\documentclass[11pt]{article} \usepackage{amstex,amscd} \usepackage{theorem} \title{Non-Hermitian Yang-Mills connections.} \author{D. Kaledin, M. Verbitsky.} %\date{June 29, 1996} %\email{kaledin@@ium.ips.ras.ru,verbit@@thelema.dnttm.rssi.ru} \hyphenation{au-to-du-al} \hyphenation{pseu-do-cur-va-ture} %\hyphenation{hy-per-k\"ah-ler} \setlength{\unitlength}{1pt} \def\cdot{{\:\raisebox{3.5pt}{\text{\circle*{1.5}}}}} %% % Sorry for the hack above - \cdot _is_ too small... -- D. 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Kaledin, M. Verbitsky}\footnote{\tt kaledin@@ium.ips.ras.ru, verbit@@thelema.dnttm.rssi.ru} \vskip 1.5em% {\large June 29, 1996} \\ \end{center} \hfill \hspace{0.1\linewidth}\begin{minipage}[t]{0.8\linewidth} {\bf Abstract}.{ \small We study Yang-Mills connections on holomorphic bundles over complex K\"ahler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk\"ahler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk\"ahler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K\"ahler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk\"ahler manifold and the space of rational curves in the twistor space of the ``Mukai dual'' hyperk\"ahler manifold. } \end{minipage} \hfill \tableofcontents \section{Introduction.} \subsection{An overview} In this paper we study non-Hermitian Yang-Mills (NHYM) connections on a complex vector bundle ${\cal B}$ over a K\"ahler manifold. By definition, a connection $\nabla$ in ${\cal B}$ is Yang-Mills if its curvature $\Theta$ satisfies % \begin{equation} \label{intro-Yang-Mills_Equation_} \begin{cases} \Lambda(\Theta)&=\text{const}\cdot\id\\ \Theta \phantom{\Lambda()}&\in \Lambda^{1,1}(M, \End({\cal B})), \end{cases} \end{equation} where $\Lambda$ is the standard Hodge operator, and $\Lambda^{1,1}(M, \End({\cal B})$ is the space of $(1,1)$-forms with coefficients in $\End({\cal B})$ (see Definition \ref{_NHYM_Definition_} for details). This definition is standard \cite{UY}, \cite{Donaldson:surfa}. However, usually $\nabla$ is assumed to be compatible with some Hermitian metric in $\cal B$. This is why we use the term ``non-Hermitian Yang-Mills'' to denote Yang-Mills connections which are not necessarily Hermitian. An important analogy for our construction is the one with flat connections on a complex vector bundle ${\cal B}$. Recall that when $c_1({\cal B}) = c_2({\cal B}) = 0$, Hermitian Yang-Mills connections are flat (L\"ubcke's principle; see \cite{S}). The moduli of flat, but not necessary unitary bundles is a beautiful subject, well studied in literature (see, e. g. \cite{S2}). This space has dimension twice the dimension of the moduli space of unitary flat bundles and has a natural holomorphic symplectic form. Also, generic part of the moduli of non-unitary flat bundles is equipped with a holomorphic Lagrangian\footnote{Lagrangian with respect to the holomorphic symplectic form.} fibration over the space of unitary flat connections. When $c_1({\cal B}) = c_2({\cal B}) = 0$, the flat connections are in one-to-one correspondence with those holomorphic structures on ${\cal B}$ which make it a polystable\footnote{By {\em polystable} we will always mean ``a direct sum of stable''. Throughout the paper, stability is understood in the sense of Mumford-Takemoto.} holomorphic bundle \cite{UY}, \cite{S}. For arbitrary bundle ${\cal B}$, a similar statement holds if we replace ``flat unitary'' by ``Hermitian Yang-Mills''. Thus, it is natural to weaken the flatness assumption and consider instead all Hermitian Yang-Mills connections. The non-Hermitian Yang-Mills connections that we define correspond then to connections that are flat but not necessarily unitary. The basic properties listed above for non-unitary flat bundles hold here as well. We show that the moduli space of NHYM connections has dimension twice the dimension of the moduli of Hermitian Yang-Mills connections and is naturally equipped with a holomorphic symplectic form. As in the case of flat bundles, generic part of the moduli of NHYM connections has a holomorphic Lagrangian fibration over the space of Hermitian Yang-Mills connections. Let us give a brief outline of the paper. Fix a compact K\"ahler manifold $M$ with a complex vector bundle ${\cal B}$. Let $\M$ be the set of equivalence classes of NHYM connections on ${\cal B}$, and let $\M_0 \subset \M$ be the subset of connections admitting a compatible Hermitian metric. Both sets turn out to have natural structures of complex analytic varieties. Recall that $\M_0$ is a moduli space of stable holomorphic bundles (\cite{UY}; see also \ref{_Uhle-Yau_Theorem_}). After giving the relevant definitions, in Section 1 we study the structure of $\M$ in the neighborhood of $\M_0$. We prove that $\dim\M = 2 \dim \M_0$ in a neighborhood of $\M_0$. Moreover, we identify the ring of germs of holomorphic functions on $\M$ near $\M_0$ with the ring of real-analytic complex-valued functions on $\M_0$. Thus an open neighborhood $U \supset \M$ is a complexification of $\M_0$ in the sense of Grauert. We also construct a holomorphic $2$-form on $\M$ which is symplectic in a neighborhood of $\M_0$. This picture is completely analogous to that for the space of flat connections, studied by Hitchin, Simpson and others (\cite{H},\cite{S2}). For concrete examples and applications of our theory, we consider the case of NHYM-connections over a hyperk\"ahler manifold (Definition \ref{defn.hyperkahler}). In this case, it is natural to modify the NHYM condition. Every hyperk\"ahler manifold is equipped with a quaternion action in its tangent space. Since the group of unitary quaternions is isomorphic to $SU(2)$, a hyperk\"ahler manifold has the group $SU(2)$ acting on its tangent bundle. Consider the corresponding action of $SU(2)$ on the space $\Lambda^\cdot(M)$ of differential forms over a hyperk\"ahler manifold $M$. Then all $SU(2)$-invariant $2$-forms satisfy \eqref{intro-Yang-Mills_Equation_} (Lemma \ref{primitive}). Thus, if the curvature $\Theta$ of a bundle $({\cal B}, \nabla)$ is $SU(2)$-invariant, ${\cal B}$ is NHYM. Converse is {\it a priori} non-true when $\dim_\R M >4$: there are $2$-forms satisfying \eqref{intro-Yang-Mills_Equation_} which are not $SU(2)$-invariant. However, as the discussion at the end of Section \ref{_autodu_Section_} shows, for bundles over compact manifolds $SU(2)$-invariance of the curvature is a good enough approximation of the NHYM property. A connection in ${\cal B}$ is called {\bf autodual} if its curvature is $SU(2)$-invariant (Definition \ref{_autodual_Definition_}). For $\dim_\R M=4$, the autoduality, in the sense of our definition, is equivalent to the anti-autoduality in the sense of 4-dimensional Yang-Mills theory. Hermitian autodual bundles were studied at great length in \cite{Vb}.\footnote{In \cite{Vb}, the term {\bf hyperholomorphic} was used for ``Hermitian autodual''.} Most of this paper is dedicated to the study of non-Hermitian autodual bundles over compact hyperk\"ahler manifolds. Consider the natural action of $SU(2)$ in the cohomology of a compact hyperk\"ahler manifold (see the beginning of Section \ref{_autodu_Section_}). Let ${\cal B}$ be a bundle with the first two Chern classes $c_1({\cal B})$, $c_2({\cal B})$ $SU(2)$-invariant. In \cite{Vb} we proved that every Hermitian Yang-Mills connection in ${\cal B}$ is autodual. It is natural to conjecture that in such a bundle every NHYM connection is autodual. In Theorem~\ref{_NHYM-are-autodu_Theorem_}, we prove a weaker form of this statement: namely, in a neighbourhood of the space of Hermitian Yang-Mills connections, every NHYM connection is autodual, assuming the first two Chern classes are $SU(2)$-invariant. This is done by constructing an explicit parametrization of this neighbourhood (Proposition~\ref{series}). Throughout the rest of this paper (starting from Section \ref{_twistors_Section_}) we study algebro-geometrical aspect of autodual connections. Two interdependent algebro-geometric interpretations of autoduality arise. Both of these itterpretations are related to the twistor formalism, which harks back to the works of Penrose and Salamon \cite{Sal}. Twistor contruction is explained in detail in Section \ref{_twistors_Section_}; here we give a brief outline of this formalism. Every hyperk\"ahler manifold $M$ has a whole 2-dimensional sphere of integrable complex structures, called {\bf induced complex structures}; these complex structures correspond bijectively to $\R$-algebra embeddings from complex numbers to quaternions (see Definition \ref{_induced_co_str_Definition_}). We identify this 2-dimensional sphere with $\C P^1$. Gluing all induced complex structures together with the complex structure in $\C P^1$, we obtain an almost complex structure on the product $M \times \C P^1$ (Definition \ref{twistor}). As proven by Salamon \cite{Sal}, this almost complex structure is integrable. The complex manifold obtained in this way is called {\bf the twistor space for $M$}, denoted by $\Tw(M)$. Consider the natural projections \[ \sigma:\; \Tw(M)= M\times \C P^1 \arrow M, \ \ \ \pi:\; \Tw(M)= M\times \C P^1 \arrow \C P^1;\] the latter map is holomorphic. The key statement to the twistor transform is the following lemma. \begin{lemma}\label{_autodua_(1,1)-on-twi-intro_Lemma_} Let $({\cal B}, \nabla)$ be a bundle with a connection over a hyperk\"ahler manifold $M$, and \[ (\sigma^*{\cal B}, \sigma^* \nabla)\] be the pullback of $({\cal B}, \nabla)$ to the twistor space. Then $(\sigma^*{\cal B}, \sigma ^* \nabla)$ is holomorphic if and only if $({\cal B}, \nabla)$ is autodual. \end{lemma} \proof This is a restatement of Lemma \ref{_autodua_(1,1)-on-twi_Lemma_}. \endproof This gives a natural map from the space of autodual connections on $M$ to the space of holomorphic bundles on $\Tw(M)$. We prove that this map is injective, and describe its image explicitly. For every points $x\in M$, the set $\sigma^{-1}(x)$ is a complex analytic submanifold of the twistor space. The projection $\pi\restrict{\sigma^{-1}(x)}:\; \sigma^{-1}(x) \arrow \C P^1$ gives a canonical identification of $\sigma^{-1}(x)$ with $\C P^1$. The rational curve $\sigma^{-1}(x) \subset \Tw(M)$ is called {\bf a horisontal twistor line in $\Tw(M)$} (Definition \ref{twistor}). The following proposition provides an inverse to the map given by Lemma \ref{_autodua_(1,1)-on-twi-intro_Lemma_}. \begin{prop} Let $M$ be a hyperk\"ahler manifold, $\Tw(M)$ its twistor space and ${\cal E}$ a holomorphic bundle over $\Tw(M)$. Then ${\cal E}$ comes as a pullback of an autodual bundle $({\cal B}, \nabla)$ if and only if restriction of ${\cal E}$ to all horisontal twistor lines is trivial as a holomorphic vector bundle. Moreover, this autodual bundle is unique, up to equivalence. \end{prop} \proof This is a restatement of Theorem \ref{_twisto_transfo_equiva_Theorem_}. \endproof We obtained an identification of the set of equivalence classes of autodual bundles with a subset of the set of equivalence classes of bundles over the twistor space. We would like to interpret this identification geometrically, as an identification of certain moduli spaces. The autodual bundles are NHYM. The set of equivalence classes of stable NHYM bundles is equipped with a natural complex structure and is finite-dimensional, as we prove in Section \ref{_NHYM_Section_}. This general construction is used to build the moduli space of autodual bundles. It remains to define the notion of stability for holomorphic bundles over the twistor space and to construct the corresponding moduli space. The usual (Mumford-Takemoto) notion of stability does not work, because twistor spaces are not K\"ahler.\footnote{Moreover, as can be easily shown, the twistor space of a compact hyperk\"ahler manifold admits no K\"ahler metric.} We apply results of Li and Yau \cite{yl}, who define a notion of stability for bundles over complex manifolds equipped with a Hermitian metric satisfying a certain condition (see \eqref{yl}). The twistor space $\Tw(M)$ is isomorphic as a smooth manifold to $\C P^1 \times M$ and as such is equipped with the product metric. This metric is obviously Hermitian. We check the condition of Li and Yau for twistor spaces by computing the terms of \eqref{yl} explicitly. This enables us to speak of stable and semistable bundles over twistor spaces. Let ${\cal E}$ be a holomorphic bundle over $\Tw(M)$ obtained as a pullback of an autodual bundle on $M$. We prove that ${\cal E}$ is semistable. This gives a holomorphic interpretation of the moduli of autodual bundles on $M$. This is the first of our algebro-geometric interpretations. The second interpretation involves significantly more geometry, but yields a more explicit moduli space. Let $M$ be a compact hyperk\"ahler manifold, and ${\cal B}$ a complex vector bundle with first two Chern classes invariant under the natural $SU(2)$-action. Let $\widehat M$ be the moduli space for the Hermitian Yang-Mills connections on ${\cal B}$.\footnote{Such connections are always autodual, \cite{Vb}.} Then $\widehat M$ is equipped with a natural hyperk\"ahler structure (\cite{Vb}). The first result of this type was obtained by Mukai \cite{_Mukai:K3_} in the context of his duality between K3-surfaces; we use the term ``Mukai dual'' for $\widehat M$ in this more general situation. Let $X\stackrel{\pi}{\arrow} \C P^1$, $\widehat X\stackrel{\hat \pi}{\arrow} \C P^1$ be the twistor spaces for $M$, $\widehat M$, equipped with the natural holomorphic projections to $\C P^1$. For an induced complex structure $L$ on the hyperk\"ahler manifold $M$, we denote by $(M, L)$ the space $M$ considered as a complex K\"ahler manifold with $L$ as a complex structure. Identifying the set of induced complex structures with $\C P^1$, we consider $L$ as a point in $\C P^1$. Then, the complex manifold $(M, L)$ is canonically isomorphic to the pre-image $\pi^{-1}(L) \subset X$. By $i_L$ we denote the natural embedding $(M, L) = \pi^{-1}(L) \stackrel{i_L}{\hookrightarrow} X$. Let $B$ be a stable holomorphic bundle over $(M, L)$, with the complex vector bundle ${\cal B}$ as underlying complex vector space. In \cite{Vb}, we produce a canonical identification between the moduli space of such stable bundles and the space $\widehat M$ of autodual connections in ${\cal B}$. Let ${\cal F}_{B} = {i_L}_* {B}$ be the coherent sheaf direct image of ${B}$ under $i_L$. The moduli space of such sheaves ${\cal F}_{B}$ is naturally identified with $\widehat X$ (Section \ref{_lines_Section_}; see also \cite{Vb}). Consider a holomorphic section $s$ of the map $\hat \pi:\; \widehat X \arrow \C P^1$, that is, a holomorphic embedding $s:\; \C P^1 \arrow \widehat X$ such that $s\circ \hat \pi = \id$. The image of such embedding is called {\bf a twistor line in $\widehat X$} (Section \ref{_twistors_Section_}). Let ${\cal E}$ be a vector bundle over $X$ such that the pullback ${i_L}^* {\cal E}$ is stable for all induced complex structures $L\in \C P^1$. Such a bundle ${\cal E}$ is called {\bf fiberwise stable} (Definition \ref{fib.st}). {}From Lemma \ref{gen.st} it follows that fiberwise stable bundles are also stable, in the sense of Li--Yau. We restrict our attention to those bundles ${\cal E}$ which are, as $C^\infty$-vector bundles, isomorphic to $\sigma^*({\cal B})$, where ${\cal B}$ is our original complex vector bundle on $M$. Every fiberwise stable bundle ${\cal E}$ on $X$ gives a twistor line $s_{\cal E}:\; \C P^1 \arrow \widehat X$ in $\widehat X$, where $s_{\cal E}$ associates a sheaf ${i_L}_* {i_L}^* {\cal E}$ to $L\in \C P^1$. Since points of $\widehat X$ are identified with isomorphism classes of such sheaves, the sheaf ${i_L}_* {i_L}^* {\cal E}$ can be naturally considered as a point in $\widehat X$. Clearly, the moduli $St_f(X)$ space of fiberwise stable bundles is open in the moduli $St(X)$ of stable bundles on $X$. This gives a complex structure on $St_f(X)$. The set $Sec(\widehat X)$ of twistor lines in $\widehat X$ is equipped with a complex structure as a subset of the Douady space of rational curves in $\widehat X$.\footnote{Douady spaces are analogues of Chow schemes, defined in the complex-analytic (as opposed to algebraic) setting.} We constructed a holomorphic map from $St_f(X)$ to $Sec(\widehat X)$. We prove that this map is in fact an isomorphism of complex varieties (Theorem \ref{iso}). The direct and inverse twistor transform give a canonical identification between the moduli of autodual bundles on $M$ and an open subset of the moduli of semi-stable bundles on $X$. Thus, we obtain an identification of the moduli of autodual bundles on $M$ and the space of twistor lines in $\widehat X$. We must caution the reader that in this introduction we mostly ignore the fact that all our constructions use different notions of stability; thus, all identifications are valid only locally in the subset where all the flavours of stability hold. The precise statements are given in Sections \ref{_twisto-tra_Section_}--\ref{_lines_Section_}. \subsection{Contents} \begin{itemize} \item Here are the contents of our article. \item The Introduction is in two parts: the first part explains the main ideas of this paper, and the second gives an overview of its content, section by section. These two parts of Introduction are independent. The introduction is also formally independent from the main part and vice versa. The reader who prefers rigorous discourse might ignore the introduction and start reading from Section \ref{_NHYM_Section_}. \item Section \ref{_NHYM_Section_} contains the definition of NHYM (non-\-Her\-mi\-tian Yang-\-Mills) connection. We give the definition of $(0,1)$-sta\-bi\-lity for NHYM connections and consider the natural forgetful map \begin{equation} \label{_(0,1_stable_to_holo-Equation_} \pi:\; {\cal M}^s \arrow {\cal M}^s_0 \end{equation} from the space of $(0,1)$-stable NHYM-connections to the moduli space of stable holomorphic bundles. The fiber of this map is described explicitly through a power series and the Green operator (Proposition \ref{series}). This map is also used to show that the moduli space of $(0,1)$-stable NHYM-connections is correctly defined and finite-dimensional (Corollary \ref{_NHYM_finite-dim_Corollary_}). Uhlenbeck--Yau theorem (Theorem \ref{_Uhle-Yau_Theorem_}; see also \cite{UY}) provides a compatible Hermitian Yang-Mils connection for every stable holomorphic bundle. This gives a section ${\cal M}^s_0 \stackrel i \hookrightarrow {\cal M}^s$ of the map \eqref{_(0,1_stable_to_holo-Equation_}. We study the structure of ${\cal M}^s$ in the neighbourhood of $i\left({\cal M}^s_0\right)$, and prove that this neighbourhood is isomorphic to the complexification of ${\cal M}^s_0$ in the sense of Grauert (Proposition \ref{_complexi_Graue_Theorem_}). \item In Section \ref{_autodu_Section_}, we recall the definition of a hyperk\"ahler manifold and consider NHYM bundles over a complex manifold with a hyperk\"ahler metric. We define autodual bundles over hyperk\"ahler manifolds (Definition \ref{_autodual_Definition_}) and show that all autodual bundles are NHYM (Proposition \ref{_autodual_is_NHYM_Proposition_}). We cite the result of \cite{Vb}, which shows that all Hermitian Yang-Mills connections on a bundle ${\cal B}$ are autodual, if the first two Chern classes of ${\cal B}$ satisfy a certain natural assumption ($SU(2)$-invariance; see Theorem \ref{HYM.inv}). We also prove that, for a NHYM connection $\nabla$ sufficiently close to Hermitian, $\nabla$ is autodual (Theorem \ref{_NHYM-are-autodu_Theorem_}). \item Further on, we restrict our attention to autodual connection over hy\-per\-k\"ah\-ler manifolds. \item Section \ref{_twistors_Section_} gives a number of definition and preliminary results from algebraic geometry of the twistor spaces. We define the twistor space for an arbitrary hyperk\"ahler manifold (Definition \ref{twistor}). The twistor space is a complex manifold equipped with a holomorphic projection onto $\C P^1$. For most hyperk\"ahler manifolds (including all compact ones), the twistor space does not admit a K\"ahler metrics. This makes it difficult to define stability for bundles over twistor spaces. We overcome this difficulty by applying results of Li and Yau (\cite{yl}). We consider the differential form which is an imaginary part of the natural Hermitian metric on the twistor space. To apply \cite{yl}, we compute explicitly the de Rham differential of this form (Lemma \ref{_differe_of_Hermi_on_twistors_Lemma_}). \item In Section \ref{_twisto-tra_Section_} we define the direct and inverse twistor transform relating autodual bundles over a hyperk\"ahler manifold $M$ and holomorphic bundles over the corresponding twistor space $X$. There is a map from the set of isomorphism classes of autodual bundles on $M$ to the set of isomorphism classes of holomorphic bundles on $X$ (Lemma \ref{_autodua_(1,1)-on-twi_Lemma_}). We show that this map is an embedding and describe its image explicitly (Theorem \ref{_twisto_transfo_equiva_Theorem_}). \item In Section \ref{_stabi_of_twi_tra_Section_}, we consider holomorphic bundles on the twistor space obtained as a result of a twistor transform. We prove semistability of such bundles. Thus, twistor transform is interpreted as a map between moduli spaces. \item In Section \ref{_lines_Section_} we return to the study the algebro-geometric properties of the twistor space. For a compact hyperk\"ahler manifold $M$ and a stable holomorphic bundle $B$ on $M$, we consider the space $\widehat M$ of deformations of $B$. When the first two Chern classes of $B$ are $SU(2)$-invariant, the space $\widehat M$ has a natural hyperk\"ahler structure; this space is called then {\bf Mukai dual} to $M$. Let $X$, $\widehat X$ be the twistor spaces for $M$ and $\widehat M$. We interpret the space of stable bundles on $X$ in terms of rational curves on $\widehat X$ (Theorem \ref{iso}). \item In Section \ref{_conje_Section_}, we relate a number of conjectures and open questions from the geometry of NHYM and autodual bundles. \end{itemize} \section{The general case.} \label{_NHYM_Section_} \subsection{Definition of NHYM connections} Let $M$ be a K\"ahler manifold of dimension $n$ with the real valued K\"ahler form $\omega$. Consider a complex vector bundle $\E$ on $M$. Denote by $\A^n(\E)$ the bundle of smooth $\E$-valued $n$-forms on $\E$. Let $$ \A^n(\E) = \bigoplus_{i+j=n} \A^{i,j}(\E) $$ be the Hodge type decomposition. The bundle $\End\E$ of endomorphisms of $\E$ is also a complex vector bundle. As usual, let $$ L:\A^{\cdot,\cdot}(\End\E) \to \A^{\cdot+1,\cdot+1}(\End\E) $$ be the operator given by multiplication $\omega$. Let $$ \Lambda:\A^{\cdot+1,\cdot+1}(\End\E) \to \A^{\cdot,\cdot}(\End\E) $$ be the adjoint operator with respect to the trace form on $\End\E$. \begin{defn}\label{_NHYM_Definition_} A connection $\nabla:\E \to \A^1(\E)$ is called {\emp non-Hermitian Yang-Mills} (NHYM for short) if its curvature $R \in \A^2(\End\E)$ is of Hodge type $(1,1)$ and satisfies $$ \Lambda \circ R = c \Id $$ for a certain constant $c \in \C$. \end{defn} \begin{rem} This terminology is perhaps unfortunate, in that a NHYM connection can (but need not) be Hermitian. We use the term for lack of better one. \end{rem} To simplify exposition, we will always consider only NHYM connections with the constant $c=0$. Let $\nabla$ be a NHYM connection on $\E$. Since the $(0,2)$-component of its curvature vanishes, the $(0,1)$-component $\nabla^{0,1}:\E \to \A^{0,1}(\E)$ defines a holomorphic structure on $\E$. We will call this {\emp the holomorphic structure associated to $\nabla$}. Let $\bE$ be the dual to the complex-conjugate to the complex bundle $\E$. Every NHYM-connection $\nabla$ obviously induces a NHYM connection $\nabla^*$ on the dual bundle $\E^*$. Let $\bnabla$ be the connection on $\bE$ complex-conjugate to $\nabla^*$. The connection $\bnabla$ is also obviously NHYM. We will call it {\emp the adjoint connection to $\nabla$}. The holomorphic structure on $\bE$ associated to $\bnabla$ will be called {\emp the adjoint holomorphic structure associated to $\nabla$}. Note that the adjoint holomorphic structure depends only on the $(1,0)$-part $\nabla^{1,0}$ of the connection $\nabla$. \subsection{Stability and moduli of NHYM connections.} Fix a compact K\"ahler manifold $M$ and a complex vector bundle $\E$ on $M$. Consider the space $\A$ of all connections on $\E$ and let $\A_0$ be the subspace of NHYM connnections. The space $\A$ is a complex-analytic Banach manifold, and $\A_0 \subset \A$ is an analytic subspace of $\A$. Let $\G = \Maps(M,\Aut\E)$ be the complex Banach-Lie group of automorphisms of $\E$. The group $\G$ acts on $\A$ preserving the subset $\A_0$. In order to obtain a good moduli space for NHYM connections, we need to impose some stability conditions. \begin{defn} A NHYM connection $\nabla$ is called {\emp $(0,1)$-stable} if the bundle $\E$ with the associated holomorphic structure is a stable holomorphic bundle. \end{defn} Further on, we sometimes use the term {\it stable} to denote $(0,1)$-stable connections. \begin{rem}\label{too.strong} This definition is sufficient for our present purposes. However, it is unnaturally restrictive. See \ref{_stabili_for_hyper_redu-Definition_} for a more natural definition. \end{rem} Let $\A^s \subset \A_0$ be the open subset of $(0,1)$-stable NYHM connections and let $$ \M = \A_s / \G $$ be the set of {\em equivalence classes} of $(0,1)$-stable NHYM connections on $\E$ endowed with the quotient topology. Choose a connection $\nabla \in \A_0, \nabla:\E \to \A^1(\E)$. Let $\nabla = \nabla^{1,0} + \nabla^{0,1}$ be the type decomposition and extend both components to differentials \begin{align*} D:\A^{\cdot,0}(\End\E) &\to \A^{\cdot+1,0}(\End\E)\\ \overline{D}:\A^{0,\cdot} &\to \A^{0,\cdot+1}(\End\E) \end{align*} The tangent space $T_\nabla(\A)$ equals $T_\nabla(\A) = \A^1(\End\E)$. The NHYM equations define a complex-analytic map $$ YM:\A \to \A^{2,0}(\End\E) \oplus \A^{0,2} \oplus \A^0(\End\E). $$ It is easy to see that the differential of $YM$ at the point $\nabla$ is given by $$ YM_\nabla = D + \overline{D} + \Lambda \nabla: \A^1(\End\E) \to \A^{2,0}(\End\E) \oplus \A^{0,2}(\End\E) \oplus \A^0(\End\E) $$ On the other hand, the differential at $\nabla$ of the $\G$-action on $\A$ is given by $$ \nabla:\A^0(\End\E) \to \A^1(\End\E) $$ \begin{defn} The complex $$ 0 \to \A^0(\End\E) \to \A^1(\End\E) \to \A^{2,0}(\End\E) \oplus \A^{0,2}(\End\E) \oplus \A^0(\End\E) $$ is called {\emp the deformation complex} of the NHYM connection $\nabla$. \end{defn} \begin{rem} The deformation complex has a natural structure of a differential graded Lie algebra. \end{rem} \begin{prop} The deformation complex is elliptic. \end{prop} \proof Indeed, the complex $$ 0 \to \A^0(\End\E) \to \A^{0,1}(\End\E) \to \A^{0,2}(\End\E) $$ is the Dolbeault complex for the holomorphic bundle $\End\E$ and is therefore elliptic. Hence it is enough to prove that \begin{equation}\label{kernel} 0 \to \A^{1,0}(\End\E) \to \A^{2,0}(\End\E) \oplus \A^0(\End\E) \end{equation} is elliptic. By Kodaira identity $\Lambda D = \sqrt{-1} D^*$ on $\A^{1,0}(\End\E)$, and this complex is the same as $$ 0 \to D \oplus D^*:\A^{1,0}(\End\E) \to \A^{2,0}(\End\E) \oplus \A^0(\End\E), $$ where $D^*$ is defined by means of the trace form on $\End\E$. This complex is obviously elliptic. \endproof \begin{corr} Let $\wt{G} \subset \G$ be the stabilizer of $\nabla \in \A$. Then \begin{enumerate} \item $\wt{G}$ is a finite dimensional complex Lie group. \item There exists a finite dimensional locally closed complex-analytic Stein subspace $\wt{\M} \subset \A$ containing $\nabla$ and invariant under $\wt{G}$ such that the natural projection $$ \wt{\M} / \wt{G} \to \M $$ is an open embedding. \end{enumerate} \end{corr} \proof This is the standard application of the Luna's slice theorem, see \cite{Kod}. \endproof \begin{corr} \label{_NHYM_finite-dim_Corollary_} The topological space $\M$ has a natural structure of a comp\-lex\--analytic space. \end{corr} \proof Indeed, since $\wt{\M}$ is Stein, the quotient $\wt{\M} / \wt{G}$ is a Stein complex-analytic space. Now by the standard argument (\cite{Kod}) the induced complex analytic charts on $\M$ glue together to give a complex-analytic structure on the whole of $\M$. \endproof \subsection[Hermitian Yang-Mills bundles and the theorem of Uhlenbeck--Yau]{Hermitian Yang-Mills bundles and the theorem of \\ Uhlenbeck--Yau} For every complex bundle $\E$ on $M$ denote by $\M_0(\E)$ the moduli space of stable holomorphic structures on $\E$. Fix $\E$ and consider the space $\M$ of $(0,1)$-stable NHYM connections on $\E$. Taking the associated holomorphic structure defines a map $\pi:\M \to \M_0(\E)$. \begin{lemma} The map $\pi$ is holomorphic. \end{lemma} \proof Clear. \endproof Since every complex vector bundle admits an Hermitian metric, the complex vector bundles $\E$ and $\bE$ are isomorphic. Therefore the moduli spaces $\M_0(\E)$ and $\M_0(\bE)$ are naturally identified. Denote the space $\M_0(\E) = \M_0(\bE)$ simply by $\M_0$, and let $\bM_0$ be the complex-conjugate space. Consider the open subset $\gM \subset \M$ of $(0,1)$-stable NHYM connections on $\E$ such that the adjoint connection on $\bE$ is also $(0,1)$-stable. Taking the adjoint holomorphic structure defines a map $\bpi:\gM \to \bM_0$. This map is also obviously holomorphic. \begin{lemma} \label{_compa_Hermi_pi=barpi_Lemma_} A NHYM connection $\nabla \in \gM$ satisfies $\pi(\nabla) = \bpi(\nabla)$ if and only if it admits a compatible Hermitian metric. \end{lemma} \proof Indeed, $\pi(\nabla) = \bpi(\nabla)$ if and only if there exists an isomorphism $h:\E \to \bE$ sending $\nabla$ to $\bnabla$. This isomorphism defines an Hermitian metric on $\E$ compatible with the connection $\nabla$. \endproof To proceed further we need to recall the following fundamental theorem. \begin{theorem}[Uhlenbeck,Yau] \label{_Uhle-Yau_Theorem_} Every stable holomorphic bundle $\E$ on a K\"ahler manifold $M$ admits a unique Hermitian Yang-Mills connection $\nabla$. Vice versa, every holomorphic bundle admitting such a connection is polystable. \end{theorem} We will call such a metric {\emp a Uhlenbeck-Yau metric} for the holomorphic bundle $\E$. \begin{corr} Let $\M_u \in \gM$ be subset of equivalence classes of Hermitian connections. The product map $\pi \times \bpi:\gM \to \M_0 \times \bM_0$ identifies $\M_u$ with the diagonal in $\M_0 \times \bM_0$. \end{corr} \proof Clear. \endproof \begin{rem} Note that the subset $\M_u \subset \gM$ is not complex-analytic, but only real-analytic. \end{rem} \subsection{Moduli of NHYM connections as a complexification of the moduli of stable bundles} Let $\nabla \in \M_u \subset \M$ be an Hermitian Yang-Mills connection, and let $\F_\nabla = \pi^{-1}(\pi(\nabla)) \subset \M$ be the fiber of $\pi$ over $\nabla$. In order to study the map $\pi \times \bpi:\gM \to \M_0 \times \bM_0$ in a neighborhood of $\nabla$, we first study the restriction of the map $\bpi$ to $\F_\nabla$. We begin with the following. \begin{theorem}\label{zhopa} Let $\overline{D}:\E \to \A^{0,1}(\E)$ be a representative in the equivalence class $\pi(\nabla) \in \M_0$ of holomorphic structures on $\E$ and let $D:\E \to \A^{1,0}(\E)$ be the operator adjoint to $\overline{D}$ with respect to the Uhlenbeck-Yau metric. The fiber $\F_\nabla$ is isomorphic to the set of all $\End\E$-valued $(1,0)$-forms $\theta$ satisfying \begin{equation}\label{formula} \begin{cases} D\theta + \theta \wedge \theta &= 0,\\ D^*\theta &= 0. \end{cases} \end{equation} \end{theorem} \proof To define the desired isomorphism, choose for any equivalence class $\nabla_1 \in \pi^{-1}(\pi(\nabla))$ a representative $\nabla_1 = \nabla^{1,0}_1 + \overline{D}:\E \to \A^1(\E)$. Every two representatives must differ by a gauge transformation $g:\E \to \E$. The map $g$ must preserve the holomorphic structure $\overline{D}$. However, this holomorphic structure is by assumption stable. Therefore $g = c\Id$ for $c \in \C$, and the operator $\nabla^{1,0}_1:\E \to \A^{1,0}(\E)$ is defined uniquely by its class in $\F_\nabla$. Take $\theta = \nabla^{1,0}_1 - D$; the equations~\eqref{formula} follow directly from the definition of NHYM connections. \endproof In order to apply this Theorem, note that by the second of the equations~\eqref{formula} every NHYM connection $\nabla_1 \in \F_\nabla$ defines a $D^*$-closed $\End\E$-valued $(1,0)$-form $\theta$. Complex conjugation with respect to the Uhlenbeck-Yau metric $h_\nabla$ identifies the space of $D^*$-closed $\End\E$-valued $(1,0)$-forms with the space of $\overline{D}^*$-closed $(0,1)$-forms, and it also identifies the respective cohomology spaces. But the cohomology spaces of the Dolbeault complex $\A^{0,\cdot}(\End\E)$ with respect to $\overline{D}$ and $\overline{D}^*$ are both equal to the space of harmonic forms, hence naturally isomorphic. Collecting all this together, we define a map $\rho:\F_\nabla \to \overline{H^1(M,\End\E)}$ by the rule \begin{equation} \label{_from_NHYM_to_classes_Equation_} \nabla_1 \mapsto \langle \text{ class of } \theta \text{ in } H^1(M,\End\E) \rangle. \end{equation} \begin{prop}\label{kur} The map $\rho$ is a closed embedding in a neighborhood of $\nabla \in \M_u \in \F_\nabla$. \end{prop} This Proposition can be deduced directly from Theorem~\ref{zhopa}. However, we prefer to prove a stronger statement. To formulate it, consider the adjoint holomorphic structure $\bpi(\nabla)$ on the complex bundle $\bE \cong \E$. Recall the following standard fact from the deformation theory of holomorphic bundles. \begin{theorem} Let $\bar\6:\E \to \A^{0,1}(\E)$ be a stable holomorphic structure on a complex Hermitian bundle $\E$. There exists a neighborhood $U \subset \M_0$ of $\bar\6$ such that every $\bar\6_1 \in U$ can be represented uniquely by an operator $\bar\6 + \theta:\E \to \A^{0,1}(\E)$ satisfying \begin{equation}\label{kuranishi} \begin{cases} \bar\6\theta = \theta \wedge \theta\\ \bar\6^*\theta = 0 \end{cases} \end{equation} The {\emp Kuranishi map} $U \to H^1(M,\End\E)$ defined by $$ \bar\6_1 \to \langle \text{ class of } \theta \text{ in } H^1_{\bar\6^*}(\A^{0,\cdot}(\End\E)) \cong H^1(M,\End\E) \rangle $$ is a locally closed embedding. \end{theorem} \begin{corr}\label{compl} The map $\bpi:\F_\nabla \to \bM_0$ is biholomophic in a neighborhood $V$ of $\nabla \in \M_u \subset \gM$, and the map $\rho:V \to H^1(M,\End\E)$ is the composition of $\bpi:\F_\nabla \to \bM_0$ and the Kuranishi map. \end{corr} \proof Complex conjugation sends equations~\eqref{formula} precisely to \eqref{kuranishi}, and thus establishes a bijection between neighborhoods of $\nabla \in \F_\nabla$ and $\bpi(\nabla) \in \bM_0$. \endproof This statement, in turn, implies the following. \begin{prop} \label{_complexi_Graue_Theorem_} The product map $\pi \times \bpi:\gM \to \M_0 \to \bM_0$ is biholomorphic on an open neighborhood $U$ of the subset $\M_u \in \gM$. \end{prop} \proof Consider both $\gM$ and $\M_0 \times \bM_0$ as spaces over $\M_0$. The map $\pi \times \bpi$ is a map over $\M_0$, and it is locally biholomorphic on every fiber of the natural projections $\gM \to \M_0$, $\M_0 \times \bM_0 \to \M_0$. \endproof Thus $\dim\M = 2\dim\M_0$, and an open neighborhood $U$ of the subspace $\M_u \subset \gM$ is the complexification of $\M_0$ is the sense of Grauert. \subsection{Holomorphic symplectic form on the moduli of NHYM bundles} In order to construct a holomorphic symplectic $2$-form on $\M$, we need to restrict our attention to a smooth open subset of $\M$. \begin{defn} A NHYM connection $\nabla$ is called {\emp smooth} if both $\nabla \in \M$ and $\pi(\nabla) \in \M_0$ are smooth points. \end{defn} Let $\nabla \in \M$ be a smooth NHYM connection and denote by $\CC_\nabla$ its deformation complex. By construction the holomorphic tangent space $T_\nabla(\M)$ is identified with a subspace of the first cohomology space $H^1(\CC_\nabla)$. By definition $\pi(\nabla)$ is a smooth point, and the tangent space $T_{\pi(\nabla)}(\M_0)$ is a subspace of the first cohomology space $H^1(M,\End\E)$ of $M$ with coefficients in $\E$ equipped with the induced holomorphic structure. Consider the natural projection map $\CC_\nabla \to \A^{0,i}(M,\End\E)$ from $\CC_\nabla$ to the Dolbeault complex of the bundle $\End\E$. Denote by $$ \pr:H^1(\CC_\nabla) \to H^1(M,\End\E) $$ the induced map on the cohomology spaces and let $$ W = \Ker\pr \subset H^1(\CC_\nabla). $$ \begin{prop}\label{diagramma} Let $d\pi:T_\nabla(\M) \to T_{\pi(\nabla)}(\M_0)$ be the differential of the map $\pi:\M \to \M_0$ in the smooth point $\nabla \in \M_0$. The diagram $$ \begin{CD} 0 @>>> T_\nabla(\F_\nabla) @>>> T_\nabla(\M) @>d{\pi}>> T_{\pi(\nabla)} @>>> 0 \\ @VVV @VVV @VVV @VVV @VVV \\ 9 @>>> W @>>> H^1(\CC_\nabla) @>{\pr}>> H^1(M,\End\E) @>>> 0 \end{CD} $$ is commutative. \end{prop} \proof Clear. \endproof We first construct a symplectic form on the space $H^1(\CC_\nabla)$. To do this, we first identify the space $W \subset H^1(\CC_\nabla)$. \begin{lemma} The space $W$ is naturally isomorphic to $H^{n-1}(M, \End\E \otimes \K)$, where $K$ is the canonical line bundle on $M$. \end{lemma} \proof Indeed, the space $W$ is isomorphic to the space of $\End\E$-valued $(1,0)$-forms satisfying \begin{equation}\label{pipupa} \begin{cases} D\theta = 0\\ \Lambda \overline{D} \theta = 0 \end{cases} \end{equation} Consider the map $$ \bullet \wedge \omega^{n-1}:\A^{1,0}(\End\E) \to \A^{n,n-1}(\End\E). $$ By Kodaira identities a form $\theta \in \A^1(M,\End\E)$ satisfies \eqref{pipupa} if and only if $\theta \wedge \omega^{n-1}$ is harmonic. Hence $\bullet \wedge \omega^{n-1}$ identifies $W$ with $H^{n-1}(M,\End\E \otimes \K)$. \endproof Consider now two $\End\E$-valued $1$-forms $\theta_0,\theta^1 \in \A^1(M,\End\E)$ and let $$ \Omega(\theta_0,\theta_1) = \int_M \tr(\theta_0 \wedge \theta_1 \wedge \omega^{n-1}), $$ where $\tr$ is the trace map, $n = \dim M$ and $\omega$ is the K\"ahler form on $M$. \begin{lemma} If $\theta_0 = \nabla g$ for some section $g \in \A^0(M,\End\E)$, then $$ \Omega(\theta_0,\theta_1) = 0 $$ for any $\theta_1 \in \A^1(M,\End\E)$ satisfying $\Lambda \nabla(\theta_1) = 0$. \end{lemma} \proof Indeed, \begin{multline*} \Omega(\theta_0,\theta_1) = \int_M \tr( \nabla(\omega^{n-1}g)) \wedge \theta_1) = \\ = \int_M \tr(\omega^{n-1} \wedge \nabla(g\theta_1)) - \int_M \tr( g \omega^{n-1} \wedge \nabla(\theta_1)) = \int_M \tr(g \Lambda \nabla(\theta_1)) \omega^n = 0 \end{multline*} \endproof \begin{corr}\label{bububuj} \begin{enumerate} \item The form $\Omega$ defines a complex $2$-form on the space $H^1(\CC_\nabla)$. \item The subspace $W$ is isotropic, and the induced pairing $$ (W \cong H^{n-1}(M,\End\E\otimes\K))\otimes H^1(M,\End\E) \to \C $$ is non-degenerate. \end{enumerate} \end{corr} \proof The first statement is clear. It is easy to see that the pairing induced by $\Omega$ is exactly the one defined by the Serre duality, which proves the second statement. \endproof Restricting to the subspace $T_\nabla(\M)$, we get a $2$-form $\Omega$ on $\M$. This form is obviously holomorphic. \begin{prop}\label{symplectic} Assume that either $\nabla \in \M$ is Hermitian, or $T_\nabla(\M) = H^1(\CC_\nabla)$. Then the form $\Omega$ on $T_\nabla(\M)$ is non-degenerate. The map $\M \to \M_0$ is a Lagrangian fibration in the neghrborhood of $\nabla$. \end{prop} \proof It is easy to see that $T_\nabla(\F_\nabla) \subset T_\nabla(\M)$ is isotropic. If the connection $\nabla$ is {\em unobstructed}, that is, the embedding $T_\nabla(\M) \hookrightarrow H^1(\CC_\nabla)$ is actually an isomorphism, then the statement follows from Corollary~\ref{bububuj}. Suppose now that the inclusion $T_\nabla(\M) \subset H^1(\CC_\nabla)$ is proper. By assumption the connection $\nabla$ is Hermitian in this case, therefore the complex conjugation map $\conj:\A^{0,1}(M,\End\E) \to \A^{1,0}(\End\E)$ is defined. It is easy to see that this map identifies $\Ker\pr$ with $\overline{H^1(M,\End\E)}$. By Corollary~\ref{compl} it also identifies $T_\nabla(\F_\nabla)$ with $\overline{T_{\pi(\nabla)}}$. The form $\Omega(\bullet,\overline{\bullet})$ is a non-degenerate Hermitian form on $\Ker\pr$. Therefore its restriction to $T_\nabla(M_\nabla)$ is also non-degenerate. The last statement now follows directly from Proposition~\ref{diagramma} and Corollary~\ref{bububuj}. \endproof \subsection{Local parametrization of the moduli of NHYM connections} In the last part of this section we give a more explicit description of the embedding $\F_\nabla \to H^1(\End\E)$ for an Hermitian Yang-Mills connection $\nabla$ in the spirit of \cite{Vb}. This description is of independent interest, and we will also use it in the next section in the study of NHYM connections on hyperk\"ahler manifolds. Fix an Uhlenbeck-Yau metric on $\E$ compatible with $\nabla$ and let $\Delta = DD^* + D^*D$ be the associated Laplace operator on $\A^\cdot(\End\E)$. Let $G$ be the Green operator provided by the Hodge theory. Recall that we have the Hodge decomposition $$ \A^\cdot(\End\E) = \HH^\cdot(\End\E) \oplus \A^\cdot_\ex(\End\E) $$ into the space of harmonic form $\HH^\cdot$ and its orthogonal complement $\A^\cdot_\ex$. This complement is further decomposed as $$ \A^\cdot_\ex(\End\E) = D\A^{\cdot-1}(\End\E) \oplus D^*\A^{\cdot+1}(\End\E) $$ The composition $DD^*G$ is by definition the projection onto $D\A^{\cdot-1}(\End\E) \subset \A^\cdot(\End\E)$. Take now a small neighborhood $U \subset \F_\nabla$ of the Hermitian connection $\nabla \in \F_\nabla$ and a NHYM connection $\nabla_1 \in U$. Let $\theta = \nabla_1 - \nabla$, $\theta \in \A^{1,0}(M,\End\E)$ and let $K(\theta) \in \overline{H^{1,0}(M,\End\E)}$ be the associated cohomology class. Shrinking $U$ if neccesary, we see that by Proposition~\ref{kur} the connection $\nabla_1$ is uniquely determined by the class $K(\theta)$. Let $\theta_0 \in \A^{1,0}(M,\End\E)$ be the harmonic form representing the class $K(\theta)$. Define by induction $$ \theta_n = D^*G \sum_{0 \leq k < n} \theta_k \wedge \theta_{n-1-k}. $$ \begin{prop}\label{series} Let $\nabla$ be an Hermitian Yang-Mills connection. There exists a neighborhood $V \subset \F_\nabla$ of $\nabla \in \F_\nabla$ such that for every $\nabla_1 \in V \subset \F_\nabla$ the series \begin{equation}\label{rjad} \sum_{0 \leq k} \theta_k \end{equation} converges to the form $\theta = \nabla_1 - \nabla$. \end{prop} \proof The metric on $\End\E$ defines a norm $\|\bullet\|$ on $\A^{\cdot,0}(\End\E)$. We can assume that $\left\| \theta_0 \right\| < \eps$ for any fixed $\eps > 0$. Since the Hodge decomposition is orthogonal, \begin{multline*} \left\|D \theta_n\right\| = \left\|DD^*G\left(\sum_{0 \leq k < n} \theta_k \wedge \theta_{n-k}\right)\right\| \\ \leq \left\|\sum_{0 \leq k < n} \theta_k \wedge \theta_{n-k}\right\| \leq \sum_{0 \leq k < n} \left\|\theta_k\right\| \cdot \left\|\theta_{n-k}\right\|. \end{multline*} Since $D:D^*(\A^{2,0}(\End\E)) \to \A^{2,0}(\End\E)$ is injective and elliptic, there exists a constant $C > 0$ such that $$ \|Df\| > C\|f\| $$ for all $f \in D^*(\A^{2,0}(\End\E))$. Let $a_n = \frac{(2n)!}{(n!)^2}$ be the Catalan numbers. By induction $$ \left\|\theta_n\right\| < a_n \left| \frac{\eps}{C} \right|. $$ Since $A(z) = \sum a_n z^n$ satisfies $A(z) = 1 + z(A(z))^2$, it equals $$ A(z) = \frac{1 - \sqrt{1-4z}}{2} $$ and converges for $z < \frac{1}{4}$. Therefore the series~\eqref{rjad} converges for $4\eps < C$. To prove that it converges to $\theta$, let $\chi_0 = \theta$ and let $$ \chi_n = \theta - \sum_{0 \leq k < n} \theta_k $$ for $n \geq 1$. Since both $\nabla$ and $\nabla + \theta$ are NHYM, we have $D\chi_1 = \chi_0 \wedge \chi_0$. Therefore $\chi_0 \wedge \chi_0 \in \A^{2,0}(\End\E)$ and $$ D \chi_1 = D D^* G (\chi_0 \wedge \chi_0) = \chi_0 \wedge \chi_0. $$ By induction $$ D \chi_n = \chi_0 \wedge \chi_{n-1} + \sum_{0 \leq k \leq n} \chi_k \wedge \theta_{n - 1 - k}, $$ and $\chi_n \in D^*(\A^{2,0}(M,\End\E))$ for all $n > 0$. Again by induction, $$ \left\|\chi_n\right\| < a_{n+1} \left(\frac{\eps}{C}\right)^n. $$ Therefore $\chi_n \to 0$ if $4\eps < C$, which proves the Proposition. \endproof \begin{rem} This Proposition can be strengthened somewhat. Namely, for any harmonic $\End_\E$-valued $(1,0)$-form $\theta_0$ in a small neighborhood of $0$ the series~\eqref{rjad} converges to a form $\theta$. As follows from \cite{Vb}, the connection $\nabla + \theta$ is NHYM provided the following holds. \begin{description} \item[*] All the forms $\theta_p \wedge \theta_q \in \A^{2,0}(M,\End\E)$ lie in $\A^{2,0}_\ex(M,\End\E)$. \end{description} This condition is also knows as vanishing of all of the so-called Massey products $[\theta_0 \wedge \ldots \wedge \theta_0]$. \end{rem} \section{Autodual and NHYM connections in the hy\-per\-k\"ah\-ler case.} \label{_autodu_Section_} We now turn to the study of NHYM connections on hyperk\"ahler manifolds. First we recall the definitions and some general facts. \begin{defn}[\cite{Cal}] \label{defn.hyperkahler} A {\emp hyperk\"ahler manifold} is a Riemannian manifold $M$ equipped with two integrable almost complex structures $I,J$ which are parallel with respect to the Levi-Civita connection and satisfy $$ I \circ J = - J \circ I. $$ \end{defn} Let $M$ be a hyperk\"ahler manifold. The operators $I,J$ define an action of the quaternion algebra $\h$ on the tangent bundle $TM$. This action is also parallel. Every imaginary quaternion $a \in \h$ satisfying $a^2 = -1$ defines an almost complex structure on $M$. This almost complex structure is parallel, hence integrable and K\"ahler. \begin{defn} \label{_induced_co_str_Definition_} A complex structure on $M$ corresponding to an imaginary quaternion $a \in \h$ with $a^2 = -1$ is said to be {\emp induced by $a$}. \end{defn} For every such $a \in \h$ we will denote by $\omega_a$ the K\"ahler form in the complex structure induced by $a$. We will always assume fixed a preferred complex K\"ahler structure $I$ on $M$. Recall that every hyperk\"ahler manifold is equipped with a canonical holomorphic symplectic $2$-form $\Omega$. If $J,K \in \h$ satisfy $J^2 = -1$, $IJ=K$ then this form equals $$ \Omega = \omega_J + \sqrt{-1}\omega_K. $$ The group $U(\h)$ of all unitary quaternions is isomorphic to $\su$. Thus every hyperk\"ahler manifold comes equipped with an action of $\su$ on its tangent bundle, and, {\it a posteriori}, with an action of its Lie algebra $\ssu$. Extend these actions to the bundles $\Lambda^\cdot$ of differential forms and let $\Lambda^\cdot_\inv \subset \Lambda^\cdot$ be the subbundle of $\su$-invariant forms. The $\su$-action does not commute with the de Rham differential. However, it does commute with the Laplacian (see \cite{Vl}). Therefore it preserves the subspace of harmonic forms. Identifying harmonic forms with coholomogy classes, we get an action of $\su$ on the cohomology spaces $H^\cdot(M)$. Let $\Lambda:\Lambda^{\cdot+2} \to \Lambda^\cdot$ be the Hodge operator associated to the K\"ahler metric on $M$. \begin{defn} A differential form $\theta$ on $M$ is called {\emp primitive} if $\Lambda\theta = 0$. \end{defn} \begin{lemma}\label{primitive} \begin{enumerate} \item All $\su$-invariant forms are primitive. All $\su$-in\-va\-ri\-ant $2$-forms are of Hodge type $(1,1)$ for every one of the induced complex structures on $M$. Vice versa, if a form is of type $(1,1)$ for all the induced complex structures, it is $\su$-invariant. \item The same statements hold for de Rham cohomology classes instead of forms. \end{enumerate} \end{lemma} \proof See \cite{Vb}, Lemma 2.1 \endproof \begin{rem} The converse is true for $\dim_\C M = 2$, but in higher dimensiona there are primitive forms that are not $\su$-invariant. \end{rem} Consider a complex bundle $\E$ on $M$ and let $\nabla:\E \to \A^1(\E)$ be a connection on $\E$. \begin{defn}\label{_autodual_Definition_} The connection $\nabla$ is called {\emp autodual} if its curvature $R \in \A^2(M,\End\E)$ is $\su$-inavariant. \end{defn} \begin{rem} The terminology comes from the $4$-dimensional topology: autodual connections on hyperk\"ahler surfaces are anti-selfdual in the usual topological sense. \end{rem} We will call an autodual connection $\nabla$ {\emp $(0,1)$-stable} if its $(1,0)$-part defines a stable holomorphic structure on the bundle $\E$. Denote by $\M_\inv$ the set of equivalence classes of $(0,1)$-stable autodual connections on the bundle $\E$. \begin{rem} Like in Remark~\ref{too.strong}, this $(0,1)$-stability condition may be too restrictive. \end{rem} Let $\nabla$ be an autodual connection on $\E$. By Lemma~\ref{primitive} for every $J \in \cp$ the $(0,1)$-component of the connection $\nabla$ with respect to the complex structure induced by $J$ defines a holomorphic structure on $\E$. We will call it {\emp the holomorphic structure induced by $J$}. \begin{prop} \label{_autodual_is_NHYM_Proposition_} Let $M$ be a hyperk\"ahler manifold and let $\E$ be a complex bundle on $M$. Every autodual connection $\nabla$ on $\E$ is NHYM. \end{prop} \proof Immediately follows from Lemma~\ref{primitive}. \endproof Therefore there exists a natural embedding $\M_\inv \hookrightarrow \M$ from $\M_\inv$ to the moduli space $\M$ of NHYM connections on $\E$. In the rest of this section we give a partial description of the image of this embedding. We will use the following. \begin{theorem}\label{HYM.inv} Assume that the first two Chern classes $$ c_1(\E),c_2(\E) \in H^\cdot(M) $$ of the bundle $\E$ are $\su$-invariant. Then every Hermitian Yang-Mills connection $\nabla$ on $\E$ is autodual. \end{theorem} \proof See \cite{Vb}. \endproof Therefore, if the Chern classes $c_1(\E),c_2(\E)$ are $\su$-invariant, then the closed subset $\M_u \in \M$ of Hermitian Yang-Mills connections lies in $\M_\inv$. \begin{theorem} \label{_NHYM-are-autodu_Theorem_} Let $M$ be a hyperk\"ahler manifold and let $\E$ be a complex vector bundle on $M$ such that the Chern classes $c_1(\E),c_2(\E)$ are $\su$-invariant. Then subset $\M_\inv \subset \M$ of autodual connections contains an open neighborhood of the subset $\M_u \subset \M$ of connections admitting a compatible Hermitian metric. \end{theorem} \proof Let $\nabla$ be an Hermitian Yang-Mills connection. It is autodual by Theorem~\ref{HYM.inv}, and it is enough to show that every $\nabla_1 \in \F_\nabla$ sufficiently close to $\nabla$ is also autodual. Let $\theta = \nabla_1 - \nabla$ and let $\theta_n, n \geq 0$ be as in \eqref{rjad}. By Propostion~\ref{series} we can assume that $$ \theta = \sum_k \theta_k. $$ It is enough to prove that $\overline{D}\theta$ is $\su$-invariant. We will prove that $\overline{D}\theta_k$ is $\su$-invariant for all $k \geq 0$. To do this, we use results of \cite{Vb}. Consider the operator $$ L^\Omega:\A^{\cdot,\cdot}(\End\E)\to\A^{\cdot+2,\cdot}(\End\E) $$ given by multiplication by the canonical holomorphic $2$-form $\Omega$ on $M$. By \cite{Vl} the operator $$ [L_\Omega,\Lambda]:\A^{\cdot,\cdot}(\End\E) \to \A^{\cdot+1,\cdot-1}(\End\E) $$ coincides with the action of a nilpotent element in the Lie algebra $\ssu$. Therefore it is a derivation with respect to the algebra structure on the complex $\A^{\cdot,\cdot}(\End\E)$. Moreover, a $(1,1)$-form $\alpha$ is $\su$-invariant if and only if $[L_\Omega,\Lambda]\alpha = 0$. Let $\6^J:\A^{\cdot,\cdot}(\End\E) \to \A^{\cdot+1,\cdot}(\End\E)$ be the commutator $$ \6^J = \left[\overline{D} , \left[L_\Omega,\Lambda\right]\right]. $$ This map again is a derivation with respect to the algebra structure on $\A^{\cdot,\cdot}(\End\E)$, and it is enough to prove that $\6^J\theta_n=0$ for all $n \geq 0$. Moreover, $$ \6^J = \left[\overline{D}, \left[L_\Omega,\Lambda\right]\right] = \left[L_\Omega, \left[\overline{D}, \Lambda\right]\right] + \left[\Lambda, \left[L_\Omega,\overline{D}\right]\right], $$ The second term is zero since $\Omega$ is holomorphic, the first term is $[L_\Omega,\sqrt{-1}D^*]$ by Kodaira identity. Hence $D^*$ and $\6^J$ anticommute. Finally, the Laplacian $\Delta_J = \6_J\6_J^* + \6_J^*\6_J$ is proportional by \cite{Vb} to the Laplacian $\Delta = DD^* + D^*D$. Therefore the Laplacian $\Delta$ and the Green operator $G$ also commute with $\6^J$. Now, the form $\theta_0$ is by definition $\Delta$-harmonic. Therefore it is also $\Delta_J$-harmonic, and $\6^J\theta_0 = 0$. To prove that $\6^J\theta_n = 0$, use induction on $n$. By definition \begin{multline*} \6^J\theta_n = \6^JD^*G\left(\sum_{0 \leq k < n} \theta_k \wedge \theta_{n-1-k}\right) = \\ = -D^*G\left(\sum_{0 \leq k 1$. Note that $\sigma_*\A^{0,i}_M$ is equipped with an $\su$-action by the Borel-Weyl theory. The corresponding representation is of highest weight $i$. The map $P$ is obviously compatible with this action, which proves the first statement. To prove the second, it is enough to prove that $P$ is invertible on the subbundles of highest vectors. But both these subsundles equal $\A^{0,i}(M)$. \endproof \subsection{The complex ${{\cal A}}^{\bullet}_{{\mathrm top}}(M)$ and autodual bundles.} The complex $\A^\cdot_\topp(M) \cong \sigma_*\A^{0,1}_M$ plays the same role for autodual bundles as the Dolbeault resp. de Rham complexes play for holomorphic resp. flat ones. Precisely, let $\E$ be a complex vector bundle on $M$ equipped with a connection $\nabla:\E \to \A^1(\E)$. Extend the operator $\nabla$ to a differential operator $D:\A^\cdot_\topp(\E) \to \A^{\cdot+1}_\topp(\E)$ by means of the embedding $\A^\cdot(\E) \hookrightarrow \A^\cdot(\E)$ and the natural $\su$-invariant projection $\A^{\cdot+1}(\E) \to \A^{\cdot+1}_\topp(\E)$. \begin{lemma}\label{autodual.complex} The connection $\nabla$ is autodual if and only if its extension $D$ satisfies $D^2 = 0$. \end{lemma} \proof The operator $D^2$ is the multiplication by the $\A^2_\topp$-part of the curvature $R$ of the bundle $\E$ with respect to the decomposition $$ \A^2(\End\E) = \A^2_\topp(\End\E) \oplus \A^2_\inv(\End\E). $$ Thus it vanishes if and only if $R$ is $\su$-invariant, which by definition means that $\nabla$ is autodual. \endproof Let $\E$ be a complex vector bundle equipped with an autodual connection $\nabla$. Since $\A^1(\E) \cong \sigma_*\A^{0,1}_M(X) \times \E$, the map $\nabla:\E \to \E \otimes \sigma_*\A^{0,1}_M(X)$ defines a $\cp$-holomorphic map $$ \bar\6_M:\sigma^*\E \to \sigma^*\E \otimes \A^{0,1}_M(X) $$ of $\cp$-holomorphic bundles on $X$. By Lemmas~\ref{iso.complex} and \ref{autodual.complex} the map $\bar\6_M$ extends to a map $\A^{0,\cdot}_M(X) \otimes \sigma^*\E \to \A^{0,\cdot+1}_M(X) \otimes \E$ satisfying $\bar\6_M^2 = 0$. By Lemma~\ref{rel.holo} this map defines a holomorphic structure on the bundle $\sigma^*\E$. \begin{lemma} The holomorphic bundle $\langle\sigma^*\E,\bar\6_M\rangle$ on $X$ is isomorphic to the twistor transform of the autodual bundle $\E$. \end{lemma} \proof Clear. \endproof Let now $\E$ be an arbitrary $\cp$-constant holomorphic bundle on $X$. Then the sheaf $\sigma_*\E$ is the sheaf of sections of a vector bundle. Since $\E$ is $\cp$-constant, $\E \cong \sigma^*\sigma_*\E$. The operator $$ \6_M:\E \cong \sigma^*\sigma_*\E \to \E \otimes \A^{0,1}_M(X) $$ gives by adjuction an operator $$ \nabla:\sigma_*\E \to \sigma_*\left(\E \otimes \A^{0,1}_M(X)\right) \cong \sigma_*\E \otimes \A^1_\topp(M). $$ By Lemmas~\ref{iso.complex} and \ref{autodual.complex} the operator $\nabla$ is an autodual connection on $\sigma_*\E$. \begin{defn} The autodual bundle $\langle \sigma_*,\nabla \rangle$ on $M$ is called {\emp the inverse twistor transform} of the $\cp$-constant holomorphic bundle $\E$ on $X$. \end{defn} \begin{theorem} \label{_twisto_transfo_equiva_Theorem_} The direct and inverse twistor transforms are mutually inverse equivalences between the category of autodual bundles on $M$ and the category of $\cp$-constant holomorphic bundles on $X$. \end{theorem} \proof Clear. \endproof \section{Stability of the twistor transform.} \label{_stabi_of_twi_tra_Section_} \subsection{Introduction} Let $M$ be a hyperk\"ahler manifold and let $X$ be its twistor space. Consider a semistable autodual bundle $\langle \E,\nabla \rangle$ on $M$ and let $\sigma^*\E$ be its twistor transform. The bundle $\sigma^*\E$ is a holomorphic bundle on $X$. In this section we prove under certain conditions that $\sigma^*\E$ is semistable in the sense of \ref{ss.twistor}. More precisely, we have the following. \begin{prop} Let $M$ be a hyperk\"ahler manifold. Denote its twistor space by $X$. Let $\langle \E, \nabla \rangle$ be a semistable autodual bundle on $M$ and let $\sigma^*\E$ be its twistor transform. Then for every coherent subsheaf $\F \subset \sigma^*\E$ we have $$ \frac{\deg c_1(\F)}{\rank\F} \leq \frac{\deg c_1(\sigma^*\E)}{\rank\sigma^*\E}, $$ where $\rank\F$ is the rank of the generic fiber of $\F$ and $\deg$ is understood in the sense of subsection~\ref{ss.twistor}. \end{prop} \subsection{Semistability for $\C P^1$-constant bundles} Before we give a proof of this Proposition, we prove the following. \begin{lemma}\label{constant.ss} Let $\E$ be a holomorphic bundle on the twistor space $X$. Assume that $\E$ is $\cp$-constant (that is, for every horizontal twistor line $\wt{m}:\cp \to X$ the restriction $\wt{m}^*\E$ is trivial). Then $\deg \E = 0$, and the bundle $\E$ is semistable. \end{lemma} \proof Since $H^1(\cp,\R) = 0$, \begin{equation}\label{ff} H^2(X,\R) = H^2(M,\R) \oplus H^2(\cp,\R). \end{equation} Since $\E$ is $\cp$-constant, $c_1(\E) \in H^2(M,\R)$. For every $I \in \cp$ let $X_I = \pi^{-1}(I) \subset X$ be the fiber over $I$. Since $$ c_1(\E)_{X_I} = c_1(\E|_{X_I}) \in H^{1,1}_I(M) $$ is of Hodge type $(1,1)$ for every $I \in \cp$, the class $c_1(\E) \in H^2(M,\R)$ is $\su$-invariant by Lemma~\ref{primitive}. Therefore $\deg c_1(\E) = \Lambda c_1(\E) = 0$. To prove semistability, let $\F \subset \E$ be a coherent subsheaf. It is enough to prove that $\deg c_1(\F) \leq 0$. Let $c_1(\F) = c_M + c_\cp$ be the decomposition associated with \eqref{ff}. Again, by Lemma~\ref{primitive} $c_M$ is $\su$-invariant, and $\deg c_1(\F) = \deg c_\cp$. For a generic horizontal twistor line $\wt{m}:\cp \to X$ we have $c_\cp = c_1(\wt{m}^*\F) \in H^2(\cp,\R)$. Since $\wt{m}^*\E$ is trivial, it is semistable, and $\deg c_1(\wt{m}^*\F) \leq 0$. \endproof Let ${\cal M}^{ss}_X$ be the moduli space of semistable holomorphic bundles on $X$. Lemma~\ref{constant.ss} implies that the set $\M_{const}$ of equivalence classes of $\cp$-constant holomorphic bundles on $X$ is a subset of ${\cal M}^{ss}_X$. The subset $\M_{const} \subset {\cal M}^{ss}_X$ is open. \subsection{Conclusion} The Propostion now follows directly from Lemma~\ref{constant.ss}. Moreover, the twistor transform provides an isomorphism $\M_\inv \to \M_{const} \subset {\cal M}^{ss}_X$ from the moduli space $\M_\inv$ of autodual bundles on $M$ to the open subset of $\cp$-constant bundles in the moduli space ${\cal M}^{ss}_X$ \section[Stable bundles and projective lines in twistor spaces.] {Stable bundles and projective lines \\ in twistor spaces.} \label{_lines_Section_} \subsection{Hyperk\"ahler structure on the Mukai dual space} \label{_Mukai_dual_Subsection_} Let $M$ be a compact hyperk\"ahler manifold and let $\E$ be a complex vector bundle on $M$ with $\su$-invariant Chern classes $c_1(\E)$ and $c_2(\E)$. Consider the moduli space $\M_0$ of stable holomorphic structures on $\E$ and let ${\cal M}^{reg}_0 \subset \M_0$ be the dense open subset of smooth points in $\M_0$. Recall that the subset ${\cal M}^{reg}_0$ is equipped with a natural K\"ahler metric, called {\em the Weil-Peterson metric}. It was proved in \cite{Vb} that the Weil-Peterson metric on ${\cal M}^{reg}_0$ is actually hyperk\"ahler. Moreover, the complex manifold $\left({\cal M}^{reg}_0\right)_J$ with the complex structure induced by a quaternion $J \in \cp \subset \h$ was naturally identified with the subset of smooth points in the moduli space of stable holomorphic structures on $\E$ with respect to the complex structure $J$ on $M$. Let $\X_{reg}$ be the twistor space of the hyperk\"ahler manifold ${\cal M}^{reg}_0$. Consider the topological space $\X = \M_0 \times \cp$. We have a natural embedding $\X_{reg} \subset X$. In \cite{Vb} the complex structure on $\X_{reg}$ and the real structure $\iota:\X_{reg} \to \X_{reg}$ were naturally extended to the whole of $\X$. The complex-analytic space $\X$ is in general singular. However, the fundamental Theorem~\ref{inv} still holds for the natural projection $\pi:\X \to \cp$. We will call holomorphic sections $\cp \to \X$ of the projection $\pi:\X \to \cp$ {\emp twistor lines} in $\X$. The space $\M_0$ is then naturally isomorphic to the subset of real twistor lines in $\X$. These data define {\bf singular hy\-per\-k\"ah\-ler structure} on $\M_0$ (see \cite{Vb} for details). The space $\M_0$ with this hyperk\"ahler structure is called {\bf Mukai dual} to $M$ (results of \cite{Vb} generalise Mukai's work about duality of K3 surfaces). We must caution the reader that this version of Mukai duality is not involutive, as the term ``dual'' might erroneously imply. \subsection{Fiberwise stable bundles} Let $X$ be the twistor space of the hyperk\"ahler manifold $M$. Let $\M_{const} \subset {\cal M}^{ss}_X$ be the open subset of $\cp$-constant holomorphic structures in the moduli space ${\cal M}^{ss}_X$ of semistable holomorphic structures on the bundle $\sigma^*\E$. In the last section we have identified $\M_{const}$ with the space $\M_\inv$ of $(0,1)$-stable autodual connections on the bundle $\E$. In this section we will need still another notion of stability for holomorphic bundles over $X$. \begin{defn}\label{fib.st} Call a stable holomorphic structure $\bar\6$ on $\sigma^*\E$ {\emp fiberwise stable} if for any $L \in \cp$ the restriction of $\langle \sigma^*\E, \bar\6 \rangle$ to the fiber $X_L = \pi^{-1}(L) \subset X$ is stable. \end{defn} Let $\M_\fib \subset {\cal M}^{ss}_X$ be the subset of fiberwise stable holomorphic structures. The intersection $\left(\M_{const} \cap \M_\fib\right) \subset {\cal M}^{ss}_X$ is, then, isomorphic to the moduli space of autodual connections on $\E$ inducing a stable holomorphic structure on $\E$ for every $I \in \cp$. The goal of this section is to prove the following. \begin{theorem}\label{iso} The space $\M_\fib$ is naturally isomorphic to the space $\Sec$ of twistor lines in the manifold $\X$. \end{theorem} \subsection{Stability of fiberwise-stable bundles} We begin by noting that one of the conditions in Definition~\ref{fib.st} is in fact redundant. \begin{lemma}\label{gen.st} Let $\E$ be a holomorphic bundle on the twistor space $X$. If the restriction $i^*\E$ is stable for a generic\footnote{In the sense of \cite{Vsym}} point $I \in \cp$, then the bundle $\E$ is stable. \end{lemma} \proof Indeed, it was proved in \cite{Vsym} that for a generic point $I \in \cp$ every rational $(1,1)$-cohomology class for the fiber $X_I$ is of degree zero. Therefore a stable holomorphic bundle on $X_I$ has no proper subsheaves. Hence for a proper subsheaf $\F \subset \E$ either $\F$ or $\E/\F$ is supported on non-generic fibers of $\pi:X \to \cp$. In particular, either $\F$ or $\E/\F$ is a torsion sheaf. This implies that the bundle $\E$ is stable. \endproof \subsection{Modular interpretation of the Mukai dual twistor space} We now construct a map $\M_\fib \to \Sec$. To do this, we give a modular interpretation of the space $\X$. For any point $I \in \cp$ let $i:X_I \hookrightarrow X$ be the natural embedding of the fiber $X_I = \pi^{-1}(I) \subset X$. For a stable holomorphic bundle $\E$ on the fiber $X_I$ call the coherent sheaf $i_*\E$ on $X$ {\emp a stable sheaf on $X$ supported in $I$}, or simply a {\emp fiber-supported stable sheaf}. More generally, for a complex analytic space $Z$ call a coherent sheaf $\EE$ on $Z \times X$ {\emp a family of fiber-supported stable sheaves on $X$} if there exists a holomorphic map $f_Z:Z \to \cp$ such that $\E$ and a holomorphic bundle $\EE_0$ on the subspace $Z \times_\cp X \subset Z \times X$ such that \begin{enumerate} \item $\EE \cong i_*\EE_0$, where $i:Z \times_\cp X \to Z \times X$ is the natural embedding. \item For every point $z \in Z$ the restriction of $\EE$ to $z \times \pi^{-1}(f_Z(z)) \subset Z \times X$ is a stable holomorphic bundle. \end{enumerate} The space $\X$ is obviously the moduli space for families of fiber-supported stable sheaves on $X$. The holomorphic map $f_\X:\X \to \cp$ is the natural projection. Let now $\E$ be a fiberwise stable holomorphic bundle on $X$. For every $I \in \cp$ the coherent sheaf $i_*i^*\E$ on $X$ is a stable sheaf supported in $I$. The correspondence $I \mapsto i_*i^*\E$ defines a holomorphic map $\cp \to \X$. This map is a section of the projection $\X \to \cp$, hence defines a point $\psi(\E) \in \Sec$. The correspondence $\E \mapsto \psi(\E)$ comes from a holomorphic map $\psi:\M_\fib \to \Sec$ of the corresponding moduli spaces. \subsection{Coarse and fine moduli spaces: a digression} In order to prove that the map $\psi:\M_\fib \to \Sec$ is an isomorphism, we need to make a digression about universal objects and coarse moduli spaces. Let $\Var$ be the category of complex-analytic varieties and let $\F:\Var \to \Sets$ be a functor. Recall that a complex-analytic space $Y$ is said to be a fine moduli space for the functor $\F$ if $\F \cong \Hom(\bullet, Y)$. This implies that there exists an element $C \in \F(Y)$ such that for every complex-analytic space $U$ and an element $a \in \F(U)$ there exists a unique map $f:U \to Y$ such that $a =\F(f)(C)$. Such an element $C$ is called {\em the universal solution} to the moduli problem posed by $\F$. It is well-known that geometric moduli problems only rarely admit fine moduli spaces. The common way to deal with this is to introduce a weaker notion of a {\em coarse moduli space}. For the purposes of this paper the following notion suffices. \begin{defn} A complex-analytic space $Y$ is called a {\emp coarse moduli space for the problem posed by $\F$} if for any complex-analytic space $Z$ and an element $a \in \F(Z)$ there exist a unique map $f:Z \to Y$, an open covering $U_\alpha$ of the space $Y$ and a collection $C_\alpha \in \F(U_\alpha)$ such that for every index $\alpha$ $$ \F(f)(C_\alpha) = a|_{f^{-1}(U_\alpha)}. $$ \end{defn} Heuristically, a coarse moduli space $Y$ admits locally a universal solution for the moduli problem $\F$, but these solutions need not come from a single global solution in $\F(Y)$. All the moduli spaces constructed as infinite-dimensional quotients by means of the slice theorem are coarse moduli spaces in the sense of this definition. This applies to all the moduli spaces considered in this paper, and to the space $\X$ in particular. Therefore for every point $x \in \X$ there exists a neighborhood $U_x \subset X$ and a coherent sheaf $\EE_x$ on $U \times X$ which is a family of fiber-supported stable sheaves on $X$ universal for the moduli problem. Let $U$ be such a neighborhood. Then the universality of the sheaf $\EE$ implies that $$ \Aut\EE = \Gamma(U,\calo^*). $$ \begin{lemma}\label{coarse} Let $\cp \to \X$ be a section of the projection $\pi:\X \to \cp$. There exists a coherent sheaf\/ $\EE$ on $\cp \times X$ such that for every $x \in \cp \subset \X$ the restriction\/ $\EE|_{U_x \times X}$ is isomorphic to the universal sheaf\/ $\EE_x$. \end{lemma} \proof Indeed, cover $\cp$ by open subsets of the form $U_x$ and choose a finite subcovering $U_\alpha$. In order to define a sheaf $\EE$, it is enough to choose a system of isomorphisms $$ g_{\alpha\beta}:\EE_\alpha|_{(U_\alpha \cap U_\beta) \times X} \to \EE_\beta|_{(U_\alpha \cap U_\beta) \times X} $$ for every intersection $U_\alpha \cap U_\beta$ so that $g_{\alpha\beta} \circ g_{\beta\gamma} = g_{\alpha\gamma}$ for every three indices $\alpha,\beta,\gamma$. Since $\Aut\EE_\alpha = \calo^*_{U_\alpha}$, the obstruction to finding such a system of isomorphisms lies in the second \v{C}ech cohomology group $H^2(\cp,\calo^*)$. Consider the long exact sequence $$ H^2(\cp ,\calo) \longrightarrow H^2(\cp, \calo^*) \longrightarrow H^3(\cp, \Z) $$ associated to the exponential exact sequence $$ 0 \longrightarrow \Z \longrightarrow \calo \longrightarrow \calo^* \longrightarrow 0. $$ Since $H^2(\cp, \calo) = H^3(\cp, \Z) = 0$, the group $H^2(\cp ,\calo^*)$ vanishes. \endproof \subsection{Conclusion} We can now finish the proof of Theorem~\ref{iso}. It remains to prove that the map $\psi:\M_\fib \to \X$ is an isomorphism. We will construct an inverse map $\psi^{-1}:\X \to \M_\fib$. Let $x \in \Sec$ be a point and let $\wt{x}:\cp \to \X$ be the corresponding section. Let $\EE$ be the coherent sheaf on $\wt{x}(\cp) \times X$ constructed in Lemma~\ref{coarse}. Let $\Delta = \pi \times \id:X \hookrightarrow \cp \times X$ be the embedding of $X$ into $\cp \times X$ as the preimage of the diagonal under the natural projection $\id \times \pi:\cp \times \X \to \cp \times \cp$. The sheaf $\EE$ is by definition isomorphic to the direct image of a holomorphic vector bundle $\E$ on $X$: $\EE \cong \Delta_*\E$. For every point $I \in \cp$ the coherent sheaf $i_*i^*\E$ on $X$ is canonically isomorphic to the restriction of $\EE$ to $I \times X \subset \cp \times X$. Therefore the bundle $\E$ is stable by Lemma~\ref{gen.st}. Let $\psi^{-1}(x) \in \M_\fib$ be the corresponding point in the moduli space $\M_\fib$. By construction $\psi(\psi^{-1}(x)) = x$. To prove that $\psi^{-1} \circ \psi = \id$, consider a stable bundle $\E \in \M_\fib$. Let $\EE$ be the coherent sheaf on $\cp \times X$ constructed in Lemma~\ref{coarse} and let $p:\cp \times X \to X$ be the projection onto the second factor. By definition $$ \EE \cong \Delta_*\Delta^*(p^*\E) \cong \Delta_*(\Delta \circ p)^*\E \cong \Delta_*\E. $$ Therefore $\psi^{-1}(\psi(\E)) = \E \in \M_\fib$. This finishes the proof of Theorem~\ref{iso}. \section{Conjectures and open questions.} \label{_conje_Section_} \subsection{NHYM moduli spaces and hyperk\"ahler reduction} \label{_Hyperkae_redu_Subsection_} \subsubsection{} Let $M$ be a K\"ahler manifold and let $\M$ be the moduli space of NHYM connections on a complex bundle $\E$ over $M$. We have shown in Section~\ref{_NHYM_Section_} % % This a good example of how _not_ to name your references. The label is % totally undescriptive (could have applied to the whole paper). The % trailing underlines serve no function whatsoever. % that the space $\M$ is equipped with a natural closed holomorphic $2$-form $\Omega$ which is is symplectic at least in a neighborhood of the subset of Hermitian connections. In fact one could hope for a much stronger statement. \begin{conjecture}\label{hyp.nhymspace} There exists a hyperk\"ahler metric on $\M$ such that $\Omega$ is the associated holomorphic symplectic from. \end{conjecture} Note that the construction of the form $\Omega$ is completely parallel to a construction of a holomorphic symplectic form on the Hitchin-Simpson moduli space $\M_{DR}$ of flat connections on $\E$ (\cite{S2}). The analog of Conjecture~\ref{hyp.nhymspace} for $\M_{DR}$ is known. \subsubsection{} To provide some evidence for Conjecture~\ref{hyp.nhymspace}, we give an interpretation of the NHYM equation in the context of hyperk\"ahler reduction. Let $\A$ be the space of all connections on the complex vector bundle $\E$. The space $\A$ is an affine space over the complex vector space $\A^1(M,\End\E)$ of $\End\E$-valued $1$-forms on $M$. Choose an Hermitian metric $h$ on the bundle $\E$. The decomposition $$ \A^1(M,\End\E) = \A^{1,0}(M,\End\E) \oplus \A^{0,1}(M,\End\E) $$ allows one to define a quaternionic structure on the space $\A^1(M,\End\E)$. Together with the natural trace metric, this structure makes the space $\A$ an (infinite-dimesional) hyperk\"ahler manifold. The complex gauge group $\G = \Maps(M,\Aut\E)$ acts on the space $\A$. This action is compatible with the hyperk\"ahler structure on $\A$. Therefore one can apply to the space $\A$ the machinery of {\emp hyperk\"ahler reduction} (see \cite{HKLR}). It turns out that the complex moment map $\A \to \C$ is equal to the map $YM:\A \to \Gamma(M,\End\E)$, $\nabla \mapsto \Lambda\nabla^2$. Vanishing of this map is precisely the NHYM condition. Let $\A_0 = YM^{-1}(0) \subset \A$ be the subset of connections with $\Lambda\nabla^2 = 0$. By the general principles of hyperk\"ahler reduction the quotient $\A_0 / \G$ should be hyperk\"ahler. The NHYM moduli space $\M$ is the closed subset $\M \subset \A_0 / \G$ of equivalence classes of connections with curvature $R=\nabla^2$ which satisfy $\Lambda R = 0$ and are, in addition, of Hodge type $(1,1)$. We expect that the embedding $\M \hookrightarrow \A_0 / \G$ is compatible with the hyperk\"ahler structure on $\A_0 /\ G$ and gives a hyperk\"ahler structure on $\M$ by restriction. \subsubsection{} The hyperk\"ahler reduction construction of the NHYM moduli space also allows to formulate an analog of the Uhlenbeck-Yau Theorem for NHYM-bundles. We first give a new definition of stability for NHYM bundles, more natural than the $(0,1)$-stability used in the body of the paper. Let $\overline{M}$ be the complex-conjugate complex manifold to $M$. Since $M$ and $\overline{M}$ are the same as smooth manifolds, the bundle $\E$ can be also considered as a complex vector bundle on $\overline{M}$. For every connection $\nabla$ on $\E$ the $(1,0)$-part $\nabla^{1,0}$ defines a holomorphic structure on the complex bundle $\E$ on $\overline{M}$. \begin{defn} Let $\langle\E, \nabla\rangle$ be a bundle with a $(1,1)$-connection over a complex manifold $X$. Let $U\subset X$ be a Zariski open subset in $X$, and let $\F\subset \E\restrict{U}$ be a subbundle which is preserved by $\nabla$. Then $\F$ is called {\bf a subsheaf of $\langle\E, \nabla\rangle$} if the following two conditions hold: \begin{description} \item[(i)] Consider $\E$ as a holomorphic bundle over $M$, with a holomorphic structure defined by the $(0,1)$-part of the connection. Then there exist a coherent subsheaf $\widetilde \F\subset \E$ on $M$ such that the restriction $\widetilde \F\restrict{U}$ is a sub-bundle of $\E$ which coinsides with $\F$. \item[(ii)] Consider $\E$ as a holomorphic bundle over $\overline{M}$, with a holomorphic structure defined by the $(1,0)$-part of the connection. Then there exist a coherent subsheaf $\widetilde \F\subset \E$ on $\overline{M}$ such that the restriction $\widetilde \F\restrict{U}$ is a sub-bundle of $\E$ which coinsides with $\F$. \end{description} \end{defn} For a subsheaf $\F\subset \E$, it is straightforward to define the Chern classes and the degree. As usually, $\F$ is called {\bf destabilizing} if \[ \frac{\deg \F}{\rank \F} \geq \frac{\deg {\E}}{\rank {\E}} \] \begin{defn}\label{_stabili_for_hyper_redu-Definition_} Let $\langle\E, \nabla\rangle$ be a bundle with $(1,1)$-connection over a compact K\"ahler manifold $M$. Then $\langle\E, \nabla\rangle$ is called $\nabla$-stable if there are no destabilizing subsheaves $\F\subset \langle\E, \nabla\rangle$. \end{defn} This definition generalizes the definition \cite{_Simpson:harmonic_} of stability for flat bundles. \begin{rem} Clearly, for NHYM bundles, $(0,1)$-stability implies the stability in the sense of Definition \ref{_stabili_for_hyper_redu-Definition_} \end{rem} An analogy with the Kempf-Ness Theorem suggests that every stable $\G$-orbit in $\A_0$ has non-trivial intersection with the zero set of the real moment map $\A_0 \to \Gamma(M,\End_\R\E)$ from $\A_0$ to the space of anti-Hermitian endomorphisms of the bundle $\E$. This moment map can be described more explicitly. \begin{defn}[pseudocurvature] Let $\langle\E, \nabla\rangle$ be a bundle with $(1,1)$-connection and a Hermitian metric $h$, not necessary compatible. Let $\nabla = \nabla' + \nabla''$ be the decomposition of $\nabla$ onto $(1,0)$ and $(0,1)$-parts. Consider the connection in $\overline{\E}^*$ associated with $\nabla$. Since $h$ identifies $\E$ and $\overline{\E}^*$, this gives another connection in $\E$, denoted by $\nabla_h$. The average $\nabla_h= \frac{\nabla +\nabla_h}{2}$ is again a connection, and is compatible with $h$. Let $\theta$ be the difference $\theta:= \frac{\nabla -\nabla_h}{2}$, which is a tensor. Applying $\nabla_h$ to $\theta$, we obtain a 2-form $\Xi$ with coefficients in $\End(\E)$. This form $\Xi$ is called {\bf the pseudocurvature} of the triple $\langle \E, \nabla, h\rangle$. \end{defn} It turns out that the real moment map on a NHYM connection $\nabla$ is given by $$ \nabla \mapsto \Lambda(\Xi), $$ where $\Xi$ is the pseudocurvature. \begin{defn} \label{_harmonic_me_Definition_} Let $\langle\E, \nabla\rangle$ be a bundle with a NHYM $(1,1)$-connection, and let $h$ be an Hermitian metric on $\E$, not necessarily compatible with $\nabla$ Then $h$ is called {\bf harmonic} if $\Lambda \Xi =0$, where $\Xi$ is the pseudocurvature of $\langle \E, \nabla, h \rangle$. \end{defn} \begin{conjecture} Let $\langle\E, \nabla\rangle$ be a bundle with NHYM connection $\nabla$. Then there exists a harmonic metric $h$ on $\E$ if and only if $\E$ is a direct sum of $\nabla$-stable bundles. Also, if $\E$ itself is $\nabla$-stable, then $h$ is unique, up to a constant factor. \end{conjecture} An analogous statement is known for flat connections. See \cite{_Simpson:harmonic_} for a discussion. \subsection{K\"ahler base manifold: open questions.} In this subsection, we relate questions pertaining to the case of base manifold $M$ compact and K\"ahler, but not necessarily hyperk\"ahler. \begin{question} \label{_NHYM_flat?_Question_} Let $(B, \nabla)$ be a NHYM-connection in a bundle with zero Chern classes. Is it true that $\nabla$ is necessarily flat? \end{question} In Hermitian case, the answer is affirmative by L\"ubcke \cite{_Lubcke_} and Simpson \cite{S}. In a neighbourhood of Hermitian Yang-Mills connection, all NHYM connections on a bundle with zero Chern classes are also flat, at one can see, e. g., from Proposition \ref{series}. The hyperk\"ahler analogue of this question is Question \ref{_NHYM-are-autodu_Question_}. \hfill Let $B$ be a stable holomorphic bundle over $M$ and let $St(B)$ be the deformation space of stable holomorphic structures on $B$. In Section \ref{_NHYM_Section_}, we defined a Kuranishi map $\phi:\; U \hookrightarrow H^1(\End(B))$, where $U$ is a neighbourhood of $[B]$ in $St(B)$. The map $\phi$ is, locally, a closed embedding, and its image in a neighbourhood of zero in $H^1(\End(B))$ is an algebraic subvariety, defined by the zeroes of so-called Massey products. Let $C$ be the Zariski closure of the image $\phi(U)$ in $H^1(\End(B))$. Let $\operatorname{NHYM}(B)$ be the space of NHYM connections inducing the same holomorphic structure. In \eqref{_from_NHYM_to_classes_Equation_}, we construct the map $\operatorname{NHYM}(B) \stackrel{\rho}{\arrow} \overline C$, where $\overline C$ is a complex conjugate manifold to $C$, and prove that in a neighbourhood of zero $\rho$ is isomorphism. Two questions arise: \begin{question}\label{_rho_surjec_Question_} Is the map $\rho$ surjective? \end{question} \begin{question} \label{_rho_etale_Question_} Is the map $\rho$ etale? Bijective? \end{question} These two questions might be reformulated in a purely algebraic way. Let $B$ be a stable holomorphic bundle equipped with a Hermitian Yang-Mills metric, and $X$ be the space of all $(1,0)$-forms $\theta\in \Lambda^{1,0}(\End (B))$ satisfying \begin{equation} \begin{cases} \6 \theta &= \theta\wedge \theta \\ \6^* \theta& =0, \end{cases} \end{equation} where $\6$ is the $(1,0)$-part of the connection, and $\6^*$ the adjoint operator. The middle cohomology space of the complex \[ \Lambda^{2,0}(\End(B)) \stackrel{\6^*}{\arrow} \Lambda^{1,0}(\End(B)) \stackrel{\6^*}{\arrow} \End(B) \] is naturally isomorphic to the complex conjugate space to $H^1(\End(B))$. This gives a map \[ X \stackrel{\rho} {\arrow} \overline{H^1(\End(B))}, \] associating to $\theta$ its cohomology class. Then, Proposition \ref{kur} implies that the image of $\rho$ lies in $\overline C$ and locally in a neighbourhood of zero, $\rho: \; X \arrow \overline C$ is an isomorphism. Question \ref{_rho_surjec_Question_} asks whether $\rho$ is surjective onto $\overline C$, and Question \ref{_rho_etale_Question_} asks whether $\overline \pi$ is etale, or even invertible. \subsection{Autodual and NHYM connections over a hyperk\"ahler base.} \subsubsection{} The first and foremost question (partially answered in Theorem \ref{_NHYM-are-autodu_Theorem_}; see also Question \ref{_NHYM_flat?_Question_}): \begin{question} \label{_NHYM-are-autodu_Question_} Let $(B, \nabla)$ be a NHYM bundle over a hyperk\"ahler base manifold. Is $(B, \nabla)$ necessarily autodual? \end{question} \subsubsection{} \label{compa_smoo_Mukai_du_Subsubsection_} Let $M$ be a compact hyperk\"ahler manifold, and let $\cal S$ be a connected component of the moduli of autodual connections on a complex vector bundle ${\cal B}$. Assume that $\cal S$ contains a point $B$ which is Hermitian autodual. Consider the ``Mukai dual'' space $\widehat M$, that is, the moduli space of Hermitian autodual connections on ${\cal B}$ (Subsection \ref{_Mukai_dual_Subsection_}). Assume that the connected component of $\widehat M$ containing $B$ is smooth and compact. Clearly, then, all connections from $\cal S$ are fiberwise stable, in the sense of Definition \ref{fib.st}. Thus, Theorem \ref{iso} gives an isomorphism between $\cal S$ and the space $Sec(\widehat M)$ of twistor lines in $\Tw(\widehat M)$. In such situation, we are going to give a conjectural description of the space $\cal S$, assuming that the answer to \ref{_rho_surjec_Question_}---\ref{_rho_etale_Question_} is affirmative. \subsubsection{} \begin{defn}[Twisted cotangent bundle] Let $M$ be a K\"ahler manifold, $\Omega^1M$ its holomorphic cotangent bundle. The K\"ahler class \[ \omega \in H^1(\Omega^1 M)= Ext^1(\calo (M), \Omega^1 M) \] gives by Yoneda an exact sequence \[ 0 \arrow \Omega ^1 M \arrow E \stackrel e \arrow \calo(M) \arrow 0, \] where $\calo(M)$ is the trivial one-dimensional bundle. Let $\nu$ be a non-zero section of $\calo(M)$, and $E_\nu$ be the set of all vectors \[ \left\{ v\in E\restrict m \;\; \left|\vphantom{\bigcup\limits_M^M}\right. \;\; e(v) = \nu\restrict m\right\} \] where $m$ runs through all points of $M$. Consider $E_\nu$ as a submanifold in the total space of $E$. Then $E_\nu$ is called {\bf a twisted cotangent bundle of $M$}, denoted by $\Omega_\omega M$. \end{defn} The space $\Omega_\omega M$ has a natural action of $\Omega^1 M$ considered as a group scheme over $M$, and as such is a torsor over $\Omega ^1 M$ \subsubsection{} The affirmative answer to the stronger form of \ref{_rho_etale_Question_} would give the proof of the following conjecture. \begin{conjecture} \label{_sec_to_twi_cota_Conjecture_} Under assumptions of \ref{compa_smoo_Mukai_du_Subsubsection_}, there exists a natural isomorphism of complex manifolds \[ Sec(\widehat M) \cong \Omega_\omega M, \] where $\Omega_\omega M$ is the twisted cotangent bundle. \end{conjecture} In the general situation, there is a natural map from the space of twistor lines $Sec(M)$ of a compact hyperk\"ahler manifold to $\Omega_\omega M$. However, in general there are no approaches to the proof of surjectivity. \begin{example} Let $M$ be a compact complex torus, $\dim_\C M = 2n$, and $B$ a trivial line bundle. Clearly, $M$ is hyperk\"ahler. Then $\widehat M$ is the dual torus, and $\cal S$ is the space of local systems on $M$, which is isomorphic to $(\C^*)^{2n}$. The space $C$ of \ref{_rho_etale_Question_} is isomorphic to $H^1(\calo M)$, and the answer to \ref{_rho_etale_Question_} is obviously affirmative. Thus, $Sec(M)$ and $\Omega_\omega M$ are also isomorphic to $(\C^*)^{2n}$ and are Stein. \end{example} In the following subsection, we shall see that this is indeed a general phenomenon -- the space of twistor lines is equipped with a canonical plurisubharmonic function and is likely to be Stein. However, we don't know a general argument constructing plurisubharmonic functions on the twisted cotangent bundle -- this is one more mystery. \subsection{Plurisubharmonic functions on moduli spaces.} Let $M$ be a hyperk\"ahler manifold $\Tw \stackrel \pi \arrow \C P^1$ its twistor space, and $Sec(M)$ the space of sections $s:\; \C P^1 \arrow \Tw$ of the map $\pi$, also called {\it twistor lines} (Section \ref{_twistors_Section_}). It is easy to equip $Sec(M)$ with a natural plurisubharmonic function. Recall that $\Tw$ is isomorphic as a $C^\infty$-manifold to $M \times \C P^1$. This decomposition gives a natural (non-K\"ahler) Hermitian metric on $\Tw$. \begin{prop} \label{_volume_twi_line_plurisubharmo_Proposition_} Consider the function $v:\; Sec(M) \arrow \R ^+$ which maps a line $s\in \Tw$ to its Hermitian volume, taken with respect to the Hermitian metric on $\Tw$. Then $v$ is strictly plurisubharmonic. \end{prop} \proof Let $\omega$ be the differential 2-form which is the symplectic part of the Hermitian metric on $\Tw$. Since the twistor lines are complex subvarieties in $\Tw$, $v(s) = \int_s \omega$ for all twistor lines $s$. Then, for all bivectors $x, \bar x$ in $T_s Sec(M)$, we have \begin{equation}\label{_6bar6omega_Equation_} \6\bar\6 v(x, \bar x) = \int_s \6\bar\6 \omega({\bf x},\bar{\bf x}), \end{equation} where ${\bf x},\bar{\bf x}$ are the sections of $T \Tw\restrict{s}$ corresponding to $x, \bar x$. Then, to prove that $v$ is plurisubharmonic it suffices to show that $\6\bar\6 \omega({\bf x},\bar{\bf x})$ is positive. {}From Lemma \ref{_differe_of_Hermi_on_twistors_Lemma_}, it is easy to see that $\6\bar\6 \omega = \omega\wedge \pi^* \operatorname{FS}(\C P^1)$, where $\operatorname{FS}(\C P^1)$ is the Fubini-Study form on $\C P^1$. Clearly, then, $\6\bar\6\omega({\bf x},\bar{\bf x})$ is positive, and $v$ is plurisubharmonic. This proves Proposition \ref{_volume_twi_line_plurisubharmo_Proposition_}. \endproof One of the most intriguing questions of hyperk\"ahler geometry is to learn whether the function $v$ is exhausting. \hfill In notation and assumptions of \ref{_sec_to_twi_cota_Conjecture_}, consider the space $Sec(\widehat M)$ which is isomorphic to the space $\cal S$ of autodual connections. There is the canonical Weil-Petersson metric on $\cal S$, coming from results of Subsection \ref{_Hyperkae_redu_Subsection_}. This metric is hyperk\"ahler. This metric is given by a potential, which is equal to the integral of the square of the absolute value of the curvature. \begin{question} Is the Weil--Petersson metric related to the metric given by $v$? \end{question} \subsubsection*{Acknowledgements:} The autors are grateful to S.-T. Yau, who stimulated the interest to the problem, D. Kazhdan and T. Pantev for valuable discussions, S. Arkhipov, M. Finkelberg and L. 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