Lower estimates for the expected Betti numbers of random real hypersurfaces

TL;DR

This paper provides lower bounds for the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds, showing they grow proportionally to the volume of the real locus and the power d^{n/2}.

Contribution

It introduces new lower bounds for Betti numbers of random hypersurfaces, complementing previous upper bounds, and demonstrates the positive probability of embedding any affine real algebraic hypersurface.

Findings

Lower bounds proportional to volume and d^{n/2} growth

Any affine real algebraic hypersurface appears with positive probability

Bounds depend only on the dimension and volume of the real locus

Abstract

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a constant which only depends on the dimension n of the manifold with the K\"ahlerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).