Bounds on the denominators in the canonical bundle formula  [ Bornes sur les dénominateurs dans la formule du fibré canonique ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, p. 1951-1969
Dans cet article on considère la partie modulaire dans la formule du fibré canonique pour une fibration lc-triviale dont la fibre générique est une courbe rationnelle. Soit r l’indice de Cartier de la fibre. Il avait été conjecturé que 12r est une borne sur les dénominateurs de la partie modulaire. Nous démontrons qu’une telle borne ne peut même pas être polynômiale en r, nous calculons une borne N(r) et nous fournissons un exemple où la borne optimale sur les dénominateurs est N(r)/r. De plus nous montrons que même localement les dénominateurs dépendent quadratiquement de r.
In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If r is the Cartier index of the fibre, it was expected that 12r would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in r, we provide a bound N(r) and an example where the smallest integer that clears the denominators of the moduli part is N(r)/r. Moreover we prove that even locally the denominators depend quadratically on r.
DOI : https://doi.org/10.5802/aif.2819
Classification:  14J10 14J26
Mots clés: fibration lc-triviale, partie modulaire, dénominateurs
@article{AIF_2013__63_5_1951_0,
     author = {Floris, Enrica},
     title = {Bounds on the denominators in the canonical bundle formula},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     pages = {1951-1969},
     doi = {10.5802/aif.2819},
     mrnumber = {3186513},
     zbl = {1295.14034},
     language = {en},
     url = {http://http://www.numdam.org/item/AIF_2013__63_5_1951_0}
}
Floris, Enrica. Bounds on the denominators in the canonical bundle formula. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1951-1969. doi : 10.5802/aif.2819. http://www.numdam.org/item/AIF_2013__63_5_1951_0/

[1] Ambro, F. The Adjunction Conjecture and its applications, The Johns Hopkins University (1999) (Ph. D. Thesis) | MR 2698988

[2] Ambro, F. Shokurov’s boundary property, J. Differential Geom., Tome 67 (2004), pp. 229-255 | MR 2153078 | Zbl 1097.14029

[3] Barth, W.; Peters, C.; De Ven, A. Van Compact Complex Surfaces, Springer Verlag (1984) | MR 749574 | Zbl 1036.14016

[4] Corti, A. Flips for 3-folds and 4-folds, Oxford University Press, Oxford Lecture Series in Mathematics and Its Applications, Tome 35 (2007) | MR 2352762

[5] Fujino, O.; Mori, S. A canonical bundle formula, J. Differential Geom., Tome 56 (2000), pp. 167-188 | MR 1863025 | Zbl 1032.14014

[6] Jiang, X. On the pluricanonical maps of varieties of intermediate Kodaira dimension, arXiv:1012.3817 (2012), pp. 1-21 | MR 3063904

[7] Kawamata, Y. Subadjunction of log canonical divisors for a variety of codimension 2, Contemporary Mathematics, Tome 207 (1997), pp. 79-88 | Article | MR 1462926 | Zbl 0901.14004

[8] Kawamata, Y. Subadjunction of log canonical divisors, II, Amer. J. Math., Tome 120 (1998), pp. 893-899 | Article | MR 1646046 | Zbl 0919.14003

[9] Kollár, J.; Mori, S. Birational Geometry of Algebraic Varieties, Cambridge University Press, Cambridge, Cambridge Tracts in Math, Tome 134 (1998) | MR 1658959 | Zbl 0926.14003

[10] Prokhorov, Yu. G.; Shokurov, V. V. Towards the second theorem on complements, J. Algebraic Geom., Tome 18 (2009), pp. 151-199 | Article | MR 2448282 | Zbl 1159.14020

[11] Todorov, G. T. Effective log Iitaka fibrations for surfaces and threefolds, Manuscripta Math., Tome 133 (2010), pp. 183-195 | Article | MR 2672545 | Zbl 1200.14032