Effective faithful tropicalizations and embeddings for abelian varieties
Shu Kawaguchi, Kazuhiko Yamaki
TL;DR
The article develops a tropical analogue of Lefschetz theory for abelian varieties over nonarchimedean fields by proving that the canonical skeleton can be faithfully tropicalized using nonarchimedean theta functions. It introduces and analyzes faithful embeddings of tropical abelian varieties via tropical theta functions, and shows how to lift these tropical objects to genuine nonarchimedean theta functions through Fourier expansions on Raynaud extensions. Key contributions include a precise bound on polarization data ensuring faithful embeddings, a detailed tropical Appell–Humbert framework for real tori, and a surjectivity result for tropicalizations of gamma-rational theta functions. The results unify tropical and nonarchimedean perspectives, enabling explicit tropical models of abelian skeleta with potential applications in tropical geometry and nonarchimedean analytic geometry.
Abstract
Let $A$ be an abelian variety over an algebraically closed field $k$ that is complete with respect to a nontrivial nonarchimedean absolute value. Let $A^{\mathrm{an}}$ denote the analytification of $A$ in the sense of Berkovich, and let $Σ$ be the canonical skeleton of $A^{\mathrm{an}}$. In this paper, we obtain a faithful tropicalization of $Σ$ by nonarchimedean theta functions, giving a tropical version of the classical theorem of Lefschetz on abelian varieties. Key ingredients of the proof are (1) faithful embeddings of tropical abelian varieties by tropical theta functions and (2) lifting of tropical theta functions to nonarchimedean theta functions, and they will be of independent interest. For (1), we use some arguments similar to the case of complex abelian varieties as well as Voronoi cells of lattices. For (2), we use Fourier expansions of nonarchimedean theta functions over the Raynaud extensions of abelian varieties.