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• W2516475744 abstract "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." @default.
• W2516475744 created "2016-09-16" @default.
• W2516475744 creator A5067715129 @default.
• W2516475744 creator A5084852309 @default.
• W2516475744 date "2019-01-01" @default.
• W2516475744 modified "2023-10-14" @default.
• W2516475744 title "Variance of the volume of random real algebraic submanifolds II" @default.
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• W2516475744 doi "https://doi.org/10.1512/iumj.2019.68.7830" @default.
• W2516475744 hasPublicationYear "2019" @default.
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