Intrinsic Stabilizer Reduction and Generalized Donaldson-Thomas Invariants

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.