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Logarithmic Bergman Kernel and Conditional Expectation of Gaussian Holomorphic Fields

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Abstract

We prove the asymptotic of the logarithmic Bergman kernel. And as an application, we calculate the conditional expectation of density of zeros of Gaussian random sections of powers of a positive line bundle that vanish along a fixed smooth subvariety.

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Acknowledgements

The author would like to thank Professor Bernard Shiffman for his continuous and unconditional support. The author would also like to thank Professor Chengjie Yu and Professor Song Sun for many very helpful discussions. The author would also like to thank the referee whose many suggestions help substantially improve the writing of this article.

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  1. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China

    Jingzhou Sun

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  1. Jingzhou Sun
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Correspondence to Jingzhou Sun.

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The author is partially supported by NNSF of China No. 11701353 and the STU Scientific Research Foundation for Talents No. 130/760181.

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Sun, J. Logarithmic Bergman Kernel and Conditional Expectation of Gaussian Holomorphic Fields . J Geom Anal 31, 8520–8538 (2021). https://doi.org/10.1007/s12220-020-00602-z

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