实双截曲率为零的三维紧Hermite流形  

Hermitian threefolds with vanishing real bisectional curvature

作　　者：周武 郑方阳 Wu Zhou;Fangyang Zheng

机构地区：[1]西南民族大学数学学院,成都610041 [2]重庆师范大学数学科学学院,重庆401331

出　　处：《中国科学：数学》2022年第7期757-764,共8页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12071050)资助项目。

摘　　要：复几何中的一个经典猜想是,任何全纯截曲率为常数的紧Hermite流形必为Kahler或Chern平坦的.该猜想在2维时已被证明.本文对该猜想在3维时的一个特殊情形给出证明:实双截曲率为0的紧3维Hermite流形必为Chern平坦的.实双截曲率是全纯截曲率概念的推广,由Yang和Zheng(2019)引入.该曲率量在Kahler时与全纯截曲率等价,在非Kahler时比后者稍强.We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature.Such manifolds are conjectured to be Kahler(hence a complex space form)when the constant is non-zero and Chern flat(hence a quotient of a complex Lie group)when the constant is zero.The conjecture is known in complex dimension two but open in higher dimensions.In this paper,we establish a partial solution in complex dimension three by proving that any compact Hermitian threefold with zero real bisectional curvature must be Chern flat.The real bisectional curvature is a curvature notion introduced by Xiaokui Yang and the second author in 2019,generalizing the holomorphic sectional curvature.It is equivalent to the latter in the Kahler case and is slightly stronger in general.

关 键 词：Hermite流形 Chern联络 全纯截曲率 实双截曲率 

分 类 号：O186.1[理学—数学]