On the topology of random real complete intersections

Abstract

Given a real projective variety $X$ and $m$ ample line bundles $L_1,\dots L_m$ on $X$ also defined over $\mathbb{R}$, we study the topology of the real locus of the complete intersections defined by global sections of $L_1^{\otimes d}\oplus\cdots\oplus L^{\otimes d}_m$. We prove that the Gaussian measure of the space of sections defining real complete intersections with high total Betti number (for example, maximal complete intersections) is exponentially small, as $d$ grows to infinity. This is deduced by proving that, with very high probability, the real locus of a complete intersection defined by a section of $L_1^{\otimes d}\oplus\dots\oplus L^{\otimes d}_m$ is isotopic to the real locus of a complete intersection of smaller degree.