Special Lagrangian fibrations, Berkovich retraction, and crystallographic groups

TL;DR

This paper constructs special Lagrangian fibrations on quotients of degenerating abelian varieties, integrates them with Berkovich retraction in non-Archimedean geometry, and analyzes their symmetries as crystallographic groups, solving a conjecture in the process.

Contribution

It introduces a hybrid technique to explicitly build special Lagrangian fibrations and connects them with Berkovich retraction, advancing understanding of symmetries and confirming a conjecture.

Findings

Explicit construction of special Lagrangian fibrations on quotients of abelian varieties

Integration of fibrations with Berkovich retraction in non-Archimedean geometry

Solution of a conjecture by Kontsevich-Soibelman for finite quotients in any dimension

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.