The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface

TL;DR

This paper characterizes the weight-two Hodge structure of moduli spaces of sheaves on K3 surfaces, showing they are irreducible symplectic varieties and computing related Donaldson invariants.

Contribution

It proves the Hodge structure matches Mukai's description for primitive first Chern class and computes higher-rank Donaldson polynomials for K3 surfaces.

Findings

Hodge structure aligns with Mukai's description

Moduli spaces are irreducible symplectic varieties

Computed higher-rank Donaldson polynomials

Abstract

We prove that the weight-two Hodge structure of moduli spaces of torsion-free sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we assume the first Chern class is primitive). We prove the moduli space is an irreducible symplectic variety (by Mukai's work it was known to be symplectic). By work of Beauville, this implies that its $H^2$ has a canonical integral non-degenrate quadratic form; Mukai's recepee for $H^2$ includes a description of Beauville's quadratic form. As an application we compute higher-rank Donaldson polynomials of $K3$ surfaces.