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

2025/07/21 by Philip Engel, Engel, Philip, Olivier de Gaay Fortman +3 · 2 citations
Mathematics · #05B35 #14C30 #14K10 #14K25 #14M25 #32G20 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2507.15695

openalex publication_date 2025/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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