Exponential rarefaction of real curves with many components

TL;DR

This paper investigates the asymptotic behavior of real holomorphic sections of line bundles over real projective manifolds, showing that the probability of sections having many real components diminishes exponentially with increasing degree.

Contribution

It provides the first exponential estimates for the volume of sections with many real components in the context of real algebraic geometry.

Findings

Volume of sections with many real components decreases exponentially as degree increases

Quantitative estimates for the cone of maximal real sections

Analysis focused on curves and surfaces

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 positive integer d 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.