Tropical Geometric Compactification of Moduli, II - A_g case and holomorphic limits -

2017/05/16 by Yuji Odaka, Odaka, Yuji · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1705.05545

openalex publication_date 2017/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compactify the classical moduli variety A_g of principally polarized abelian varieties of complex dimension g by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and show that they form special non-trivial subsets of the whole boundary. We also do it for algebraic curves case and observe a crucial difference with the case of abelian varieties.

Citations

• On the Singularities of Degenerate Abelian Varieties

Cited by

• On multiscale aspects of Kähler-Einstein metrics and algebraic geometry

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