Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences

TL;DR

This paper demonstrates that certain moduli spaces of torsion sheaves on K3 surfaces admit autoequivalences of their derived categories constructed via P-functors, involving geometric transformations like flops and fibrations, and explores their derived equivalences.

Contribution

It introduces a new method to construct autoequivalences of derived categories of moduli spaces using P-functors and geometric transformations, and establishes derived equivalences between non-birational hyperkähler manifolds.

Findings

Constructed autoequivalences of derived categories via P-functors.

Established derived equivalences between non-birational hyperkähler manifolds.

Linked geometric transformations to derived category autoequivalences.

Abstract

We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivalent.