Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

TL;DR

This paper investigates the stability properties of vector bundles on Hilbert schemes of points on K3 surfaces, revealing new relationships between the geometry of the surface and the moduli spaces of stable bundles.

Contribution

It demonstrates that in specific cases, the universal family of stable vector bundles on a K3 surface induces a flat family on the Hilbert scheme, linking the surface to a component of a moduli space of sheaves.

Findings

Universal family forms a complete flat family of stable bundles on the Hilbert scheme.

Identifies the K3 surface with a component of a moduli space of sheaves on the Hilbert scheme.

Establishes stability properties of vector bundles in the context of Hilbert schemes.

Abstract

Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $\mathcal{E}$ on $X\times M$ can be understood as a complete flat family of stable vector bundles on $M$ parametrized by $X$, which identifies $X$ with a smooth connected component of some moduli space of stable sheaves on $M$.