On The Two Types Of Affine Structures For Degenerating Kummer Surfaces -Non-Archimedean VS Gromov-Hausdorff Limits-

TL;DR

This paper proves that the integral affine structures with singularities arising from non-Archimedean degenerations and Gromov-Hausdorff limits are equivalent for maximal degenerations of polarized Kummer surfaces, confirming a key conjecture in mirror symmetry.

Contribution

We establish the equivalence of two types of affine structures for degenerating Kummer surfaces, confirming a conjecture in the context of mirror symmetry.

Findings

Proved the conjecture for Kummer surfaces.

Showed the equivalence of affine structures in the specific case.

Confirmed the expected geometric correspondence in degenerations.

Abstract

Kontsevich and Soibelman constructed integral affine manifolds with singularities (IAMS, for short) for maximal degenerations of polarized Calabi-Yau manifolds in a non-Archimedean way. On the other hand, for each maximally degenerating family of polarized Calabi-Yau manifolds, we can consider the Gromov-Hausdorff limit of the fibers. It is expected that this Gromov-Hausdorff limit carries an IAMS-structure. Kontsevich and Soibelman conjectured that these two types of IAMS are the same. This conjecture is believed in the mirror symmetry context. In this paper, we prove the above conjecture for maximal degenerations of polarized Kummer surfaces.