On the ampleness of the cotangent bundles of complete intersections

Xie, Song-Yan

doi:10.1007/s00222-017-0783-8

Cited by 29 publications

(37 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This example was constructed by intersecting (many) particular deformations of Fermat type hypersurfaces, on which we were able to produce explicit symmetric differential forms. Afterwards, in [60] (see also [61]), Xie was able to prove the Debarre conjecture (with an explicit bound on the degree) by, among other things, generalizing the symmetric differential forms constructed in [6] to a wider class of complete intersections. Independently, in a joint work with Darondeau [7], we gave a geometric interpretation of the cohomological computations of [6], in order to give a short proof of the Debarre conjecture.…”

Section: Introductionmentioning

confidence: 99%

On the hyperbolicity of general hypersurfaces

Brotbek

2017

Publ.math.IHES

View full text Add to dashboard Cite

In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in P n are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. To do so, we introduce a Wronskian construction which associates a (twisted) jet differential to every finite family of global sections of a line bundle.

show abstract

Section: Introductionmentioning

confidence: 99%

On the hyperbolicity of general hypersurfaces

Brotbek

2017

Publ.math.IHES

View full text Add to dashboard Cite

show abstract

“…The present work also finds its roots in [Bro16], and pushes further the ideas initiated there to complete the study of families of complete intersection varieties. In contrast with the cited works [Bro16,Xie15], we do not manipulate (1) for a vector bundle E on a variety X we denote by π E : (E) → X the projectivization of rank one quotients of E explicit expressions of some symmetric differential forms. Although our intuitions grew from the geometric interpretation of a generalization of the cohomological computations arising in [Bro16], in the present text we choosed to emphasize the intrinsic simplicity and the geometric nature of the proof.…”

mentioning

confidence: 99%

Complete intersection varieties with ample cotangent bundles

Brotbek

Darondeau

2017

Invent. math.

View full text Add to dashboard Cite

Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.2010 Mathematics Subject Classification. -14M10. Key words and phrases. -Ample cotangent bundle, symmetric differential forms, complete intersection varieties.The research of the second author was supported by the USIAS project "Rational Points, Rational Curves and Automorphisms of Special Varieties" of Carlo Gasbarri and Gianluca Pacienza (http://www.usias.fr).arXiv:1511.04709v2 [math.AG] 8 Dec 2017 J and θ p J involve only a p J , the linear map ϕ p ξ is diagonal by blocks. Each block H 0 (M, O M (ε p )) → k × k corresponds to the map a p J → (α p J ) V (a p , ξ), (θ p J ) V (a p , ξ) for a certain J. Recall from (6) that locally (for readability, we now drop the "local" notation)There are thus two cases, depending on J: Either (ζ(ξ)) J 0, or not.Before discussing these two cases, let us note that (since ε p 1 and since sections of very ample line bundles separate first order jets) there exists b 1 ∈ H 0 (M, O M (ε p )) such that b 1 (ξ) = 0 and db 1 (ξ) 0, and there also exists b 2 ∈ H 0 (M, O M (ε p )) such that b 2 (ξ) 0.In the first case, when (ζ(ξ)) J 0, the respective images (0, * 0 ) and ( * 0 , * ) of b 1 and b 2 cannot be colinear. Thus the rank of the block is 2. Observe that there are exactly k 0 +δ p

show abstract

“…We give here a simple proof of the Kobayashi [Deng16], in chronological order. Related ideas had been used earlier in [Xie15] and [BrDa17] to establish Debarre's conjecture on the ampleness of the cotangent bundle of generic complete intersections of codimension at least equal to dimension.…”

Section: Proof Of the Kobayashi Conjecture On Generic Hyperbolicitymentioning

confidence: 99%

Recent results on the Kobayashi and Green-Griffiths-Lang conjectures

Demailly

2020

Jpn. J. Math.

View full text Add to dashboard Cite

The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -especially through the concepts of curvature and positivity which are central themes in Kodaira's contributions to mathematics. The aim of these lectures is to present recent results concerning the geometric side of the problem. The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over C, there exists a proper algebraic subvariety Y of X containing all non constant entire curves f : C → X. Using the formalism of directed varieties and jet bundles, we show that this assertion holds true in case X satisfies a strong general type condition that is related to a certain jet-semistability property of the tangent bundle TX . It is possible to exploit similar techniques to investigate a famous conjecture of Shoshichi Kobayashi (1970), according to which a generic algebraic hypersurface of dimension n and of sufficiently large degree d dn in the complex projective space P n+1 is hyperbolic: in the early 2000's, Yum-Tong Siu proposed a strategy that led in 2015 to a proof based on a clever use of slanted vector fields on jet spaces, combined with Nevanlinna theory arguments. In 2016, the conjecture has been settled in a different way by Damian Brotbek, making a more direct use of Wronskian differential operators and associated multiplier ideals; shortly afterwards, Ya Deng showed how the proof could be modified to yield an explicit value of dn. We give here a short proof based on a drastic simplification of their ideas, along with a further improvement of Deng's bound, namely dn = ⌊ 1 5 (en) 2n+2 ⌋.em

show abstract

On the ampleness of the cotangent bundles of complete intersections

Cited by 29 publications

References 37 publications

On the hyperbolicity of general hypersurfaces

On the hyperbolicity of general hypersurfaces

Complete intersection varieties with ample cotangent bundles

Recent results on the Kobayashi and Green-Griffiths-Lang conjectures

Contact Info

Product

Resources

About