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