A natural topological manifold structure of phase tropical hypersurfaces

Abstract

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (C∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin’s decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.