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Asymptotic Expansion of the Variance of Random Zeros on Complex Manifolds

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Abstract

Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the variance of linear statistics of the zero divisors of random holomorphic sections of powers of a positive line bundle on a compact Kähler manifold. This expansion extends the leading-order asymptotics (in the codimension one case) given by Shiffman–Zelditch in 2010.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Bernard Shiffman

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  1. Bernard Shiffman
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Correspondence to Bernard Shiffman.

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Research partially supported by NSF Grant CCF-1640970.

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Shiffman, B. Asymptotic Expansion of the Variance of Random Zeros on Complex Manifolds. J Geom Anal 31, 8607–8631 (2021). https://doi.org/10.1007/s12220-020-00604-x

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