Counting equivariant sheaves on K3 surfaces

TL;DR

This paper investigates the counting of equivariant sheaves on K3 surfaces with finite group actions, establishing invariance of Joyce invariants under stability conditions and proving a multiple cover formula for local K3 surfaces.

Contribution

It demonstrates the invariance of Joyce invariants for Gieseker semistable sheaves on quotient stacks of K3 surfaces and proves a multiple cover formula for local K3 surfaces with symplectic group actions.

Findings

Joyce invariants are independent of Bridgeland stability conditions.

Established the multiple cover formula for counting invariants.

Applied results to local K3 surfaces with symplectic group actions.

Abstract

We study the equivariant sheaf counting theory on K3 surfaces with finite group actions. Let $\sS=[S/G]$ be a global quotient stack, where $S$ is a K3 surface and $G$ is a finite group acting as symplectic homomorphisms on $S$. We show that the Joyce invariants counting Gieseker semistable sheaves on $\sS$ are independent on the Bridgeland stability conditions. As an application we prove the multiple cover formula of Y. Toda for the counting invariants for semistable sheaves on local K3 surfaces with a symplectic finite group action.