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RC-positive metrics on rationally connected manifolds

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

Published online by Cambridge University Press:  16 November 2020

Xiaokui Yang
Xiaokui Yang*
Affiliation:
Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing100084, China; E-mail: xkyang@mail.tsinghua.edu.cn

Abstract

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In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric $\omega $ such that $(T_X,\omega )$ is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on $T_X$ .

Keywords

rationally connectedRC-positivityvanishing theorem

MSC classification

Primary: 53C55: Hermitian and Kählerian manifolds
Secondary: 14E08: Rationality questions 14F17: Vanishing theorems 32L20: Vanishing theorems
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e53
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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