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Equivalence of plurisubharmonic singularities and Siu-type metrics

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Abstract

We show by an example that the (equivalence class of) singularity of a plurisubharmonic function cannot be determined by the data of its Lelong numbers, in a nontrivial sense. Such an example is provided by Siu-type singular hermitian metrics associated to a line bundle. We also show that a Siu-type metric has analytic singularities if and only if the section ring of the line bundle is finitely generated.

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Acknowledgments

The author wishes to thank Sébastien Boucksom for his crucial suggestion of using an argument of [2] toward Theorem 3.5 and Mihai Păun for showing him the proof of Proposition 2.3 in 2009. He also thanks Mattias Jonsson for telling him Example 2.5, Nikolay Shcherbina for raising a question on using multiplier ideal sheaves to distinguish psh functions and the anonymous referee for careful reading and helpful comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 151-747, Korea

    Dano Kim

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  1. Dano Kim
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Correspondence to Dano Kim.

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Communicated by A. Cap.

This work was supported by the National Research Foundation of Korea Grants NRF-2012R1A1A1042764 and No. 2011-0030795, funded by the Republic of Korea government.

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Kim, D. Equivalence of plurisubharmonic singularities and Siu-type metrics. Monatsh Math 178, 85–95 (2015). https://doi.org/10.1007/s00605-015-0742-7

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