K\"ahlerness of moduli spaces of stable sheaves over non-projective K3 surfaces

TL;DR

This paper characterizes when moduli spaces of stable sheaves over non-projective K3 surfaces are irreducible hyperkähler manifolds based on their Betti and Hodge numbers.

Contribution

It provides a necessary and sufficient condition linking the Betti number and Hodge numbers for the hyperkähler property of these moduli spaces.

Findings

Moduli space is hyperkähler iff second Betti number equals sum of Hodge numbers.

Establishes a criterion for hyperkähler structure in non-projective K3 surface moduli.

Connects topological invariants with geometric structure of moduli spaces.

Abstract

We show that a moduli space of slope-stable sheaves over a K3 surface is an irreducible hyperk\"ahler manifold if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.