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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Expected local topology of random complex submanifolds

Author: Damien Gayet
Journal: J. Algebraic Geom.
Published electronically: May 3, 2023
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Abstract | References | Additional Information

Abstract: Let $n\geq 2$ and $r\in \{1, \cdots , n-1\}$ be integers, $M$ be a compact smooth Kähler manifold of complex dimension $n$, $E$ be a holomorphic vector bundle with complex rank $r$ and equipped with a Hermitian metric $h_E$, and $L$ be an ample holomorphic line bundle over $M$ equipped with a metric $h$ with positive curvature form. For any $d\in \mathbb N$ large enough, we equip the space of holomorphic sections $H^0(M,E\otimes L^d)$ with the natural Gaussian measure associated to $h_E$, $h$ and its curvature form. Let $U\subset M$ be an open subset with smooth boundary. We prove that the average of the $(n-r)$-th Betti number of the vanishing locus in $U$ of a random section $s$ of $H^0(M,E\otimes L^d)$ is asymptotic to ${n-1 \choose r-1} d^n\int _U c_1(L)^n$ for large $d$. On the other hand, the average of the other Betti numbers is $o(d^n)$. The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and contrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.

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Additional Information

Damien Gayet
Affiliation: Université Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France
MR Author ID: 662257

Received by editor(s): February 18, 2022
Received by editor(s) in revised form: July 21, 2022, and July 28, 2022
Published electronically: May 3, 2023
Dedicated: This paper is dedicated to the memory of Steve Zelditch
Article copyright: © Copyright 2023 University Press, Inc.