Algebraic families of smooth hyperbolic surfaces of low degree in \(\mathbb{P}_ \mathbb{C}^ 3\)
type: Item
instance of: scholarly article
title: Algebraic families of smooth hyperbolic surfaces of low degree in \(\mathbb{P}_ \mathbb{C}^ 3\)
zbMATH Open document ID: 0887.32007
DOI: 10.1007/BF02568323
author: Jawher El Goul
published in: Manuscripta Mathematica
publication date: 1998-05-06T00:00:00Z
full work available at URL: https://eudml.org/doc/156210
review text: In [\textit{S. Kobayashi}, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York (1970; Zbl 0207.37902)], Kobayashi conjectured that a generic hypersurface of \(\mathbb{P}_{\mathbb{C}}^n\) of sufficiently high degree \(d\) (expected bound being \(d \geq 2n-1\)) is hyperbolic. It is true for \(n=2\), but for \(n>2\) only some examples are known. For \(n=3\), Brody and Green obtained the first example of a smooth hyperbolic surface in \(\mathbb{P}_{\mathbb{C}}^3\) of any degree \(d=2p \geq 50\) [\textit{R. Brody} and \textit{M. Green}, Duke Math. J. 44, 873-874 (1977; Zbl 0383.32009)], and Nadel obtained such examples of any degree \(d=6p+3\geq 21\) [\textit{A. M. Nadel}, Duke Math. J. 58, No. 3, 749-771 (1989; Zbl 0686.32015)]. In the present paper, the author improves Nadel's technique and produces examples of smooth hyperbolic surfaces in \(\mathbb{P}_{\mathbb{C}}^3\) of arbitrary degree \(d \geq 14\), which is closer to the expected bound \(5\). Furthermore, if \(H_{3,d}\) is the subset of all hyperbolic surfaces in the projective space \(P_{3,d}\) of all surfaces of degree \(d\) in \(\mathbb{P}_{\mathbb{C}}^3\), and if \(d>9+ \sum_{i=0}^3 k_i\) where at least two of the integers \(k_i\) are \(\geq 2\), then the author proves that \(H_{3,d}\) contains a Zariski open subset of \(P_{3,d}\) of dimension \(\sum_{i=0}^3 \binom{k_i+3}{k_i} -1\).
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 980712
zbMATH Keywords: hyperbolic surfaces
zbMATH Keywords: complex projective 3-space
reviewed by: Aleksandar T. Lipkovski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 980712