In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

中文翻译：

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关于具有伪有效切线束的投影流形

在本文中，我们开发了向量丛上的奇异 Hermitian 度量理论。作为一个应用，我们给出了一个射影流形X的结构定理，它具有赝有效切丛；X允许平滑纤维化 $X \to Y$ 到一个平面射影流形Y，使得它的一般纤维是有理连接的。此外，通过应用这个结构定理，我们用伪有效切丛对所有最小曲面进行分类，并研究一般非最小曲面，这些曲面提供了（可能是奇异的）正弯曲切丛的示例。