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Hermitian threefolds with constant holomorphic sectional curvature
An old conjecture in non-Kähler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant $c$, then the metric must be Kähler when $c\neq 0$ and be Chern flat when $c=0$. The conjecture is known to be true in dimension two by the work of
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