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Algebraic Geometry Seminar - Jun Wang

Algebraic Geometry Seminar
January 14, 2020
3:00PM - 4:00PM
Math Tower 154

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Add to Calendar 2020-01-14 15:00:00 2020-01-14 16:00:00 Algebraic Geometry Seminar - Jun Wang Title: A mirror theorem for Gromov-Witten theory without convexity Speaker: Jun Wang - The Ohio State University Abstract: One central question in Gromov-Witten (GW) theory is to relate the GW invariants of a hypersurface to the GW invariants of the ambient space such as smooth projective variety or orbifold. In genus zero, this is usually done by the so-called quantum hyperplane principle, which uses the twisted GW invariants of the ambient space. This is analogous to the classical theorem that the number of lines inside a cubic surface can be obtained by computing the Euler number of a certain vector bundle on the space of lines inside \mathbb P^3 (which is the Grassmannian G(2,4)). But this approach requires a technical assumption called convexity for the line bundle over the ambient space defining the hypersurface, which can fail for hypersurfaces in orbifolds. In this talk, I will present a way to obtain the genus zero GW invariants of a positive hypersurface in Toric stacks for which the convexity may fail. One key ingredient in the proof is to resolve the genus zero quasimap Wall-Crossing (WC) conjecture proposed by Ionuţ Ciocan-Fontaine and Bumsig Kim, where we don't require the target to be carried with a good torus action as opposed to all previously proven WC examples (or hypersurfaces for which the convexity holds thereof). Seminar Link Math Tower 154 Department of Mathematics math@osu.edu America/New_York public

Title: A mirror theorem for Gromov-Witten theory without convexity

Speaker: Jun Wang - The Ohio State University

Abstract: One central question in Gromov-Witten (GW) theory is to relate the GW invariants of a hypersurface to the GW invariants of the ambient space such as smooth projective variety or orbifold. In genus zero, this is usually done by the so-called quantum hyperplane principle, which uses the twisted GW invariants of the ambient space. This is analogous to the classical theorem that the number of lines inside a cubic surface can be obtained by computing the Euler number of a certain vector bundle on the space of lines inside \mathbb P^3 (which is the Grassmannian G(2,4)). But this approach requires a technical assumption called convexity for the line bundle over the ambient space defining the hypersurface, which can fail for hypersurfaces in orbifolds. In this talk, I will present a way to obtain the genus zero GW invariants of a positive hypersurface in Toric stacks for which the convexity may fail. One key ingredient in the proof is to resolve the genus zero quasimap Wall-Crossing (WC) conjecture proposed by Ionuţ Ciocan-Fontaine and Bumsig Kim, where we don't require the target to be carried with a good torus action as opposed to all previously proven WC examples (or hypersurfaces for which the convexity holds thereof).

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