Towards global homological mirror symmetry for genus 2 curves

Catherine Cannizzo (Simons Center)

29-Oct-2020, 15:00-16:00 (3 years ago)

Abstract: The first part of the talk will discuss work in arXiv:1908.04227 [math.SG] on constructing a Donaldson-Fukaya-Seidel type category for the generalized SYZ mirror of a genus $2$ curve. We will explain the categorical mirror correspondence on the cohomological level. The key idea uses that a $4$-torus is SYZ mirror to a $4$-torus. So if we view the complex genus $2$ curve as a hypersurface of a $4$-torus $V$, a mirror can be constructed as a symplectic fibration with fiber given by the dual $4$-torus $V^\vee$. Hence on categories, line bundles on $V$ are restricted to the genus $2$ curve while fiber Lagrangians of $V^\vee$ are parallel transported over $U$-shapes in the base of the mirror. Next we describe ongoing work with H. Azam, H. Lee, and C-C. M. Liu on extending the result to a global statement, namely allowing the complex and symplectic structures to vary in their real six-dimensional families. The mirror statement for this more general result relies on work of A. Kanazawa and S-C. Lau.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )


Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

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Organizers: Alexander Kasprzyk*, Johannes Hofscheier*, Erroxe Etxabarri Alberdi
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