Bounding the Betti numbers of real hypersurfaces near the tropical limit

Arthur Renaudineau, Kristin Shaw

Published 2018-05-05Version 1

We prove an upper bound on the Betti numbers of real algebraic hypersurfaces obtained by primitive combinatorial patchworking. These bounds are given in terms of the dimensions of tropical homology groups with $\mathbb{Z}_2$-coefficients of a tropical hypersurface. To establish these bounds, we introduce a real variant of tropical homology and define a filtration on the corresponding chain complex inspired by Kalinin's filtration. The terms of the first page of the spectral sequence associated to this filtration are the tropical homology groups with $\mathbb{Z}_2$-coefficients. Tropical homology groups with rational coefficients give Hodge numbers of complex projective varieties by a theorem of Itenberg, Mikhalkin, Katzarkov, and Zharkov. We conjecture that, in the case of tropical hypersurfaces, the dimensions of the rational and $\mathbb{Z}_2$-tropical homology groups are equal. This would imply a bound conjectured by Itenberg on the Betti numbers of the real part of a patchworked hypersurface in terms of Hodge numbers. Using our techniques we also recover Haas' combinatorial criterion for the maximality of patchworked plane curves.