zbMATH — the first resource for mathematics

Manifolds polarized by vector bundles. (English) Zbl 1222.14091
Summary: Let \(X\) be a complex projective manifold of dimension \(n\) and let \(\mathcal E\) be an ample vector bundle of rank \(r\). Let also \(\tau = \tau(X,\mathcal E) = \min \{t\in\mathbb{R}: K_X + t \det \mathcal E \text{ is nef}\}\) be the nef value of the pair \((X,\mathcal E)\). In this paper we classify the pairs \((X,\mathcal E)\) such that \(\tau(X,\mathcal E)\geq \frac{n-2}{r}\).

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J40 \(n\)-folds (\(n>4\))
14E30 Minimal model program (Mori theory, extremal rays)
Full Text: DOI
[1] Andreatta, M.; Ballico, E.; Wiśniewski, J. A., Vector bundles and adjunction, Int. J. Math., 3, 331-340 (1992) · Zbl 0770.14008
[2] Andreatta, M.; Mella, M., Contractions on a manifold polarized by an ample vector bundle, Trans. AMS, 349, 4669-4683 (1997) · Zbl 0885.14004
[3] Andreatta, M.; Occhetta, G., Special rays in the Mori cone of a projective variety, Nagoya Math. J., 168, 127-137 (2002) · Zbl 1055.14015
[4] Andreatta, M.; Wiśniewski, J. A., On contractions of smooth varieties, J. Algebraic Geometry, 7, 253-312 (1998) · Zbl 0966.14012
[5] Andreatta, M.; Wiśniewski, J. A., On manifolds whose tangent bundle contains an ample subbundle, Invent. Math., 146, 209-217 (2001) · Zbl 1081.14060
[6] Beltrametti, M.C., Sommese, A.J.: The Adjunction Theory of Complex Projective Varieties. Exp. Math., vol. 16. de Gruyter, Berlin (1995) · Zbl 0845.14003
[7] Cho, K., Miyaoka, Y., Shepherd Barron, N.I.: Characterization of projective space and applications to complex symplectic manifolds. Higher dimensional birational geometry, Kyoto. Adv. Stud. Pure Math. 35, 1-88 (1997); Math. Soc. Japan, Tokyo (2002) · Zbl 1063.14065
[8] Fujita, T.: On polarized manifolds whose adjoint bundles are not semipositive. Algebraic Geometry, Senday, 1985. Adv. Stud. Pure Math. 10, 167-178. North-Holland, Amsterdam (1987)
[9] Fujita, T.: On adjoint bundle of ample vector bundles. In: Proceedings of the Conference in Alg. Geometry, Bayreuth (1990)
[10] Fujita, T., On Kodaira energy and reduction of polarized manifolds, Manuscripta Math., 76, 59-84 (1992) · Zbl 0766.14027
[11] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the Minimal Model Problem. Algebraic Geometry, Sendai, 1985. Adv. Stud. Pure Math. 10, 283-360. North-Holland, Amsterdam (1987) · Zbl 0672.14006
[12] Maeda, H., Nefness of adjoint bundles for ample vector bundles, Le Matematiche (Catania), 50, 73-82 (1995) · Zbl 0865.14024
[13] Mori, S., Projective manifolds with ample tangent bundle, Ann. Math., 110, 593-606 (1979) · Zbl 0423.14006
[14] Ohno, M., On nef values of determinants of ample vector bundles, Free resolutions of coordinate rings of projective varieties and related topics, Kyoto, 1998. Surikaisekikenkyusho Kokyuroku, 1078, 75-85 (1999) · Zbl 0954.14006
[15] Ohno, M.: Classification of generalized polarized manifolds by their nef values. e-print ArXiv Math. AG/0503119 · Zbl 1138.14026
[16] Peternell, T., A characterization of \[\mathbb P^n\] by vector bundles, Math. Z., 205, 487-490 (1990) · Zbl 0726.14034
[17] Peternell, T., Ample vector bundles on Fano manifolds, Int. J. Math., 2, 311-322 (1991) · Zbl 0744.14009
[18] Peternell, T.; Szurek, M.; Wiśniewski, J. A., Fano manifolds and vector bundles, Math. Ann., 294, 151-165 (1992) · Zbl 0786.14027
[19] Wiśniewski, J. A., On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math., 417, 141-157 (1991) · Zbl 0721.14023
[20] Wiśniewski, J.A.: A report on Fano manifolds of middle index and \(b_2\geq 2\). Projective geometry with applications, pp. 19-26. Lecture Notes in Pure and Applied Mathematics, vol. 166. Dekker, New York (1994) · Zbl 0839.14032
[21] Ye, Y. G.; Zhang, Q., On ample vector bundle whose adjunction bundles are not numerically effective, Duke Math. J., 60, 671-687 (1990) · Zbl 0709.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.