紧致平衡三维簇与具有常全纯截面曲率的LCK流形
Compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature
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中文总结 AI 辅助
该研究验证了$c\leq0$时紧致平衡三维簇满足埃尔米特几何中关于常陈全纯截面曲率的长期猜想，还将该猜想推广到典范度量联络情形并对连通紧致LCK流形成立。
中文摘要 AI 辅助
埃尔米特几何中一个长期存在的猜想指出，具有常陈全纯截面曲率$c$的紧致埃尔米特流形，当$c\neq0$时是凯勒流形，当$c=0$时是陈平坦的。尽管该猜想在复二维情形已被证明，但在更高维数下仍普遍未解决。我们验证了当$c\leq0$时，紧致平衡三维簇满足该猜想。对于紧致局部共形凯勒（LCK）流形，Chen、Chen和Nie已证明$c\leq0$的情形，Huang和Wan近期解决了剩余情形。受Huang和Wan方法的启发，我们研究了该猜想对典范度量联络的推广，并证明其对连通紧致局部共形凯勒流形成立。
英文摘要
A long-standing conjecture in Hermitian geometry says that a compact Hermitian manifold with constant Chern holomorphic sectional curvature $c$ is Kähler for $c\neq 0$ and Chern flat for $c=0$. Although the conjecture has been established in complex dimension two, it remains open in general in higher dimensions. We verify the conjecture for compact balanced threefolds when $c\leq 0$. For compact locally conformally Kähler manifolds, Chen, Chen, and Nie established the case $c\leq 0$, while Huang and Wan recently settled the remaining case. Inspired by the approach of Huang and Wan, we investigate a generalization of the conjecture for canonical metric connections and establish it for connected compact locally conformally Kähler manifolds.
