Lower estimates for the expected Betti numbers of random real hypersurfaces (Q2874651)

Given a smooth complex projective variety \(X\) with a real structure such that the real point set \({\mathbb R}X\) is non-empty, and given a real holomorphic line bundle \({\mathcal L}\to X\) with a Hermitian metric, one can define a Gaussian probability measure on the space of \(L^2\)-sections of the bundle \({\mathcal L}^d\), \(d\geq1\). The main result of the paper of a lower bound for the expected number of connected components of the zero locus of a random section of \({\mathcal L}^d\), \(d\to \infty\), that have a prescribed diffeomorphism type. As a consequence the authors derive a lower bound for the expected Betti numbers of the zero locus of a random section of \({\mathcal L}^d\), \(d\to\infty\). Both lower bounds have the form \(c\cdot{\mathrm{Vol}}({\mathbb R}X)\cdot d^{n/2}\), \(n=\dim X\), \(c=\)const. These results supplement the previous authors' work [``Betti numbers of random real hypersurfaces and determinants of random symmetric matrices'', Preprint \url{arXiv:1207.1579}], where an upper bound with a similar asymptotics was provided. The proofs use the techniques developed by \textit{F. Nazarov} and \textit{M. Sodin} [Am. J. Math. 131, No. 5, 1337--1357 (2009; Zbl 1186.60022)].