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A partial converse to the Andreotti–Grauert theorem

Part of: Global differential geometry (Co)homology theory Holomorphic fiber spaces

Published online by Cambridge University Press:  19 November 2018

Xiaokui Yang
Xiaokui Yang*
Affiliation:
Morningside Center of Mathematics, HCMS, CEMS, NCNIS, HLM, UCAS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China email xkyang@amss.ac.cn
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Abstract

Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $\unicode[STIX]{x1D714}$ on $X$ such that its Ricci curvature $\text{Ric}(\unicode[STIX]{x1D714})$ has at least one positive eigenvalue everywhere.

Keywords

vanishing theorem(n - 1)-ample(n - 1)-positivepseudoeffectiveuniruled manifolds

MSC classification

Primary: 14F17: Vanishing theorems 32L20: Vanishing theorems 53C55: Hermitian and Kählerian manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 1 , January 2019 , pp. 89 - 99
Copyright
© The Author 2018 

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