Hyperbolicity of the Complements of Plane Algebraic Curves

Dethloff, Gerd; Schumacher, Georg; Wong, Pit-Mann

doi:10.2307/2375081

Cited by 23 publications

(26 citation statements)

References 14 publications

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“…Zaidenberg [Zai89] showed the existence of curves with hyperbolic complement for every degree d 5. In response to a conjecture of [Zai89], [DSW92,94] showed that the complement of the union of at least 3 generic curves is hyperbolic, when the sum of degrees is at least 5. More recently, Siu and Yeung [SiYe96a] proved the harder fact that the complement of a generic irreducible curve of high degree in P 2 is hyperbolic.…”

Section: Corollarymentioning

confidence: 99%

“…Especially noticeable in this respect is the work of Dethloff-Wong-Schumacher [DSW92,94] on the hyperbolicity of complements of 3 or more generic curves in the projective plane, and the construction by Masuda-Noguchi [MaNo93] of hyperbolic hypersurfaces of large degree in P n . Also, in a more algebraic setting, there is an extensive literature dealing with the question of computing genus of curves in algebraic surfaces, bearing an intimate connection with hyperbolicity ( [Bog77], [Cle86], [CKM88], [LuYa90], [Lu91], [LuMi95], [Lu96] [Xu94]).…”

Section: §0 Introductionmentioning

confidence: 99%

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Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Demailly¹

1997

Proceedings of Symposia in Pure Mathematics

190

416

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Abstract. These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995. The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi. A convenient framework for this is the category of "directed manifolds", that is, the category of pairs (X, V ) where X is a complex manifold and V a holomorphic subbundle of T X . If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody's criterion). We describe a construction of projectivized kjet bundles P k V , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature. More precisely, π k : P k V → X is a tower of projective bundles over X and carries a canonical line bundle O P k V (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on O P k V (−1) for k large enough. The direct images (π k ) ⋆ O P k V (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization. Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when O P k V (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric. This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus. Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus. Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves. We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections. These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of "partial projective projection" and the associated Wronskian operators, substantial improvements on Nadel's degree estimate will be achieved here.

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Section: Corollarymentioning

confidence: 99%

Section: §0 Introductionmentioning

confidence: 99%

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Demailly¹

1997

Proceedings of Symposia in Pure Mathematics

190

416

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show abstract

“…We now prove that the property of having order at most 2 is preserved under rational maps (see also [11] for a similar result).…”

Section: Brody Curves Maps Of Order 2 and Limit Sets Of Entire Curvesmentioning

confidence: 60%

“…Another consequence of Theorem 1.2 is the "best possible" result for algebraic degeneracy in the three component case of complements of plane curves (see Dethloff-Schumacher-Wong '95 [11] and [12], BerthelootDuval '01 [2]): …”

Section: Theorem 12 Gives Us the Following Corollaries On Hyperbolicmentioning

confidence: 99%

Logarithmic Surfaces and Hyperbolicity

Dethloff¹,

Lu²

2007

Ann. inst. Fourier

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“…Le théorème de linéarisation des courbes entières dans les espaces projectifs et son applicationà la preuve du théorème de Green sont dûsà Berteloot et Duval [17], ce théorème de linéarisation estégalement utilisé pour retrouver (de façonélémentaire) un résultat de Dethloff, Schumacher et Wong [29] sur l'hyperbolicité du complémentaire d'une courbeà trois composantes dans l'espace projectif bi-dimensionnel. L'article de Berteloot-Duval contient aussi une preuve du théorème de Green basée sur la stratégie de Ros.…”

Section: Linéarisation De Courbes Entières Dans P K Et Théorème De Greenunclassified

Méthodes de changement d’échelles en analyse complexe

Berteloot¹

2008

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Hyperbolicity of the Complements of Plane Algebraic Curves

Cited by 23 publications

References 14 publications

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Logarithmic Surfaces and Hyperbolicity

Méthodes de changement d’échelles en analyse complexe

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