The Mukai conjecture for log Fano manifolds

• Autorzy: Kento Fujita
• Języki publikacji: EN

Abstrakty

EN

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D 1) of Picard groups is injective for any irreducible component D 1 ⊂ D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).

Słowa kluczowe

EN

Fano manifold; Mukai conjecture; Log Fano manifold; Mori dream space; Simple normal crossing Fano variety

Kategorie tematyczne

• 14E30: Minimal model program (Mori theory, extremal rays)
• 14J45: Fano varieties

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 1
• Strony: 14-27

Daty

wydano

2014-01-01

online

2013-10-30

Twórcy

autor

Kento Fujita

• Kyoto University

Bibliografia

• [1] Andreatta M., Chierici E., Occhetta G., Generalized Mukai conjecture for special Fano varieties, Cent. Eur. J. Math., 2004, 2(2), 272–293 http://dx.doi.org/10.2478/BF02476544
• [2] Birkar C., Cascini P., Hacon C.D., McKernan J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 2010, 23(2), 405–468 http://dx.doi.org/10.1090/S0894-0347-09-00649-3
• [3] Bonavero L., Casagrande C., Debarre O., Druel S., Sur une conjecture de Mukai, Comment. Math. Helv., 2003, 78(3), 601–626 http://dx.doi.org/10.1007/s00014-003-0765-x
• [4] Casagrande C., On Fano manifolds with a birational contraction sending a divisor to a curve, Michigan Math. J., 2009, 58(3), 783–805 http://dx.doi.org/10.1307/mmj/1260475701
• [5] Casagrande C., On the Picard number of divisors in Fano manifolds, Ann. Sci. Éc. Norm. Supér., 2012, 45(3), 363–403
• [6] Fujino O., Introduction to the log minimal model program for log canonical pairs, preprint available at http://arxiv.org/abs/0907.1506/
• [7] Fujita K., Simple normal crossing Fano varieties and log Fano manifolds, preprint available at http://arxiv.org/abs/1206.1994/
• [8] Fujita T., On polarized manifolds whose adjoint bundles are not semipositive, In: Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, 167–178
• [9] Hu Y., Keel S., Mori dream spaces and GIT, Michigan Math. J., 2000, 48, 331–348 http://dx.doi.org/10.1307/mmj/1030132722
• [10] Ishii S., Quasi-Gorenstein Fano 3-folds with isolated nonrational loci, Compositio Math., 1981, 77(3), 335–341
• [11] Kollár J., Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb., 32, Springer, Berlin, 1996 http://dx.doi.org/10.1007/978-3-662-03276-3
• [12] Kollár J., Miyaoka Y., Mori S., Rational connectedness and boundedness of Fano manifolds, J. Differential Geom., 1992, 36(3), 765–779
• [13] Kollár J., Mori S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998 http://dx.doi.org/10.1017/CBO9780511662560
• [14] Maeda H., Classification of logarithmic Fano threefolds, Compositio Math., 1986, 57(1), 81–125
• [15] Mori S., Mukai S., Classification of Fano 3-folds with B 2 ≥ 2, Manuscr. Math., 1981, 36(2), 147–162, Erratum: Manuscr. Math., 2003, 110(3), 407 http://dx.doi.org/10.1007/BF01170131
• [16] Mukai S., Problems on characterization of the complex projective space, In: Birational Geometry of Algebraic Varieties, Open Problems, Katata, August 22–27, 1988, 57–60
• [17] Novelli C., Occhetta G., Rational curves and bounds on the Picard number of Fano manifolds, Geom. Dedicata, 2010, 147, 207–217 http://dx.doi.org/10.1007/s10711-009-9452-4
• [18] Wiśniewski J.A., On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math., 1991, 417, 141–157

Identyfikatory

DOI

10.2478/s11533-013-0326-5