具有常Chern全纯截面曲率的Bismut挠平行埃尔米特流形
Bismut-Torsion-Parallel Hermitian Manifolds With Constant Chern Holomorphic Sectional Curvature
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中文总结 AI 辅助
本文证明了所有复维数下平衡Bismut挠平行埃尔米特流形的非零Chern全纯截面曲率情形，确认该类紧流形若Chern全纯截面曲率非零则必为凯勒流形，完善了相关猜想的证明。
中文摘要 AI 辅助
复几何中一个著名猜想指出，具有常Chern全纯截面曲率的紧埃尔米特流形，当常数非零时必为凯勒流形，当常数为零时必为Chern平坦流形。该猜想在复二维情形及高维的若干特殊类中已获证明。对于带Bismut平行挠率的埃尔米特度量，Chen-Zheng已建立非平衡情形与平衡三维流形情形，Wang-Zheng近期解决了平衡四维流形情形。本文中，我们证明了所有复维数下平衡Bismut挠平行埃尔米特流形的非零情形。作为推论，我们确认：若具有非零常数Chern全纯截面曲率的紧BTP埃尔米特流形，则其度量g为凯勒度量。
英文摘要
A well-known conjecture in complex geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature must be Kähler when the constant is nonzero and Chern flat when the constant is zero. The conjecture is known in complex dimension two and in several special classes in higher dimensions. For Hermitian metrics with Bismut-parallel torsion, the non-balanced case and the balanced threefold case were established by Chen--Zheng, while the balanced fourfold case was settled recently by Wang--Zheng. In this article, we prove the nonzero case for balanced Bismut-torsion-parallel Hermitian manifolds in every complex dimension. As a corollary, we confirm that every BTP Hermitian manifold with Chern holomorphic sectional curvature is a nonzero constant, then $g$ is Kähler.
