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Comparison of two notions of subharmonicity on non-archimedean curves

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Abstract

We show that a continuous function on the analytification of a smooth proper algebraic curve over a non-archimedean field is subharmonic in the sense of Thuillier if and only if it is psh, i.e. subharmonic in the sense of Chambert-Loir and Ducros. This equivalence implies that the property psh for continuous functions is stable under pullback with respect to morphisms of curves. Furthermore, we prove an analogue of the monotone regularization theorem on the analytification of \(\mathbb {P}^{1}\) and Mumford curves using this equivalence.

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References

  1. Berkovich, V.G.: Spectral Theory and Analytic Geometry Over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence (1990)

    MATH  Google Scholar 

  2. Berkovich, V.G.: Integration of One-Forms on \(p\)-Adic Analytic Spaces. Annals of Mathematics Studies, vol. 162. Princeton University Press, Princeton (2007)

    Book  MATH  Google Scholar 

  3. Boucksom, S., Favre, C., Jonsson, M.: Singular semipositive metrics in non-archimedean geometry (2012). arXiv:1201.0187

  4. Boucksom, S., Favre, C., Jonsson, M.: Solution to a non-Archimedean Monge-Ampère equation. J. Am. Math. Soc. 28(3), 617–667 (2015)

    Article  MATH  Google Scholar 

  5. Bieri, R., Groves, J.R.J.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math. 347, 168–195 (1984)

    MathSciNet  MATH  Google Scholar 

  6. Bosch, S., Lütkebohmert, W.: Stable reduction and uniformization of abelian varieties. I. Mathematische Annalen 270, 349–379 (1985)

  7. Bosch, S., Lütkebohmert, W.: Formal and rigid geometry. I. Rigid Spaces Mathematische Annalen 295, 291–317 (1993)

  8. Baker, M., Payne, S., Rabinoff, J.: On the structure of non-Archimedean analytic curves. In: Tropical and non-Archimedean geometry, volume 605 of Contemp. Math., pp 93–121. Amer. Math. Soc., Providence, RI (2013)

  9. Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line. Mathematical Surveys and Monographs, vol. 159. American Mathematical Society, Providence (2010)

  10. Chambert-Loir, A., Ducros, A.: Formes différentielles réelles et courants sur les espaces de Berkovich (2012). arXiv:1204.6277

  11. Gubler, W., Künnemann, K.: A tropical approach to non-archimedean Arakelov theory. Algebra Number Theory 11(1), 77–180 (2017). arXiv:1406.7637

    Article  MathSciNet  MATH  Google Scholar 

  12. Gubler, W., Künnemann, K.: Positivity properties of metrics and delta-forms (2015). arXiv:1509.09079

  13. Gubler, W.: Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros). In: Matthew, B., Sam, P. (eds.) Nonarchimedean and Tropical Geometry, Simons Symposia, pp 1–30. Springer, Switzerland (2016)

  14. Jell, P., Wanner, V.: Poincaré duality for tropical Dolbeault cohomology of non-archimedean Mumford curves. J. Number Theory. (2018). https://doi.org/10.1016/j.jnt.2017.11.004

  15. Katz, E., Rabinoff, J., Zureick-Brown, D.: Uniform bounds for the number of rational points on curves of small Mordell–Weil rank. Duke Math. J. 16, 3189–3240 (2016). https://doi.org/10.1215/00127094-3673558

    Article  MathSciNet  MATH  Google Scholar 

  16. Lagerberg, A.: Super currents and tropical geometry. Math. Z. 270(3–4), 1011–1050 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Thuillier, A.: Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov. Thése del’Université de Rennes 1 (2005)

  18. Wanner, V.: Harmonic functions on the Berkovich projective line. Master Thesis. (2016). https://d-nb.info/1136471502/34

  19. Zhang, S.: Admissible pairing on a curve. Invent. Math. 112(1), 171–193 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Walter Gubler for very carefully reading drafts of this work and for the helpful discussions. The author is also grateful to Philipp Jell for providing a simplification of the proof of Proposition 4.4 and further useful comments. Finally, the author would like to thank the referee for their very precise report and helpful suggestions.

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Authors and Affiliations

  1. Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Veronika Wanner

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  1. Veronika Wanner
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Correspondence to Veronika Wanner.

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The author was supported by the collaborative research center SFB 1085 ‘Higher Invariants’ funded by the Deutsche Forschungsgemeinschaft.

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Wanner, V. Comparison of two notions of subharmonicity on non-archimedean curves. Math. Z. 293, 443–474 (2019). https://doi.org/10.1007/s00209-018-2205-z

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