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Séminaire de Géométrie

La courbure totale des varietes algebriques affines reelles

Jean-Jacques Risler

( IM Jussieu )

Salle 2

le 22 février 2013 à 10:45

\section{Total curvature} Let XRn+1X\subset\mathbb{R}^{n+1} be a smooth algebraic hypersurface. Then the total curvature of XX is the "volume" of the Gauss map g:XRPng:X\rightarrow \mathbb{R}P^n.\ The total curvature of the real Amoeba is then the volume of the image of the Logarithmic Gauss map. \section{Simple Harnack curves} I will recall the definition of G. Mikhalkin, and the theorem of Mikhalkin- Rullgard which characterize plane Simple Harnack curves by the fact that the Amoeba has maximal area. \section{Total Curvature of the Real Amoeba} I will give a bound for the total curvature of the real Amoeba of a real plane curve XX (in term of its Newton Polygon) and prove that this bound is reached if and only if XX is a (smooth) simple Harnack curve. \section{Total curvature of tropical hypersurfaces} If time , I will quote a recent result about total curvature of Real tropical hypersurfaces.