Betti numbers of random real hypersurfaces and determinants of random symmetric matrices (Q277525)

The main theorem of the present paper is an upper bound to the suitably normalized asymptotics of the expectation of Betti numbers of a random real hypersurface in a given smooth real projective variety $X$ when the degree of the hypersurface grows to infinity. Here the hypersurfaces of degree $d$ are understood as zero sets $Z(\sigma)$ of real sections $\sigma\in H^0(X,L^{\otimes d})$, $L$ being a real ample line bundle. This strengthens the previous authors' result [J. Reine Angew. Math. 689, 137--168 (2014; Zbl 1348.14138)] with a slightly weaker upper bound for the asymptotics of the total Betti number of a random real hypersurface. The authors' approach is close to that by \textit{M. Shub} and \textit{S. Smale} [Prog. Math. 109, 267--285 (1993; Zbl 0851.65031)], who showed that a random real univariate polynomial of degree $d$ has $\sqrt{d}$ real roots for a suitable choice of the probability measure of the space of polynomials. More precisely, the authors estimate the expectation of the minimal number $m_i$ of critical points of index $i$ over all Morse functions on the random hypersurface: \[ \lim_{d\to\infty}\sup\frac{1}{\sqrt{d}^n}E(m_i)\le\frac{\mathrm{Vol}({\mathbb R}X)}{\sqrt{\pi}}e_{\mathbb R}(i,n-1-i)\ , \] where $e_{\mathbb R}(i,n-1-i)$ is the expectation of the absolute value of the determinant of a random real symmetric $(n-1)\times(n-1)$ matrix of signature $(i,n-1-i)$. The key ingredient is the equidistribution result, which roughly states that the density of the critical points of index $i$ of a fixed Morse function on ${\mathbb R}X$ restricted to a real hypersurface of degree $d$ normalized by $\frac{1}{\sqrt{d}^n}$ weakly converges (with respect to the volume form on ${\mathbb R}X$) to $\frac{1}{\sqrt{\pi}}e_{\mathbb R}(i,n-1-i)$. A similar equidistribution result is obtained in the complex setting as well.