Constant Chern Holomorphic Sectional Curvature: Rigidity and Counterexamples to the Flatness Conjecture
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.