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Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of
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