Mirror Symmetry for log Calabi-Yau Surfaces II

TL;DR

This paper establishes a canonical basis for the ring of regular functions on smooth affine log Calabi-Yau surfaces, linking tropical geometry, mirror symmetry, and algebraic structures through geometric and period computations.

Contribution

It provides a canonical basis parametrized by tropical points and describes the algebra structure using geometric constructions, advancing the understanding of mirror symmetry for log Calabi-Yau surfaces.

Findings

Basis parametrized by tropical points

Algebra structure with geometric coefficients

Computed periods for the mirror family

Abstract

We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb{C}$-algebra structure with structure coefficients given by the geometric construction of Keel-Yu. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross-Hacking-Keel where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from work of Ruddat-Siebert and use this to give a modular interpretation of the compactified mirror family.