Contributions to the study of random submanifolds

Abstract

We study the volume and Euler characteristic of codimension $r \in \{1,\dots,n\}$ random submanifolds in a dimension $n$ manifold $M$. First, we consider Riemannian random waves. That is $M$ is a closed Riemannian manifold and we study the common zero set $Z_\lambda$ of $r$ independent random linear combinations of eigenfunctions of the Laplacian associated to eigenvalues smaller than $\lambda\geq 0$. We compute estimates for the mean volume and Euler characteristic of $Z_\lambda$ as $\lambda$ goes to infinity. We also consider a model of random real algebraic manifolds. In this setting, $M$ is the real locus of a projective manifold defined over the reals. Then, we consider the real vanishing locus $Z_d$ of a random real global holomorphic section of $\E \otimes \L^d$, where $\E$ is a rank $r$ Hermitian vector bundle, $\L$ is an ample Hermitian line bundle and both these bundles are defined over the reals. We compute the asymptotics of the mean volume and Euler characteristic of $Z_d$ as $d$ goes to infinity. In this real algebraic setting, we also compute the asymptotic of the variance of the volume of $Z_d$, when $1 \leq r < n$. In this case, we prove asympotic equidistribution results for $Z_d$ in $M$.

Locations

• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF