From Logarithmic Limit Sets to Algebraicity

Abstract

We study a converse problem to the theorem of Bergman and Bieri--Groves on logarithmic limit sets of algebraic subvarieties of the complex algebraic torus $(\mathbb C^*)^n$. We introduce the notion of \emph{finite logarithmic type}, a boundary condition formulated on toric compactifications in terms of coherent meromorphic extensions with uniformly bounded pole orders along toric boundary divisors. We prove that a closed reduced analytic subvariety of $(\mathbb C^*)^n$ whose logarithmic limit set is a finite rational spherical polyhedral complex of the expected dimension is algebraic whenever it is of finite logarithmic type. The proof relies on toric compactifications, coherent extension theory, Serre's GAGA theorem, and Chow's theorem. A principal result of the paper is the complete treatment of the one-dimensional case. We prove that every closed analytic curve in $(\mathbb C^*)^n$ with finite logarithmic limit set is algebraic. Consequently, the finiteness of the logarithmic limit set completely characterizes algebraicity for analytic curves, yielding a genuine converse to Bergman's theorem in dimension one. These results establish new links between logarithmic limit sets, tropical geometry, toric geometry, and complex analytic geometry.