Moduli Spaces of Sheaves on K3 Surfaces and Symplectic Stacks

TL;DR

This paper studies the moduli space of semistable sheaves on K3 surfaces as a symplectic stack, providing a new perspective and computing its cotangent complex using universal sheaves and Atiyah classes.

Contribution

It introduces the concept of symplectic stacks and demonstrates that all moduli stacks of semistable sheaves on K3 surfaces are examples, with explicit cotangent complex computations.

Findings

Moduli stacks of sheaves on K3 surfaces are symplectic stacks.

Cotangent complex is computed via universal sheaves on Quot schemes.

The notion of symplectic stacks generalizes classical symplectic geometry in algebraic stacks.

Abstract

We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.