Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems

Deng, Ya

doi:10.1093/imrn/rnaa018

Cited by 9 publications

(6 citation statements)

References 32 publications

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“…In the same paper and also in a work of Iwai [10], slightly better quadratic bound was shown for klt Q-pairs, improving the results of the current paper in high dimensions. In the situation of Theorem C, Iwai showed this generation at regular values without any assumptions on relative freeness of ω ⊗k X , improving a similar statement by Deng [3]. The algebraic methods in this paper rely on Kawamata's arguments in [13], which in turn uses the arguments stemming from the work of Bombieri [2], Kawamata [12] and Shokurov [20], involving the problem of finding suitable singular divisors passing through the point at which one aims to show global generation.…”

Section: Remark 14 (A Discussion On Recent Resultssupporting

confidence: 57%

“…This suffices however in order to deduce the next Theorem, where assuming semiampleness of the canonical bundle along the smooth fibres, we prove that the global generation holds at the smooth (regular) values of f in Y . The relative semiampleness hypothesis was removed by Deng [3], later was improved by Iwai [10] when dim Y 5 (see 1.4 below).…”

Section: Introductionmentioning

confidence: 99%

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On the Effective Freeness of the Direct Images of Pluricanonical Bundles

Dutta¹

2021

Annales De l'Institut Fourier

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We give effective bounds on the number of twists by ample line bundles, for global generations of pushforwards of log-pluricanonical bundles on klt pairs. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove two types of statements: first, more in the spirit of the general conjecture, we show generic global generation with the predicted bound when the dimension of the variety is less than or equal to 4 and more generally, with a quadratic Angehrn-Siu type bound. Secondly, assuming that the relative canonical bundle is relatively semi-ample, we make a very precise statement. In particular, when the morphism is smooth, it solves the conjecture with the same bounds, for certain pluricanonical bundles.Résumé. -Nous donnons des limites effectives sur le nombre de torsions par fibrés en droites amples pour des générations globales de faisceaux log-pluricanoniques sur des paires de klt. Cela donne une réponse partielle à une hypothèse proposée par Popa et Schnell. Nous démontrons deux types d'énoncés: premièrement, plus dans l'esprit de la conjecture générale, nous démontrons la génération globale générique avec la borne annoncée quand la dimension de la variété est inférieure ou égale à 4 et plus généralement, avec une limite de type Angehrn-Siu. Deuxièmement, en supposant que le fibré canonique relatif soit relativement semi-ample, nous donnons un énoncé très précis. En particulier, quand le morphisme est lisse, ceci résout la conjecture avec les mêmes limites, pour certains faisceaux pluricanoniques.

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Section: Remark 14 (A Discussion On Recent Resultssupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

On the Effective Freeness of the Direct Images of Pluricanonical Bundles

Dutta¹

2021

Annales De l'Institut Fourier

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“…For the proof, refer to [34,Corollary 2.10]. Just remark that: in [34] this theorem is only stated for f a projective morphism.…”

Section: Then For Any Holomorphic Sectionmentioning

confidence: 99%

“…For the proof, refer to [34,Corollary 2.10]. Just remark that: in [34] this theorem is only stated for f a projective morphism. The above Kähler version holds because the proof of [34, Corollary 2.10] depends only on [31, (2.8) Theorem] (cf.…”

Section: Then For Any Holomorphic Sectionmentioning

confidence: 99%

On the Iitaka Conjecture C n,m for Kähler Fibre Spaces

Wang¹

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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By applying the positivity theorem of direct images and a pluricanonical version of the structure theorem on the cohomology jumping loci à la Green-Lazarsfeld-Simpson, we show that the klt Kähler version of the Iitaka conjecture Cn,m (Ueno, 1975) for f : X Y (surjective morphism between compact Kähler manifolds with connected general fibre) holds true when the determinant of the direct image of some power of the relative canonical bundle is big on Y or when Y is a complex torus. These generalize the corresponding results of Viehweg ( 1983) and of Cao-Păun (2017) respectively. We further generalize the later case to the geometric orbifold setting, i.e. prove that C orbifold n,m (Campana, 2004) holds when Y is a complex torus. RÉSUMÉ. -En appliquant la positivité des images directes et une version pluricanonique du théorème de structure des lieux de saut cohomologique à la Green-Lazarsfeld-Simpson, nous démontrons que la version klt kählérienne de la conjecture d'Iitaka Cn,m (Ueno, 1975) pour f : X Y (morphisme surjectif entre variétés kählériennes compactes à fibre générale connexe) est vraie si le déterminant de l'image directe d'une certaine puissance du fibré canonique relative est gros sur Y ou si Y est un tore complexe. Ceci généralisent les résultats correspondants de Viehweg (1983) et de Cao-Păun (2017) respectivement. De plus nous généralisons le deuxième résultat ci-dessus au cadre des orbifoldes géométriques, c-à-d., nous démontrons que C orbifold n,m (Campana, 2004) est vraie quand Y est un tore complexe. (*) Reçu le 16 juillet 2019, accepté le 24 janvier 2020.

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“…For the proof, refer to [Den17, Corollary 2.10]. Just remark that: in [Den17] this theorem is only stated for f a projective morphism. The above Kähler version holds because the proof of [Den17, Corollary 2.10] depends only on [Dem15, (2.8)Theorem] (c.f.…”

Section: Let (L H L ) Be Any Holomorphic Line Bundle On X Equipped Wi...mentioning

confidence: 99%