Equivariant Moduli Theory on $ K3 $ Surfaces

TL;DR

This paper investigates equivariant moduli spaces of sheaves on K3 surfaces with symplectic group actions, showing they are deformation equivalent to Hilbert schemes through advanced geometric correspondences.

Contribution

It establishes a link between equivariant moduli spaces and Hilbert schemes on K3 surfaces using Gieseker, Bridgeland, and derived McKay correspondence techniques.

Findings

Equivariant moduli spaces are irreducible symplectic manifolds.

They are deformation equivalent to Hilbert schemes of points.

The connection is made via Gieseker, Bridgeland, and derived McKay correspondence.

Abstract

In this paper we study equivariant moduli spaces of sheaves on a $ K3 $ surface $ X $ under a symplectic action of a finite group. We prove that under some mild conditions, equivariant moduli spaces of sheaves on $ X $ are irreducible symplectic manifolds deformation equivalent to Hilbert schemes of points on $ X $ via a connection between Gieseker and Bridgeland moduli spaces, as well as the derived McKay correspondence.