Moduli of sheaves on hyperk\"ahler manifolds

TL;DR

This paper surveys recent progress in understanding the moduli spaces of stable sheaves on hyperk"ahler manifolds, extending well-known theories from K3 surfaces to higher dimensions.

Contribution

It reviews techniques from the K3 surface case and discusses their extension to higher-dimensional hyperk"ahler manifolds.

Findings

Summarizes recent advances in moduli space theory for hyperk"ahler manifolds.

Highlights techniques extendable from K3 surfaces to higher dimensions.

Provides a comprehensive overview of current research in the area.

Abstract

We survey recent advances in the theory of moduli spaces of stable sheaves on hyperk\"ahler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the techniques which can be extended to higher dimensions.