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Transactions of the American Mathematical Society

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Curve counting invariants for crepant resolutions


Authors: Jim Bryan and David Steinberg
Journal: Trans. Amer. Math. Soc. 368 (2016), 1583-1619
MSC (2010): Primary 14N35
Published electronically: June 15, 2015
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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$.


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Additional Information

Jim Bryan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: jbryan@math.ubc.ca

David Steinberg
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Address at time of publication: Department of Mathematics, Fenton Hall, University of Oregon, Eugene, Oregon 97403-1222
Email: dsteinbe@math.ubc.ca, dcstein@uoregon.edu

DOI: https://doi.org/10.1090/tran/6377
Received by editor(s): August 20, 2012
Received by editor(s) in revised form: December 18, 2013
Published electronically: June 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society