Abstract

Almost 150 years ago Harnack proved a tight upper bound on the number of connected components of a real planar algebraic curve of degree d. However, in higher dimensions we know very little about the topology of real algebraic hypersurfaces. For example, we do not know the maximal number of connected components of real quintic surfaces in projective space.

In this talk I will explain the proof of a conjecture of Itenberg which, for a particular class of real algebraic projective hypersurfaces, bounds all Betti numbers, not only the number of connected components, in terms of the Hodge numbers of the complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is an effective and combinatorial method for constructing topological types of real algebraic varieties. Today these real hypersurfaces can be thought of as near the tropical limit. To prove the bounds conjectured by Itenberg we develop a real analogue of tropical homology and use a spectral sequence to relate these groups to tropical homology and their dimensions to Hodge numbers. Lurking in the spectral sequence of the proof are the keys to having combinatorial control of the topology of the real hypersurfaces near the tropical limit in any toric variety.

This is joint work with Arthur Renaudineau.