Birational symplectic manifolds and their deformations

TL;DR

This paper investigates the relationship between birational hyperkahler manifolds and their moduli spaces, providing conditions under which they define non-separated points and establishing deformation equivalences for certain moduli spaces.

Contribution

It proves that birational hyperkahler manifolds often correspond to non-separated points in moduli space and shows deformation equivalence of certain moduli spaces of sheaves to Hilbert schemes.

Findings

Birational hyperkahler manifolds can define non-separated points in moduli space.

Moduli spaces of rank two sheaves on K3 surfaces are deformation equivalent to Hilbert schemes.

The techniques apply to cases of Mukai's elementary transformations and codimension two isomorphisms.

Abstract

The known counterexamples to the global Torelli theorem for higher-dimensional hyperkahler manifolds are provided by birational manifolds. We address the question whether two birational hyperkahler manifolds (i.e. irreducible symplectic) manifolds always define non-separated points in the moduli space of marked manifolds. An affirmative answer is given for the cases of Mukai's elementary transformations and birational correspondences which are isomorphic in codimension two. The techniques are applied to show that the moduli spaces of rank two sheaves on a K3 surface are deformation equivalent to appropriate Hilbert schemes.