Abstract
Let \(f:X\rightarrow Y\) be an algebraic fibre space between normal projective varieties and F be a general fibre of f. We prove an Iitaka-type inequality \(\kappa (X,-K_X)\le \kappa (F,-K_F)+\kappa (Y,-K_Y)\) under some mild conditions. We also obtain results relating the positivity of \(-K_X\) and \(-K_Y\).
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03 May 2024
A Correction to this paper has been published: https://doi.org/10.1007/s00208-024-02846-4
Notes
In this case, since the pullback of \(\mathbb {Q}\)-Weil divisors is well-defined, we do not need to assume \(-K_Y\) is \(\mathbb {Q}\)-Cartier.
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Acknowledgements
I would like to thank my advisor Jungkai Alfred Chen for his valuable advice and encouragement. This work can not be done without his consistent support and enlightening suggestions. I want to thank Yoshinori Gongyo for his inspiring classes during his visit to National Taiwan University in 2019, and for his help and detailed explanations of his works. I thank Meng Chen and Qi Zhang for answering my questions. I would like to thank David Wen for his helpful comments about writing techniques. Next, I want to thank Yoshinori Gongyo and Sho Ejiri for reading my preprint, this article was inspired by their work in [14]. Also, I want to thank the reviewer gives many fruitful comments on this paper. I would also like to thank the National Taiwan University, National Center of Theoretical Sciences, and the Ministry of Science and Technology of Taiwan for their generous support. Finally, I want to thank Ching-Jui Lai, Jheng-Jie Chen, Chih-Wei Chang, Hsin-Ku Chen, Hsueh-Yung Lin, Bin Nguyen, Iacopo Brivio, and Shi-Xin Wang for their helpful suggestions and encouragements.
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The author is supported by the National Taiwan University and the Ministry of Science and Technology of Taiwan.
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Chang, CK. Positivity of anticanonical divisors in algebraic fibre spaces. Math. Ann. 385, 787–809 (2023). https://doi.org/10.1007/s00208-021-02282-8
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DOI: https://doi.org/10.1007/s00208-021-02282-8