Almost 150 years ago, Harnack gave a tight upper bound on the number of connected components of a non-singular real algebraic curve of fixed degree in the plane. In higher dimensions, there are still no known tight upper bounds on the number of connected components of real projective algebraic hypersurfaces of a fixed degree, nor are there any tight bounds on their higher Betti 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, yet it is not known if this inequality is tight for hypersurfaces in toric varieties.
In this talk I will consider real algebraic hypersurfaces near a so-called non-singular tropical limit. These hypersurfaces arise from Viro’s patchworking construction and have the advantage that they can be studied combinatorially. I will explain the proof of a conjecture of Itenberg that bounds the individual Betti numbers of the real part of such a hypersurface in terms of the Hodge numbers of its complexification. The proof relies on a spectral sequence which relates the homology of the real hypersurface to the tropical homology of Itenberg, Katzarkov, Mikhalkin and Zharkov. Turning the pages of the spectral sequence could lead to a proof of the tightness of the Smith-Thom inequality for hypersurfaces in toric varieties.
This talk is based on joint with with Arthur Renaudineau and work in progress with Erwan Brugallé, Benoît Bertrand, and Renaudineau.