Comparison of non-archimedean and logarithmic mirror constructions via the Frobenius structure theorem

TL;DR

This paper establishes a connection between two mirror symmetry constructions for log Calabi-Yau pairs, proving their equivalence under certain conditions and confirming a mirror conjecture for Fano pairs.

Contribution

It proves the equality of mirror algebras from Gross-Siebert and Keel-Yu constructions under specific assumptions and verifies Mandel's mirror conjecture for certain Fano pairs.

Findings

Mirror algebras from different constructions are equal under certain conditions.

Structure constants are given by naive curve counts.

Confirmed Mandel's mirror conjecture for specific Fano pairs.

Abstract

For a log Calabi Yau pair (X,D) with X\D smooth affine, satisfying either assumption 1.1 of "The canonical wall structure and intrinsic mirror symmetry" or contains a Zariski dense torus, we prove under the condition that D is the support of a nef divisor, that the structure constants defining a trace form on the mirror algebra constructed by Gross-Siebert are given by the naive curve counts defined by Keel-Yu in definition 1.1 of "The Frobenius structure theorem for log Calabi-Yau varieties containing a torus". As a corollary, we deduce the equality of the mirror algebras constructed by Gross-Siebert and Keel-Yu in the case X\D contains a Zariski dense torus. In addition, we use this result to prove a mirror conjecture proposed by Mandel in "Fano mirror periods from the Frobenius structure conjecture" for Fano pairs satisfying assumption 1.1.