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Neighborhoods of rational curves without functions

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Abstract

We prove the existence of (non compact) complex surfaces with a smooth rational curve embedded such that there does not exist any formal singular foliation along the curve. In particular, at arbitrary small neighborhood of the curve, any meromorphic function is constant. This implies that the Picard group is not countably generated.

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Authors and Affiliations

  1. Universidade Federal Fluminense, Rua Mário Santos Braga S/N, Niterói, RJ, Brazil

    Maycol Falla Luza

  2. Univ Rennes, CNRS, IRMAR-UMR 6625, 35000, Rennes, France

    Frank Loray

Authors
  1. Maycol Falla Luza
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  2. Frank Loray
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Correspondence to Frank Loray.

Additional information

Communicated by Ngaiming Mok.

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Frank Loray is supported by CNRS, Henri Lebesgue Center and ANR-16-CE40-0008 project “Foliage”. The authors also thank Brazilian-French Network in Mathematics and CAPES-COFECUB Project Ma 932/19 “Feuilletages holomorphes et intéractions avec la géométrie”. We finally thank Jorge Vitório Pereira for helpfull discussions on the subject.

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Falla Luza, M., Loray, F. Neighborhoods of rational curves without functions. Math. Ann. 382, 1047–1058 (2022). https://doi.org/10.1007/s00208-020-02126-x

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  • DOI: https://doi.org/10.1007/s00208-020-02126-x

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