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

A tropical characterization of complex analytic varieties to be algebraic