AI 中文总结
本文证明有理连通射影流形切丛存在一致RC正度量，回答Yang的问题并刻画有理连通性；同时证明该性质在爆破下保持，并在非Kähler情形下对Hopf和Kato曲面构造度量并给出分类。
AI 中文摘要
本文证明了每个有理连通射影流形在其切丛上允许一个光滑的一致RC正厄米度量，回答了Yang的问题，并给出了用一致RC正性刻画有理连通性的特征。我们找到一个例子表明仅凭RC正性不能刻画有理连通性。我们还证明了在紧复流形上沿连通光滑中心作爆破时，切丛的一致RC正性得以保持。在非Kähler情形下，我们在所有Hopf曲面和Kato曲面上构造了这样的度量，并获得了紧复曲面的分类结果。
英文摘要
In this paper, we prove that every rationally connected projective manifold admits a smooth uniformly RC-positive Hermitian metric on its tangent bundle, answering Yang's question and giving a characterization of rational connectedness by uniform RC-positivity. We find an example to show that RC-positivity alone does not characterize rational connectedness. We also prove that uniform RC-positivity of the tangent bundle is preserved under blow-ups along connected smooth centers on compact complex manifolds. In the non-Kähler setting, we construct such metrics on all Hopf and Kato surfaces and obtain classification results for compact complex surfaces.
