On the Effective Freeness of the Direct Images of Pluricanonical Bundles

Dutta, Yajnaseni

Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1545-1561.

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.

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.

Classification : 14C20, 14F05, 14Q20, 14J17
Keywords: pluricanonical bundles, Fujita’s conjecture, effective results.
Mot clés : faisceaux pluricanoniques, conjecture de Fujita, résultats effectifs.

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