New examples of Kobayashi hyperbolic surfaces in \(\mathbb C\mathbb P^3\). (English. Russian original) Zbl 1095.32009

Shiffman, Bernard; Zaidenberg, Mikhail

Funct. Anal. Appl. 39, No. 1, 76-79 (2005); translation from Funkts. Anal. Prilozh. 39, No. 1, 90-94 (2005).

The authors construct new examples of hyperbolic surfaces in \(\mathbb{P}_3(\mathbb{C})\) of any given degree \(d\) with \(d\geq 8\). Let \(C\) be an algebraic curve in a plane \(H\subset\mathbb{P}_3(\mathbb{C})\). Denote by \(X = \langle C,p\rangle\) the cone formed by the lines passing through a given point \(p\in\mathbb{P}_3(\mathbb{C})\setminus H\) and the points of \(C\). Notice that \(\deg X=\deg C\). Then the main result in this paper can be stated as follows: For arbitrary given integers \(m,n\geq 4\), a generic small deformation of the union \(X= X'\cup X''\) of two generic cones in \(\mathbb{P}_3(\mathbb{C})\) of degree \(m\) and \(n\), respectively, is a hyperbolic surface of degree \(m+ n\).
It is noticed that a hyperbolic surface of degree 6 was constructed by J. Duval [Math. Ann. 330, 473–476 (2004; Zbl 1071.14045)].

Reviewer: Yoshihiro Aihara (Shizuoka)

MSC:

32Q45

Hyperbolic and Kobayashi hyperbolic manifolds

Keywords:

projective surface; Kobayashi hyperbolic surface; deformation

Citations:

Zbl 1071.14045

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