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\stepcounter{Mycounter} {\bf \thedefinition:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \theexample:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \theremark:\ }} \newcounter{problem}[section] \setcounter{problem}{0} \renewcommand{\theproblem}{{Problem \thesection.\arabic{problem}}} \newcommand{\problem}{% \setcounter{problem}{\value{Mycounter}} \refstepcounter{problem} \stepcounter{Mycounter} {\bf \theproblem:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \thequestion:\ }} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % equations % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\theequation}{\thesection.\arabic{equation}} \@addtoreset{equation}{section} \@addtoreset{footnote}{section} \makeatother \begin{document} \begin{center} {\Large\bf Algebraic structures on hyperk\"ahler manifolds.}\\[4mm] Misha Verbitsky,\\[4mm] {\tt verbit@@thelema.dnttm.rssi.ru} \end{center} \hfill {\small \hspace{0.2\linewidth} \begin{minipage}[t]{0.7\linewidth} Let $M$ be a compact hyperk\"ahler manifold. The hyperk\"ahler structure equips $M$ with a set $\c R$ of complex structures parametrized by $\C P^1$, called {\bf the set of induced complex structures.} It was known previously that induced complex structures are {\it non-algebraic}, except may be a countable set. We prove that a {\it countable} set of induced complex structures is {\it algebraic}, and this set is dense in $\c R$. A more general version of this theorem was proven by A. Fujiki. \end{minipage} } \hfill \hfill The structure of this paper is following. In Subsection \ref{_hype_defi_Subsection_} we define hyperk\"ahler manifolds and induced complex structures. In Subsection \ref{_gene_appe_Subsection_}, we define induced complex structures of general type. The main result of this paper and its proof are given in Section \ref{_algebra_Section_}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction.} \label{_Intro_Section_} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We give the basic definitions and cite the results relevant to this paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Hyperk\"ahler manifolds} \label{_hype_defi_Subsection_} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition \label{_hyperkahler_manifold_Definition_} %%%%%%%%%% (\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ endowed with three complex structures $I$, $J$ and $K$, such that the following holds. \begin{description} \item[(i)] the metric on $M$ is K\"ahler with respect to these complex structures and \item[(ii)] $I$, $J$ and $K$, considered as endomorphisms of a real tangent bundle, satisfy the relation $I\circ J=-J\circ I = K$. \end{description} \hfill The notion of a hyperk\"ahler manifold was introduced by E. Calabi (\cite{_Calabi_}). \hfill Clearly, a hyperk\"ahler manifold has the natural action of quaternion algebra ${\Bbb H}$ in its real tangent bundle $TM$. Therefore its complex dimension is even. For each quaternion $L\in \Bbb H$, $L^2=-1$, the corresponding automorphism of $TM$ is an almost complex structure. It is easy to check that this almost complex structure is integrable (\cite{_Besse:Einst_Manifo_}). \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition \label{_indu_comple_str_Definition_} Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying $L^2=-1$. The corresponding complex structure on $M$ is called {\bf an induced complex structure}. The $M$ considered as a complex manifold is denoted by $(M, L)$. \hfill Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$ with the group of unitary quaternions. This gives a canonical action of $SU(2)$ on the tangent bundle, and all its tensor powers. In particular, we obtain a natural action of $SU(2)$ on the bundle of differential forms. \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \lemma \label{_SU(2)_commu_Laplace_Lemma_} The action of $SU(2)$ on differential forms commutes with the Laplacian. {\bf Proof:} This is Proposition 1.1 of \cite{Verbitsky:Symplectic_II_}. \endproof Thus, for compact $M$, we may speak of the natural action of $SU(2)$ in cohomology. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Appendix: Induced complex structures of general type.} \label{_gene_appe_Subsection_} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We cite results useful for understanding of behaviour of induced complex structures. The function of this appendix is illustrative. We do not refer to this subsection in the body of the article. This appendix is perfectly safe to skip. \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition \label{_generic_manifolds_Definition_} %%%%%%%%% Let $M$ be a hyperk\"ahler manifold, $\c R$ the set of all induced complex structures. With each $I\in \c R$, we associate the Hodge decomposition $H^*(M) = \oplus H^{p,q}_I(M)$ on the cohomology of $M$. We say that $I$ is {\bf of general type} when all elements of the group % \[ H^{p,p}_I(M)\cap H^{2p}(M,\Z)\] % are $SU(2)$-invariant. \hfill As \ref{_gene_type_co_div_by2_Remark_} below implies, the manifolds $(M, I)$ have no Weil divisors when $I$ is of general type. In particular, induced complex structures of general type are never algebraic. \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \proposition \label{_generic_are_dense_Proposition_} %%%%%%% Let $M$ be a hyperk\"ahler manifold and $\c R$ be the set of induced complex structures over $M$. Let $\c R_{ng}\subset \c R$ be the set of all induced complex structures {\bf not} of general type. Then $\c R_{ng}$ is no more than countable. {\bf Proof:} This is Proposition 2.2 from \cite{Verbitsky:Symplectic_II_} \endproof \hfill Let $M$ be a compact hyperk\"ahler manifold, $dim_\R M =2m$. \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition\label{_trianalytic_Definition_} %%%%%%%%%% Let $N\subset M$ be a closed subset of $M$. Then $N$ is called {\bf trianalytic} if $N$ is an analytic subset of $(M,L)$ for any induced complex structure $L$. \hfill Let $I$ be an induced complex structure on $M$, and $N\subset(M,I)$ be a closed analytic subvariety of $(M,I)$, $dim_\C N= n$. Denote by $[N]\in H_{2n}(M)$ the homology class represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote the Poincare dual cohomology class. Recall that the hyperk\"ahler structure induces the action of the group $SU(2)$ on the space $H^{2m-2n}(M)$. \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theorem\label{_G_M_invariant_implies_trianalytic_Theorem_} %%%%% Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic. {\bf Proof:} This is Theorem 4.1 of \cite{Verbitsky:Symplectic_II_}. \endproof %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \remark \label{_triana_dim_div_4_Remark_} Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4. \hfill \ref{_G_M_invariant_implies_trianalytic_Theorem_} has the following immediate corollary, also proven by a Fujiki (\cite{_Fujiki_}, Theorem 4.8 (1)): %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \corollary \label{_hyperkae_embeddings_Corollary_} %%%%%%%%%% Let $M$ be a compact hyperk\"ahler manifold, $I$ induced complex structure of general type, and $S\subset (M,I)$ its complex analytic subvariety. Then $S$ is trianalytic. \endproof %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \remark \label{_gene_type_co_div_by2_Remark_} {}From \ref{_hyperkae_embeddings_Corollary_} and \ref{_triana_dim_div_4_Remark_}, it follows that a holomorphically symplectic manifold of general type has no closed complex analytic subvarieties of odd dimension; in particular, such a manifold has no divisors. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Induced complex structures which are algebraic} \label{_algebra_Section_} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a compact hyperk\"ahler manifold. Then $M$ is called {\bf simple} if $M$ is simply connected and cannot be non-trivially represented as a direct product of hyperk\"ahler manifolds. \hfill The general version of the following theorem was proven by A. Fujiki (\cite{_Fujiki_}, Theorem 4.8 (2)). Let $\pi:\; \c M \arrow S$ be a deformation of a simple holomorphically simplectic manifold, with arbitrary base of positive dimension. Assume that $\c M\arrow S$ is not isotrivial (not trivial on periods). Fujiki proves that for a dense subset $S_a\subset S$, the fibers $\pi^{-1}(s_a)$ are algebraic for all $s_a\in S_a$. I am grateful to Daniel Huybrechts, who provided me with this reference. Also, a similar (but weaker) result was proven by F. Campana (\cite{_Campana_}). \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theorem \label{_alge_dense_Theorem_} Let $M$ be a compact simple hyperk\"ahler manifold and $\c R$ be the set of induced complex structures $\c R \cong \C P^1$. Let $\c R_{alg}\subset \c R$ be the set of all $L\in \c R$ such that the complex manifold $(M, L)$ is algebraic. Then $\c R_{alg}$ is countable and dense in $\c R$. \hfill {\bf Proof:} For each $L\in R$, consider the K\"ahler cone of $(M,L)$, denoted by $K(L)\subset H^2(M, \R)$. By definition, $K(L)$ is the set of all cohomology classes $\omega\subset H^2(M,\R)$ which are K\"ahler classes with respect to some metric on $(M,L)$. Let \[ K:= \bigcup\limits_{L\in \c R} K(L). \] By \cite{_main_}, Lemma 5.6, $K$ is open in $H^2(M, \R)$. Therefore, the intersection $H^2(M, \Q) \cap K$ is dense in $K$. By Kodaira, a compact K\"ahler manifold is algebraic if and only if there exist a rational K\"ahler class on $M$ (\cite{_Griffi_Harri_}). By \ref{_K_proje_on_R_Lemma_} below, every cohomology class $\omega \in K$ corresponds to a unique induced complex structure $I(\omega)\in \c R$ such that $\omega$ is K\"ahler with respect to $I(\omega)$. We also prove that thus obtained map $\pi:\; K \arrow \c R$ is continuous. Thus, $\pi\left( H^2(M, \Q \cap K)\right)$ is dense in $\c R$. On the other hand, by Kodaira, $\pi\left( H^2(M, \Q \cap K)\right)$ coinsides with $\c R_{alg}$. This proves \ref{_alge_dense_Theorem_}. \endproof \hfill The following lemma is implicit from \cite{_main_}. We decided to spell out its proof, for clarity; for missing details the reader is referred to \cite{_main_}. \hfill %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \lemma \label{_K_proje_on_R_Lemma_} Let $\omega\in K$ be a cohomology class which is K\"ahler with respect to $I\in \c R$. Then % \begin{description} \item [(i)] such $I$ is unique, \item[(ii)] and the obtained map $\pi:\; K \arrow \c R$ is continuous. \end{description} \hfill {\bf Proof:} Consider the positively defined scalar product on the cohomology space $H^2(M)$ induced by the standard scalar product on harmonic forms. This scalar product is clearly $SU(2)$-invariant. For an induced complex structure $L$, denote by $\omega_L\in H^2(M, \R)$ the K\"ahler class of the K\"ahler structure associated with $L$ and the hyperk\"ahler structure. The corresponding harmonic form can be expressed as $\omega_L(x, y) = (x, L(y))$, where $(\cdot,\cdot)$ is the Riemannian form on $M$. Let $V\subset H^2(M, \R)$ be the 3-dimensional subspace generated by $\omega_L$, for all $L\in \c R$ (see \cite{_main_}, Section 4), and $p:\; H^2(M) \arrow V$ be the orthogonal projection to $V$. For a K\"ahler class $\omega$ on $(M, I)$, the product $(\omega, \omega_I)$ is positive, by Hodge--Riemann relations (\cite{_Griffi_Harri_}). Thus, for all $\omega\in K$, the vector $p(\omega)\in V$ is non-zero. Now, for each $I\in \c R$, the intersection $H^{1,1}_I(M) \cap V$ is one-dimensional, because in notation of \ref{_hyperkahler_manifold_Definition_}, the cohomology class $\omega_J+\1 \omega_K\in V$ belongs to $H^{2,0}_I(M)$ and $\omega_J-\1 \omega_K\in V$ belongs to $H^{0,2}_I(M)$. Therefore, $\omega$ and $\omega_I$ are collinear. Clearly, a complex structure $L$ is uniquely defined by the form $\omega_L$, and for all $L\in \c R$, the vectors $\omega_L\in V$ all have the same length. Thus, for each line $l\in V$, there exist no more than two induced complex structures $L\in \c R$ satisfying $l\in H^{1,1}_L(M)$. It is easy to check that these induced complex structures are opposite: we have $H^{1,1}_{L}(M) = H^{1,1}_{-L}(M)$. This implies that in $\c R$, only $L=I$ and $L=-I$ satisfy $\omega\in H^{1,1}_L(M)$. On the other hand, $\omega_{L} = - \omega_{-L}$. Thus, the numbers $(\omega, \omega_I)$ and $(\omega, \omega_{-I})$ cannot be positive simultaneously. This implies that $\omega$ cannot be a K\"ahler class for $I$ and $-I$ at the same time. We proved \ref{_K_proje_on_R_Lemma_} (i). \hfill To prove \ref{_K_proje_on_R_Lemma_} (ii), consider the composition $s$ of $p:\; K \arrow V\backslash 0$ and the natural projection map from $V\backslash 0 $ to the sphere $S^2\subset V \cong \R^3$. If we identify $S^2$ with $\c R\cong\C P^1$, we find that $\pi$ is equal to $s$ (see, for instance, \cite{_main_}, the proof of Sublemma 5.6). On the other hand, $s$ is continous by construction. This proves \ref{_K_proje_on_R_Lemma_} (ii). \endproof \hfill I am grateful to Daniel Huybrechts, who provided me with reference to \cite{_Campana_} and \cite{_Fujiki_} after the preliminary version of this paper appeared in alg-geom preprint server. \hfill \begin{thebibliography}{666} \bibitem[Bes]{_Besse:Einst_Manifo_} Besse, A., {\em Einstein Manifolds}, Springer-Verlag, New York (1987) \bibitem[C]{_Calabi_} E. Calabi, {\em Metriques k\"ahleriennes et fibr\`es holomorphes}, Ann. Ecol. Norm. Sup. {\bf 12} (1979), 269-294. \bibitem[Ca]{_Campana_} Campana, F., {\it Densite des varietes hamiltoniennes primitives projectives}, C. R. Acad. Sci. Paris Ser. 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