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

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.