Logarithmic Orbifold Euler Numbers of Surfaces With Applications

Langer, Adrián

doi:10.1112/s0024611502013874

Cited by 71 publications

(85 citation statements)

References 26 publications

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“…Applying the Bogomolov-Gieseker inequality to the corresponding stable parabolic bundle E and using (7-6) we get 0 par-ch 2 .E / D Inequality (7-5) was proven previously by Langer [11] using different methods. We finish with the following lemma.…”

Section: Proof Of Stabilitymentioning

confidence: 54%

Polyhedral Kähler manifolds

Panov

2009

Geom. Topol.

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In this article we introduce the notion of polyhedral Kähler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. A parabolic version of the Kobayshi-Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kähler metrics of nonnegative curvature on CP 2 with singularities at complex line arrangements.53C56; 32Q15, 53C55

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Section: Proof Of Stabilitymentioning

confidence: 54%

Polyhedral Kähler manifolds

Panov

2009

Geom. Topol.

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“…We shall briefly explain how smooth Kummer covers can be used to get locally free inverse image sheaves which are everywhere defined by the above formulae, when (X, ∆) is smooth. Such covers have also be introduced by A. Langer for similar purposes in the surface case ( [26]), and also in [18], §.2, in the case of integral multiplicities.…”

Section: Local Constructionmentioning

confidence: 99%

Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

Campana¹,

Păun²

2015

Ann. inst. Fourier

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Let X be a normal projective manifold, equipped with an effective 'orbifold' divisor ∆, such that the pair (X, ∆) is logcanonical. We first define the notion of 'orbifold cotangent bundle' Ω 1 (X, ∆), living on any suitable ramified cover of X. We are then in position to formulate and prove (in a completely different way) an orbifold version of Y. Miyaoka's generic semi-positivity theorem:Using the deep results of the LMMP, we immediately get a statement conjectured by E. Viehweg: if X is smooth, and if ∆ is a reduced divisor with simple normal crossings on X such that some tensor power of Ω 1 (X, ∆) = Ω 1 X (Log(∆)) contains the injective image of a big line bundle, then K X + ∆ is big.This implies, by fundamental results of Viehweg-Zuo, the 'Shafarevich-Viehweg hyperbolicity conjecture': if an algebraic family of canonically polarized manifolds parametrised by a quasiprojective manifold B has 'maximal variation', then B is of loggeneral type.Résumé: Nous définissons la notion de 'fibré cotangent orbifolde' Ω 1 (X, ∆) pour une paire (X, ∆) log-canonique: ce fibré est défini sur des revêtement cycliques adéquats. Nous formulons et démontrons ensuite une version orbifolde du théorème de semi-positivité générique de Y. Miyaoka: Ω 1 (X, ∆) est génériquement semi-positif si K X + ∆ est pseudo-effectif. Nous en déduisons, à l'aide des résultats récents du PMML, un énoncé conjecturé par E. Viehweg: si X est lisse, et si ∆ est un diviseur réduit à croisements normaux simples sur X tel qu'une puissance tensorielle de Ω 1 X (Log(∆)) contienne un fibré en droites 'big', alors K X + ∆ est lui-même 'big'. Les travaux de Viehweg-Zuo impliquent alors la conjecture d'hyperbolicité de V.I. Shafarevich: si une famille algébrique de variétés projectives canoniquement polarisées et paramétrée par une variété quasi-projective irréductible lisse B a une 'variation' maximale, égale à dim(B), alors B est de type log-général.Mots-clé: Fibré cotangent orbifolde, semi-positivité générique, variété canoniquement polarisées.

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“…Assume first that (X, D) is not almost minimal. Since D is connected, by [3,6.20] S contains a curve isomorphic to C 1 such that κ(S \ ) = κ(S) = 0. Since Pic(S) is torsion, there is a rational map f : S C 1 such that (f ) = m for some positive integer m. Because S is affine we may assume that f is regular on S, so we get a morphism f :…”

Section: Rational Cuspidal Curvesmentioning

confidence: 99%

The Coolidge–Nagata conjecture, part I

Palka

2014

Advances in Mathematics

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Let E ⊆ P 2 be a complex rational cuspidal curve contained in the projective plane and let (X, D) → (P 2 , E) be the minimal log resolution of singularities. Applying the log Minimal Model Program to (X, 1 2 D) we prove that if E has more than two singular points or if D, which is a tree of rational curves, has more than six maximal twigs or if P 2 \ E is not of log general type then E is Cremona equivalent to a line, i.e. the Coolidge-Nagata conjecture for E holds. We show also that if E is not Cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in D.

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Logarithmic Orbifold Euler Numbers of Surfaces With Applications

Cited by 71 publications

References 26 publications

Polyhedral Kähler manifolds

Polyhedral Kähler manifolds

Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

The Coolidge–Nagata conjecture, part I

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