Bounding the Betti numbers of real hypersurfaces near the tropical limit

We prove a bound conjectured by Itenberg on the Betti numbers of real algebraic hypersurfaces near non-singular tropical limits. These bounds are given in terms of the Hodge numbers of the complexification. To prove the conjecture we introduce a real variant of tropical homology and define a filtration on the corresponding chain complex inspired by Kalinin's filtration. The spectral sequence associated to this filtration converges to the homology groups of the real algebraic variety and we show that the terms of the first page are tropical homology groups with $\mathbb{Z}_2$-coefficients. The dimensions of these homology groups correspond to the Hodge numbers of complex projective hypersurfaces. The bounds on the Betti numbers of the real part follow, as well as a criterion to obtain a maximal variety. We also generalise a known formula relating the signature of the complex hypersurface and the Euler characteristic of the real algebraic hypersurface, as well as Haas' combinatorial criterion for the maximality of plane curves near the tropical limit.