On the Topology of Random Real Complete Intersections

Abstract

Given a smooth real projective variety X and m ample line bundles \(L_1,\dots L_m\) on X also defined over \(\mathbf {R}\), 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.