On the Topology of Random Real Complete Intersections

Given a smooth real projective variety X and m ample line bundles L₁, ⋯ L_m on X also defined over R, we study the topology of the real locus of the complete intersections defined by global sections of L1⊗d⊕⋯⊕Lm⊗d. 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 L1⊗d⊕⋯⊕Lm⊗d is isotopic to the real locus of a complete intersection of smaller degree.