Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-27T08:13:00.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple normal crossing Fano varieties and log Fano manifolds

Published online by Cambridge University Press:  11 January 2016

Kento Fujita
Kento Fujita*
Affiliation:
Research Institute for Mathematical Sciences Kyoto UniversityKyoto [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

Keywords

14J4514E30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 214 , June 2014 , pp. 95 - 123
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[A] Ambro, F., Quasi-log varieties (in Russian), Tr. Mat. Inst. Steklova 240 2003, 220–239; English translation in Proc. Steklov Inst. Math. 240 2003, 214–233. MR 1993751.Google Scholar
[ABW] Andreatta, M., Ballico, E., and Wisniewski, J. A., Two theorems on elementary contractions, Math. Ann. 297 1993, 191–198. MR 1241801. DOI 10.1007/ BF01459496.CrossRefGoogle Scholar
[AW] Andreatta, M. and Wisniewski, J. A., A note on nonvanishing and applications, Duke. Math. J. 72 1993, 739–755. MR 1253623. DOI 10.1215/ S0012–7094-93–07228-6.Google Scholar
[CMS] Cho, K., Miyaoka, Y., and Shepherd-Barron, N. I., “Characterizations of projective spaces and applications to complex symplectic manifolds” in Higher Dimensional Birational Geometry (Kyoto, 1997), Adv. Stud. Pure Math. 35, Math. Soc. Japan, Tokyo, 2002, 1–88. MR 1929792.Google Scholar
[F] Fujino, O., Introduction to the log minimal model program for log canonical pairs, preprint, arXiv:0907.1506v1 [math.AG]Google Scholar
[Ft1] 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. MR 0946238.Google Scholar
[Ft2] Fujita, T., Classification Theories of Polarized Varieties, London Math. Soc. Lec ture Note Ser. 155, Cambridge University Press, Cambridge, 1990. MR 1162108. DOI 10.1017/CBO9780511662638.Google Scholar
[GM] Goresky, M. and MacPherson, R., Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3) 14, Springer, Berlin, 1988. MR 0932724.CrossRefGoogle Scholar
[I] Iskovskikh, V. A., Fano threefolds, I, Izv. Ross. Akad. Nauk Ser. Mat. 41 1977, 516–562, 717; II, 42 1978, 506–549. MR 0463151.Google Scholar
[KO] Kobayashi, S. and Ochiai, T., Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 1973, 31–47. MR 0316745.Google Scholar
[Kol1] Kollär, J., “Adjunction and discrepancies” in Flips and Abundance for Algebraic Threefolds (Salt Lake City, 1991), Astérisque 211, Soc. Math. France, Paris, 1992, 183–192. MR 1225842.Google Scholar
[Kol2] Kollär, J., Singularities of the Minimal Model Program, Cambridge Tracts in Math. 200, Cambridge University Press, Cambridge, 2013. MR 3057950.Google Scholar
[Kol3] Kollär, J., New examples of terminal and log canonical singularities, preprint, arXiv:1107.2864v1 [math.AG]Google Scholar
[KolM] Kolläar, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge, 1998. MR 1658959. DOI 10.1017/CBO9780511662560.Google Scholar
[M] Maeda, H., Classification of logarithmic Fano threefolds, Compos. Math. 57 1986, 81–125. MR 0817298.Google Scholar
[Mat] Matsumura, H., Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge, 1986. MR 0879273.Google Scholar
[MMu] Mori, S. and Mukai, S., Classification of Fano 3-folds with B2 ≥ 2, Manuscripta Math. 36 (1981/82), 147–162; Erratum, Manuscripta Math. 110 2003, 407. MR 0641971. DOI 10.1007/BF01170131.Google Scholar
[Mu1] Mukai, S., “Problems on characterization of the complex projective space” in Birational Geometry of Algebraic Varieties, Open Problems (Katata, 1988), Taniguchi Foundation, Katata, 1988, 57–60.Google Scholar
[Mu2] Mukai, S., Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 1989, 3000–3002. MR 0995400. DOI 10.1073/ pnas.86.9.3000.Google Scholar
[NO] Novelli, C. and Occhetta, G., Rational curves and bounds on the Picard number of Fano manifolds, Geom. Dedicata 147 2010, 207–217. MR 2660578. DOI 10. 1007/s10711–009-9452–4.Google Scholar
[W1] Wiśniewski, J., Fano 4-folds of index 2 with b2 ≥ 2: A contribution to Mukai classification, Bull. Pol. Acad. Sci. Math. 38 1990, 173–184. MR 1194261.Google Scholar
[W2] Wiśniewski, J., On a conjecture of Mukai, Manuscripta Math. 68 1990, 135–141. MR 1063222. DOI 10.1007/BF02568756.Google Scholar
[W3] Wiśniewski, J., On contractions of extremal rays of Fano manifolds, J. Reine Angew.Math. 417 1991, 141–157. MR 1103910. DOI 10.1515/crll.1991.417.141.Google Scholar
[W4] Wiśniewski, J., On Fano manifolds of large index, Manuscripta Math. 70 1991, 145–152. MR 1085628. DOI 10.1007/BF02568366.Google Scholar