A smooth but non-symplectic moduli of sheaves on a hyperk\"ahler variety

TL;DR

This paper constructs a smooth, non-symplectic component of the moduli space of stable sheaves on a hyperk"ahler variety, specifically a generalized Kummer variety, revealing new geometric structures.

Contribution

It provides the first explicit example of a smooth, non-symplectic component in the moduli space of sheaves on a hyperk"ahler variety, characterized as a blowup of the dual abelian surface.

Findings

The moduli space component is isomorphic to the blowup of the dual abelian surface.

This component is smooth but has a non-trivial canonical bundle.

The result is specific to stable vector bundles on generalized Kummer varieties.

Abstract

For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles of degree $0$ is isomorphic to the blowup of the dual abelian surface in one point. We believe that this is the first explicit example of a component which is smooth with a non-trivial canonical bundle.