Tropical Theta Functions and Log Calabi-Yau Surfaces

TL;DR

This paper extends combinatorial methods from toric geometry to log Calabi-Yau surfaces using tropical and mirror symmetry techniques, introducing theta functions and their tropicalizations to generalize classical dualities and polytopes.

Contribution

It introduces a framework replacing lattices with integral linear manifolds and defines tropicalized theta functions, broadening the scope of toric geometry methods to log Calabi-Yau surfaces.

Findings

Described tropicalizations of theta functions.

Generalized dual pairings and polytopes.

Extended Fourier series expansions.

Abstract

We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described by Gross, Hacking, and Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher-rank cluster varieties.