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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2311.09679 |
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| _version_ | 1866929739303747584 |
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| author | Ancona, Michele Gayet, Damien |
| author_facet | Ancona, Michele Gayet, Damien |
| contents | For any integer $n\geq 2$, we prove that for any large enough integer $d$, with large probability the injectivity radius of a random degree $d$ complex hypersurface in $\C P^n$ is larger than $d^{-\frac{1}2(3n+2)}$. Here the hypersurface is endowed with the restriction of the ambient Fubini-Study metric, and the probability measure is induced by the Fubini-Study $L^2$-Hermitian product on the space of homogeneous complex polynomials of degree $d$ in $(n+1)$-variables. We also prove that with high probability, the sectional curvatures of the random hypersurface are bounded by $d^{\frac{3}2(n+2)}$, and that its spectral gap is bounded below by $\exp(-d^{\frac{1}4(3n+15)})$. These results extend to random submanifolds of higher codimension in any complex projective manifold. Independently, we prove that the diameter of a degree $d$ divisor is bounded by $Cd^3$, which generalizes and amends the bound given in~\cite{feng1999diameter} for planar curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09679 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Metric and spectral aspects of random complex divisors Ancona, Michele Gayet, Damien Algebraic Geometry Probability For any integer $n\geq 2$, we prove that for any large enough integer $d$, with large probability the injectivity radius of a random degree $d$ complex hypersurface in $\C P^n$ is larger than $d^{-\frac{1}2(3n+2)}$. Here the hypersurface is endowed with the restriction of the ambient Fubini-Study metric, and the probability measure is induced by the Fubini-Study $L^2$-Hermitian product on the space of homogeneous complex polynomials of degree $d$ in $(n+1)$-variables. We also prove that with high probability, the sectional curvatures of the random hypersurface are bounded by $d^{\frac{3}2(n+2)}$, and that its spectral gap is bounded below by $\exp(-d^{\frac{1}4(3n+15)})$. These results extend to random submanifolds of higher codimension in any complex projective manifold. Independently, we prove that the diameter of a degree $d$ divisor is bounded by $Cd^3$, which generalizes and amends the bound given in~\cite{feng1999diameter} for planar curves. |
| title | Metric and spectral aspects of random complex divisors |
| topic | Algebraic Geometry Probability |
| url | https://arxiv.org/abs/2311.09679 |