Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

Damian Brotbek Centre de Mathématiques Laurent Schwartz, École polytechnique Ya Deng Institut de Recherche Mathématique Avancée, Université de Strasbourg

Complex Variables and Complex Analysis mathscidoc:1906.01008

Geometric And Functional Analysis, 29, 690-750, 2019.6
In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct. As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.
Kobayashi hyperbolicity, Orbifold hyperbolicity, Logarithmic-orbifold Kobayashi conjecture, Second Main Theorem, Jet differentials, Logarithmic Demailly tower, Higher order log connections, Logarithmic Wronskians
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@inproceedings{damian2019kobayashi,
  title={Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree},
  author={Damian Brotbek, and Ya Deng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190626223324656023374},
  booktitle={Geometric And Functional Analysis},
  volume={29},
  pages={690-750},
  year={2019},
}
Damian Brotbek, and Ya Deng. Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree. 2019. Vol. 29. In Geometric And Functional Analysis. pp.690-750. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190626223324656023374.
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