Modular sheaves with many moduli

TL;DR

The paper constructs large-dimensional moduli spaces of slope stable vector bundles on general polarized hyperkähler varieties of type K3^{[2]}, using specific cases involving elliptic K3 surfaces and exploring their birational relations to sheaf moduli spaces.

Contribution

It demonstrates the existence of high-dimensional moduli spaces of stable vector bundles on hyperkähler varieties of type K3^{[2]}, extending known constructions and proposing a conjectural relationship with moduli spaces of sheaves.

Findings

Existence of irreducible components of dimension 2a^2+2 for moduli spaces on HK varieties.

Construction of these components via cases on elliptic K3 surfaces.

Conjecture relating the closure of stable vector bundle moduli to smooth HK varieties of type K3^{[a^2+1]].

Abstract

We exhibit moduli spaces of slope stable vector bundles on general polarized HK varieties $(X,h)$ of type $K3^{[2]}$ which have an irreducible component of dimension $2a^2+2$, with $a$ an arbitrary integer greater than $1$. This is done by studying the case $X=S^{[2]}$ where $S$ is an elliptic $K3$ surface. We show that in this case there is an irreducible component of the moduli space of stable vector bundles on $S^{[2]}$ which is birational to a moduli space of sheaves on $S$. We expect that if the moduli space of sheaves on $S$ is a smooth HK variety (necessarily of type $K3^{[a^2+1]}$) then the following more precise version holds: the closure of the moduli space of slope stable vector bundles on $(X,h)$ in the moduli space of Gieseker-Maruyama semistable sheaves with its GIT polarization is a general polarized HK variety of type $K3^{[a^2+1]}$.