold}} if for every
structure\index{terms}{formal quaternionic structure} on $\tm$ and
manifold\index{terms}{linear Hodge manifold} structures
connection\index{terms}{formal Hodge connection}} on $\tm$ along $M
\eqref{conn.eq}. A formal Hodge connection\index{terms}{formal Hodge
connection\index{terms}{formal Hodge connection} is called {\em
linear\index{terms}{linear Hodge connection}} if it satisfies the
f = \half\sigma\left((D-D^\iota\index{terms}{involution $\iota$})f\right), 
automorphism $\iota\index{terms}{involution $\iota$}:\tm \to \tm$ is
$D^\iota\index{terms}{involution $\iota$}:\Lambda^0(\tm,\C) \to
operator $\iota\index{terms}{involution $\iota$}^*$-conjugate to
Linear formal Hodge manifold\index{terms}{formal Hodge manifold}
connections\index{terms}{formal Hodge
connection}\index{terms}{linear Hodge connection} on $\tm$ along
\subsection{The Weil algebra}\index{terms}{Weil algebra}
connections\index{terms}{formal Hodge connection} on $\tm$ along $M$
$\tm$. We call such operators {\em extended\item{terms}{extended
manifolds\index{terms}{regular Hodge manifolds} ``in the formal
neighborhood\index{terms}{formal neighborhood} of the subset of
\punkt Before we define extended\item{terms}{extended connection}
{\em the Weil algebra\index{terms}{Weil algebra}}.  We begin with
to the $M$-adic completion\index{terms}{adic completion}
bundle\index{terms}{Hodge bundle} of weight\index{terms}{weight of a Hodge bundle} $0$. Therefore
weight\index{terms}{weight of a Hodge bundle} $0$.  Moreover, it is a commutative algebra bundle in
$\llim\prehodge_0(M)$. Let $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ be the conormal bundle to the
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\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$. Then the algebra
$S^\cdot(M,\C)$ of the Hodge bundle\index{terms}{Hodge bundle}
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ with respect to the augmentation ideal $S^{>0}(M,\C)$.
$\prehodge(M)$ of Hodge bundles\index{terms}{Hodge bundle} on $M$ is
bigrading\index{terms}{Hodge type bigrading}
bograding $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) = S^{1,-1}(M) \oplus S^{-1,1}(M)$ on the
generators subbundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \subset \B^0(M,\C)$, which was
\rem The complex vector bundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ is canonically isomorphic
bundle\index{terms}{Hodge bundle}
different weight\index{terms}{weight of a Hodge bundle}s).
$\Lambda^i(M,\C)$ 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
the Hodge type bigrading\index{terms}{Hodge type bigrading} on $\B^i(M,\C)$. 
The Hodge bundle\index{terms}{Hodge bundle} structures on
the Weil algebra\index{terms}{Weil algebra}} of the complex manifold $M$. 
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
\iota\index{terms}{involution $\iota$}^* = -\id:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \qquad 
\iota\index{terms}{involution $\iota$}^* =  \id:\Lambda^1(M,\C) \to \Lambda^1(M,\C).  
N^\iota\index{terms}{involution $\iota$} = \iota\index{terms}{involution $\iota$}^* \circ N \circ \iota\index{terms}{involution $\iota$}^*:\B^p(M,\C) \to
the operator $\iota\index{terms}{involution $\iota$}^*$-conjugate to $N$. 
by setting $\sigma = 0$ on $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \subset \B^0(M,\C)$.  The
Hodge\index{terms}{weakly Hodge map}. However, it is real with
respect to the real structure on the Weil algebra\index{terms}{Weil
\B^1(M,\C)$ to the subbundle $S^1\index{terms}{Hodge bundle
\to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$. To describe
sections of the subbundle $S^1\index{terms}{Hodge bundle
\to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ induces an
C = \sigma^{-1}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \Lambda^1(M,\C)
vector bundle isomorphism $\sigma:\Lambda^1(M,\C) \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ is
\Lambda^1(M,\C)$ to $S^{1,-1}(M) \subset S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$, and it sends
isomorphism $C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \Lambda^1(M,\C)$ is weakly
Hodge\index{terms}{weakly Hodge map}. It coincides with the
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$, and it equals minus the tautological isomorphism on the
subbundle $S^{-1,1} \subset S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$.
extended\item{terms}{extended connection} connections. Keep the
\defn \label{ext.con} An {\em extended\item{terms}{extended
operator $D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
\B^1(M,\C)$ which is weakly Hodge\index{terms}{weakly Hodge map} in
Let $D$ be an extended\item{terms}{extended connection} connection
Therefore the operator $D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}
D = \sum_{p \geq 0}D_p, \quad D_p:S^1\index{terms}{Hodge bundle
weakly Hodge bundle maps\index{terms}{weakly Hodge map} on $M$,
D_1:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \otimes \Lambda^1(M,\C) 
is a connection\index{terms}{connection on a vector bundle} in the
usual sense on the Hodge bundle\index{terms}{Hodge bundle}
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$.
\defn The weakly Hodge bundle map\index{terms}{weakly Hodge map}
$D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
part} of the extended\item{terms}{principal part of extended
connection\index{terms}{connection on a vector bundle} $D_1$ is called
{\em the reduction} of the extended\item{terms}{reduction of
connections\index{terms}{formal Hodge connection} on the total space
to the generators subbundle $S^1\index{terms}{Hodge bundle
extended\item{terms}{extended connection} connection on $M$ in the
$D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)
connection\index{terms}{Hodge connection} $D$ in the sense of
linearity conditions\index{terms}{linearity condition} on a Hodge
connection\index{terms}{Hodge connection} on $\tm$ for the
associated extended connection\item{terms}{extended connection} on
$D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^1(M,\C)$
D^\iota\index{terms}{involution $\iota$}:\B^\cdot(M,\C) \to
be the operator $\iota\index{terms}{involution $\iota$}^*$-conjugate
An extended\item{terms}{linear extended connection} connection $D$
is called {\em linear\index{terms}{linear extended connection}} if
for every local section $f$ of the bundle $S^1\index{terms}{Hodge
f = \half \sigma((D-D^\iota\index{terms}{involution $\iota$})f).
connection\index{terms}{formal Hodge connection} $D$ on $\tm$ is
linear\index{terms}{linear Hodge connection} if and only if so is
the extended\item{terms}{extended connection} connection $\rho_*D$
D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^1(M,\C)
be an extended\item{terms}{extended connection} connection on $M$.
subbundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \subset
\B^1(M,\C)$, by \eqref{e.c} the operator $D:S^1\index{terms}{Hodge
derivation of the Weil algebra\index{terms}{Weil algebra} 
connection\item{terms}{derivation associated to extended connection}
Vice versa, the Weil algebra\index{terms}{Weil algebra}
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C),\Lambda^1(M,\C)
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
Hodge\index{terms}{weakly Hodge map}, then this restriction is an
extended\item{terms}{extended connection} connection on $M$.
The extended\item{terms}{flat extended connection} connection $D$ is
If a formal Hodge connection\index{terms}{formal Hodge connection}
extended\item{terms}{extended connection} connection
$\rho_*D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
connection\index{terms}{Hodge connection} $D$ on $\tm$ from the
corresponding extended\item{terms}{extended connection} connection
set of formal Hodge connections\index{terms}{formal Hodge
extended\item{terms}{extended connection} connections on $M$. A
that every extended\item{terms}{extended connection} connection on
$M$ comes from a unique formal Hodge connection\index{terms}{formal
connection\index{terms}{Hodge connection} if and only if
$D'=\rho_*D$ is weakly Hodge\index{terms}{weakly Hodge map} and
of all extended\item{terms}{extended connection} connections on $M$,
Analogously, for every extended\item{terms}{extended connection}
Hodge\index{terms}{weakly Hodge map} differential operator
extended\item{terms}{extended connection} connection $D'$ is flat,
the Hodge connection\index{terms}{Hodge connection} $D$ is
0$, which means that the extended\item{terms}{extended connection}
conditions\index{terms}{linearity condition} on the Hodge
connection\index{terms}{Hodge connection} $D$ and on the
extended\item{terms}{extended connection} connection $D' = \rho_*D$
classification of linear formal Hodge manifold\index{terms}{formal
Hodge manifold}\index{terms}{linear Hodge manifold} structures on
of extended\item{terms}{extended connection} connections on the
algebra}\label{Weil.section}\index{terms}{Weil algebra} 
complex}\label{de.rham.sub}\index{terms}{total de Rham complex}
extended\item{terms}{extended connection} 
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ defined in
bundle\index{terms}{Hodge bundle} algebra on $M$ which we call {\em
the total Weil algebra\index{terms}{total Weil algebra}}. This is
Rham complex\index{terms}{de Rham complex} of a complex manifold $M$
which we call {\em the total de Rham complex\index{terms}{total de
that by \ref{de.Rham} the de Rham complex\index{terms}{de Rham
canonically a Hodge bundle\index{terms}{Hodge bundle} algebra on
the weight\index{terms}{weight of a Hodge bundle} $0$ Hodge bundle obtained by applying the functor
Hodge\index{terms}{weakly Hodge map}. Therefore it induces an
\defn The weight\index{terms}{weight of a Hodge bundle} $0$ Hodge bundle\index{terms}{Hodge bundle} algebra
complex\index{terms}{total de Rham complex}} of the complex manifold
structure\index{terms}{$\R$-Hodge structure} $\W^*_1$, as in
\punkt The category of complexes of Hodge bundles\index{terms}{Hodge
The total de Rham complex\index{terms}{total de Rham complex}
\punkt \label{S} We can describe the Hodge bundle\index{terms}{Hodge
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
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) &= \Lambda^{1,0}(M)(1) \oplus \Lambda^{0,1}(M)(-1) \subset
\Lambda^1_{tot}(M) = S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \oplus \Lambda^1_{ll}(M) \oplus
\iota\index{terms}{involution $\iota$}^*\overline{\Lambda^1_{tot}(M)}$ preserves the subbundle 
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \subset \Lambda^1_{tot}(M,\C)
is trivial, then Hodge bundles\index{terms}{Hodge bundle} are the
this case the Hodge bigrading\index{terms}{Hodge type bigrading} on
\rem The Hodge bundle\index{terms}{Hodge bundle} $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ is
complex\index{terms}{total de Rham complex}
\Lambda^1_l(M) &= S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \oplus \Lambda^1_{ll}(M) \subset \Lambda^1_{tot}(M),\\
\Lambda^1_r(M) &= S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \oplus \Lambda^1_{rr}(M) \subset \Lambda^1_{tot}(M).
is the subalgebra in the total de Rham complex\index{terms}{total de
\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\index{terms}{total de Rham complex}.  By \eqref{cap.cup} we
the usual de Rham complex\index{terms}{de Rham complex}
$\Lambda^\cdot(M,\C)$. As a Hodge bundle\index{terms}{Hodge bundle}
Hodge bundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ of weight\index{terms}{weight of a Hodge bundle} $0$ on the manifold $M$.
\punkt \rem The total de Rham complex\index{terms}{total de Rham
bundles\index{terms}{Higgs bundle} (see \cite{Shiggs}) in the
$\nabla$ admits a unique Hermitian metric\index{terms}{Hermitian
complex\index{terms}{de Rham complex} $\Lambda^\cdot(M,\C)$.  It
complex\index{terms}{total de Rham complex}
$\R$-Hodge structure\index{terms}{$\R$-Hodge structure} on $\E$ if
and only if there exists a Hodge bundle\index{terms}{Hodge bundle}
algebra}\label{t.W.sub}\index{terms}{total Weil algebra}
studying the Weil algebra\index{terms}{Weil algebra} of the manifold
$M$. Let $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(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
S^\cdot &= \widehat{S}^\cdot(S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C))\\
be the Weil algebra\index{terms}{Weil algebra} of the complex
$\B^\cdot$ carries a natural Hodge bundle\index{terms}{Hodge bundle}
bigrading\index{terms}{Hodge type bigrading}
Weil algebra\index{terms}{Weil algebra} $\B^\cdot$. The commutative
S^1\index{terms}{Hodge bundle $S^1(M,\C)$} = S^{1,-1} \oplus S^{-1,1} \subset \B^0 \quad\text{ and }\quad
bigrading\index{terms}{augmentation bigrading} on the algebra
\defn The {\em augmentation bigrading\index{terms}{augmentation
algebra\index{terms}{Weil algebra} of augmentation
bidegree\index{terms}{augmentation bidegree} $(p,q)$. For any linear
grading\index{terms}{augmentation grading}} on $\B^\cdot$, defined
augmentation degree\index{terms}{augmentation degree} $k$.
\punkt Note that the Hodge bidegree\index{terms}{Hodge bidegree} and
the augmentation bidegree\index{terms}{augmentation bidegree}
grading\index{terms}{augmentation grading} is compatible
degree\index{terms}{augmentation degree} $k$-component
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}
\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}{total Weil algebra}.  Recall that we
\Gamma(\B^\cdot)$ of weight\index{terms}{weight of a Hodge bundle} $0$ on $M$.  By \ref{gamma.tensor} the
\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}{total Weil algebra}}
the total Weil algebra\index{terms}{total Weil algebra}, see
complex\index{terms}{total de Rham complex} introduced in
Subsection~\ref{de.rham.sub} Hodge bundle\index{terms}{Hodge bundle}
algebra\index{terms}{total Weil algebra} $\B_{tot}^\cdot$ and let
de Rham complex\index{terms}{de Rham complex} $\Lambda^\cdot$,
isomorphic to the usual Weil algebra\index{terms}{Weil algebra}
Hodge\index{terms}{weakly Hodge map}.
\punkt The total Weil algebra\index{terms}{total Weil algebra}
carries a canonical weight\index{terms}{weight of a Hodge bundle} $0$ Hodge bundle\index{terms}{Hodge
grading\index{terms}{Hodge type bigrading} by upper indices:
augmentation bigrading\index{terms}{augmentation bigrading} on the
Weil algebra\index{terms}{Weil algebra} introduced in \ref{aug}
extends to a bigrading of the total Weil algebra\index{terms}{total
degree\index{terms}{augmentation degree} $1$ and Hodge
degree\index{terms}{Hodge degree} $\pm 1$, so that the sum $n+k$ of
degree\index{terms}{augmentation degree} $1$,
By \ref{S} we have $\Lambda^1_o \cong S^1\index{terms}{Hodge bundle $S^1(M,\C)$} = S^{1,-1} \oplus
S^{-1,1}$ as Hodge bundles\index{terms}{Hodge bundle}, so that the
Hodge degrees\index{terms}{Hodge degree} on $\Lambda^1_o \subset
\ref{S.Hodge.type} of Hodge bidegree\index{terms}{Hodge bidegree}
\subsection{Derivations of the Weil algebra}\index{terms}{Weil algebra}
Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ which will
type\index{terms}{Hodge type} decomposition. The following fact is
\punkt \label{C.and.sigma} Let $C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \Lambda^1$ be the
canonical weakly Hodge map\index{terms}{weakly Hodge map} introduced
generators $S^1\index{terms}{Hodge bundle $S^1(M,\C)$},\Lambda^1 \subset \B^\cdot$. Therefore $C \circ C =
Hodge\index{terms}{weakly Hodge map}; however, it is real and admits
components of Hodge types\index{terms}{Hodge type} $(-1,0)$ and
algebra\index{terms}{Weil algebra} $\B^\cdot$. We obviously have
\sigma^{0,-1} \circ \sigma^{0,-1} = 0$ on generators $S^1\index{terms}{Hodge bundle $S^1(M,\C)$},\Lambda^1
algebra\index{terms}{Weil algebra}.
bidegree\index{terms}{Hodge bidegree} components coincide with the
so-called {\em Koszul differentials\index{terms}{Koszul
differential}} on the Weil algebra\index{terms}{Weil algebra}
is by definition weakly Hodge\index{terms}{weakly Hodge
algebra\index{terms}{total Weil algebra} $\B_{tot}^\cdot$ preserving
the weight\index{terms}{weight of a Hodge bundle} $0$ Hodge bundle\index{terms}{Hodge bundle} structure on
algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}$ equipped
with the derivation $C$ is a complex of Hodge bundles of weight\index{terms}{weight of a Hodge bundle} $0$.
algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}$ of the
extended\item{terms}{flat extended connection} connections on $M$ is
of the total Weil algebra\index{terms}{total Weil algebra}
bidegrees\index{terms}{augmentation bidegree} $(p,q)$ with $p,q \geq
of Hodge bundles\index{terms}{Hodge bundle} of weight\index{terms}{weight of a Hodge bundle} $0$ on $M$.
the subbundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \subset \B_0 \cong \B^0_{tot}$ induces a Hodge
bundle\index{terms}{Hodge bundle} isomorphism
C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \Lambda^1_o \subset \Lambda^1_{tot} \subset \B^1_{tot}.
Define a map $\sigma_{tot}:\Lambda^1_{tot} \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ by 
The map $\sigma_{tot}:\Lambda^1_{tot} \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ preserves the Hodge
bundle\index{terms}{Hodge bundle} structures of weight\index{terms}{weight of a Hodge bundle} $0$ on both
map $\sigma:\Lambda^1 \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ introduced in \ref{C.and.sigma}.
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$, the map $\sigma_{tot}:\Lambda^1_{tot} \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ does {\em not}
\B_{tot}^\cdot$ of the total Weil algebra\index{terms}{total Weil
and set $\sigma_l = 0$ on $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$. By \ref{gamma.use} we have
The map $\sigma_l:\Lambda^1_l \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ vanishes on the second
S^{1,-1} \subset S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ on the first summand. The restriction of the map
type\index{terms}{Hodge type}-$(0,-1)$ component
$\sigma^{0,-1}:\Lambda^1 \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ of the canonical map
$\sigma:\Lambda^1 \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$.
bundles $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ and $\Lambda^1_l$, and the ideal of relations is
Since the map $\sigma_l:\Lambda^1_l \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ vanishes on
derivation of the total Weil algebra\index{terms}{total Weil
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \subset 
bidegree\index{terms}{augmentation bidegree}
decomposition of $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ is by
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}_{1,0} = S^{1,-1} \qquad\qquad S^1\index{terms}{Hodge bundle $S^1(M,\C)$}_{0,1} = S^{-1,1},
By the definition of the map $\sigma_l:\Lambda^1_l \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ (see
we have $h_l=p\id$ on the generator subbundles $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}_{p,q}$ and on
derivation and the augmentation bidegree\index{terms}{augmentation
algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}$, which
extended\item{terms}{flat extended connection} connections given in
bidegree\index{terms}{augmentation bidegree}
connections}\label{main.section}\item{terms}{flat extended connection}
\subsection{K\"ah\-le\-ri\-an connections}\index{terms}{K\"ahelrian
manifold\index{terms}{formal Hodge manifold} structures on the
flat extended connections\index{terms}{linear extended
connection}\item{terms}{flat extended connection} on the manifold
connections\index{terms}{K\"ahlerian connection} {\em
extended connections\item{terms}{extended connection} on $M$ and
Hodge type\index{terms}{Hodge type}.  
\defn The connection\index{terms}{K\"ahlerian connection} $\nabla$
\ex The Levi-Civita connection\index{terms}{Levi-Civita connection}
of the connection\index{terms}{connection on a vector bundle}
holomorphic\index{terms}{holomorphic
connection}\index{terms}{K\"ahlerian connection}.
extended\item{terms}{extended connection} connection $D$ on $M$ a
the reduction\item{terms}{reduction of extended connection} of
linear\index{terms}{linear extended connection}\item{terms}{flat
\item Every K\"ah\-le\-ri\-an connection\index{terms}{K\"ahlerian
connection\index{terms}{K\"ahlerian connection} $\nabla$ on the
flat linear extended connection\index{terms}{linear extended
connection}\item{terms}{flat extended connection} $D$ on $M$ with
$C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ of the manifold
connection\item{terms}{extended connection} $D_{\leq 1}$ is 
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$, called the
reduced Weil algebra\index{terms}{reduced Weil algebra}. The reduced
Weil algebra\index{terms}{reduced Weil algebra} is defined in such a
way that for every extended\item{terms}{extended connection}
the connection\index{terms}{K\"ahlerian connection} $D_1$ is
Hodge\index{terms}{weakly Hodge map} derivation of the quotient
connection\index{terms}{linear extended connection} $D_{\leq 1}$ and
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ such that $D
Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$, called the
augmentation grading\index{terms}{augmentation grading}, so that the
augmentation degree\index{terms}{augmentation degree} $k$. In order
degree\index{terms}{augmentation degree} $k$ in the composition
Hodge\index{terms}{weakly Hodge map}, and the
extended\item{terms}{extended connection} connection $D_{\leq k} =
D_{\leq k-1} + D_k$ must be linear\index{terms}{linear extended
\punkt In order to analyze weakly Hodge maps\index{terms}{weakly
Hodge map} from $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$
to the Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$, we
obtain the total Weil algebra\index{terms}{total Weil algebra}
$\B_{tot}^\cdot(M,\C) = \Gamma(\B^\cdot(M,\C))$ of weight\index{terms}{weight of a Hodge bundle} $0$, which
bundle\index{terms}{Hodge bundle} $S^1\index{terms}{Hodge bundle
$S^1(M,\C)$}(M,\C)$ on the manifold $M$ is of weight\index{terms}{weight of a Hodge bundle} $0$, and, by
from $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ to
maps from $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ to the
total Weil algebra\index{terms}{total Weil algebra}
$\B^\cdot_{tot}(M,\C)$. The canonical map $C:S^1\index{terms}{Hodge
$R_k:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^2(M,\C)$ defines a Hodge bundle map
$R_k^{tot}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^2_{tot}(M,\C)$, and solving \eqref{tslv}
is equivalent to finding a Hodge bundle map $D_k:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}(M,\C)$
\B^\cdot_{tot}(M,\C)$ of the total Weil algebra\index{terms}{total
which implies that the Hodge bundle\index{terms}{Hodge bundle} map
D_k = -h^{-1} \circ \sigma_{tot} \circ R_k^{tot}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^1_{tot}(M,\C)
linear\index{terms}{linear extended connection}, and and we have to
condition\index{terms}{linear condition} on an
extended\item{terms}{extended connection} connection $D$ in terms of
\B^{\cdot+1}_{tot}$ of the total Weil algebra\index{terms}{total
algebra\index{terms}{reduced Weil algebra} $\wB^\cdot(M,\C)$ and
algebra\index{terms}{reduced Weil algebra}. Finally, in
\to \prehodge$ and of the total Weil algebra\index{terms}{total Weil
algebra}\label{pf.first.sub}\index{terms}{total Weil algebra}
\punkt Assume given an extended\item{terms}{extended connection}
connection $D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \B^1$
\B^{\cdot+1}$ of the Weil algebra\index{terms}{Weil algebra} as in
degree\index{terms}{augmentation degree} decomposition.  The
derivation $D$ is weakly Hodge\index{terms}{weakly Hodge map} and
algebra\index{terms}{total Weil algebra} $\B_{tot}^\cdot$.
condition\index{terms}{linearity condition} \ref{lin.ext.con} on
the extended\item{terms}{extended connection} connection $D$ in
terms of the total Weil algebra\index{terms}{total Weil algebra}.
The extended\item{terms}{extended connection} connection $D$ is
linear\index{terms}{linear extended connection} if and only if
$D_0=C$ and $\sigma_{tot} \circ D_k = 0$ on $S^1\index{terms}{Hodge
\B_{tot}^1$ vanishes. Therefore the map $D_k:S^1\index{terms}{Hodge
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ regardless of the
extended\item{terms}{extended connection} connection $D$. On the
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ we have $\sigma_{tot}
$\sigma:\Lambda^1 \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ is
equivalent to $\sigma \circ D_0 = \id:S^1\index{terms}{Hodge bundle
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$. Therefore the condition
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 $S^1(M,\C)$} \subset \B^0$ and as $\id$ on $\Lambda^1
$(-1)^{i+k}$ on $\B_k^i \subset \B^\cdot$. Therefore $D_k^\iota\index{terms}{involution $\iota$} =
\sigma \circ \half(D - D^\iota\index{terms}{involution $\iota$}) =
\id:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \B^1 \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$},  
connection\index{terms}{linear extended connection}. 
algebra}\label{pf.third.sub}\index{terms}{reduced Weil algebra}
step is to reduce the classification of linear\index{terms}{linear
extended connection} flat extended\item{terms}{flat extended
connection} connections $D:S^1\index{terms}{Hodge bundle
algebra\index{terms}{Weil algebra} $\B^\cdot$. We introduce this
algebra\index{terms}{reduced Weil algebra}. We then show that every
extended\item{terms}{extended connection} connection $D$ on $M$
reduced Weil algebra\index{terms}{reduced Weil algebra}, and that a
algebra\index{terms}{total Weil algebra} $\B_{tot}^\cdot$ is
augmentation bidegree\index{terms}{augmentation bidegree} $(p,q)$
algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}$, and it is
\geq 1}\B^\cdot_{p,q}$ in the Weil algebra\index{terms}{Weil
of the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$. The
algebra\index{terms}{reduced Weil algebra} $\wB^\cdot =
of the full Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ by
The reduced Weil algebra\index{terms}{reduced Weil algebra}
with respect to the augmentation bigrading\index{terms}{augmentation
bigrading} on the Weil algebra\index{terms}{Weil algebra}
algebra\index{terms}{reduced Weil algebra} carries a canonical Hodge
bundle\index{terms}{Hodge bundle} structure compatible with the
bigrading\index{terms}{augmentation bigrading}, and defines an ideal
\subset \B_{tot}^\cdot$ in the total Weil algebra\index{terms}{total
derivation associated to the extended\item{terms}{derivation
bidegree\index{terms}{augmentation bidegree}, it preserves the ideal
Hodge\index{terms}{weakly Hodge map} derivation of the reduced Weil
algebra\index{terms}{reduced Weil algebra} $\wB^\cdot$, which we
denote by $\wD$. If the extended\item{terms}{extended connection}
of this type comes from a linear\index{terms}{linear extended
connection} flat extended\item{terms}{flat extended connection}
Let $D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \B^1$ be a
linear\index{terms}{linear extended connection} but not necessarily
flat extended\item{terms}{flat extended connection} connection on
\wB^{\cdot+1}$ be the associated weakly Hodge\index{terms}{weakly
algebra\index{terms}{reduced Weil algebra} $\wB^\cdot$. Assume that
There exists a unique weakly Hodge\index{terms}{weakly Hodge map}
bundle map $P:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \I^1$
such that the extended\item{terms}{extended connection} connection
$D' = D + P:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \B^1$ is
\punkt \proof Assume given a linear\index{terms}{linear extended
construct a weakly Hodge\index{terms}{weakly Hodge map} map
$P:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to
\I^1$ such that the extended\item{terms}{extended connection}
degree\index{terms}{augmentation degree}, that is, we construct
Assume given a linear extended connection\index{terms}{linear
extended connection} $D:S^1\index{terms}{Hodge
D:\B^\cdot \to \B^{\cdot+2}$ maps $S^1\index{terms}{Hodge bundle
There exists a unique weakly Hodge\index{terms}{weakly Hodge map}
bundle map $P_k:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to
\I^1_{k+1}$ such that the extended\item{terms}{extended connection}
connection $D' = D + P_k:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}
\to \B^1$ is linear\index{terms}{linear extended connection}, and for the
$D' \circ D'$ maps $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ into
of the total Weil algebra\index{terms}{total Weil algebra}
associated to the extended\item{terms}{derivation associated to
be the component of augmentation degree\index{terms}{augmentation
degree} $k$ of the composition $D \circ D:S^1\index{terms}{Hodge
subbundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \subset
\B^0_{tot}$. Indeed, since $C$ maps $S^1\index{terms}{Hodge bundle
equal to the commutator $[C,R]:S^1\index{terms}{Hodge bundle
by assumption linear\index{terms}{linear extended connection}, we
augmentation degree\index{terms}{augmentation degree} $k$ in the
commutator $[C,\Theta \circ \Theta]:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \B^3_{tot}$. By
The set of all weakly Hodge\index{terms}{weakly Hodge map} maps
$P:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \I_k^1$ coincides with the set of all maps $P:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to
bundle\index{terms}{Hodge bundle} structures. Let $P$ be such a map,
associated to the extended\item{terms}{derivation associated to
linear\index{terms}{linear extended connection}, by
Lemma~\ref{lin.aug} the extended\item{terms}{extended connection}
Moreover, since the augmentation degree\index{terms}{augmentation
augmentation degree-$k$ component $Q:S^1\index{terms}{Hodge bundle
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \subset \B_{tot}^0$ and vanishes on $\Lambda^1_{tot} \subset
\B_{tot}^1$. Since $C$ maps $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$ into $\Lambda^1_{tot} \subset \B_{tot}^1$, we
P = -h^{-1} \circ \sigma_{tot} \circ R:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \I^1_{k+1}.
connections}\label{pf.last.sub}\item{terms}{reduction of extended connection}
bidegree\index{terms}{Hodge bidegree} components
of the reduced Weil algebra\index{terms}{reduced Weil algebra}
\Lambda^1 \oplus \left(S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \otimes \Lambda^1\right) \subset \wB^1\\ 
\punkt Let now $\nabla:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}
\to S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \otimes \Lambda^1$ be
an arbitrary real connection\index{terms}{connection on a vector
bundle} on the bundle $S^1\index{terms}{Hodge bundle
D = C + \nabla:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to
\Lambda^1 \oplus \left(\Lambda^1 \otimes S^1\index{terms}{Hodge
is then automatically weakly Hodge\index{terms}{weakly Hodge map}
and defines therefore an extended\item{terms}{extended connection}
is linear\index{terms}{linear extended connection} by
algebra\index{terms}{reduced Weil algebra}.
the connection\index{terms}{K\"ahlerian connection} $\nabla$ is
Hodge\index{terms}{weakly Hodge map}, hence factors through a bundle
\wD \circ \wD: S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \wB^{3,-1} \oplus \wB^{2,0} \oplus \wB^{1,1}
\{C, \nabla\} = T \circ C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}
where $T$ is the torsion of the connection\index{terms}{connection
where $R^{2,0}$, $R^{0,2}$ are the Hodge type\index{terms}{Hodge
augmentation degree\index{terms}{augmentation degree}-$2$ component
$D_2$ of a flat linear\index{terms}{linear extended connection}
extended\item{terms}{flat extended connection} connection $D$ on the
Let $D = \sum_{k \geq 0}D_k:S^1\index{terms}{Hodge bundle
$S^1(M,\C)$} \to \B^1$ be a flat linear\index{terms}{linear extended
connection} extended\item{terms}{flat extended connection}
is a K\"ah\-le\-ri\-an connection\index{terms}{K\"ahlerian
D_1:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}
algebra\index{terms}{total Weil algebra}. By the construction used
\circ \sigma_{tot} \circ R_{tot}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \left(\B^1_{tot}\right)_3$,
and $R^{tot}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to \left(\B^2_{tot}\right)_3$ is the square
\left(B^2_{tot}\right)_3 = \left(S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \otimes \Lambda^2\right) \oplus
and the map $R^{tot}:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to
\left(B^2_{tot}\right)_3$ sends $S^1\index{terms}{Hodge bundle
coincides with the curvature $R:S^1\index{terms}{Hodge bundle
$S^1(M,\C)$} \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \otimes
connection\index{terms}{K\"ahlerian connection} $D_1$. Therefore
manifolds}\index{terms}{Hodge manifold}\index{terms}{hyperk\"ahler
K\"ah\-le\-ri\-an connection\index{terms}{K\"ahlerian connection}
linear formal Hodge manifold\index{terms}{formal Hodge
manifold}\index{terms}{linear Hodge manifold} structure on the
polarizations\index{terms}{polarization} on the Hodge 
manifold\index{terms}{Hodge manifold} $\tm$ in the sense of
metrics\index{terms}{K\"ahler metric} on $M$ compatible with the
metric\index{terms}{hyperk\"ahler metric} on $\tm$, or, more
structure\index{terms}{hypercomplex manifold} and
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric} in the sense of
to $h$ in the preferred\index{terms}{preferred complex structure} complex structure $\tm_I$ on $\tm$.
particular, it is closed, and the metric\index{terms}{K\"ahler
\item K\"ahler metrics\index{terms}{K\"ahler metric} on $M$
connection\index{terms}{K\"ahlerian connection} $\nabla$, and
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric} hyperk\"ahler
metrics\index{terms}{hyperk\"ahler metric} on $\tm$ compatible with
the given formal Hodge manifold\index{terms}{formal Hodge manifold}
terms of polarizations\index{terms}{polarization} rather than
the formal Hodge manifold\index{terms}{formal Hodge manifold} $\tm$
complementary\index{terms}{complementary complex structure} complex
$H$-type\index{terms}{$H$-type} $(1,1)$ with respect to the
canonical Hodge bundle\index{terms}{Hodge bundle} structure on
Hermitian-Hodge\index{terms}{Hermitian-Hodge metric} hyperk\"ahler
metrics\index{terms}{hyperk\"ahler metric} on $\tm$ are in one-to-one
correspondence with polarizations\index{terms}{polarization}. Let
$h$ be a K\"ahler metric\index{terms}{K\"ahler metric} on $M$, and
\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
$H$-type\index{terms}{$H$-type} $(1,1)$, the form $\Res\Omega \in
type\index{terms}{Hodge type} $(1,1)$. By
polarization\index{terms}{polarization} $\Omega$ of the formal Hodge
manifold\index{terms}{formal Hodge manifold} $\tm$ the restriction
with the connection\index{terms}{connection on a vector bundle}
the holomorphic de Rham algebra\index{terms}{holomorphic de Rham
which would be independent of the Hodge manifold\index{terms}{Hodge
the relative de Rham complex\index{terms}{relative de Rham complex}
$M$ 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
compatible with the natural Hodge bundle\index{terms}{Hodge bundle}
structure\index{terms}{quaternionic structure} on $\tm$. For such
Since the Hodge manifold\index{terms}{Hodge manifold} structure on
$\tm$ is linear\index{terms}{linear Hodge manifold}, this equals
the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ of $M$. All
bundle\index{terms}{Hodge bundle} structures and with the
manifold\index{terms}{Hodge manifold} structure on $\tm$. Moreover,
bundle\index{terms}{Hodge bundle} structures on both sides.
\subsection{The Dolbeult differential\index{terms}{Dolbeault differential} on $\protect\tm_J$}
differential\index{terms}{Dolbeault differential} $\bar\6_J$ of the
complex\index{terms}{de Rham complex} $\Lambda^{\cdot,\cdot}(\tm_J)$
operator $D$ is weakly Hodge\index{terms}{weakly Hodge map}. It
satisfies the Leibnitz rule\index{terms}{Leibnitz rule} with respect
manifold\index{terms}{Hodge manifold} structure on $\tm$. For $k >
differential graded module over the Weil algebra\index{terms}{Weil
The derivation $d^r$ also is weakly Hodge\index{terms}{weakly Hodge
rule\index{terms}{Leibnitz rule}, so it suffices to prove that it
augmentation degree\index{terms}{augmentation degree} $0$. By
\punkt Let now $\nabla=D_1:S^1\index{terms}{Hodge bundle
$S^1(M,\C)$} \to S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \otimes
\Lambda^1$ be the reduction of the extended\item{terms}{reduction of extended
connection\index{terms}{connection on a vector bundle} on the bundle
$L^1 \cong S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$, and this
connection extends by the Leibnitz rule\index{terms}{Leibnitz rule}
satisfies the Leibnitz rule\index{terms}{Leibnitz rule} with respect
\punkt Introduce the augmentation grading\index{terms}{augmentation
degree\index{terms}{augmentation degree}, and we have the
rule\index{terms}{Leibnitz rule} with respect to the multiplication
L^1 \otimes \B^0$ restricted to $S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \subset \B^0$ becomes an
isomorphism $d^r:S^1\index{terms}{Hodge bundle $S^1(M,\C)$} \to L^1$. Since $\{D_1,d^r\} = \{D_2,d^r\} =
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}$. The first one then becomes the definition of $\nabla$, and
rule\index{terms}{Leibnitz rule} with respect to the multiplication
\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}{Hodge type} $(1,1)$ and
degree\index{terms}{augmentation degree} decomposition.
of augmentation degree\index{terms}{augmentation degree}
connection\index{terms}{connection on a vector bundle} $\nabla$, so
\otimes \B^0)$ which is of Hodge type\index{terms}{Hodge type}
$\Omega$ is of Hodge type\index{terms}{Hodge type} $(1,1)$, we must
type\index{terms}{Hodge type} $(p,q)$ such that
\B_k)$ of the same Hodge type\index{terms}{Hodge type} $(p,q)$ and
algebra\index{terms}{total Weil algebra} introduced in
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
bundle\index{terms}{Hodge bundle} structure.  The commutator $h =
\B^{\cdot+1}$ is weakly Hodge\index{terms}{weakly Hodge map}, it
degree\index{terms}{augmentation degree} $k$. Since $D^{tot} \circ
type\index{terms}{Hodge type} $(p,q)$ and of augmentation degree
structure\index{terms}{formal hyperk\"ahler structure} on the total
for the preferred\index{terms}{preferred complex structure} complex structure $\tm_I$ on $M$. 
obtain a hyperk\"ahler metric\index{terms}{hyperk\"ahler metric} of
the formal neighborhood\index{terms}{formal neighborhood} of the
of the Darboux Theorem\index{terms}{Darboux-Weinsten Theorem}, which
obtain a hyperk\"ahler metric\index{terms}{hyperk\"ahler metric} on
holomorphic de Rham complex\index{terms}{holomorphic de Rham
Assume given either a formal Hodge manifold\index{terms}{formal
manifold\index{terms}{Hodge manifold} structure on an open
$\Omega^\cdot_k(U) \subset \Omega^\cdot(U)$ of forms of weight\index{terms}{weight of an $U(1)$-action} $k$
M$ is holomorphic for the preferred\index{terms}{preferred complex structure} complex structure $\tm_I$ on
Dolbeult differential\index{terms}{Dolbeault differential} $\6_I$.
Assume given a formal polarized Hodge manifold\index{terms}{formal
$\rho:\tm_I \to M$ is holomorphic for the preferred\index{terms}{preferred complex structure} complex
in a formal neighborhood\index{terms}{formal neighborhood} of the
polarized Hodge manifold\index{terms}{Hodge manifold} structure on
$1$-form $\alpha \in \Omega^1(U)$ is of weight\index{terms}{weight of an $U(1)$-action} $1$ with respect to
$\alpha \in C^\infty(U,\rho^*\Lambda^{1,0}(M)$ which is of weight\index{terms}{weight of an $U(1)$-action}
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
weight\index{terms}{weight of an $U(1)$-action} $1$ with respect to the $U(1)$-action and such that
function $f$ is of weight\index{terms}{weight of an $U(1)$-action} $1$ with respect to the $U(1)$-action.  On
K\"ah\-le\-ri\-an connection\index{terms}{K\"ahlerian connection}
defines a flat linear extended\item{terms}{flat extended connection}
connection\index{terms}{linear extended connection}
$D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^1(M,\C)$
on $M$ and therefore a formal Hodge connection\index{terms}{formal
connection\index{terms}{formal Hodge connection} defines, in turn, a
formal Hodge manifold structure\index{terms}{formal Hodge manifold}
on $\tm$ in the formal neighborhood\index{terms}{formal
connection\index{terms}{K\"ahlerian connection}
manifold structure\index{terms}{formal Hodge manifold} on $\tm$ is
the completion of an actual Hodge manifold\index{terms}{Hodge
$\nabla$ comes from a K\"ahler metric\index{terms}{K\"ahler metric}
polarization\index{terms}{polarization} $\Omega$ of the 
K\"ah\-le\-ri\-an connection\index{terms}{K\"ahlerian connection}
manifold\index{terms}{Hodge manifold} structure on $U \subset \tm$
defines a linear\index{terms}{linear extended connection} flat
extended\item{terms}{flat extended connection} connection $D$ on $M$
metric\index{terms}{K\"ahler metric}
polarization\index{terms}{formal polarization} of the Hodge
manifold\index{terms}{Hodge manifold} structure on $U \subset
metrics\index{terms}{Hermitian metric} on the
algebra\index{terms}{de Rham complex} $\Lambda^\cdot(M,\C)$ of the
manifold $M$ and, further, to the Weil algebra\index{terms}{Weil
which is by definition the $\J$-adic completion\index{terms}{adic
sections of the Weil algebra\index{terms}{Weil algebra}. This vector
bundle\index{terms}{Hodge bundle} structure on $\B^\cdot(M,\C)$
induces an $\R$-Hodge structure\index{terms}{$\R$-Hodge structure}
\punkt The $\J$-adic completion\index{terms}{adic completion}
algebra\index{terms}{de Rham complex}
\Lambda^1(V)$ and by $V_2 = V \subset S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(V)$. It is convenient to
\punkt The complex vector bundle $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$ on $M$ is also
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \cong \V$ in such a way that the canonical map
$C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(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 $S^1(M,\C)$}(M,\C)) \subset \B^0 
the subset of constant sections in $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \cong \V$. Then the
Weil algebra\index{terms}{Weil algebra} $\B^\cdot$ becomes
Weil algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ which we
call {\em the augmentation grading\index{terms}{augmentation
algebra\index{terms}{Weil algebra} $\B^\cdot$. The augmentation
which we will call {\em the total grading\index{terms}{total
\subset \B^\cdot$ of the Weil algebra\index{terms}{Weil algebra}
component of augmentation degree\index{terms}{augmentation degree}
$k$ and total degree\index{terms}{total degree} $n$. Note that
bigrading\index{terms}{augmentation bigrading}} on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ \, and it this
connection\index{terms}{Hodge connection} on the pair $\langle \tm,M
\rangle$. The corresponding extended\item{terms}{extended
connection} connection $D^{const}:S^1\index{terms}{Hodge bundle
\nabla^{const}_1:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)
\to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \otimes 
on $S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \cong \V$ and
C = \id:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
algebra\index{terms}{Weil algebra} associated to the
extended\item{terms}{derivation associated to extended connection}
degree\index{terms}{total degree}.
\punkt Let now $D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)
\to \B^1(M,\C)$ be the an arbitrary linear\index{terms}{linear
be the derivation of the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot$ associated to the extended\item{terms}{derviation
degree\index{terms}{total degree}\index{terms}{augmentation degree}
\punkt Since the extended\item{terms}{extended connection}
connection $D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
\B^1(M,\C)$ is linear\index{terms}{linear extended connection}, its
component $D_0:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
\Lambda^1(M,\C)$ of augmentation degree\index{terms}{augmentation
$C:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to
S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \subset \B^0$ satisfies
\punkt The fixed Hermitian metric\index{terms}{Hermitian metric} on
metric\index{terms}{standard metric on the Weil algebra} on the
whole Weil algebra\index{terms}{Weil algebra} such that the
derivation associated to an extended\item{terms}{derivation
metric\index{terms}{standard metric on the Weil algebra} on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot$. If for certain
then the formal Hodge connection\index{terms}{formal Hodge
actual real-analytic Hodge connection\index{terms}{Hodge connection}
\subset \tm$. Conversely, if the extended\item{terms}{extended
\proof The constant Hodge connection\index{terms}{Hodge conection}
formal Hodge connection\index{terms}{formal Hodge connection} on
neighborhood\index{terms}{formal neighborhood} of $M
connection\index{terms}{Hodge connection} $D$ converges on a subset
total degree\index{terms}{total degree} $n$ of the formal power
extended\item{terms}{extended connection} connection $D$. First, let
polarizations\index{terms}{polarization} of Hodge 
manifold\index{terms}{Hodge manifold} structures on $\tm$, we will
\B^{\cdot+1}$ be a derivation of the Weil algebra\index{terms}{Weil
algebra} $\B^\cdot$ associated to a flat linear\index{terms}{linear
extended connection} extended\item{terms}{flat extended connection} connection
degree\index{terms}{augmentation degree} $p$ and total
degree\index{terms}{total degree} $q$. Since both $\B^\cdot_{p,q}$
metric\index{terms}{standard metric on the Weil algebra} on
manifold\index{terms}{Hodge manifold} structure on $\tm$
corresponding to the extended\item{terms}{extended connection}
\ref{total.Weil} the total Weil algebra\index{terms}{total Weil
structure\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $k$ universal
for weakly Hodge maps\index{terms}{weakly Hodge map}, as in
metric\item{terms}{Hermitian metric} on $\W_k$ such
and all the Hodge degree\index{terms}{Hodge degree} components
$w_k^{p,q}$ of the universal weakly Hodge map\index{terms}{universal
\defn The {\em standard metric\index{terms}{standard metric on the
Weil algebra}} on the total Weil algebra\index{terms}{total Weil
algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}$ is
map\index{terms}{weakly Hodge map} $w_1:\R(0) \to \W_1$ defines a
with respect to the standard metrics\index{terms}{standard metric on
respect to this metric\index{terms}{standard metric on the Weil
with respect to the standard metric\index{terms}{standard metric on
\punkt The total and augmentation gradings\index{terms}{total
grading}\index{terms}{augmentation grading} on the Weil
algebra\index{terms}{Weil algebra} $\B^\cdot$ extend to gradings on
the total Weil algebra\index{terms}{total Weil algebra}
connection $D$ on $M$ induces a derivation\item{terms}{derivation
metric\index{terms}{standard metric on the Weil algebra} on
$D_{k,n}^{tot}$ of the total Weil algebra\index{terms}{total Weil
\punkt Since the extended\item{terms}{flat extended connection}
connection $D$ is linear\index{terms}{linear extended connection}
and $h$ preserve the augmentation degree\index{terms}{augmentation
augmentation degree\index{terms}{augmentation degree}, hence it maps
degree\index{terms}{total degree}, we can rewrite \eqref{indu} as
rule\index{terms}{Leibnitz rule} and the triangle inequality show
for the complementary\index{terms}{complementary complex structure}
extended\item{terms}{extended connection} connection $D$ on $M$. To
obtain estimates on the Dolbeult differential\index{terms}{Dolbeault
the de Rham complex\index{terms}{de Rham complex}
algebra\index{terms}{Weil algebra} $\B^\cdot(M,\C)$ generated by a
differential\index{terms}{Dolbeault differential} $\bar\6_J$ for the
complementary\index{terms}{complementary complex structure} complex structure $\tm_J$ induces an algebra
algebra\index{terms}{Weil algebra}.
Rham algebra\index{terms}{de Rham complex} $\Lambda^\cdot(M,\C)$. In
the $\J$-adic completion\index{terms}{adic completion} of the space
metric\index{terms}{standard metric on the Weil algebra}}. For every
gradings\index{terms}{total grading}\index{terms}{augmentation
total and the augmentation degrees\index{terms}{augmentation
degree}\index{terms}{total degree}. Denote by $\|D_{k,n}\|^p_q$ the
$\Lambda^\cdot(V_4)$ with the total Weil algebra\index{terms}{total
gradings\index{terms}{total grading}\index{terms}{augmentation
grading} and the metric\index{terms}{standard metric on the Weil
augmentation degrees\index{terms}{total
degree}\index{terms}{augmentation degree}.  Denote by
rule\index{terms}{Leibnitz rule}, it suffices to prove the estimate
\to \LL^{1,\cdot}$ satisfies the Leibnitz rule\index{terms}{Leibnitz
augmentation degree\index{terms}{augmentation degree} $k$ does not
$D:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \to \B^1(M,\C)$
be a flat linear\index{terms}{linear extended connection} extended
connection on $M$. Assume that its reduction\item{terms}{reduction
of extended connection} $D_1=\nabla:S^1\index{terms}{Hodge bundle
$S^1(M,\C)$}(M,\C) \to S^1\index{terms}{Hodge bundle
connection\index{terms}{connection on a vector bundle} on the bundle
$S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)$.
The operator $D_1:S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C)
\to S^1\index{terms}{Hodge bundle $S^1(M,\C)$}(M,\C) \otimes
extended\item{terms}{extended connection} connection on $M$ defines
a Hodge connection\index{terms}{Hodge connection}
connection\index{terms}{Hodge connection} is also
derivation of the Weil algebra\index{terms}{Weil algebra} $\B^\cdot$
associated to the extended\item{terms}{derivation associated to
to the Hodge connection\index{terms}{Hodge connection} $D_1$ proves
formal Hodge connection\index{terms}{formal Hodge connection} $D$ on
extended\item{terms}{extended connection} connection $D$ converges
to a real-analytic Hodge connection\index{terms}{Hodge connection}
connection\index{terms}{K\"ahlerian connection} $\nabla$, so that
type\index{terms}{Hodge type} $(p,q)$ with respect to the
complementary\index{terms}{complementary complex structure} complex structure $\tm_J$. The spaces
gradings\index{terms}{total grading}\index{terms}{augmentation
degree\index{terms}{total degree} decomposition.  The decomposition
\subset \LL^{2,0}$ be the formal polarization\index{terms}{formal
polarization} of the Hodge manifold\index{terms}{Hodge manifold}
to the extended\item{terms}{derivation associated to extended
Weil algebra\index{terms}{total Weil algebra} $\B^\cdot_{tot}$
formal polarization\index{terms}{formal polarization} $\Omega$ taken
with respect to the standard metric\index{terms}{standard metric on
polarization\index{terms}{formal polarization} $\Omega$ at $0
the polarization\index{terms}{polarization} $\Omega$ is indeed
spaces\index{terms}{quaternionic vector space} (see,
manifolds\index{terms}{Hodge manifold}. In particular, we establish,
structures\index{terms}{$\R$-Hodge structure}. For the sake of
neighborhood\index{terms}{formal neighborhood} of $0 \in \R^{4n}$
$\SB$. Say that a sheaf $\E \in \Ob\Shv(\SB)$ is {\em of \index{terms}{weight of a sheaf on $\C P^1$} $p$}
Consider a quaternionic vector space\index{terms}{quaternionic
quaternionic vector spaces\index{terms}{quaternionic vector space}
\index{terms}{weight of a sheaf on $\C P^1$} $1$. Call $\loc{V}$
space\index{terms}{quaternionic vector space} $V$. For every algebra
spaces\index{terms}{equivariant quaternionic vector space} and the
\index{terms}{weight of a sheaf on $\C P^1$} $1$. For an equivariant
respectively, with the preferred\index{terms}{preferred complex structure} and the complementary\index{terms}{complementary complex structure} complex
of sheaves of \index{terms}{weight of a sheaf on $\C P^1$} $n$ is equivalent to the category of pure
$\R$-Hodge structures\index{terms}{$\R$-Hodge structures} of weight\index{terms}{weight of a Hodge structure}
W_n\E/W_{n-1}\E \quad \text{ is a sheaf of \index{terms}{weight of a sheaf on $\C P^1$} } \quad n \quad
structures\index{terms}{mixed $\R$-Hodge structure}. (In particular,
\punkt For every pure $\R$-Hodge structure\index{terms}{$\R$-Hodge
$\R(0)$ of weight\index{terms}{weight of a Hodge structure} $0$ the sheaf $\loc{\R(0)}$ coincides with the
with the space of weakly Hodge maps\index{terms}{weakly Hodge map}
For every pure $\R$-Hodge structure\index{terms}{$\R$-Hodge
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
formal neighborhood\index{terms}{formal neighborhood} of $0 \in
V$. Let $\B^\cdot$ be the Weil algebra\index{terms}{Weil algebra} of
structure\index{terms}{$\R$-Hodge structure} of weight\index{terms}{weight of a Hodge structure} $n$, so that
algebra\index{terms}{localized Weil algebra}}.
The augmentation grading\index{terms}{augmentation grading} on
structures\index{terms}{$\R$-Hodge structure}.  Therefore it defines
algebra\index{terms}{localized Weil algebra} $\loc{\B^\cdot}$. The
finer augmentation bigrading\index{terms}{augmentation bigrading} on
\punkt Assume now given a flat extended\item{terms}{flat extended
Hodge\index{terms}{weakly Hodge map}, it corresponds to a derivation
algebra\index{terms}{localized Weil algebra} $\loc{\B^\cdot}$.  It
grading\index{terms}{augmentation grading} on
on the extended\item{terms}{extended connection} connection $D$. To
structure\index{terms}{$\R$-Hodge structure} $W$ of
weight\index{terms}{weight of a Hodge structure} $-1$ by setting $W
structure $W$. In particular, it is a sheaf of \index{terms}{weight
the extended\item{terms}{extended connection} connection $D$. The
\index{terms}{weight of a sheaf on $\C P^1$} $0$ on $\SB$. Moreover,
the localized Weil algebra\index{terms}{localized Weil algebra}
complex\index{terms}{relative de Rham complex} of $\loc{\B^0}$ over
extended\item{terms}{extended connection} connection $D$, solely
extended\item{terms}{flat extended connection} connections on $M$
$\R$-Hodge structures\index{terms}{mixed $\R$-Hodge structure}
filtration\index{terms}{weight filtration} on $\HH^0$ and the pure
$\R$-Hodge structure\index{terms}{$\R$-Hodge structure} $W$ defined
twistor space\index{terms}{twistor space} of the manifold $\tm$ with
the hypercomplex structure\index{terms}{hypercomplex structure}
given by the extended\item{terms}{extended connection} connection
hypercomplex manifolds\index{terms}{hypercomplex manifold} and mixed
$\R$-Hodge structures\index{terms}{mixed $\R$-Hodge
structures\index{terms}{$\R$-Hodge structure} with weakly Hodge
maps\index{terms}{weakly Hodge map} as morphisms is identified by
\punkt Consider the localized Weil algebra\index{terms}{localized
Hodge\index{terms}{weakly Hodge map} derivation introduced in
$U(1)$-equivariant sheaf of \index{terms}{weight of a sheaf on $\C P^1$} $1$ on $\SB$ corresponding to the
$\R$-Hodge structure\index{terms}{$\R$-Hodge structure} given by
bidegrees\index{terms}{augmentation bidegree}
the components in the Weil algebra\index{terms}{Weil algebra}
$\B^\cdot$ of augmentation bidegree\index{terms}{augmentation
structure\index{terms}{$\R$-Hodge structure}. The crucial point in
bidegree\index{terms}{augmentation bidegree} $(p,q)$ in the
localized Weil algebra\index{terms}{localized Weil algebra}
\index{terms}{weight of a sheaf on $\C P^1$}. Therefore the complex $\langle \Gamma(\I^\cdot),C \rangle$
manifold\index{terms}{Hodge manifold} structures on $\tm$ in the
formal completion\index{terms}{formal completion} of the local ring
structure\index{terms}{mixed $\R$-Hodge structure}. (In particular,
If the Hodge manifold\index{terms}{Hodge manifold} structure on
for the preferred\index{terms}{preferred complex structure} complex structure $\tm_I$ on $\tm$, then it is
connection\index{terms}{K\"ahlerian connection} $\nabla$ on $M$
which corresponds to the Hodge manifold\index{terms}{Hodge manifold}
bundles\index{terms}{Higss bundle} and local