Intrinsic Stabilizer Reduction and Generalized Donaldson-Thomas Invariants

TL;DR

This paper introduces a generalized framework for Donaldson-Thomas invariants on Calabi-Yau threefolds, utilizing intrinsic stabilizer reduction and Kirwan blowups to produce deformation-invariant virtual counts of semistable objects.

Contribution

It constructs a new intrinsic stabilizer reduction stack with a semi-perfect obstruction theory and defines generalized DT invariants via virtual cycles, extending previous approaches to broader stability conditions.

Findings

Defines generalized Donaldson-Thomas invariants for various stability conditions.

Constructs a proper Deligne-Mumford stack with a semi-perfect obstruction theory.

Invariants remain stable under complex structure deformations.

Abstract

Let $\sigma$ be a stability condition on the bounded derived category $D^b({\mathop{\rm Coh}\nolimits} W)$ of a Calabi-Yau threefold $W$ and $\mathcal{M}$ a moduli stack parametrizing $\sigma$-semistable objects of fixed topological type. We define generalized Donaldson-Thomas invariants which act as virtual counts of objects in $\mathcal{M}$, fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne-Mumford stack $\widetilde{\mathcal{M}}^{\mathbb{C}^\ast}$, called the $\mathbb{C}^\ast$-rigidified intrinsic stabilizer reduction of $\mathcal{M}$, with an induced semi-perfect obstruction theory of virtual dimension zero, and define the generalized Donaldson-Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde{\mathcal{M}}^{\mathbb{C}^\ast}]^{\mathrm{vir}} \in A_0 (\widetilde{\mathcal{M}}^{\mathbb{C}^\ast})$. This stays invariant under deformations of the complex structure of $W$. Examples of applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.