On the topology of random real complete intersections

TL;DR

This paper investigates the topology of real complete intersections in algebraic geometry, showing that as the degree increases, the probability of complex topologies diminishes exponentially, and most are isotopic to simpler intersections.

Contribution

It proves that high-degree real complete intersections rarely have maximal Betti numbers, and most are topologically similar to lower-degree intersections.

Findings

Gaussian measure of complex topologies is exponentially small

Most high-degree intersections are isotopic to lower-degree ones

Maximal topologies are extremely rare as degree increases

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.