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Hyperbolicity in complex geometry. (English) Zbl 1076.32011
Laudal, Olav Arnfinn (ed.) et al., The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer (ISBN 3-540-43826-2/hbk). 543-566 (2004).
A complex manifold \(X\) is said to be hyperbolic if there exists no nonconstant holomorphic map \(\mathbb C \to X\). The hyperbolicity problem in complex geometry studies the conditions for a given complex manifold \(X\) to be hyperbolic. Hyperbolicity problems trace back to the small Picard theorem, the hyperbolicity of compact Riemann surfaces of genus \(\geq 2\) from Liouville’s theorem and the uniformization theorem. The hyperbolicity of compact Riemann surfaces of genus \(\geq 2\) led to the study of compact complex manifolds \(X\) with positive canonical line bundle \(K_X\). The small Picard theorem led to the study of the complement of a hypersurface \(Y\) in a compact algebraic manifold \(X\) with conditions related to the positivity of \(Y + K_X\). For a long time a great part of the research on the hyperbolicity problems has been focussed on the following two environments. (1) The setting of abelian varieties. The hyperbolicity problem concerns a subvariety in an abelian variety or concerns the complement of a hypersurface in an abelian variety. (2) The setting of the complex projective space \(\mathbb P_n\). The hyperbolicity problem concerns a generic hypersurface of sufficiently high degree or concerns the complement of a generic hypersurface of sufficiently high degree.
In this survey article, the author presents the essential techniques from function theory developed over the years and especially recently for the investigation of hyperbolicity problems for these two settings. There are three main parts in this survey paper: the techniques of Bloch for the setting of abelian varieties, the most recent methods for the setting of the complex projective space which were developed from the techniques of Clemens-Ein-Voisin from algebraic geometry and the techniques of McQuillan for hyperbolicity problems for surfaces of general type.
For the entire collection see [Zbl 1047.00019].

MSC:
32H30 Value distribution theory in higher dimensions
14J29 Surfaces of general type
14K05 Algebraic theory of abelian varieties
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
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