Constant Chern holomorphic sectional curvature conjecture

Establish whether, on every compact complex manifold of complex dimension at least two, a Hermitian metric with constant nonzero Chern holomorphic sectional curvature must be Kähler, and whether a Hermitian metric with zero Chern holomorphic sectional curvature must be Chern-flat.

Background

The paper studies the conjectural rigidity of Hermitian metrics whose Chern holomorphic sectional curvature is constant. For nonpositive curvature, it proves the conjectured Kähler and flatness conclusions for arbitrary Hermitian metrics on compact manifolds in Fujiki’s class $\mathcal C$. It also proves the positive-curvature conclusion for compact balanced complex threefolds.

The general conjecture remains unresolved beyond the settings treated in the paper: the results do not address arbitrary Hermitian manifolds in positive curvature, balanced manifolds of dimension greater than three, or zero curvature without the Fujiki-class hypothesis. The zero-curvature conjectural conclusion is Chern flatness, meaning vanishing of the Chern curvature tensor, which is weaker than Kähler flatness in general.