Stable vector bundles on generalized Kummer varieties

TL;DR

This paper constructs new stable vector bundles on generalized Kummer varieties derived from abelian surfaces, revealing complex geometric structures and moduli space components that differ from K3 surfaces.

Contribution

It introduces two explicit families of stable bundles on generalized Kummer varieties, expanding understanding of their moduli spaces and geometric properties.

Findings

Two new families of stable vector bundles constructed

Moduli space components are smooth, connected, and holomorphic symplectic

These components are not simply connected, unlike K3 surfaces

Abstract

For an abelian surface $A$, we explicitly construct two new families of stable vector bundles on the generalized Kummer variety $K_n(A)$ for $n\geqslant 2$. The first is the family of tautological bundles associated to stable bundles on $A$, and the second is the family of the "wrong-way" fibers of a universal family of stable bundles on the dual abelian surface $\widehat{A}$ parametrized by $K_n(A)$. Each family exhibits a smooth connected component in the moduli space of stable bundles on $K_n(A)$, which is holomorphic symplectic but not simply connected, contrary to the case of K3 surfaces.