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

TL;DR

This paper establishes an upper bound on the expected total Betti number of high degree random real hypersurfaces in smooth real projective manifolds, using techniques involving critical point equirepartition and advanced analytical tools.

Contribution

It introduces a novel approach to bounding Betti numbers by proving equirepartition of critical points for real Lefschetz pencils on random hypersurfaces.

Findings

Bound on expected total Betti number derived

Equirepartition of critical points established

Application of H"ormander's peak sections and Poincaré-Martinelli formula

Abstract

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.