On the curvatures of random complex submanifolds
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909414163742720 |
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| author | Ancona, Michele Gayet, Damien |
| author_facet | Ancona, Michele Gayet, Damien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the curvatures of random complex submanifolds Ancona, Michele Gayet, Damien Probability 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. |
| title | On the curvatures of random complex submanifolds |
| topic | Probability |
| url | https://arxiv.org/abs/2412.02486 |