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A NATURAL TOPOLOGICAL MANIFOLD STRUCTURE OF PHASE TROPICAL HYPERSURFACES

  • Kim, Young Rock (Major in Mathematics Education Graduate School of Education Hankuk University of Foreign Studies) ;
  • Nisse, Mounir (Department of Mathematics Xiamen University Malaysia)
  • Received : 2020.03.14
  • Accepted : 2020.10.28
  • Published : 2021.03.01

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.