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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 $p<q$.
For even weight, {\bf \purple the period space for (weakly) polarized Hodge structures
is again a complex manifold}.



\newpage

{\bf \blue Hodge structures and homogeneous spaces}


\example
The {\bf\blue  Hodge structure of weight 1}
is a decomposition $V_\C = V^{1,0}\oplus V^{0,1}$, 
with $\overline{V^{1,0}}= V^{0,1}$. Clearly, the Hodge structures
of weight 1 are in bijective correspondence with complex structures on $V$.
Therefore, {\bf \purple the period space of Hodge structures of weight 1
is identified with $GL(2n, \R)/GL(n, \C)$.} 


\example
The {\bf\blue  Hodge structure of K3 type}
is a (weakly polarized) Hodge structure
$V_\C= \bigoplus_{ p+q=2 \atop p,q \geq 0} V^{p, q}$
of weight 2 with $\dim V^{2,0}=1$.

\remark {\bf \purple 
The period space of weakly  polarized Hodge structures of K3 type}
is identified with the quadric of lines $Q:=\{l\in {\Bbb P}V_\C\ \ |\ \ h(l, l)=0, h(l, \bar l)\neq 0\}$.

\theorem {\bf \blue (Kuga-Satake)} \\
Let $Q$ be the space of weakly polarized Hodge structures of K3 type on $(W,h)$.
Then {\bf \red there exists a vector space $V$ equipped with $SO(W)$-action and
an $SO(W)$-equivariant embedding from $Q$ to the space of Hodge structures
of weight 1 on $V$.}



\newpage

{\bf \blue Kuga-Satake embedding and Clifford modules}


\theorem {\bf \blue (Kuga-Satake)} 
Let $Q$ be the space of weakly polarized Hodge structures of K3 type on $(W,h)$.
Then {\bf \red there exists a vector space $V$ equipped with $SO(W)$-action and
an $SO(W)$-equivariant embedding from $Q$ to the space of Hodge structures
of weight 1 on $V$.}

\pstep
For any Hodge structure of K3 type, the corresponding action of
$\goth{u}(1)$ is generated by a skew-symmetric
matrix $\mu$ of rank 2, acting trivially on the orthogonal
complement to a 2-dimensional plane $l=\langle \Re \Theta, \im \Theta\rangle$,
where $\Theta$ is a generator of $V^{2,0}$, and acting
as $\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$ on $l$.

{\bf \green Step 2:} Let $\Cl(W)$ be the Clifford algebra of $W$, and
$V$ a space with $\Cl(W)$-action (such space is called {\bf \blue Clifford module}.
Using the standard embedding $\goth{so}(W)\subset \Cl(W)$,
we can consider $\mu$ as an element of $\Cl(W)$. Then
 $\mu^2=-1$ in the Clifford algebra, and this gives
a complex structure on $V$.
\endproof

\remark
Kuga and Satake were interested in {\bf \green constructing an embedding of the
symmetric spaces} associated with polarized Hodge structures of 
weight 1 and of K3 type.


\newpage

{\bf \blue Hyperk\"ahler manifolds}

\definition (E. Calabi, 1978)\\ Let $(M, g)$ be a Riemannian
manifold equipped with three complex structure operators
$I, J, K:\; TM\arrow TM$, satisfying the quaternionic relation
\[ I^2=J^2=K^2=IJK=-\Id.\]  Suppose that $I$, $J$, $K$ are
K\"ahler. Then $(M, I, J, K, g)$ is called {\bf \blue hyperk\"ahler}.


\remark A hyperk\"ahler manifold $M$ is
equipped with 3 symplectic forms $\omega_I$, $\omega_J$, 
$\omega_K$. The form 
$\Omega:= \omega_J+\1\omega_K$ 
{\bf \purple is a holomorphic symplectic 2-form on
$(M,I)$.} \endproof


\theorem (Calabi-Yau)
Let $M$ be a compact, holomorphically symplectic K\"ahler
manifold. Then {\bf \red $M$ admits a hyperk\"ahler metric,} which is
uniquely determined by the cohomology class of its 
K\"ahler form $\omega_I$.

{\green \it Hyperk\"ahler geometry is essentially
the same as holomorphic symplectic geometry}


\newpage

{\bf \blue Kuga-Satake construction in arbitrary dimension}

\remark
Let $M$ be a hyperkahler manifold
Kuga-Satake construction {\bf \purple gives an embedding from $H^2(M)$
to the second cohomology of a torus, 
compatible with the Hodge structure.}
Indeed, $W$ is embedded to $\Lambda^2(V)$,
where $V$ is a $\Cl(W)$-module.

\theorem
For any hyperkahler manifold $M$ of complex dimension $n$,
{\bf \red there exists a torus $T$ of dimension $n+l$ 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 $M$.

\remark The corresponding map
from the period space of $M$ to the period space of $T$
{\bf \purple coinsides with the Kuga-Satake map.}



\newpage

{\bf \blue Holomorphically symplectic manifolds}

{\bf\green DEFINITION:} {\bf \blue A holomorphically symplectic manifold }
is a complex manifold equipped with non-degenerate, holomorphic
$(2,0)$-form.

\remark Hyperk\"ahler manifolds are holomorphically symplectic.
Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic
form on $(M,I)$.

{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
{\bf \red admits a unique hyperk\"ahler metric in any K\"ahler class.}

{\bf\green DEFINITION:}  For the rest of this talk, 
{\bf \red a hyperk\"ahler manifold
is a compact, K\"ahler, holomorphically symplectic manifold.}

{\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called
{\bf \blue of maximal holonomy}, or {\bf \blue simple},
or {\bf \blue IHS}, if $\pi_1(M)=0$, $H^{2,0}(M)=\C$.

{\bf \purple Bogomolov's decomposition:} Any 
hyperk\"ahler manifold admits a finite covering
which is a product of a torus and several 
maximal holonomy hyperk\"ahler manifolds.

{\bf \red Further on, all hyperk\"ahler manifolds
are assumed to be of maximal holonomy}.

\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 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}

