Expected topology of random real algebraic submanifolds

TL;DR

This paper investigates the expected topological complexity of real algebraic submanifolds in complex projective manifolds, providing bounds on Betti numbers and probabilities of certain topologies in vanishing loci of sections.

Contribution

It offers new estimates for Betti numbers of real algebraic submanifolds and proves the positive probability of complex topologies appearing in vanishing loci, independent of degree d.

Findings

Bounds on expected Betti numbers of real vanishing loci

Positive probability of complex topologies in vanishing loci

Existence of components diffeomorphic to any given submanifold S

Abstract

Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k <=n, that X, E and L are defined over the reals and denote by RX the real locus of X. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in RX of holomorphic real sections of E tensored with L^d, where d is a large enough integer. Moreover, given any closed connected codimension k submanifold S of R^n with trivial normal bundle, we prove that a real section of E tensored with L^d has a positive probability, independent of d, to contain around the square root of d^n connected components diffeomorphic to S in its vanishing locus.