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

### Damien Gayet

Université Joseph Fourier Grenoble 1, Saint-Martin-d'Hères, France### Jean-Yves Welschinger

Université Lyon 1, Villeurbanne, France

## 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 Kählerian 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 random real hypersurfaces.

## Cite this article

Damien Gayet, Jean-Yves Welschinger, Betti numbers of random real hypersurfaces and determinants of random symmetric matrices. J. Eur. Math. Soc. 18 (2016), no. 4, pp. 733–772

DOI 10.4171/JEMS/601