Generalized Donaldson-Thomas Invariants via Kirwan Blowups

TL;DR

This paper introduces a new virtual cycle approach to generalized Donaldson-Thomas invariants for Calabi-Yau threefolds, utilizing Kirwan blowups to define invariants that remain stable under complex structure deformations.

Contribution

It develops a novel method using Kirwan blowups and semi-perfect obstruction theories to define deformation-invariant generalized Donaldson-Thomas invariants.

Findings

Constructed a Deligne-Mumford stack with a semi-perfect obstruction theory.

Defined invariants via the degree of the virtual cycle on the blown-up stack.

Proved invariance of the invariants under complex structure deformations.

Abstract

We develop a virtual cycle approach towards generalized Donaldson-Thomas theory of Calabi-Yau threefolds. Let $\mathcal{M}$ be the moduli stack of Gieseker semistable sheaves of fixed topological type on a Calabi-Yau threefold $W$. We construct an associated Deligne-Mumford stack $\widetilde{\mathcal{M}}$ with an induced semi-perfect obstruction theory of virtual dimension zero and define the generalized Donaldson-Thomas invariant of $W$ via Kirwan blowups to be the degree of the virtual cycle $[\widetilde{\mathcal{M}}]^{\mathrm{vir}}$. We show that it is invariant under deformations of the complex structure of $W$.