On the curvatures of random complex submanifolds

TL;DR

This paper investigates the curvature properties of random complex submanifolds in projective spaces, showing that certain curvature measures tend to be negative with high probability as the degree increases.

Contribution

It establishes that for large degrees, the expected curvature fractions of random submanifolds approach one under specific conditions, extending results to general complex projective manifolds.

Findings

Expected volume fraction of negative bisectional curvature tends to one as degree increases.

Provides estimates for holomorphic sectional, Ricci, and scalar curvatures.

Results apply broadly to submanifolds in any complex projective manifold.

Abstract

For any integers $n\geq 2$ and $1\leq r\leq n-1$ satisfying $3r\geq 2n-1$, we show that the expected volume fraction of a random degree $d$ complex submanifold of $\C\mathbb{P}^n$ of codimension $r$ where the bisectional holomorphic curvature (for the induced ambient metric) is negative tends to one when $d$ goes to infinity. Here, the probability measure is the natural one associated with the Fubini--Study metric. We provide similar estimates for the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature. Our results hold more generally for random submanifolds within any complex projective manifold.