Abstract
In this first chapter we state and describe the basic definitions of complex hyperbolic geometry, basically following [35] and [17]. Then, we state and prove the classical Brody’s lemma and Picard’s theorem. We conclude by giving a brief account of elementary examples and describing the case of Riemann surfaces.
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Notes
- 1.
We recall here that the usual Schwarz-Pick lemma says that for \(f: \Delta \rightarrow \Delta\) a holomorphic map, one has the following inequality:
$$\displaystyle{ \frac{\vert f'(\zeta )\vert } {1 -\vert f(\zeta )\vert ^{2}} \leq \frac{1} {1 -\vert \zeta \vert ^{2}}.}$$This means exactly that holomorphic maps contract the Poincaré metric.
Bibliography
Theodore J. Barth. Convex domains and Kobayashi hyperbolicity. Proc. Amer. Math. Soc., 79(4):556–558, 1980.
Julien Duval. Sur le lemme de Brody. Invent. Math., 173(2):305–314, 2008.
Serge Lang. Introduction to complex hyperbolic spaces. Springer-Verlag, New York, 1987.
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Diverio, S., Rousseau, E. (2016). Kobayashi hyperbolicity: basic theory. In: Hyperbolicity of Projective Hypersurfaces. IMPA Monographs, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-32315-2_1
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DOI: https://doi.org/10.1007/978-3-319-32315-2_1
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