Curve-counting invariants for crepant resolutions

Abstract

We construct curve counting invariants for a Calabi-Yau threefold $Y$ equipped with a dominant birational morphism $\pi:Y \to X$. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when $\pi:Y\to Y$ is the identity. Our main result is a PT/DT-type formula relating the partition function of our invariants to the Donaldson-Thomas partition function in the case when $Y$ is a crepant resolution of $X$, the coarse space of a Calabi-Yau orbifold $\mathcal{X}$ satisfying the hard Lefschetz condition. In this case, our partition function is equal to the Pandharipande-Thomas partition function of the orbifold $\mathcal{X}$. Our methods include defining a new notion of stability for sheaves which depends on the morphism $\pi $. Our notion generalizes slope stability which is recovered in the case where $\pi $ is the identity on $Y$. Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence via the Hall algebra and Joyce's integration map.