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Bounding Betti numbers of patchworked real hypersurfaces by Hodge numbers

Time: Wed 2019-03-13 10.15 - 11.15

Location: Room 3418, Lindstedtsvägen 25. Department of Mathematics, KTH

Participating: Kristin Shaw, Department of Mathematics, University of Oslo

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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. For hypersurfaces in higher dimensional projective space we can ask analogous questions, yet we know very little about the topology of real algebraic hypersurfaces.

In this talk I will explain the proof of a conjecture of Itenberg which, for a particular class of real algebraic projective hypersurfaces, bounds their individual Betti numbers in terms of the Hodge numbers of the 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 a version of tropical homology with coefficients in \mathbb{Z}_2. This is joint work with Arthur Renaudineau. To establish upper bounds in terms of Hodge numbers, we show that the dimensions of these tropical homology groups are equal to Hodge numbers by establishing versions of the Lefschetz hyperplane section theorem for tropical homology with integral coefficients (this is done together with Arnal and Renaudineau).