Complete intersection varieties with ample cotangent bundles

Brotbek, Damian; Darondeau, Lionel

doi:10.1007/s00222-017-0782-9

Cited by 30 publications

(70 citation statements)

References 15 publications

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“…We are now in position to introduce the main geometric framework used during the proof of our main result. As in [BD18,Bro17,Den17,BD17] we rely on the universal complete intersection variety associated to our problem defined by…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…By "general position" we mean that the divisors defined by (τ j = 0) j=0,...,N are all smooth and meet transversally. For any a ∈ A, write H a := (T − p * σ(a) = 0) ⊂ L. By [BD18] there exists a non-empty Zariski open subset A sm ⊂ A such that D a is a smooth hypersurface for any a ∈ A sm , and so is H a . Let us now shrink the family H (resp.…”

Section: Main Constructionsmentioning

confidence: 99%

“…Next, we follow the general strategy in [BD18,Bro17,BD17] which consists in reducing the general case to the construction of a particular example (P n , D) satisfying a certain ampleness property, which implies Brody hyperbolicity and which is a Zariski open property. Such examples are indeed suitable deformations of Fermat type hypersurfaces.…”

Section: Introductionmentioning

confidence: 99%

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Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

Brotbek

Deng

2019

Geom. Funct. Anal.

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In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct.As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.I J := {I ∈ I | Supp(I) ⊂ {0, . . . , k} \ J}, and let us also consider the restricted universal complete intersection varietiesLet us denote by pr : Y → Gr k the canonical projection, and for any J ⊂ {0, . . . , k} we set pr J := pr ↾Y J : Y J → Gr k . Observe that the pr J is generically finite. This observation is crucial in the rest of the argument which highly rests on the understanding of the geometry of the non-finite locus of pr J :E J := y ∈ Y J | dim y pr −1 J (pr J (y)) > 0 , and its image in Gr k :

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Main Constructionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

Brotbek

Deng

2019

Geom. Funct. Anal.

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“…A similar notion to V-bigness, strong bigness, has been defined by Brotbek for vector bundles, cf. [3].…”

Section: Corollary 65 If E Is V-big Then It Is L-big As Well Proofmentioning

confidence: 99%

“…As it turns out, the asymptotic base loci defined here behave well with respect to the natural map induced by the projectivization of the vector bundle E, as shown in Sect. 3.…”

Section: Introductionmentioning

confidence: 99%

On positivity and base loci of vector bundles

Bauer

Kovács

Küronya

et al. 2015

European Journal of Mathematics

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The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the literature. In particular, we define augmented and restricted base loci B + (E) and B − (E) of a vector bundle E on the variety X , as generalizations of the corresponding TB was partially supported by DFG Grant BA 1559/6-1. SJK notions studied extensively for line bundles. As it turns out, the asymptotic base loci defined here behave well with respect to the natural map induced by the projectivization of the vector bundle E.

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On singular generalizations of the Singer–Hopf conjecture

Maxim

2023

Mathematische Nachrichten

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The Singer-Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer-Hopf conjecture, formulated in terms of the Euler-Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi-positivity properties of nef vector bundles together with a topological version of the Riemann-Roch theorem, proved by Kashiwara.

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Complete intersection varieties with ample cotangent bundles

Cited by 30 publications

References 15 publications

Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

On positivity and base loci of vector bundles

On singular generalizations of the Singer–Hopf conjecture

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