Moduli of sheaves on $K3$'s and higher dimensional HK varieties

TL;DR

This paper reviews the deformation properties of moduli spaces of stable sheaves on K3 surfaces and extends these results to moduli of vector bundles on higher-dimensional hyperk"ahler varieties of Type K3^{[2]}, highlighting their geometric significance.

Contribution

The paper revisits known deformation results for moduli spaces on K3 surfaces and extends these techniques to higher-dimensional hyperk"ahler varieties, providing new insights into their moduli spaces.

Findings

Moduli spaces of stable sheaves on K3 surfaces are deformations of hyperk"ahler varieties.

Extension of deformation results to moduli of vector bundles on K3^{[2]}-type hyperk"ahler varieties.

New methods adapted for higher-dimensional hyperk"ahler moduli analysis.

Abstract

We review a proof of the well know result stating that moduli spaces of stable sheaves with fixed Chern character on a polarized $K3$ surface are deformations of a hyperk\"ahler variety of Type $K3^{[n]}$ (if a suitable numerical hypothesis is satisfied). In a recent work we have adapted that proof in order to prove results on moduli of vector bundles on polarized hyperk\"ahler varieties of Type $K3^{[2]}$ - this is the content of the second part of the paper.