2012/08/04 by Jim Bryan, Bryan, Jim, David Steinberg +1 · 1 citation
Mathematics · #14N35 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14N35
paper · pdf · doi:10.48550/arxiv.1208.0884
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
arxiv created 2014/07/01 · arxiv updated 2014/07/02
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.