Geometry and a natural symplectic structure of phase tropical hypersurfaces

TL;DR

This paper introduces phase tropical hypersurfaces, demonstrating their smooth and symplectic structures, and establishes their diffeomorphism with complex hyperplanes, enriching the understanding of degenerations in algebraic geometry.

Contribution

It defines phase tropical hypersurfaces via degeneration data and proves they have natural smooth and symplectic structures, extending Mikhalkin's decomposition methods.

Findings

Complex hyperplanes are diffeomorphic to phase tropical hyperplanes.

Phase tropical hypersurfaces with smooth tropicalization have natural smooth structures.

They also possess a natural symplectic structure.

Abstract

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.