AI 中文总结
本文证明非平坦Ricci-flat Kähler复曲面的全纯切丛具有RC正Hermite度量，并揭示RC正性在张量运算下不保持，且弱于一致RC正性，不蕴含有理连通性。
AI 中文摘要
给定一个具有非平坦Ricci-flat Kähler度量的紧致复曲面，我们证明其全纯切丛允许一个RC正Hermite度量。证明依赖于通过Weyl算子的反自对偶部分对RC正性的刻画、四维Einstein流形上的Weitzenböck公式，以及对Ricci-flat度量的共形扰动。作为推论，我们证明RC正性在取张量、外积或对称幂后不被保持。此外，我们证明RC正性是比一致RC正性严格更弱的概念，并且全纯切丛的RC正性不一定蕴含底流形的有理连通性。
英文摘要
Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.
Comments16 pages
