Equations of mirrors to log Calabi--Yau pairs via the heart of canonical wall structures

TL;DR

This paper presents an algorithm to compute explicit equations of mirror families for log Calabi--Yau pairs, especially when constructed via blow-ups of toric varieties, using a combinatorial approach based on the heart of canonical wall structures.

Contribution

It introduces a practical algorithm for computing mirror equations in complex blow-up scenarios, extending previous symplectic results to more intricate configurations.

Findings

Algorithm successfully computes mirror equations for blow-ups of toric varieties.

Results agree with known symplectic computations in simple cases.

Provides first explicit equations for complex blow-up configurations with multiple hypersurfaces.

Abstract

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi--Yau pair $(X,D)$, consisting of a smooth projective variety $X$ with a normal-crossing anti-canonical divisor $D$ in $X$. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when $X$ is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and $D$ is the strict transform of the toric boundary. The main ingredient is ``the heart of the canonical wall structure'' associated to such pairs $(X,D)$, which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux--Abouzaid--Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.