Resumen de Tropical theta functions and log Calabi–Yau surfaces

Travis Mandel

• 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 in Gross et al. (Math. Inst. Hautes Etudes Sci. 122, 65–168, 2015), and monomials on toric varieties are replaced with the canonical theta functions defined in Gross et al. (2015) 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.