An open conjecture on Hermitian space forms predicts that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kahler or Chern flat. This work proves the conjecture for all compact solvmanifolds carrying a left-invariant Hermitian metric whose underlying Lie algebra has complex commutator, extending the known nilmanifold case. The argument is carried out on Hermitian Lie algebras: the constant-holomorphic-sectional-curvature condition is translated into algebraic relations among the structure constants, which force the metric to be Kahler or Chern flat. It illustrates how homogeneous (Lie-theoretic) models turn a hard curvature problem in non-Kahler geometry into a tractable algebraic one.
A long-standing conjecture in non-Kahler geometry predicts that any compact Hermitian manifold with constant holomorphic sectional curvature must be either Kahler (hence a complex space form) or Cher…