no. 4
On the Effective Freeness of the Direct Images of Pluricanonical Bundles
[Sur la libeté effective des images directes de faisceaux pluricanoniques]
Dutta, Yajnaseni1
1 Northwestern University Department of Mathematics 2033, Sheridan Road Evanston IL-60208 (USA)
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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3351
Classification : 14C20, 14F05, 14Q20, 14J17
Keywords: pluricanonical bundles, Fujita’s conjecture, effective results.
Mot clés : faisceaux pluricanoniques, conjecture de Fujita, résultats effectifs.

Dutta, Yajnaseni 1

1 Northwestern University Department of Mathematics 2033, Sheridan Road Evanston IL-60208 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2020__70_4_1545_0,
     author = {Dutta, Yajnaseni},
     title = {On the {Effective} {Freeness} of the {Direct} {Images} of {Pluricanonical} {Bundles}},
     journal = {Annales de l'Institut Fourier},
     pages = {1545--1561},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {4},
     year = {2020},
     doi = {10.5802/aif.3351},
     zbl = {07197935},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3351/}
}
TY  - JOUR
AU  - Dutta, Yajnaseni
TI  - On the Effective Freeness of the Direct Images of Pluricanonical Bundles
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 1545
EP  - 1561
VL  - 70
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3351/
DO  - 10.5802/aif.3351
LA  - en
ID  - AIF_2020__70_4_1545_0
ER  - 
%0 Journal Article
%A Dutta, Yajnaseni
%T On the Effective Freeness of the Direct Images of Pluricanonical Bundles
%J Annales de l'Institut Fourier
%D 2020
%P 1545-1561
%V 70
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3351/
%R 10.5802/aif.3351
%G en
%F AIF_2020__70_4_1545_0
Dutta, Yajnaseni. On the Effective Freeness of the Direct Images of Pluricanonical Bundles. Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1545-1561. doi : 10.5802/aif.3351. https://aif.centre-mersenne.org/articles/10.5802/aif.3351/

[1] Angehrn, Urban; Siu, Yum-Tong Effective freeness and point separation for adjoint bundles, Invent. Math., Volume 122 (1995) no. 2, pp. 291-308 | DOI | MR | Zbl

[2] Bombieri, Enrico Canonical models of surfaces of general type, Publ. Math., Inst. Hautes Étud. Sci. (1973) no. 42, pp. 171-219 | DOI | Numdam | MR | Zbl

[3] Deng, Ya Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems, Int. Math. Res. Not. (2020), rnaa018 | DOI

[4] Dutta, Yajnaseni; Murayama, Takumi Effective generation and twisted weak positivity of direct images, Algebra Number Theory, Volume 13 (2019) no. 2, pp. 425-454 | DOI | MR | Zbl

[5] Esnault, Hélène; Viehweg, Eckart Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser, 1992, vi+164 pages | DOI | MR | Zbl

[6] Fujino, Osamu Effective base point free theorem for log canonical pairs — Kollár type theorem, Tôhoku Math. J., Volume 61 (2009) no. 4, pp. 475-481 | DOI | MR | Zbl

[7] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | MR | Zbl

[8] Helmke, Stefan On Fujita’s conjecture, Duke Math. J., Volume 88 (1997) no. 2, pp. 201-216 | DOI | MR | Zbl

[9] Helmke, Stefan On global generation of adjoint linear systems, Math. Ann., Volume 313 (1999) no. 4, pp. 635-652 | DOI | MR | Zbl

[10] Iwai, Masataka On the global generation of direct images of pluri-adjoint line bundles, Math. Z. (2017), pp. 1-8 | DOI | MR | Zbl

[11] Jouanolou, Jean-Pierre Théorèmes de Bertini et applications, Progress in Mathematics, 42, Birkhäuser, 1983, ii+127 pages | MR | Zbl

[12] Kawamata, Yujiro On the finiteness of generators of a pluricanonical ring for a 3-fold of general type, Am. J. Math., Volume 106 (1984) no. 6, pp. 1503-1512 | DOI | MR | Zbl

[13] Kawamata, Yujiro On a relative version of Fujita’s freeness conjecture, Complex geometry (Göttingen, 2000), Springer, 2002, pp. 135-146 | DOI | MR | Zbl

[14] Kollár, János Higher direct images of dualizing sheaves. I, Ann. Math., Volume 123 (1986) no. 1, pp. 11-42 | DOI | MR | Zbl

[15] Kollár, János Shafarevich maps and automorphic forms, Princeton University Press, 1995, x+201 pages | DOI | Zbl

[16] Kollár, János Singularities of pairs, Algebraic geometry (Santa Cruz, 1995) (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society, 1997, pp. 221-287 | DOI | MR | Zbl

[17] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013 (in collaboration with Sandor Kovács) | DOI | MR | Zbl

[18] Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004, xviii+387 pages | DOI | Zbl

[19] Popa, Mihnea; Schnell, Christian On direct images of pluricanonical bundles, Algebra Number Theory, Volume 8 (2014) no. 9, pp. 2273-2295 | DOI | MR | Zbl

[20] Shokurov, Vyacheslav Vladimirovich A nonvanishing theorem, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 49 (1985) no. 3, pp. 635-651 | MR

[21] Viehweg, Eckart Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) (Advanced Studies in Pure Mathematics), Volume 1, North-Holland, 1983, pp. 329-353 | DOI | MR | Zbl

Cité par Sources :