Expected volume and Euler characteristic of random submanifolds

Thomas Letendre (1, 2, 3)

1 ICJ - Institut Camille Jordan
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 MMCS - Modélisation mathématique, calcul scientifique

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.

Mots clés: ample line bundle; Bergman kernel; random polynomial; spectral function; real projective manifold; Riemannian random wave; Gaussian field; Euler characteristic

Domaines: Géométrie métrique [math.MG]; Géométrie algébrique [math.AG]; Probabilités [math.PR]

Thomas Letendre. Expected volume and Euler characteristic of random submanifolds. Journal of Functional Analysis, 2016, 270 (8), pp.3047--3110. ⟨10.1016/j.jfa.2016.01.007⟩. ⟨hal-01053847v3⟩