Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction

TL;DR

This survey explores the Mumford construction of degenerating abelian varieties, emphasizing its analytic form, connections to toric geometry, and the Hodge theory of multivariable degenerations linked to regular matroids.

Contribution

It extends fundamental results on 1-parameter degenerations to multivariable cases and highlights the interplay between combinatorics, Hodge theory, and degenerations of abelian varieties.

Findings

Extended Clemens' results to multivariable degenerations.

Connected Mumford's construction with toric geometry.

Analyzed Hodge structures in multivariable degenerations.

Abstract

We survey the Mumford construction of degenerating abelian varieties, with a focus on the analytic version of the construction, and its relation to toric geometry. Moreover, we study the geometry and Hodge theory of multivariable degenerations of abelian varieties associated to regular matroids, and extend some fundamental results of Clemens on 1-parameter semistable degenerations to the multivariable setting.