Enumerative invariants of 3-fold flops: hyperplane arrangements and wall-crossing

Abstract: 3-fold flopping contractions form a fundamental building block of the higher-dimensional Minimal Model Program. They exhibit extremely rich geometry, which has been investigated by many people over the past half-century. I will present an elegant and visually-pleasing relationship between enumerative invariants of flopping contractions and certain hyperplane arrangements constructed combinatorially from root system data. I will discuss both Gopakumar-Vafa (GV) and Gromov-Witten (GW) invariants, explaining how these are related to one another and how they are encoded in finite and infinite arrangements, respectively. Finally, I will discuss wall-crossing: our combinatorial approach allows us to explicitly construct flops from root system data, leading to a new “direct” proof of the Crepant Transformation Conjecture, with a very explicit formulation. This is joint work with Michael Wemyss.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

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