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On the hyperbolicity of surfaces of general type with small $$c_{1}^{2}$$. (English) Zbl 1276.14053
Motivated by the Green-Griffiths-Lang Conjectures, this article investigates hyperbolicity properties of some special surfaces of general type known as Horikawa surfaces. Indeed if $$X$$ is a minimal surface of general type, it seems that bigger the ratio $$c_1(X)^2/c_2(X)$$ is, the more hyperbolic $$X$$ is. It is for instance illustrated by the works of [F. A. Bogomolov, Sov. Math., Dokl. 18(1977), 1294–1297 (1978); translation from Dokl. Akad. Nauk SSSR 236, 1041–1044 (1977; Zbl 0415.14013)], [S. S. Y. Lu and S. T. Yau, Proc. Natl Acad. Sci. USA 87, 163–175 (1990)] and [M. McQuillan [Publ. Math., Inst. Hautes Étud. Sci. 87, 121–174 (1998; Zbl 1006.32020)].
In this paper, the authors study the opposite side, i.e. when the preceding ratio is minimum. The surfaces having this property are called Horikawa surfaces (following [E. Horikawa, Ann. Math. (2) 104, 357–387 (1976; Zbl 0339.14024)]) and some of them arise as (minimal resolution of) ramified covers of the plane $$\mathbb{P}^2$$ or of the Hirzebruch surfaces $$\mathbb{F}_N$$. An orbifold can then be associated with such surfaces. Using orbifolds jet differentials (as initiated by the second author in [E. Rousseau, Trans. Am. Math. Soc. 362, No. 7, 3799–3826 (2010; Zbl 1196.32018)]), they succeed in showing hyperbolicity properties of these surfaces (such as quasi-hyperbolicity or algebraic hyperbolicity).
The paper is clearly written and contains plenty of interesting examples.

##### MSC:
 14J29 Surfaces of general type 14J25 Special surfaces 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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