The constant holomorphic sectional curvature conjecture for compact Hermitian manifolds

Equivalent formulations 3

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The constant holomorphic sectional curvature conjecture for compact Hermitian manifolds

   Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. Let $H$ denote its holomorphic sectional curvature and let $R$ denote its curvature tensor. The constant holomorphic sectional curvature conjecture. If $H=c$ for a constant $c$, then, when $c\neq0$, $g$ is Kähler, and when $c=0$, $R=0$. The compactness hypothesis is essential, since there are counterexamples in the noncompact case; the conjecture is therefore refuted without that hypothesis.

   source: Kai Tang, “On mixed curvature for Hermitian manifolds”, arXiv:2501.03749 (2025).

2. The constant holomorphic sectional curvature conjecture for compact Hermitian manifolds

   Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. The holomorphic sectional curvature is $H(X)=R(X,\overline{X},X,\overline{X})/|X|^{4}$ for nonzero $(1,0)$-vectors $X$, where $R$ is the Chern curvature tensor. Constant holomorphic sectional curvature conjecture. If $H=c$ is constant, then $c\neq0$ implies that $g$ is Kähler, while $c=0$ implies $R=0$. This is a folklore conjecture concerning rigidity of compact non-Kähler Hermitian manifolds; the supplied text gives no resolution status.

   source: Weiguo Chen and Kai Tang, “Constant kth-mixed curvature”, arXiv:2510.05546 (2025).

3. Constant holomorphic sectional curvature conjecture for compact Hermitian manifolds

   Let $M$ be a compact Hermitian manifold with Hermitian metric $g$, and let its Chern or Levi-Civita connection have holomorphic sectional curvature equal to a constant $c$. Constant holomorphic sectional curvature conjecture. If $c\neq 0$, then $g$ must be Kähler, and hence $M$ is a complex space form; if $c=0$, then $g$ must be Chern-flat or Levi-Civita-flat, respectively. This conjecture proposes a compactness-driven rigidity theorem, contrasting with complete non-Kähler examples on $\mathbb{C}^n$ having vanishing Chern holomorphic sectional curvature but nonvanishing Chern curvature. The Kähler case is classical, while the general Hermitian statement remains open.

   source: Zhuzhu Huang and Xueyuan Wan, “Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature”, arXiv:2606.24425 (2026).