发表机构
Halıcıoğlu Data Science Institute, University of California San Diego; School of Mathematical Sciences, Chongqing Normal University（加州大学圣地亚哥分校哈利奥格鲁数据科学研究所; 重庆师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明复维数4的所有平衡Bismut挠率平行流形满足非凯勒几何中关于具有常全纯截面曲率的紧埃尔米特流形的古老猜想，为4维平衡BTP流形结构研究提供线索。
AI 中文摘要
非凯勒几何中的一个古老猜想指出：若紧埃尔米特流形具有常全纯截面曲率，则当该常数非零时，度量必为凯勒度量；当该常数为零时，度量必为陈平坦。1985年Balas和Gauduchon的工作证明复维数2时该猜想成立（常数为负或零），1996年Apostolov、Davidov和Muskarov证明一般情况也成立。对于维数3及更高的情况，该猜想仅在一些特殊情形下已知成立，例如Davidov、Grantcharov和Muskarov证明所有扭量空间均满足该猜想，H. Chen、L. Chen和Nie证明局部共形凯勒情形（常数为负或零）满足该猜想，S. Chen和Zheng证明所有非平衡Bismut挠率平行（BTP）流形满足该猜想，还利用Zhao和Zheng的分类结果证明所有平衡BTP三维流形满足该猜想。本文中，我们证明复维数4的所有平衡BTP流形均满足该猜想。值得注意的是，尽管平衡BTP流形具有很强的限制性，但4维及更高维数的分类仍缺失，本研究可能为4维平衡BTP流形的结构提供一些线索。
英文摘要
An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (in general). In dimension $3$ or higher, the conjecture is only known in some special cases, such as for all twistor spaces by the work of Davidov, Grantcharov, and Muskarov, or for the locally conformally Kähler case (when the constant is negative or zero) by the work of H. Chen, L. Chen and Nie, or for all non-balanced Bismut torsion parallel (BTP) manifolds by the work of S. Chen and Zheng, where they also showed that the conjecture holds for all balanced BTP threefolds, utilizing a classification result by Zhao and Zheng. In this article, we show that the conjecture is valid for all balanced BTP manifolds in complex dimension $4$. The interesting part is that while balanced BTP manifolds are highly restrictive, the classification is still lacking in dimensions $4$ or higher, and this study might shed some light on the structure of balanced BTP manifolds in dimension $4$.
Comments18 pages
