A twisted derived category of hyper-K\"ahler varieties of $K3^{[n]}$-type

TL;DR

This paper proposes a conjecture linking twisted derived categories of hyper-K"ahler varieties of $K3^{[n]}$-type to their Markman-Mukai lattice, proving it under certain numerical conditions and establishing derived equivalences of moduli spaces.

Contribution

It introduces a conjecture relating twisted derived categories to the Markman-Mukai lattice and proves it in specific cases, advancing understanding of derived equivalences in hyper-K"ahler geometry.

Findings

Conjecture relating twisted derived categories to Markman-Mukai lattice.

Proof of the conjecture under numerical constraints.

Derived equivalence of moduli spaces of stable sheaves on $K3$ surfaces.

Abstract

We conjecture that a natural twisted derived category of any hyper-K\"ahler variety of $K3^{[n]}$-type is controlled by its Markman-Mukai lattice. We prove the conjecture under numerical constraints, and our proof relies heavily on Markman's projectively hyperholomorphic bundle and a recently proven twisted version of the D-equivalence conjecture. In particular, we prove that any two fine moduli spaces of stable sheaves on a $K3$ surface are derived equivalent if they have the same dimension.