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DTSTART;TZID=America/New_York:20211102T093000
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DTSTAMP:20240213T062436Z
CREATED:20240213T062436Z
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UID:10002107-1635845400-1635849000@cmsa.fas.harvard.edu
SUMMARY:Counting invariant curves on a Calabi-Yau threefold with an involution
DESCRIPTION:Abstract: Gopakumar-Vafa invariants are integers n_beta(g) which give a virtual count of genus g curves in the class beta on a Calabi-Yau threefold. In this talk\, I will give a general overview of two of the sheaf-theoretic approaches to defining these invariants: via stable pairs a la Pandharipande-Thomas (PT) and via perverse sheaves a la Maulik-Toda (MT). I will then outline a parallel theory of Gopakumar-Vafa invariants for a Calabi-Yau threefold X with an involution. They are integers n_beta(g\,h) which give a virtual count of curves of genus g in the class beta which are invariant under the involution and whose quotient by the involution has genus h. I will give two definitions of n_beta(g\,h) which are conjectured to be equivalent\, one in terms of a version of PT theory\, and one in terms of a version of MT theory. These invariants can be computed and the conjecture proved in the case where X=SxC where S is an Abelian or K3 surface with a symplectic involution. In these cases\, the invariants are given by formulas expressed with Jacobi modular forms. In the case where S is an Abelian surface\, the specialization of n_beta(g\,h) to h=0 recovers the count of hyperelliptic curves on Abelian surfaces first computed by B-Oberdieck-Pandharipande-Yin. This is joint work with Stephen Pietromonaco.
URL:https://cmsa.fas.harvard.edu/event/counting-invariant-curves-on-a-calabi-yau-threefold-with-an-involution/
CATEGORIES:Joint Harvard-CUHK-YMSC Differential Geometry
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/Jim-Bryan_poster_3Nov2021.png
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