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arXiv · 2110.04223

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

Enrica Mazzon·Léonard Pille-Schneider
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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.

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Enrica Mazzon, Léonard Pille-Schneider. 2021-10-08. Toric geometry and integral affine structures in non-archimedean mirror symmetry. https://arxiv.org/abs/2110.04223

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