What is the total Betti number of a random real hypersurface?

Details What is the total Betti number of a random real hypersurface? What is the total Betti number of a random real hypersurface?

2011-07-12

arxiv.org/abs/1107.2288v1

Mathematics

Algebraic Geometry

Scientific paper

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve H\"ormander's theory of peak sections as well as the formula of Poincar\'e-Martinelli.

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