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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Variance of the volume of random real algebraic submanifolds
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by Thomas Letendre PDF
Trans. Amer. Math. Soc. 371 (2019), 4129-4192 Request permission

Abstract:

Let $\mathcal {X}$ be a complex projective manifold of dimension $n$ defined over the reals, and let $M$ denote its real locus. We study the vanishing locus $Z_{s_d}$ in $M$ of a random real holomorphic section $s_d$ of $\mathcal {E} \otimes \mathcal {L}^d$, where $\mathcal {L} \to \mathcal {X}$ is an ample line bundle and $\mathcal {E}\to \mathcal {X}$ is a rank $r$ Hermitian bundle. When $r\in \{1,\dots ,n-1\}$, we obtain an asymptotic of order $d^{r-\frac {n}{2}}$, as $d$ goes to infinity, for the variance of the linear statistics associated with $Z_{s_d}$, including its volume. Given an open set $U \subset M$, we show that the probability that $Z_{s_d}$ does not intersect $U$ is a $O$ of $d^{-\frac {n}{2}}$ when $d$ goes to infinity. When $n\geqslant 3$, we also prove almost sure convergence for the linear statistics associated with a random sequence of sections of increasing degree. Our framework contains the case of random real algebraic submanifolds of $\mathbb {R}\mathbb {P}^n$ obtained as the common zero set of $r$ independent Kostlan–Shub–Smale polynomials.
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Additional Information
  • Thomas Letendre
  • Affiliation: École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, UMR CNRS 5669, 46 allée d’Italie, 69634 Lyon Cedex 07, France
  • MR Author ID: 1153580
  • Email: thomas.letendre@ens-lyon.fr
  • Received by editor(s): October 11, 2016
  • Received by editor(s) in revised form: November 30, 2017
  • Published electronically: September 11, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 4129-4192
  • MSC (2010): Primary 53C40, 60G60; Secondary 14P99, 32A25, 60G57
  • DOI: https://doi.org/10.1090/tran/7478
  • MathSciNet review: 3917219