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Accueil > Agenda > Colloques et rencontres > Archives > Agenda 2010 > Mini-cours de Florin Ambro : The Canonical Bundle Formula

Mini-cours de Florin Ambro : The Canonical Bundle Formula

IRMA, les 5, 12 et 19 octobre 2010

Florin Ambro, professeur à Bucarest est invité à l’IRMA et septembre et octobre 2010. Il donnera un cours destiné à un public assez large d’étudiants et chercheurs.

Le cours aura lieu :

les mardi 5, 12 et 19 octobre 2010

de 14h-15h et 15h30-16h30

dans la salle de conférences de l’IRMA.

Contenu :

Given a fibration of a surface by elliptic curves, the classical canonical bundle formula [Kod] relates the canonical divisors of the total space and the base.
Besides the canonical divisors, the formula involves two divisors on the base : the discriminant divisor, measuring the singularities of the special fibers, and the moduli divisor, measuring the variation of the generic fiber in its moduli space. One application of this formula is to establish Castelnuovo’s criterion that a surface which is not uniruled has the 12th plurigenera non-zero.

Extending this formula to higher dimensions, for example when the fiber is a Calabi-Yau variety, or the base is a surface, is a fundamental open problem, with many applications to the classification of algebraic varieties. In dimension 3, it is due to Fujita [Fuj], and partial results in higher dimension have been obtained in [Mo], [Ka], [FM], [Ko] and [A1, A2]. As for the applications, let us mention the subadjunction for log canonical centers known (see [Ka]) and the recent study of pluricanonicanical maps (see [VZ], [P], [TX]). Moreover a weak form of the canonical bundle formula was used in the recent proof of the finite generation of the canonical ring, see [BCHM].

The aim of this mini-course is to introduce the canonical bundle formula, and to discuss its higher dimensional extensions.

Outline :

- 1) Classical canonical bundle formula. Extension to base of higher dimension.
- 2) General case : discriminant and moduli (bi)divisors. Conjectures, known results.
- 3) Some applications : adjunction to log canonical centers, classification of algebraic varieties.
- 4) Extensions : case when the fiber has non-negative Kodaira dimension.
- 5) Methods : variation of Hodge structures, cyclic covers, GIT, L^2-techniques.

Prerequisites :

Besides standard notions of Algebraic Geometry, some acquaintance with log varieties and bi-divisors would help. Before the talks, I will provide some introductory notes.

References :

- [A1] Ambro, F., The Adjunction Conjecture and its applications,
Phd Thesis Johns Hopkins Univ. (1999), preprint math.AG/9903060.

- [A2] Ambro, F., Shokurov’s boundary property,
J. Differential Geometry 67(2) (2004), 229-255

- [BCHM] C. Birkar, P. Cascini, C. Hacon, J. McKernan,
Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405-468.

- [FM] Fujino, O., Mori, S., A canonical bundle formula,
J. Differential Geom. 56 (2000), no. 1, 167—188.

- [Fuj] Fujita, T., Zariski decomposition and canonical rings of elliptic threefolds,
J. Math. Soc. Japan 38 (1986), no. 1, 19—37.

- [Kod] Kodaira, K., On the structure of compact complex analytic surfaces, I,
Amer. J. Math. 86 (1964), 751—798.

- [Ka] Kawamata, Y., Subadjunction of log canonical divisors, II,
Amer. J. Math. 120 (1998), 893—899.

- [Ko] Kollár, J., Kodaira’s canonical bundle formula and adjunction. Flips for 3-folds and 4-folds, 134—162, Oxford Lecture Ser. Math. Appl., 35, Oxford Univ. Press, Oxford, 2007.

- [Mo] Mori, S., Classification of higher-dimensional varieties.
Algebraic geometry, Bowdoin, 1985, Proc. Sympos. Pure Math. 46, Part 1, AMS, 269—331.

- [P] Pacienza, G. On the uniformity of the Iitaka fibration. Math. Res. Lett. 16, No. 4, 663-681 (2009).

- [TX] Todorov, G. ; Xu, Ch. Effectiveness of the log Iitaka fibration for 3-folds and 4-folds.
Algebra Number Theory 3, No. 6, 697-710 (2009).

- [VZ] Viehweg, E. ; Zhang, D.-Q., Effective Iitaka fibrations. J. Algebr. Geom. 18, No. 4, 711-730 (2009).

Dernière mise à jour le 5-10-2010