AI 中文总结
本文针对具有常Levi-Civita或Bismut全纯截面曲率的局部共形Kähler流形，运用已有技术解答了相关类似问题，推进了非Kähler几何中相关猜想的研究。
AI 中文摘要
非Kähler几何中有一个古老猜想：任何具有常Chern全纯截面曲率的紧致Hermitian流形必为Kähler流形或Chern平坦流形。该猜想在二维时已被证明，但三维及以上维度仍未解决，仅对几类特殊Hermitian流形成立。对于重要的局部共形Kähler流形类，H. Chen、L. Chen和Nie在2021年证明了当常全纯截面曲率非正时猜想成立，剩余情况近期由Huang和Wan利用Kamishima关于Bochner-Kähler流形的结果解决。本文运用他们的技术，解答具有常Levi-Civita或Bismut全纯截面曲率的局部共形Kähler流形的类似问题。
英文摘要
AAn old conjecture in non-Kähler geometry states that any compact Hermitian manifold with constant Chern holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but still open in dimensions 3 or higher, except for several special classes of Hermitian manifolds. For the important class of locally conformally Kähler manifolds, the conjecture was proved by H. Chen, L. Chen, and Nie in 2021 when the constant holomorphic sectional curvature is non-positive and the remaining case was solved recently by Huang and Wan using the result of Kamishima on Bochner-Kähler manifolds. In this article, we use their technique to answer similar questions for locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature.
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