Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Contractions on a manifold polarized by
an ample vector bundle


Authors: Marco Andreatta and Massimiliano Mella
Journal: Trans. Amer. Math. Soc. 349 (1997), 4669-4683
MSC (1991): Primary 14E30, 14J40; Secondary 14C20, 14J45
DOI: https://doi.org/10.1090/S0002-9947-97-01832-1
MathSciNet review: 1401760
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a generalized polarized variety. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If $K_X+detE$ is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which $dimF = r$ for every fiber and $\pi$ is birational.


References [Enhancements On Off] (What's this?)

  • [An] Andreatta, M., Contractions of Gorenstein polarized varieties with high nef value, Math. Ann. 300 (1994), 669-679. MR 96b:14007
  • [ABW1] Andreatta, M., Ballico, E., Wi\'{s}niewski, J.A., On contractions of smooth algebraic varieties, preprint UTM 344 (1991).
  • [ABW2] -, Vector bundles and adjunction, International Journal of Mathematics,3 (1992), 331-340. MR 93h:14031
  • [AW] Andreatta, M., Wi\'{s}niewski, J. A., A note on nonvanishing and applications, Duke Math.J. 72 (1993), 739-755. MR 95c:14007
  • [BS] Beltrametti, M., Sommese, A. J. On the adjunction theoretic classification of polarized varieties, J. Reine Angew. Math. 427 (1992), 157-192. MR 93d:14012
  • [Fu1] Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive, in Algebraic Geometry, Sendai, Adv. Studies in Pure Math. 10, Kinokuniya-North-Holland 1987, 167-178. MR 89d:14006
  • [Fu2] -, On adjoint bundles of ample vector bundles, in Proc. Alg. Geom. Conf. Bayreuth (1990), Lect. Notes Math., 1507, 105-112. MR 93j:14052
  • [Fu3] -On Kodaira energy and reduction of polarized manifolds, Manuscr. Math 76 (1992), 59-84. MR 93i:14032
  • [Io] Ionescu, P. Generalized adjunction and applications, Math. Proc. Camb. Phil. Soc. 99 (1986), 457-472. MR 87e:14031
  • [KMM] Kawamata, Y., Matsuda, K., Matsuki, K., Introduction to the minimal model program, in Algebraic Geometry, Sendai, Adv. Studies in Pure Math. 10, Kinokuniya-North-Holland 1987, 283-360. MR 89e:14015
  • [Ma] Maeda, H. Nefness of adjoint bundles for ample vector bundles, Le Matematiche (Catania) 50 (1995), 73-82. CMP 96:08
  • [Mo] Mori, S. Projective manifolds with ample tangent bundles, Ann. of Math. 110 (1979), 593-606. MR 81j:14010
  • [Mo1] -, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176. MR 84e:14032
  • [Mu] Mukai, S. Problems on characterizations of complex projective space, Birational Geometry of Algebraic Varieties - Open Problems, Katata, Japan (1988), 57-60.
  • [OSS] Okonek, C., Schneider, M., Spindler, H. Vector bundles on complex projective spaces, Birkhäuser, 1980. MR 81b:14001
  • [PSW] Peternell, T., Szurek, M., Wi\'{s}niewski, J. A., Fano manifolds and vector bundles, Math.Ann. 294 (1992), 151-165. MR 93h:14030
  • [So] Sommese, A. J., On the adjunction theoretic structure of projective varieties, Complex Analysis and Algebraic Geometry, Proceedings Göttingen, 1985 (ed. H. Grauert), Lecture Notes in Math., 1194 (1986), 175-213. MR 87m:14049
  • [Wi1] Wi\'{s}niewski, J. A., On a conjecture of Mukai, Manuscr. Math. 68 (1990), 135-141. MR 91f:14040
  • [Wi2] -, Length of extremal rays and generalized adjunction, Math. Z. 200 (1989), 409-427. MR 91e:14032
  • [Wi3] -, On contraction of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141-157. MR 92d:14032
  • [YZ] Ye, Y. G., Zhang, Q., On ample vector bundles whose adjunction bundles are not numerically effective, Duke Math. Journal, 60 (1990), 671-687. MR 91g:14040
  • [Zh] Zhang, Q., A theorem on the adjoint system for vector bundles, Manuscripta Math. 70 (1991), 189-201. MR 92c:14009
  • [Zh2] -, Ample vector bundles on singular varieties, Math. Zeit. 220 (1995), 59-64. MR 96i:14034

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14E30, 14J40, 14C20, 14J45

Retrieve articles in all journals with MSC (1991): 14E30, 14J40, 14C20, 14J45


Additional Information

Marco Andreatta
Affiliation: Dipartimento di Matematica,Universitá di Trento, 38050 Povo (TN), Italia
Email: andreatt@science.unitn.it

Massimiliano Mella
Affiliation: Dipartimento di Matematica,Universitá di Trento, 38050 Povo (TN), Italia
Email: mella@science.unitn.it

DOI: https://doi.org/10.1090/S0002-9947-97-01832-1
Keywords: Vector bundle, contraction, extremal ray
Received by editor(s): March 11, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society