Affiliation:
1. Institut Camille Jordan, Université Claude Bernard Lyon, Boulevard du Novembre, Villeurbanne Cedex, France
Abstract
Abstract
Let $\mathcal{L}$ be a positive line bundle over a Riemann surface $\Sigma $ defined over $\mathbb{R}$. We prove that sections $s$ of $\mathcal{L}^d$, $d\gg 0$, whose number of real zeros $\#Z_s$ deviates from the expected one are rare. We also provide asymptotics of the form $\mathbb{E}[(\#Z_s-\mathbb{E}[\# Z_s])^k]=O(\sqrt{d}^{k-1-\alpha })$ and $\mathbb{E}[\#Z^k_s]=a_k\sqrt{d}^{k}+b_k\sqrt{d}^{k-1}+O(\sqrt{d}^{k-1-\alpha })$ for all the (central) moments of the number of real zeros. Here $\alpha $ is any number in $(0,1)$, and $a_k$ and $b_k$ are some explicit and positive constants. Finally, we obtain similar asymptotics for the distribution of complex zeros of random sections. Our proof involves Bergman kernel estimates as well as Olver multispaces.
Funder
LABEX MILYON
University of Lyon
French National Research Agency
Publisher
Oxford University Press (OUP)
Cited by
2 articles.
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