Special Lagrangian fibrations, Berkovich retraction, and crystallographic groups

Abstract

We explicitly construct special Lagrangian fibrations on finite quotients of maximally degenerating abelian varieties, glue with Berkovich retraction in non-Archimedean geometry by using "hybrid" technique. We also study their symmetries explicitly which can be regarded as crystallographic groups. In particular, a conjecture of Kontsevich-Soibelman is solved at an enhanced level for finite quotients of abelian varieties in any dimension.

30 pages, v2: added an appendix which discusses a certain limiting process of CY metrics to non-Archimedean CY metric and corrected some typos, v3: corrected some typos and minor mathematical errors