Hodge numbers of moduli spaces of stable bundles on K3 surfaces

TL;DR

This paper proves that the Hodge numbers of certain moduli spaces of stable sheaves on K3 surfaces match those of specific Hilbert schemes of points, revealing a deep geometric correspondence.

Contribution

It establishes a precise equivalence of Hodge numbers between moduli spaces of stable rank two sheaves and Hilbert schemes on K3 surfaces under certain conditions.

Findings

Hodge numbers of moduli spaces match those of Hilbert schemes

The result applies to moduli spaces with dimension at least 10

Provides a formula relating second Chern class and number of points

Abstract

We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is: Theorem: Let $X$ be a K3 surface, $L$ a primitive big and nef line bundle and $H$ a generic polarization. If the moduli space of rank two $H$ semi-stable torsion-free sheaves with determiant $L$ and second Chern class $c_2$ has at least dimension 10 then its Hodge numbers coincide with those of the Hilbert scheme of $l:=2c_2-\frac{L^2}{2}-3$ points on $X$.