Obstruction classes for moduli spaces of sheaves and Lagrangian fibrations

TL;DR

This paper studies the obstruction classes of moduli spaces of sheaves on K3 surfaces, providing explicit calculations, criteria for their properties, and applications to related geometric structures.

Contribution

It explicitly determines the obstruction class in the Brauer group, establishes a sequence relating Brauer groups, and characterizes birational equivalences of Beauville-Mukai systems.

Findings

Explicit determination of the obstruction class and its order in the Brauer group.

A short exact sequence relating the Brauer groups of the moduli space and the K3 surface.

Complete characterization of birational equivalences of Beauville-Mukai systems.

Abstract

We investigate obstruction classes of moduli spaces of sheaves on K3 surfaces. We extend previous results by Caldararu, explicitly determining the obstruction class and its order in the Brauer group. Our main theorem establishes a short exact sequence relating the Brauer group of the moduli space to that of the underlying K3 surface. This provides a criterion for when the moduli space is fine, generalising well-known results for K3 surfaces. Additionally, we explore applications to Ogg-Shafarevich theory for Beauville-Mukai systems. Furthermore, we investigate birational equivalences of Beauville-Mukai systems on elliptic K3 surfaces, presenting a complete characterisation of such equivalences.