Let$X$be a smooth complex projective manifold of dimension$n$equipped with an ample line bundle$L$and a rank$k$holomorphic vector bundle$E$. We assume that$1\leqslant k\leqslant n$, that$X$,$E$and$L$are defined over the reals and denote by$\mathbb{R}X$the real locus of$X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in$\mathbb{R}X$of holomorphic real sections of$E\otimes L^{d}$, where$d$is a large enough integer. Moreover, given any closed connected codimension$k$submanifold${\it\Sigma}$of$\mathbb{R}^{n}$with trivial normal bundle, we prove that a real section of$E\otimes L^{d}$has a positive probability, independent of$d$, of containing around$\sqrt{d}^{n}$connected components diffeomorphic to${\it\Sigma}$in its vanishing locus.
Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
