Generalized Donaldson-Thomas Invariants via Kirwan Blowups

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 $\tilde{\mathcal{M}}$ 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 $[\tilde{\mathcal{M}}{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$. We show that it is invariant under deformations of the complex structure of $W$.