Generalized Donaldson–Thomas invariants via Kirwan blowups

Abstract

$\def\M{\mathcal{M}}\def\Mtilde{\widetilde{\M}}$We develop a virtual cycle theoretic approach towards generalized Donaldson–Thomas theory of Calabi–Yau threefolds. Let $\M$ 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 $\Mtilde$ 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 $[\Mtilde]^{\mathrm{vir.}}$ We show that it is invariant under deformations of the complex structure of $W$.