Compact locally conformal K\"ahler manifolds with constant Chern holomorphic sectional curvature

Abstract

We prove the Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal K\"ahler manifolds. More precisely, let $(M^n,h)$, $n\geq2$, be a compact locally conformal K\"ahler manifold whose Chern holomorphic sectional curvature is a constant $c$. We show that $h$ is necessarily K\"ahler and therefore is a complex space form metric of holomorphic sectional curvature $c$. In particular, when $c=0$, the metric is K\"ahler flat. This removes the nonpositivity assumption from a theorem of Chen, Chen, and Nie. The proof derives a curvature identity on the universal K\"ahler cover and shows that the covering metric is Bochner--K\"ahler. The globally conformally K\"ahler case is then treated by compact Bochner--K\"ahler rigidity, while the strict LCK case is excluded by Kamishima's uniformization theorem and the automorphy of the conformal factor.