Expected volume and Euler characteristic of random submanifolds

TL;DR

This paper estimates the expected volume and Euler characteristic of random submanifolds in both Riemannian and complex projective manifolds, providing asymptotic formulas as parameters grow large.

Contribution

It introduces a unified approach to asymptotically compute expected geometric invariants of random submanifolds in two different geometric settings.

Findings

Asymptotic formulas for expected volume as eigenvalue bound increases

Asymptotic formulas for expected Euler characteristic in both settings

Unified techniques applicable to Riemannian and complex algebraic geometries

Abstract

In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\lambda$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as $\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of $\mathcal{E}\otimes\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.