Teichmüller space for hyperkähler and symplectic structures

Abstract

Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmuller space of hyperkahler structures on a hyperkahler manifold, identifying any of its connected components with an open subset of the Grassmannian $SO(b_2-3,3)/SO(3)\times SO(b_2-3)$ consisting of all Beauville-Bogomolov positive 3-planes in $H^2(M, R)$ which are not orthogonal to any of the MBM classes. This is used to determine the Teichmuller space of symplectic structures of Kahler type on a hyperkahler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes $v\in H^2(M,\R)$ with $q(v,v)>0$, where $q$ is the Bogomolov-Beauville-Fujiki form on $H^2(M,\R)$.