发表机构
Chern Institute of Mathematics and LPMC, Nankai University; School of Mathematics and Statistics, Beijing Institute of Technology（南开大学陈省身数学研究所和LPMC; 北京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明紧致Hermitian流形上常Chern全纯截面曲率的刚性定理：非正曲率情形度量必为Kähler，负曲率时典范丛丰富且为Kähler-Einstein度量，正曲率平衡三维流形等距于带Fubini-Study度量的复射影空间。
AI 中文摘要
我们证明了具有常Chern全纯截面曲率的紧致Hermitian流形的两个刚性定理。在Fujiki类$\mathcal C$中的流形上，每个具有常非正Chern全纯截面曲率的Hermitian度量都是Kähler的，无需平衡或pluriclosed假设。在负曲率情形，该流形是射影的且典范丛丰富，给定的度量是归一化的负Kähler--Einstein度量，其万有覆盖为复双曲空间。在零曲率情形，度量是平坦的，且流形有一个由复环给出的有限étale覆盖。我们还证明了具有常正Chern全纯截面曲率的紧致平衡三维流形等距于带有缩放Fubini--Study度量的复射影三维空间。非正曲率论证使用了Kähler背景上的积分Chern--Lu恒等式。对于负曲率，扭曲Kähler--Einstein体积的下界给出典范丰富性，加权Stokes恒等式和体积比矩不等式识别出给定度量。在零曲率时，张量Bochner论证在Ricci平坦背景上给出平行性。对于平衡三维流形，微分相容性和挠率能量恒等式产生一个强制性估计，该估计由在每个挠率秩上有效的有理矩阵分解证明。
英文摘要
We prove rigidity results for Hermitian metrics of constant Chern holomorphic sectional curvature and construct counterexamples to the flatness conjecture. On a compact complex manifold in Fujiki's class $\mathcal C$ of dimension $n\ge2$, every such metric with nonpositive curvature is Kähler; its universal cover is complex hyperbolic in the negative case and Euclidean in the zero case. On an arbitrary compact complex threefold, nonzero constant curvature forces Kählerness, while zero curvature forces Chern flatness. For every complex dimension $n\ge7$, we construct compact Hermitian manifolds carrying balanced metrics with zero Chern holomorphic sectional curvature, vanishing first and second Chern--Ricci tensors, and nonzero full Chern curvature. The rigidity proofs use weighted integral comparison, differential compatibility in complex dimension three, and compactness obstructions from common kernels and null foliations. The counterexamples arise from a positive invariant Hermitian metric on a seven-dimensional complex quadric. Its Chern curvature is expressed by the octonion associator, whose alternating four-tensor makes holomorphic sectional curvature and both Ricci contractions vanish while retaining nonzero curvature. The metric descends to compact quotients; products with flat complex tori give the higher-dimensional counterexamples.
