Geometric Analysis and Topology Seminar

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Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces

Speaker: Nick McCleerey, Purdue University

Location: Warren Weaver Hall 512

Date: Friday, January 31, 2025, 11 a.m.

Synopsis:

Using the framework of Yang Li, we establish the weak metric SYZ conjecture on the existence of special Lagrangian fibrations for a class of maximally degenerate families of Calabi-Yau hypersurfaces in projective space. This is accomplished by studying non-Archimedean and tropical Monge-Ampère equations, on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space; this is enough to apply the method of Li. This is based on joint works with R. Andreasson, J. Hultgren, M. Jonsson, and E. Mazzon.