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

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.