Toric geometry and integral affine structures in non-archimedean mirror symmetry

TL;DR

This paper explores the structure of non-archimedean mirror symmetry using toric geometry and integral affine structures, extending previous results and constructing new retractions for complex degenerations.

Contribution

It introduces new non-archimedean retractions for degenerating Calabi-Yau families, linking them to integral affine structures in mirror symmetry.

Findings

Berkovich retraction is an affinoid torus fibration around toric strata.

Constructs new non-archimedean retractions for quartic K3 and quintic 3-folds.

Induces singular integral affine structures matching those in the Gross-Siebert program.

Abstract

We study integral dlt models of a proper C((t))-variety X along a toric stratum of the special fiber. We prove that the associated Berkovich retraction - from the non-archimedean analytification of X onto the dual complex of the model - is an affinoid torus fibration around the simplex corresponding to the toric stratum, which extends results from Nicaise-Xu-Yu. This allows us to construct new types of non-archimedean retractions for maximally degenerate families of quartic K3 surfaces and quintic 3-folds, by gluing several non-archimedean SYZ fibrations, each one toric along a codimension one stratum. We then show that the new retractions induce the same singular integral affine structures that arise on the dual complex of toric degenerations in the Gross-Siebert program, as well as on the Gromov-Hausdorff limit of the family.