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

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Abstract | References | Similar Articles | Additional Information

Abstract: A complex manifold of dimension together with an ample vector bundle on it will be called a **generalized polarized variety**. The adjoint bundle of the pair is the line bundle . We study the positivity (the nefness or ampleness) of the adjoint bundle in the case . If this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map from onto a normal projective variety with connected fiber and such that , for some ample line bundle on . We describe those contractions for which . We extend this result to the case in which has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which for every fiber and 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