Holomorphic curves in surfaces of general type.

Lu, Steven S. Y.; Yau, Shing–Tung

doi:10.1073/pnas.87.1.80

Cited by 26 publications

(14 citation statements)

References 6 publications

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“…As observed by Lu and Yau [LuYa90], one can say more if the topological index c 2 1 −2c 2 is positive, using the following result of Schneider-Tancredi [ScTa85] (the special case when E = T ⋆ X is due to Miyaoka [Miy82]). …”

Section: Of Jets In Irreducible Representationsmentioning

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]). Last but not least, there are several important questions of Number Theory which either depend on Nevanlinna theory or suggest new tools for the study of differential geometric problems.…”

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: Of Jets In Irreducible Representationsmentioning

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

View full text Add to dashboard Cite

show abstract

“…It worth to mention here that on a smooth surface in P 3 , there are no symmetric differentials, by a result of Sakai [13]. Next, as a "second order jet" generalization of some ideas of Miyaoka and Lu-Yau, (see [9,11,14]), Demailly-ElGoul equally show that if the vanishing order of the jet differential is smaller than an explicit number C(d), depending on the degree of the hypersurface, then Theorem 1 follows.…”

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confidence: 99%

Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity

Păun

2007

Math. Ann.

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The results we obtain in this article concern the hyperbolicity of very generic hypersurfaces in the 3-dimensional projective space: we show that the Kobayashi conjecture is true in this setting, as long as the degree of the hypersurface is greater than 18. IntroductionLet X be a generic hypersurface in the projective space P n . It was conjectured by S. Kobayashi in [8] that X is hyperbolic (i.e. the family of disks in X is normal), provided that deg(X ) ≥ 2n − 1.Recently, Y.-T. Siu obtained a confirmation of this conjecture, under the assumption that the degree of X is large enough, greater than a constant depending on n. As far as the degree bound is concerned, it was previously obtained by Demailly and ElGoul [5] the hyperbolicity of generic hypersurfaces X ⊂ P 3 , such that deg(X ) ≥ 21.In this paper, we would like to prove the next result.Theorem 1 Let X ⊂ P 3 be a generic hypersurface such that deg X ≥ 18. Then there exists a divisor Y ⊂ P(T X ) of the projectivized tangent bundle of X such that given any entire holomorphic curve ϕ : C → X , its derivative is contained in Y . Remark that a corollary of the previous result is the hyperbolicity of X by general results due to McQuillan [10] and Clemens [3]. Thus we get a slight improvement of the Demailly-ElGoul degree; on the other hand, our hope is that the techniques developed M. Pȃun (B)

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“…Hence, I proposed the following definition to replace the Carathéodory metric. Based on a theorem of Bogomolov, Lu and Yau [7] observed that for an algebraic surface M such that c 2 1 > c 2 , the metric · exists and that either (1) There is a divisor D ⊆ M such that for every holomorphic map f : C → M , the image f (C) ⊆ D ; or (2) There is an algebraic foliation with singularity on M such that f maps C to a leaf of such foliation.…”

Section: Generalized Carathéodory Metricmentioning

confidence: 99%

Canonical metrics on complex manifold

Yau

2008

Sci. China Ser. A-Math.

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Complex manifolds are topological spaces that are covered by coordinate charts where the coordinate changes are given by holomorphic transformations. For example, Riemann surfaces are one dimensional complex manifolds. In order to understand complex manifolds, it is useful to introduce metrics that are compatible with the complex structure. In general, we should have a pair (M, ds 2 M ) where ds 2 M is the metric. The metric is said to be canonical if any biholomorphisms of the complex manifolds are automatically isometries. Such metrics can naturally be used to describe invariants of the complex structures of the manifold.The first important examples of such metrics were constructed by Poincaré for Riemann surfaces with genus greater than one. Note that the flat metric on the torus is not quite canonical unless we require the biholomorphisms to preserve the area. (In higher dimensions, this is the same as preserving the Kähler class.) Poincaré's metrics are metrics with constant negative curvature. It requires a proof of the existence theorem for conformal deformation of metrics. It is a nonlinear differential equation and hence a difficult higher dimensional problem. Generalizations of Poincaré's work took some time.Besides Riemannian metrics that are compatible with the complex structure, there are other types of metrics which we shall discuss in the following. Carathéodory metricThis is a metric on a Riemann surface X constructed by the following procedure. Given a tangent vector v at a point p ∈ X, the Carathéodory metric is defined by v = sup{ f * (v) : f is a holomorphic map from X to the unit disk equipped with the Poincaré metric}.In the 1930s, Ahlfors proved a powerful generalization of the Schwarz lemma in complex analysis. Theorem 1. Holomorphic maps from the Poincaré disk to a Riemann surface with curvature less than −1 is distance decreasing.Grauert-Reckziegel generalized Ahlfors-Schwarz lemma in the 1960s for maps from disk into higher dimensional complex hermitian manifold with strongly negative curvature. In 1978, I [1]

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Holomorphic curves in surfaces of general type.

Cited by 26 publications

References 6 publications

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity

Canonical metrics on complex manifold

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