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Essential skeletons of pairs and Temkin’s metric

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Abstract

We study weight metrics and essential skeletons of pairs in the Berkovich analytification of a variety over a trivially-valued field of characteristic zero. In the smooth case, we show that Temkin’s canonical metric on pluricanonical bundles coincides with the weight metric defined via log discrepancies.

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Notes

  1. In loc. cit. the retraction is defined on \(X^{\textrm{bir}} \cap X^\beth \), but it extends to \(X^\beth \).

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Acknowledgements

We would like to thank Walter Gubler, Mattias Jonsson, Mircea Mustata and Michael Temkin for useful conversations on this paper, and the anonymous referees for their precious comments and specifically for suggesting the proof of Lemma 2.1.5.

Funding

Funding was provided by Deutsche Forschungsgemeinschaft (Grant Number SFB 1085). European Union’s Horizon 2020 research and innovation programme (Grant Number 101034413) and Max Planck Institute for Mathematics.

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Authors and Affiliations

  1. École Polytechnique, Rte de Saclay, 91120, Palaiseau, France

    Mirko Mauri

  2. Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 31, 93040, Regensburg, Germany

    Enrica Mazzon

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109–1043, USA

    Matthew Stevenson

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  1. Mirko Mauri
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  2. Enrica Mazzon
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Correspondence to Mirko Mauri.

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Mirko Mauri was supported by the Institute of Science and Technology Austria, and École Polytechnique in France. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101034413. Enrica Mazzon was supported by Max Planck Institute for Mathematics in Bonn during the preparation of this paper, and by the collaborative research center SFB 1085 Higher Invariants-Interactions between Arithmetic Geometry and Global Analysis funded by the Deutsche Forschungsgemeinschaft. All authors declare that they have no conflicts of interest.

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Mauri, M., Mazzon, E. & Stevenson, M. Essential skeletons of pairs and Temkin’s metric. Ann Univ Ferrara (2024). https://doi.org/10.1007/s11565-024-00504-w

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