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Article Dans Une Revue Indiana University Mathematics Journal Année : 2019

Variance of the volume of random real algebraic submanifolds II

Résumé

Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ be 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, $r \in \{1,\dots, n\}$. We establish the asymptotic of the variance of the linear statistics associated with $Z_{s_d}$, as $d$ goes to infinity. This asymptotic is of order $d^{r-\frac{n}{2}}$. As a special case, we get the asymptotic variance of the volume of $Z_{s_d}$. The present paper extends the results of [20], by the first-named author, in essentially two ways. First, our main theorem covers the case of maximal codimension ($r = n$), which was left out in [20]. And second, we show that the leading constant in our asymptotic is positive. This last result is proved by studying the Wiener–It{\=o} expansion of the linear statistics associated with the common zero set in $\mathbb{RP}^n$ of $r$ independent Kostlan–Shub–Smale polynomials.
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Dates et versions

hal-01570360 , version 1 (29-07-2017)
hal-01570360 , version 2 (17-12-2019)
hal-01570360 , version 3 (06-01-2020)

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Paternité - Pas de modifications

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Thomas Letendre, Martin Puchol. Variance of the volume of random real algebraic submanifolds II. Indiana University Mathematics Journal, 2019, 68 (6), pp.1649-1720. ⟨10.1512/iumj.2019.68.7830⟩. ⟨hal-01570360v3⟩
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