Almost-positive vector bundles on projective surfaces

Schneider, Michael; Tancredi, Alessandro

doi:10.1007/bf01450074

Cited by 6 publications

(3 citation statements)

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“…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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confidence: 99%

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

Păun

2007

Math. Ann.

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“…This result and its improvements carried out in [8] are useful in studying the ampleness of cotangent bundles [6]. They also find use in an approach of Demailly to the Green-Griffiths-Lang conjecture [1].…”

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confidence: 81%

Criteria for the ampleness of certain vector bundles

Biswas¹,

Pingali²

2022

Preprint

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We prove that certain vector bundles (of rank two over arbitrary manifolds or rank three over surfaces) are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to det(E)). This result generalises a theorem of Schneider and Tancredi for rank two vector bundles over surfaces. We also provide counterexamples indicating that our theorem is sharp for rank three vector bundles over surfaces.

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“…Those ideas were later refined by Miyaoka (5) and led to a stronger result on surfaces of positive index, claiming that their cotangent bundles are "almost" ample [cf. also Schneider and Tancredi (6)]. In this note, we also use the same ideas to demonstrate that if the minimal model of a two-dimensional projective variety S is of positive index,t then S is C-hyperbolict and in fact strongly C-hyperbolic.…”

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confidence: 83%

Holomorphic curves in surfaces of general type.

Yau²

1990

Proc. Natl. Acad. Sci. U.S.A.

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This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy cl > 2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a rinite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q -1 irrespective of the map. Bogomolov (4) played an important role in their analysis. Those ideas were later refined by Miyaoka (5) and led to a stronger result on surfaces of positive index, claiming that their cotangent bundles are "almost" ample [cf. also Schneider and Tancredi (6)]. In this note, we also use the same ideas to demonstrate that if the minimal model of a two-dimensional projective variety S is of positive index,t then S is C-hyperbolict and in fact strongly C-hyperbolic. In a future paper, we shall weaken the hypothesis of positive index and study the case when S is quasiprojective. Section 2. Some PreliminariesThis section will serve mainly to establish the notation to be used. Unless otherwise specified, objects such as maps, bundles, and their sections are assumed to be holomorphic. No Riemann surface with (Gaussian) curvature K = -Cl(g)g-.We will need to consider a slight generalization of a metric on a line bundle: g will be allowed to degenerate, but locally it will differ from a metric only by a factor of IhlI, where h # 0 is a holomorphic function and v > 0. This means that g is degenerate only along proper subvarieties and that cl(g) defines a current which is smooth aside from delta functions supported along these subvarieties. In this case, g will be called a pseudometric. Note the identity c1(frlg) = fwcl(g).We now make some observations about metrics on D and D*. The Poincar6 (punctured) disk of radius c is defined to be tImplicit is that the minimal model of S is unique. So S cannot be CP2 and, by Kodaira classification of surfaces with positive index, must in fact be of general type. tIn principle we can find the corresponding algebraic subvariety explicitly. 80The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

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Almost-positive vector bundles on projective surfaces

Cited by 6 publications

References 5 publications

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

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

Criteria for the ampleness of certain vector bundles

Holomorphic curves in surfaces of general type.

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