Stable Sheaves on K3 Surfaces via Wall-Crossing

TL;DR

This paper proves that moduli spaces of stable complexes on K3 surfaces are hyperk"{a}hler varieties of a specific type, using wall-crossing and derived equivalences to simplify the problem.

Contribution

It provides a new proof of the hyperk"{a}hler nature of these moduli spaces, leveraging wall-crossing and derived equivalences for Bridgeland stability.

Findings

Moduli spaces are hyperk"{a}hler varieties of [K3] type.

The proof reduces to the case of Hilbert schemes of points.

The approach simplifies understanding of stable complexes on K3 surfaces.

Abstract

We give a new proof of the following theorem: moduli spaces of stable complexes on a complex projective K3 surface, with primitive Mukai vector and with respect to a generic Bridgeland stability condition, are hyperk\"{a}hler varieties of $\mathrm{K3}^{[n]}$-type of expected dimension. We use derived equivalences, deformations and wall-crossing for Bridgeland stability to reduce to the case of the Hilbert scheme of points.