A new six dimensional irreducible symplectic variety

TL;DR

This paper constructs a new six-dimensional irreducible symplectic variety arising from a desingularization of a moduli space on a genus-two Jacobian, which is not deformation equivalent to known examples.

Contribution

The paper introduces a novel 6D irreducible symplectic variety from moduli space desingularization, expanding the class of known hyperkähler manifolds.

Findings

The fiber over (0,0) is a 6D irreducible symplectic variety.

It has second Betti number equal to 8.

It is not deformation equivalent to known examples.

Abstract

By results of the author there exists a projective (holomorphic) symplectic desingularization of the moduli space of rank-two torsion-free sheaves on a genus-two Jacobian with $c_1=0$ and $c_2=2$. This desingularization has a natural map to the self-product of the Jacobian. We show that the fiber over $(0,0)$ is a 6-dimensional projective irreducible symplectic variety (and hence a 12-dimensional compact Hyperkahler manifold) with second Betti number equal to 8. Thus it is not deformation equivalent to any of the (few) known examples of irreducible symplectic varieties, even up to birational equivalence.