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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Images of manifolds with semi-ample anti-canonical divisor

Authors: Caucher Birkar and Yifei Chen
Journal: J. Algebraic Geom. 25 (2016), 273-287
Published electronically: August 27, 2015
MathSciNet review: 3466352
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Abstract | References | Additional Information


We prove that if $f\colon X\to Z$ is a smooth surjective morphism between projective manifolds and if $-K_X$ is semi-ample, then $-K_Z$ is also semi-ample. This was conjectured by Fujino and Gongyo. We list several counterexamples to show that this fails without the smoothness assumption on $f$.

We prove the above result by proving some results concerning the moduli divisor of the canonical bundle formula associated to a klt-trivial fibration $(X,B)\to Z$.

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Additional Information

Caucher Birkar
Affiliation: Department of Pure Mathematics and Mathematical Statistics (DPMMS), Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom

Yifei Chen
Affiliation: Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, Chinese Academy of Sciences, No. 55 Zhonguancun East Road, Haidian District, Beijing, 100190, People’s Republic of China

Received by editor(s): April 14, 2013
Received by editor(s) in revised form: February 24, 2014
Published electronically: August 27, 2015
Article copyright: © Copyright 2015 University Press, Inc.