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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
MathSciNet review: 1401760
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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.


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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