A tropical characterization of complex analytic varieties to be algebraic

TL;DR

This paper characterizes when complex analytic varieties in algebraic tori are algebraic based on their logarithmic limit sets and amoeba volume, extending previous theorems and providing a tropical geometric criterion.

Contribution

It establishes a new tropical criterion linking the algebraicity of analytic subvarieties to their logarithmic limit sets and amoeba volume, generalizing prior results.

Findings

Finite rational spherical polyhedron of the limit set implies algebraicity.

Finite amoeba volume characterizes algebraic subvarieties.

Results extend and converse previous theorems by Bieri and Groves.

Abstract

In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-dimensional spherical polyhedron, then each irreducible component of the variety is algebraic. This gives a converse of a theorem of Bieri and Groves and generalizes a result proven in \cite{MN2-11}. More precisely, if the dimension of the ambient space is at least twice of the dimension of the generic analytic subvariety, then these properties are equivalent to the volume of the amoeba of the subvariety being finite.