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Higher convexity of coamoeba complements
Nisse, Mounir,Sottile, Frank Oxford University Press 2015 The bulletin of the London Mathematical Society Vol.47 No.5
<P>We show that the complement of the closure of the coamoeba of a variety of codimension [Formula] is [Formula]-convex, in the sense of Gromov and Henriques. This generalizes a result of Nisse for hypersurface coamoebas. We use this to show that the complement of the nonarchimedean coamoeba of a variety of codimension [Formula] is [Formula]-convex.</P>
Higher convexity for complements of tropical varieties
Nisse, Mounir,Sottile, Frank Springer-Verlag 2016 Mathematische Annalen Vol. No.
<P>We consider Gromov's homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and prove a weak form of this conjecture for the nonarchimedean amoeba of any variety over the complex Puiseux field. One of our main tools is Jonsson's limit theorem for tropical varieties.</P>
A natural topological manifold structure of phase tropical hypersurfaces
김영록,Mounir Nisse 대한수학회 2021 대한수학회지 Vol.58 No.2
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 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.