Abstract: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.
From: Xueyuan Wan [view email]
[v1]
Tue, 23 Jun 2026 11:04:05 UTC (14 KB)
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