## Variance of the volume of random real algebraic submanifolds

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- by Thomas Letendre PDF
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**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.## References

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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