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Second Chern class of Fano manifolds and anti-canonical geometry

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Abstract

Let X be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of X in terms of its index and degree. As an application, if Y is a n-dimensional Fano manifold with \(-K_Y=(n-3)H\) for some ample divisor H, we prove that \(h^0(Y,H)\ge n-2\). Moreover, we show that the rational map defined by \(\vert mH\vert \) is birational for \(m\ge 5\), and the linear system \(\vert mH\vert \) is basepoint free for \(m\ge 7\). As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most 4 are also investigated.

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Acknowledgements

I heartly thank my advisor Andreas Höring for valuable discussions, suggestions and for his careful proofreading of the rather awkward draft of this paper. I want to thank Chen Jiang and Stéphane Druel for helpful communications and comments. I also would like to thank the anonymous reviewer for pointing out several inacuracies in previous versions and for his valuable comments and suggestions to improve the explanation of the paper. This paper was written while I stayed in Institut de Recherche Mathématique de Rennes (IRMAR) and I would like to thank the institution for the hospitality and support.

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  1. Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, Université de Côte d’Azur, 06108, Nice Cedex 02, France

    Jie Liu

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  1. Jie Liu
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Communicated by Ngaiming Mok.

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Liu, J. Second Chern class of Fano manifolds and anti-canonical geometry. Math. Ann. 375, 655–669 (2019). https://doi.org/10.1007/s00208-018-1702-z

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