具有常全纯截面曲率或常实双截面曲率的 Hermitian 流形

中文摘要 AI 辅助

Hermitian 几何中有一个古老的猜想：一个紧致 Hermitian 流形若具有常全纯截面曲率，则当该常数为非零时，流形是 Kähler 的；当该常数为零时，流形是 Chern 平坦的。Yang 和 Zheng 引入了实双截面曲率作为全纯截面曲率的推广，并猜想：一个具有常实双截面曲率的紧致 Hermitian 流形，其常数必为零，且流形是 Chern 平坦的。他们证明了该常数不能为正。在本文中，我们首先证明了他们的猜想。对于负常数，我们证明度量是 pluriclosed 的，并通过与负 Kähler-Einstein 度量比较得出矛盾。对于零常数，我们将 Lin 和 Ren 的挠率恒等式与 Zhou 和 Zheng 的 Bochner 公式相结合，证明了 Chern 平坦性。我们还证明了在紧致 Kähler 流形上，具有常负全纯截面曲率的 pluriclosed 度量是 Kähler 的。

英文摘要

An old conjecture in Hermitian geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature is Kähler when the constant is nonzero and Chern flat when the constant is zero. Yang--Zheng introduced the real bisectional curvature as a generalization of the holomorphic sectional curvature and conjectured that, a compact Hermitian manifold with constant real bisectional curvature has zero constant and is Chern flat. They proved that the constant cannot be positive. In this paper, we first prove their conjecture. For a negative constant, we show that the metric is pluriclosed and use comparison with a negative Kähler--Einstein metric to obtain a contradiction. For the zero constant, we combine a torsion identity of Lin--Ren with the Bochner formula of Zhou--Zheng to prove Chern flatness. We also prove that a pluriclosed metric with constant negative holomorphic sectional curvature on a compact Kähler manifold is Kähler.

发表机构

• Zhejiang Normal University（浙江师范大学）

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