Intrinsic Stabilizer Reduction and Generalized Donaldson-Thomas Invariants

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.