Bounding the Betti numbers of patchworked real hypersurfaces by Hodge numbers

The Smith-Thom inequality bounds the sum of the Betti numbers of a real algebraic variety by the sum of the Betti numbers of its complexification. In this talk I will explain our proof of a conjecture of Itenberg which refines this bound for a particular class of real algebraic projective hypersurfaces in terms of the Hodge numbers of its complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg, we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalisation. Lurking in the spectral sequences of the proof are the keys to controlling the topology of the real hypersurface produced from a patchwork. This is joint work in preparation with Arthur Renaudineau.