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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \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:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \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:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1: }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Kuga-Satake map for arbitrary dimension} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Lille, March 9, 2017\\ Workshop on holomorphic symplectic varieties }\\[30mm] {\bf\small \red Joint work with Nikon Kurnosov and Andrei Soldatenkov} \end{center} %\newpage % %{\bf \blue Plan:} % %1. Hodge structures. Kuga-Satake construction for Hodge structures of K3 type. % %2. Hyperk\"ahler manifolds. Multi-dimensional Kuga-Satake construction: the main result. % %3. Lefschetz triples in Frobenius algebras. Explicit computation of the %algebra $\g$ generated by Lefschetz triples for a hyperkahler manifold. % %4. $k$-symplectic spaces: construction and applications to the %Lefschetz-Frobenius algebra. % %5. $\g$-invariant embedding of the cohomology algebra to the Grassmann %algebra of a $k$-symplectic space. % %6. $\g$-action and the Hodge decomposition. \newpage {\bf \blue Hodge structures} \definition Let $V_\R$ be a real vector space. {\bf \blue A (real) Hodge structure of weight $w$} on a vector space $V_\C=V_\R \otimes_\R \C$ is a decomposition $V_\C =\bigoplus_{p+q=w} V^{p,q}$, satisfying $\overline{V^{p,q}}= V^{q,p}$. It is called {\bf \blue integer Hodge structure} if one fixes an integer lattice $V_\Q$ or $V_\Z$ such that $V_\R=V_\Q \otimes_\Q \R$ or $V_\R=V_\Z \otimes_\Z \R$. A Hodge structure is equipped with $U(1)$-action, with $u\in U(1)$ acting as $u^{p-q}$ on $V^{p,q}$. {\bf\blue Morphism} of integer Hodge structures is a map which is $U(1)$-invariant and preserves the lattice. \definition {\bf \blue Weak polarization} on a Hodge structrure of weight $w$ is a $U(1)$-invariant non-degenerate 2-form $h\in V_\Q^*\otimes V^*_\Q$ (symmetric or antisymmetric depending on parity of $w$). It is called {\bf\blue polarization} if it is in addition satisfies $ -(\1)^{p-q}h(x, \bar x)>0$ for each non-zero $x\in V^{p,q}$. \definition {\bf \blue Period space} of (weakly polarized) Hodge structure with dimensions $\dim V^{p,q}=v_{pq}$ is the space of all decompositions $V_\C =\bigoplus_{p+q=w} V^{p,q}$ such that the above conditions are sattisfied. \remark The period space, for odd weight, {\bf \purple is a complex manifold.} Indeed, the $V^{q,p}$ spaces for $q >p$ are determined by $V^{p,q}$ uniquely, hence the period space is an open subspace in the space of $k$-tuples of subspaces $V^{p, q}\subset V\otimes \C$, with $p0$ 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 Multi-dimensional BBF form} This is the original motivation for the present work. 1. BBF form is a weak polarization on $H^2(M)$ compatible with all complex structures. 2. It is not hard to produce a weak polarization on $H^k(M)$ compatible with all complex structures. However, there is no canonical choice. \definition Let $a, b\in H^{2k}(M)$, $M$ K\"ahler of complex dimension $2n$, and $q\in \Sym^2(H^2(M))\subset H^4(M)$ be the element corresponding to the BBF form. Then {\bf \blue the multi-dimensional BBF form} is $a, b \arrow \int_M a\wedge b \wedge q^{n-k}$. \conjecture {\bf \red It is non-degenerate.} \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 $q_M(x, y)=1$ $x^2=y^2=0$, $x, y \bot h^2(M)$ 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 Structure of $\g$ and its graded components} % %\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: $H_m^2(M):= H_0 \oplus H^2(M) \oplus H_4$, %with $H_0$ and $H_4$ 1-dimensional. Then %$\g_0= \check \g_0\oplus H$, 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 $\check \g_0:=\goth{so}(b_1 -1,3)$ %is generated by the Weil maps $W_I$ for all complex %structures $I$ of hyperk\"ahler type obtained by deformations.} %The corresponding Lie group $G_0$ acts as $\Spin(b_1 -1,3)$ %in odd-dimensional cohomology and $SO(b_1 -1,3)$ on %even-dimensional ones. It is generated by the complex %structure action on $H^2(M)$ for all deformations of $I$. % %\remark %The space $\g_2=H^2(M)$ is the space of all $L_\alpha$, with $\alpha \in H^2(M)$. %Its dual space $\g_{-2}$ is generated by all $\lambda_\alpha$. The action on %$G_0$ on $\g_{\pm 2}$ is compatible with these identifications. \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 $\g_0':=\goth{so}(b_1 -1,3)$ is generated by the Weil maps $W_I$ for all complex structures $I$ of hyperk\"ahler type obtained by deformations.} The corresponding Lie group $G_0$ acts as $\Spin(b_1 -1,3)$ in odd-dimensional cohomology and $SO(b_1 -1,3)$ on even-dimensional ones. It is generated by the complex structure action on $H^2(M)$ for all deformations of $I$. \corollary Let $M$ be a hyperk\"ahler manifold, and $H^*(M) \mapsto H^{*+l}(T)$ an embedding to the cohomology of a torus. Suppose that this embedding is compatible with an action of the Lie algebra generated by all Lefschetz $sl(2)$-triples on $M$. {\bf \red Then it is compatible with the Hodge structures,} in the same sense as the usual Kuga-Satake map. \newpage {\bf \blue $k$-symplectic structures} \definition Let $V$ be a $4n$-dimensional vector space, and $\Psi:\; W \arrow \Lambda^2(V)$ a linear map. Assume that $\Psi(\omega)$ is a symplectic form for general $\omega\in W$, and has rank $\frac 1 2 \dim W$ for $\omega$ in a non-degenerate quadric $Q\subset W$. Then $\Psi$ is called {\bf\blue $k$-symplectic structure on $V$}, where $k=\dim W$. \example For $k=3$ this is {\bf \blue hypersymplectic structure}, which is the same as a triple of symplectic form $\omega_1, \omega_2, \omega_3$ such that $\omega_i \omega_j^{-1}$ act on $V$ as the matrices algebra $\Mat(2)$. In a sense, hypersymplectic structure is ``a complexification of a hyperk\"ahler structure''. \theorem {\bf \blue (Soldatenkov-V.)} Let $V$ be a $k$-symplectic space. {\bf \red Then $V$ is a Clifford module over a Clifford algebra $\Cl(W)$}. \theorem In this situation, {\bf \red the algebra generated by $\goth{sl}(2)$-triples associated with $\omega\in W$ acts on $\Lambda^*(V)$ as the algebra $\g= \goth{so}(\hat W)$,} where $\hat W= W \oplus \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$ is Mukai extension of $W$. \proof We need to check that the $\goth{so}(4, 1)$-relations hold for any non-degenerate 3-dimensional subspace $W_3\subset W$. However, this subspace is associated with a hypersymplectic structure, which is a complexification of a hyperk\"ahler, and satisfies the same relations. \endproof \newpage {\bf \blue Proof of the main result} \theorem For any hyperkahler manifold $M$ of complex dimension $n$, {\bf \red there exists a torus $T$ of dimension $n+k$ and an embedding of cohomology space $H^*(M) \mapsto H^{*+l}(T)$ which is compatible with the Hodge structures and the Poincare pairing. } Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz $sl(2)$-triples on $M$. \proof Let $\g$ be the Lie algebra generated by all $\goth{sl}(2)$-triples, and $W:=H^2(M)$. For any Clifford module $V$ over $\Cl(W)$, $V$ admits a $b_2$-symplectic structure which gives $\g$-action in $\Lambda^*(V)$. {\bf \purple If we manage to produce an embedding of $\g$-modules $H^*(M) \hookrightarrow \Lambda^*(V)$, we are done.} However, $\Lambda^*(V)$ is an exact representation of $\Spin(\hat W)$, hence its tensor powers contain any representation of $\Spin(\hat W)$. These tensor powers correspond to $\Lambda^*(V^n)$, which is also a Grassmann algebra for a Clifford module over $\Cl(W)$. \endproof \end{document}