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

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 in [NXY19]. 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.

Survey on the metric SYZ conjecture and non-archimedean geometry

“…Remark 13. The very recent work of Pille-Schneider and Mazzon [59] proposes gluing the retraction maps associated to several divisorial log terminal models to obtain a map X an K → Sk(X). Their map still factors through the dual intersection complex of some larger snc model, hence is compatible with the comparison property.…”