Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

TL;DR

This paper provides asymptotic upper bounds on the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds, linking geometric properties with random matrix determinants and critical point distribution.

Contribution

It introduces new asymptotic estimates for Betti numbers of random hypersurfaces, connecting topology, geometry, and random matrix theory.

Findings

Expected Betti numbers grow as the square root of the hypersurface degree.

Coefficients involve the Kähler volume and determinants of random symmetric matrices.

Large dimensions show exponential decay of Betti number bounds away from mid-dimension.

Abstract

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the K\"ahlerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces.