Curve-counting invariants for crepant resolutions

الأصل · EN

We construct curve counting invariants for a Calabi-Yau threefold Y equipped with a dominant birational morphism π:Y → X. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when π:Y→ 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 X satisfying the hard Lefschetz condition. In this case, our partition function is equal to the Pandharipande-Thomas partition function of the orbifold 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.