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The asymptotic Fubini-Study operator over general non-Archimedean fields

Abstract

Given an ample line bundle L over a projective \({{\mathbb {K}}}\)-variety X, with \({{\mathbb {K}}}\) a non-Archimedean field, we study limits of non-Archimedean metrics on L associated to submultiplicative sequences of norms on the graded pieces of the section ring R(XL). We show that in a rather general case, the corresponding asymptotic Fubini-Study operator yields a one-to-one correspondence between equivalence classes of bounded graded norms and bounded plurisubharmonic metrics that are regularizable from below. This generalizes results of Boucksom-Jonsson where this problem has been studied in the trivially valued case.

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Correspondence to Rémi Reboulet.

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The author would like to thank his advisors Catriona Maclean and Sébastien Boucksom, for their guidance, advice, and support. He also thanks the anonymous referee for many useful suggestions and remarks

This work was supported by the European Research Council Advanced Grant ALKAGE (Algebraic and Kähler Geometry).

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Reboulet, R. The asymptotic Fubini-Study operator over general non-Archimedean fields. Math. Z. (2021). https://doi.org/10.1007/s00209-021-02770-2

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Keywords

  • Non-Archimedean geometry
  • Berkovich spaces
  • Pluripotential theory
  • Monge–Ampère operators