Curve counting invariants for crepant resolutions

Bryan, Jim; Steinberg, David

doi:10.1090/tran/6377

Curve counting invariants for crepant resolutions
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by Jim Bryan and David Steinberg PDF

Trans. Amer. Math. Soc. 368 (2016), 1583-1619 Request permission

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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