Mounir Nisse, Texas A&M University, USA

A necessary and sufficient condition on analytic subvarieties of the complex torus to be algebraic

Abstract

In this talk we deal with generic analytic subvarieties of the complex algebraic torus (C^*)ⁿ. We show that a generic k-dimensional analytic subvariety of the n-dimensional complex torus is algebraic if and only if its logarithmic limit set is a finite rational complex polyhedron of dimension k - 1. This is equivalent to saying that its phase limit set contains no real torus of dimension strictly greater than k.

In particular, if the dimension of the ambient space, n, is at least 2k, then, the last conditions are equivalent to the fact that the volume of the amoeba is finite. It is a stronger, tropical version of Chow's theorem.