Bounding the betti numbers of real hypersurfaces near the tropical limit

Abstract

A bound is demonstrated by Itenberg on the Betti numbers of real algebraic hypersurfaces close to the tropical boundary. These bounds are expressed as a function of the Hodge numbers of the complexification. To demonstrate these bounds, we introduce a variant of tropical homology into the real framework, and define filtration on the associated chain complex, inspired by Kalinin filtration. The spectral sequence associated with this filtration converges towards the homology groups of the real algebraic variety, and we show that the terms of the first page are the groups of tropical homology (with coefficients in2 / 2Z). The dimensions of these homology groups correspond to the Hodge numbers of the complex projective hypersurfaces thanks to the results of Itenberg, Mikhalkin, Katzarkov, on the one hand, and authors with Arnal on the other. The bounds on the Betti numbers enmentate, as well as a criterion for obtaining a maximum variety. We also generalize the formula due to Bertrand linking the signature of a complex hypersurface and the Euler characteristic of a real hypersurface, and we will redequip the combinatorial criterion of Haas on the maximum of the flat curves close to the tropical boundary.