Pré-Publication, Document De Travail Année : 2024

On the curvatures of random complex submanifolds

Damien Gayet

Résumé

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.

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Dates et versions

hal-04798627 , version 1 (22-11-2024)

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  • HAL Id : hal-04798627 , version 1

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Michele Ancona, Damien Gayet. On the curvatures of random complex submanifolds. 2024. ⟨hal-04798627⟩
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