Teichmuller space for hyperkahler and symplectic structures

TL;DR

This paper characterizes the Teichmuller space of hyperkahler and symplectic structures on hyperkahler manifolds, linking it to Grassmannians and cohomology classes, and clarifies the structure of these moduli spaces.

Contribution

It explicitly identifies the connected components of the Teichmuller space with positive 3-planes and cohomology classes, extending understanding of hyperkahler moduli spaces.

Findings

Connected components correspond to positive 3-planes avoiding MBM classes.

Teichmuller space of symplectic structures is identified with classes having positive Bogomolov-Beauville-Fujiki form.

Provides a geometric description of the moduli space structure.

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 $H^2(M,\R)$.