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Hyperbolicity and negativity of the curvature

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Hyperbolicity of Projective Hypersurfaces

Part of the book series: IMPA Monographs ((IMPA,volume 5))

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Abstract

In this chapter we shall explain how negativity properties of the curvature of complex manifolds is connected to hyperbolicity. We start with some basic notions of curvature and then prove the classical Ahlfors-Schwarz lemma. Then, we come back to higher order jets, and prove the basic result that every entire curve automatically satisfies every global jet differential with values in an antiample line bundle; as a consequence we deduce Bloch’s theorem about entire curves on complex tori. To conclude the chapter we illustrate a general strategy to prove algebraic degeneracy of entire curves.

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Authors and Affiliations

  1. CNRS, Institut de Mathématiques de Jussieu – Paris Rive Gauche, Sorbonne Universités Univ Paris Diderot, Sorbonne Paris Cité, Paris, France

    Simone Diverio

  2. Aix Marseille Université, CNRS, Centrale Marseille, Marseille, France

    Erwan Rousseau

Authors
  1. Simone Diverio
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  2. Erwan Rousseau
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Diverio, S., Rousseau, E. (2016). Hyperbolicity and negativity of the curvature. In: Hyperbolicity of Projective Hypersurfaces. IMPA Monographs, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-32315-2_4

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