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\setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcommand{\question}{% {\bf \green QUESTION:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Teichmuller space of geometric structures 2 \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Teichm\"uller spaces for geometric structures,\\ lecture 2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize Misha Verbitsky } \\[20mm] {\tiny\bf Conference - Teichm\"uller Theory in Higher Dimension and Mirror Symmetry \\ April 24-28, 2017} \end{center} \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Gr}{\operatorname{\sf Gr}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Hyp}{\operatorname{Hyp}} \newpage {\bf \blue Plan:} 1. Hyperk\"ahler manifolds (introduction) 2. Supersymmetry on K\"ahler and hyperk\"ahler manifolds 3. Automorphisms of cohomology 4. Computation of the mapping class group \newpage {\bf \blue Hyperk\"ahler manifolds (reminder)} \definition A {\bf \blue hyperk\"ahler structure} on a manifold $M$ is a Riemannian structure $g$ and a triple of complex structures $I,J,K$, satisfying quaternionic relations $I\circ J = - J \circ I =K$, such that $g$ is K\"ahler for $I,J,K$. {\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$: the parallel translation along the connection preserves $I, J,K$. {\bf\green DEFINITION:} Let $M$ be a Riemannian manifold, $x\in M$ a point. The subgroup of $GL(T_xM)$ generated by parallel translations (along all paths) is called {\bf \blue the holonomy group} of $M$. {\bf\green REMARK:} {\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has holonomy in $Sp(n)$ } (the group of all endomorphisms preserving $I,J,K$). \claim A compact hyperk\"ahler manifold $M$ {\bf \purple has maximal holonomy of Levi-Civita connection $Sp(n)$} if and only if {\bf \purple $\pi_1(M)=0$, $h^{2,0}(M)=1$.} \theorem {\bf \blue (Bogomolov decomposition)}\\ {\bf \red Any compact hyperk\"ahler manifold has a finite covering isometric to a product of a torus and several maximal holonomy hyperk\"ahler manifolds.} \newpage {\bf \blue Holomorphically symplectic manifolds (reminder)} {\bf\green DEFINITION:} {\bf \blue A holomorphically symplectic manifold} is a complex manifold equipped with non-degenerate, holomorphic $(2,0)$-form. \remark In these lectures, all holomorphically symplectic manifolds are assumed to be K\"ahler and compact. \remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\ $\omega_I:= g(I\cdot, \cdot)$, $\omega_J:= g(J\cdot, \cdot)$, $\omega_K:= g(K\cdot, \cdot)$.} \claim In these assumptions, $\omega_J + \1\omega_K$ is holomorphic symplectic on $(M,I)$. {\bf\green THEOREM:} {\bf \blue (Calabi-Yau)}\\ A compact, K\"ahler, holomorphically symplectic manifold admits a unique hyperk\"ahler metric in any K\"ahler class. {\bf\green DEFINITION:} For the rest of these lectures, {\bf \red a hyperk\"ahler manifold is a compact, K\"ahler, holomorphically symplectic manifold of maximal holonomy.} \newpage {\bf \blue EXAMPLES.} \example An even-dimensional complex torus. \example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi). \remark $T^*\C P^1$ {\bf \blue is a resolution of a singularity $\C^2/{\pm1}$.} \example Take a 2-dimensional complex torus $T$, then the singular locus of $T/{\pm1}$ is of form $(\C^2/{\pm1}) \times T$. Its resolution $\widetilde {T/{\pm1}}$ is called {\bf \green a Kummer surface}. {\bf \red It is holomorphically symplectic}. \remark Take a symmetric square $\Sym^2 T$, with a natural action of $T$, and let $T^{[2]}$ be a blow-up of a singular divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the Kummer surface $\widetilde {T/{\pm1}}$.} \definition A complex surface is called {\bf \blue K3 surface} if it a deformation of the Kummer surface. \theorem {\bf \blue (a special case of Enriques-Kodaira classification)}\\ Let $M$ be a compact complex surface which is hyperk\"ahler. {\bf \red Then $M$ is either a torus or a K3 surface.} \newpage {\bf \blue Hilbert schemes} \definition A {\bf\blue Hilbert scheme} $M^{[n]}$ of a complex surface $M$ is a classifying space of all ideal sheaves $I\subset \calo_M$ for which the quotient $\calo_M/I$ has dimension $n$ over $\C$. \remark A Hilbert scheme {\bf \purple is obtained as a resolution of singularities} of the symmetric power $\Sym^n M$. \theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of a hyperk\"ahler surface is hyperk\"ahler.} \example {\bf\blue A Hilbert scheme of K3} is hyperk\"ahler. \example Let $T$ be a torus. Then it acts on its Hilbert scheme freely and properly by translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For $n>2$, a universal covering of $T^{[n]}/T$ is called {\bf \blue a generalized Kummer variety}. \remark There are 2 more ``sporadic'' examples of compact hyperk\"ahler manifolds, constructed by K. O'Grady. {\bf \purple All known compact hyperkaehler manifolds are these 2 and the two series:} Hilbert schemes of K3, and generalized Kummer. \newpage {\bf \blue The Bogomolov-Beauville-Fujiki form} {\bf \green THEOREM:} (Fujiki). Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is hyperk\"ahler. {\bf \red Then $\int_M \eta^{2n}=c q(\eta,\eta)^n$, for some primitive integer quadratic form $q$ on $H^2(M,\Z)$, and $c>0$ an integer number.} {\bf \green Definition:} This form is called {\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined by the Fujiki's relation uniquely, up to a sign.} The sign is determined from the following formula (Bogomolov, Beauville) \begin{align*} \lambda q(\eta,\eta) &= \int_X \eta\wedge\eta \wedge \Omega^{n-1} \wedge \bar \Omega^{n-1} -\\ &-\frac {n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right) \end{align*} where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$. {\bf \green Remark:} {\bf \red $q$ has signature $(b_2-3,3)$.} It is negative definite on primitive forms, and positive definite on $\langle \Omega, \bar \Omega, \omega\rangle$, where $\omega$ is a K\"ahler form. \newpage {\bf \blue Global Torelli theorem.} \theorem Let $M$ be a simple hyperk\"ahler manifold, and $\Gamma:= \Diff/\Diff_0$ its mapping class group. Then $H^2(M,\Z)$ admits a $\Gamma$-invariant, non-degenerate integer quadratic form $q$ such that the natural action of $\Gamma$ on $H^2(M,\Z)$ {\bf \red induces a homomorphism $\Gamma\arrow SO(H^2(M,\Z), q)$ with finite index and finite kernel.} \remark Suppose that $\phi:\; M \arrow M'$ is a bimeromorphic map of Calabi-Yau manifolds. Then the exceptional set of $\phi$ has codimension $\geq 2$, {\bf \red hence $H^2(M)=H^2(M')$.} \definition Let $\Teich$ be the Teichm\"uller space of complex structures of hyperk\"ahler type on $M$, and $\Teich_b$ the quotient of $\Teich$ by an equivalence relation induced by bimeromorphic maps $(M,I)\arrow (M,I')$ inducing identity on $H^2(M)$. Then $\Teich_b$ is called {\bf \blue birational Teichm\"uller space} of $M$. As we shall see, $\Teich_b$ is a smooth, Hausdorff complex manifold. \theorem Consider the space \[ \Perspace:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \ q(l,l)=0, q(l, \bar l) >0\} \] and let the {\bf \blue period map} $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ map a complex structure $I$ to a line $H^{2,0}(M,I)\in {\Bbb P}H^2(M, \C)$. {\bf \red Then $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ induces a bijective map $\Teich_b\arrow \Perspace$} on any connected component of $\Teich_b$. \newpage {\bf \blue Automorphisms of cohomology} We are going to prove the following theorem. {\bf \green Theorem 1:} The natural action of $\Spin(H^2(M,\R), q)$ on $H^2(M)$ {\bf \red is extended to an action on the algebra $H^*(M)$ by automorphisms.} Moreover, {\bf \red this action preserves the Chern classes.} \claim Let $M$ be a compact K\"ahler manifold. Consider {\bf \blue the Weil operator} $W\in \End(H^*(M))$ acting on $H^{p,q}(M)$ as $W\restrict{H^{p,q}(M)}=\1(p-q)$, and let $U(1)$ act on $H^*(M)$ as $e^{-\1 W}$. {\bf \red Then $U(1)$ acts by automorphisms.} Theorem 1 is implied by the following result, proven later in this lecture. \theorem Consider the Lie subalgebra $\g_W\subset \Aut(H^*(M))$ generated by the Weil operators $W$ for all K\"ahler structures of K\"ahler type. {\bf \red Then $\g_W$ is isomorphic to $\goth{so}(H^2(M,\R), q)$.} \newpage {\bf \blue Lie superalgebras} \definition A {\bf \blue graded vector space} is a space $V^* =\bigoplus_{i\in \Z} V^i$. \remark If $V^*$ is graded, the endomorphisms space $\End(V^*)=\bigoplus_{i\in \Z} \End^i(V^*)$ is also graded, with $\End^i(V^*)= \bigoplus_{j\in \Z} \Hom(V^j, V^{i+j})$ \definition An operator on a graded vector space is called {\bf \blue even} ({\bf \blue odd}) if it shifts the grading by even (odd) number. The {\bf\blue parity} $\tilde a$ of an operator $a$ is 0 if it is even, 1 if it is odd. We say that an operator is {\bf \blue pure} if it is even or odd. \definition A {\bf \blue supercommutator} of pure operators on a graded vector space is defined by a formula $\{a,b\}= ab - (-1)^{\tilde a \tilde b}ba$. \definition {\bf \blue A graded Lie algebra} (Lie superalgebra) is a graded vector space $\g^*$ equipped with a bilinear graded map $\{\cdot,\cdot\}:\; \g^*\times \g^* \arrow \g^*$ which is graded anticommutative: $\{a,b\} = - (-1)^{\tilde a \tilde b}\{b,a\}$ and satisfies {\bf \blue the super Jacobi identity} $\{c, \{a,b\}\} = \{\{c, a\},b\}+ (-1)^{\tilde a \tilde c}\{a,\{c, b\}\}$ \newpage {\bf \blue Supersymmetry in K\"ahler geometry } Let $(M, I, g)$ be a Kaehler manifold, $\omega$ its Kaehler form. {\bf \blue On $\Lambda^*(M)$, the following operators are defined.} 0. $d$, $d^*$, $\Delta$, because it is Riemannian. 1. $L(\alpha):= \omega\wedge \alpha$ 2. $\Lambda(\alpha) := * L * \alpha$. It is easily seen that $\Lambda= L^*$. 3. The Weil operator $W\restrict{\Lambda^{p,q}(M)}=\1(p-q)$ \theorem {\bf \red These operators generate a Lie superalgebra $\goth a$ of dimension $(5|4)$,} acting on $\Lambda^*(M)$. Moreover, the Laplacian $\Delta$ is central in $\goth a$, hence {\bf \purple $\goth a$ also acts on the cohomology of $M$. } \remark This is a convenient way to summarize the K\"ahler relations and the Lefschetz $\goth{sl}(2)$-action. \newpage {\bf \blue Supersymmetry in hyperk\"ahler geometry} Let $(M, I, J,K, g)$ be a hyperkaehler manifold, $\omega_I$, $\omega_J$, $\omega_K$ its Kaehler forms. {\bf \blue On $\Lambda^*(M)$, the following operators are defined.} 0. $d$, $d^*$, $\Delta$, because it is Riemannian. 1. $L_I(\alpha):= \omega_I\wedge \alpha$ 2. $\Lambda_I(\alpha) := * L_I * \alpha$. It is easily seen that $\Lambda_I= L^*_J$. 3. Three Weil operators $W_{I}\restrict{\Lambda^{p,q}(M,I)}=\1(p-q)$, $W_{ J}\restrict{\Lambda^{p,q}(M,J)}=\1(p-q)$, $W_{K}\restrict{\Lambda^{p,q}(M,K)}=\1(p-q)$ \theorem {\bf \red These operators generate a Lie superalgebra $\goth a$} of dimension $(11|8)$, acting on $\Lambda^*(M)$. Moreover, the Laplacian $\Delta$ is central in $\goth a$, hence {\bf \purple $\goth a$ also acts on the cohomology of $M$. } \remark The Weil operators form the Lie algebra $\goth{su}(2)$ of unitary quaternions. This means that {\bf \blue the quaternionic action belongs to $\goth a$}. In particular, $L_J, L_K, \Lambda_J$ and $\Lambda_K$. \remark The twisted de Rham differentials $d_I, d_J, d_K$, associated to $I,J,K$ also belong to $\goth a$: {\bf \purple $d_I= [ W_I, d]$, $d_J= [ W_J, d]$, $d_K= [ W_K, d]$} \newpage {\bf \blue $\goth{so}(4,1)$-action and the Hodge decomposition} \remark 1. $[L_I, \Lambda_J]=W_K$, $[L_J, \Lambda_K]=W_I$, $[L_I, \Lambda_K]=-W_J$. 2. The even part of $\goth a$ {\bf \red is isomorphic to $\goth{sp}(1,1, {\Bbb H})\oplus\R \cdot \Delta $.} 3. The odd part $\langle d, d_I, d_J, d_K, d,^* d_I^*, d_J^*, d_K^*\rangle$ {\bf \red generates the 9-dimensional odd Heisenberg algebra,} with the only non-trivial supercommutators being $\{d, d^*\}=\{d_I, d^*_I\}=\{d_J, d^*_J\}=\{d_K, d^*_K\}=\Delta$ 4. The action of $\goth a_{even}$ on $\goth a_{odd}$ {\bf \purple is the fundamental representation of $\goth{sp}(1,1, {\Bbb H})$ in ${\Bbb H^2}$,} with the quaternionic Hermitian metric on $\goth a_{odd}$ provided by the anticommutator. \corollary The weight decomposition of the $\goth{sp}(1,1, {\Bbb H})= \goth{so}(4,1)$-action on $H^*(M)$ {\bf \purple coincides with the Hodge decomposition.} %\newpage % %{\bf \blue Lefschetz-Frobenius algebras} % %\definition %{\bf \blue A Frobenius algebra} is %a graded commutative algebra $A =\bigoplus_{i=0}^{d} A^i$ equipped with %the Poincare-type non-degenerate product. % %\definition %A {\bf\blue Lefschetz triple} %in a Frobenius algebra $A =\bigoplus_{i=0}^{2n} A^i$ is a triple %of operators $L_\eta, H, \Lambda_\eta$ %where $\eta \in A^2$ is a fixed element, %$L_\eta(x):= \eta \wedge x$, $H\restrict{A^i} = i-n$ and %$\Lambda_\eta$ is an element such that %$L_\eta, H, \Lambda_\eta$ is an $\goth{sl}(2)$-triple. %A Frobenius algebra admitting a Lefschetz triple %is called {\bf \blue a Lefschetz-Frobenius algebra} %(Looijenga, Lunts). % %\remark %Such $\Lambda_\eta$ {\bf \purple is uniquely %determined by $H$ and $\eta$} (this statement is sometimes %called ``Morozov's lemma'', and sometimes included %in the statement of Jacobson-Morozov theorem). % % %\remark %Existence of $\Lambda_\eta$ %for given $\eta\in A^2$ is an open property in $A^2$, %hence {\bf \purple a Lefschetz-Frobenius algebra admits %many $\goth{sl}(2)$-triples. } \newpage {\bf \blue Lia algebra $\g$ generated by $\goth{sl}(2)$-triples } \theorem Let $M$ be a hyperk\"ahler manifold of maximal holonomy, $A^*$ its cohomology algebram and $\g:=\g(A)$ the Lie algebra generated by all Lefschetz $\goth{sl}(2)$-triples. {\bf \red Then $\g$ is isomorphic to $\goth{so}(b_2 -2,4)$.} {\bf \green Sketch of the proof. Step 1:} Consider the action of $\g$ on the {\bf\blue Mukai extension} $\hat H^2(M):= \R\cdot x \oplus H^2(M) \oplus \R\cdot y,$ where $x$ has grading 0, $y$ has grading 4, $H^2(M)$ has grading $2$. We equip $\hat H^2(M)$ with {\bf\blue the Mukai form} which is equal to BBF on $H^2(M)$, preserves grading, and satisfies $q_M(x, y)=1$ $q(x,x)=q(y,y)=0$, $x, y \bot H^2(M)$ and $(x,y)=1$. The action of $\g$ on $\hat H^2(M)$ is determined by the following properties: {\bf \purple 1. It is compatible with the grading. 2. For all $\alpha, \beta\in H^2(M)$, one has $L_\alpha x = \alpha$, $L_\alpha \beta = q(\alpha, \beta)y$, where $q$ is the BBF form. 3. $\Lambda_\alpha y = \alpha$, $\Lambda_\alpha \beta = q(\alpha, \beta)x$.} To see that this action is well-defined, we need to check that commutator relations hold. This follows from commutator relations in $\goth{so}(1,4)$ and Zariski density of pairs $\alpha, \beta \in \langle \omega_I, \omega_J, \omega_K\rangle$ in the set of all pairs $\alpha, \beta\in H^2(M)$. {\bf\green Step 2:} The map $\g \arrow \goth{so}(\hat H^2(M))$ is surjective, which follows from the dimension argument (dimensions are computed using the local Torelli theorem). Injectivity of $\g \arrow \goth{so}(\hat H^2(M))$ is clear, because $\goth{so}(\hat H^2(M))$ is given by generators and relations which hold true in $\g$. \endproof \newpage {\bf\blue Hodge structures and $\g$-action} \remark The Lie algebra $\g=\goth{so}(b_1 -2,4)$ is equipped with a grading $\g=\g_{-2}\oplus \g_0\oplus g_2$, induced by the grading on the Mukai space: $\hat H^2(M):= H_0 \oplus H^2(M) \oplus H_4$, with $H_0$ and $H_4$ 1-dimensional. Then $\g_0= \g_0'\oplus H$, where $H=[L_\omega,\Lambda_\omega]$ is the operator inducing the grading and commuting with the rest of $\g_0$, denoted by $\check \g_0$. \remark {\bf \purple The Lie algebra $\g_0':=\goth{so}(b_2 -3,3)$ is generated by the Weil maps $W_I$ for all complex structures $I$ of hyperk\"ahler type.} The corresponding Lie group $G_0'$ acts as $\Spin(b_2 -3,3)$ in odd-dimensional cohomology and $SO(b_2 -2,3)$ on even-dimensional ones. \corollary The natural action of $\g_0':=\goth{so}(b_2 -3,3)= \goth{so}(H^2(M,\R), q)$ on $H^2(M)$ {\bf \red is extended to an action on the algebra $H^*(M)$ by automorphisms.} Moreover, {\bf \red this action preserves the Chern classes.} {\bf \green This finishes the proof of Theorem 1.} \newpage {\bf \blue Automorphisms of cohomology.} \theorem Let $M$ be a simple hyperk\"ahler manifold, and $G\subset GL(H^*(M))$ a group of automorphisms of its cohomology algebra preserving the Pontryagin classes. Then $G$ acts on $H^2(M)$ {\bf \red preserving the BBF form.} Moreover, the map $G\arrow O(H^2(M, \R), q)$ {\bf \red is surjective on a connected component, and has compact kernel.} {\bf \green Proof. Step 1:} Fujiki formula $v^{2n}= q(v,v)^n$ implies that $\Gamma_0$ {\bf \purple preserves the Bogomolov-Beauville-Fujiki up to a sign. } The sign is fixed, if $n$ is odd. {\bf \green Step 2:} For even $n$, the sign is also fixed. Indeed, $G$ preserves $p_1(M)$, and (as Fujiki has shown) $v^{2n-2}\wedge p_1(M)= q(v,v)^{n-1} c$, for some $c\in \R$. The constant $c$ is positive, {\bf \purple because the degree of $c_2(B)$ is positive} for any Yang-Mills bundle with $c_1(B)=0$. {\bf \green Step 3:} ${\goth {so}}(H^2(M, \R), q)$ acts on $H^*(M, \R)$ by derivations preserving Pontryagin classes (Theorem 1). Therefore $\Lie(G)$ surjects to ${\goth o}(H^2(M, \R), q)$. {\bf \green Step 4:} {\bf \purple The kernel $K$ of the map $G \arrow G\restrict{H^2(M,\R)}$ is compact,} because it commutes with the Hodge decomposition and Lefschetz ${\goth sl}(2)$-action, hence preserves the Riemann-Hodge form, which is positive definite. \endproof \newpage {\bf \blue Computation of the mapping class group} {\bf \green Theorem:} {\bf \blue (Sullivan)} Let $M$ be a compact, simply connected K\"ahler manifold, $\dim_\C M\geq 3$. Denote by $\Gamma_0$ the group of automorphisms of an algebra $H^*(M, \Z)$ preserving the Pontryagin classes $p_i(M)$. Then {\bf \red the natural map $\Diff_+(M)/\Diff_0\arrow \Gamma_0$ has finite kernel, and its image has finite index in $\Gamma_0$.} {\bf \green Theorem:} Let $M$ be a simple hyperk\"ahler manifold, and $\Gamma_0$ as above. Then \\ (i) $\Gamma_0\restrict{H^2(M,\Z)}$ {\bf \blue is a finite index subgroup of $O(H^2(M, \Z), q)$.}\\ (ii) The map $\Gamma_0\arrow O(H^2(M, \Z), q)$ {\bf \blue has finite kernel.} \proof (i) is clear from the $\Spin(H^2(M,\R))$-action on $H^*(M)$, and (ii) follows because the kernel $K$ of the map $\Aut(H^*(M)) \arrow G\restrict{H^2(M,\R)}$ is compact, hence the discrete group $\Gamma_0\cap K$ is finite. \endproof \corollary {\bf \red The mapping class group of $M$ is mapped to $O(H^2(M, \Z), q)$ with finite kernel and finite index.} \end{document}