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Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
A long-lasting conjecture in non-Kähler geometry says that any compact Hermitian manifold with constant holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but remains open in dimensions 3 or higher, except in several special cases.
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