Curve-counting invariants for crepant resolutions

Abstract

We construct curve counting invariants for a Calabi-Yau threefold $Y$ equipped with a dominant birational morphism $π:Y \to X$. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when $π:Y\to Y$ 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 $π$. Our notion generalizes slope stability which is recovered in the case where $π$ 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.

In this version, Jim Bryan has been added as an author and the required boundedness result for our stability condition has been added. arXiv admin note: text overlap with arXiv:1002.4374 by other authors