Vector bundles on log terminal varieties

Mella, Massimiliano

doi:10.1090/S0002-9939-98-04596-1

Vector bundles on log terminal varieties
HTML articles powered by AMS MathViewer

by Massimiliano Mella PDF

Proc. Amer. Math. Soc. 126 (1998), 2199-2204 Request permission

Abstract:

Let $X$ be an $n$-dimensional variety and $E$ an ample vector bundle on $X$ of rank $e$. We give a complete classification of pairs $(X,E)$, with $X$ log terminal and $e\geq n$ such that $K_X+det E$ is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.

References

Some remarks on the study of good contractions

87

MR 96i:14017

M. Andreatta, E. Ballico, and J. Wiśniewski, Vector bundles and adjunction, Internat. J. Math. 3 (1992), no. 3, 331–340. MR 1163727, DOI 10.1142/S0129167X92000114
Andreatta,M. Mella,M. Contractions on a manifold polarized by ample vector bundles, Trans. A.M.S. 349 (1997).
M. Andreatta and J. A. Wiśniewski, A note on nonvanishing and applications, Duke Math. J. 72 (1993), no. 3, 739–755. MR 1253623, DOI 10.1215/S0012-7094-93-07228-6
Mauro C. Beltrametti and Andrew J. Sommese, On the adjunction-theoretic classification of polarized varieties, J. Reine Angew. Math. 427 (1992), 157–192. MR 1162435, DOI 10.1515/crll.1992.427.157
Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 167–178. MR 946238, DOI 10.2969/aspm/01010167
Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR 1162108, DOI 10.1017/CBO9780511662638
Takao Fujita, On adjoint bundles of ample vector bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 105–112. MR 1178722, DOI 10.1007/BFb0094513
Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
Hidetoshi Maeda, Nefness of adjoint bundles for ample vector bundles, Matematiche (Catania) 50 (1995), no. 1, 73–82. MR 1373571
Mella, M. Adjunction theory on terminal variety, Proceeding “Complex Analysis and Geometry" editors V.Ancona, E.Ballico, A.Silva, Pitman Research Notes in Math. 366 (1997), 153-164.
Mukai,S. Problems on characterizations of complex projective space, Birational Geometry of Algebraic Varieties - Open Problems, Katata Japan (1988), 57-60.
Thomas Peternell, Ample vector bundles on Fano manifolds, Internat. J. Math. 2 (1991), no. 3, 311–322. MR 1104121, DOI 10.1142/S0129167X91000181
Thomas Peternell, MichałSzurek, and Jarosław A. Wiśniewski, Fano manifolds and vector bundles, Math. Ann. 294 (1992), no. 1, 151–165. MR 1180456, DOI 10.1007/BF01934319
Jarosław A. Wiśniewski, 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
Yun-Gang Ye and Qi Zhang, On ample vector bundles whose adjunction bundles are not numerically effective, Duke Math. J. 60 (1990), no. 3, 671–687. MR 1054530, DOI 10.1215/S0012-7094-90-06027-2
Qi Zhang, A theorem on the adjoint system for vector bundles, Manuscripta Math. 70 (1991), no. 2, 189–201. MR 1085632, DOI 10.1007/BF02568370
Qi Zhang, Ample vector bundles on singular varieties, Math. Z. 220 (1995), no. 1, 59–64. MR 1347157, DOI 10.1007/BF02572602
—, Ample vector bundles on singular varieties II, Math. Ann. 307 (1997), 505–509.