Kähler Rigidity for Constant Chern Holomorphic Sectional Curvature

Abstract

We prove two rigidity theorems for compact Hermitian manifolds with constant Chern holomorphic sectional curvature. On a manifold in Fujiki's class $\mathcal C$, every Hermitian metric of constant nonpositive Chern holomorphic sectional curvature is Kähler, without a balanced or pluriclosed assumption. In the negative case, the manifold is projective with ample canonical bundle, and the given metric is the normalized negative Kähler--Einstein metric, with complex hyperbolic universal cover. In the zero case, the metric is flat and the manifold admits a finite étale cover by a complex torus. We also prove that a compact balanced threefold of constant positive Chern holomorphic sectional curvature is holomorphically isometric to complex projective three-space with a scaled Fubini--Study metric. The nonpositive argument uses an integrated Chern--Lu identity on a Kähler background. For negative curvature, a lower bound for twisted Kähler--Einstein volumes yields canonical ampleness, and weighted Stokes identities and volume-ratio moment inequalities identify the given metric. At zero curvature, a tensor Bochner argument gives parallelness on a Ricci-flat background. For balanced threefolds, differential compatibility and the torsion energy identities yield a coercive estimate, proved by a rational matrix decomposition valid at every torsion rank.