Exponential rarefaction of real curves with many components

Abstract

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X , the space of real holomorphic sections of the bundle L d inherits for every d ∈ℕ ∗ a L 2 -scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.