Semi-Kähler Hermitian and locally conformally Hermitian-flat manifolds (Q2746843)

Let \((M,g)\) denote a compact Hermitian manifold. If there exist a locally conformal change of \(g\) whose curvature of the Chern connection is zero, then \(M\) is locally conformally Hermitian-flat [\textit{P. Gauduchon}, Math. Ann. 267, 495-518 (1984; Zbl 0523.53059)]. If moreover certain Ricci tensors are equal \((Q=R)\), then \(M\) is Hermitian-flat [\textit{K. Matsuo}, Tokyo J. Math. 19, 449-515 (1996; Zbl 0882.53050)]. Here it is proved that \((M,g)\) is semi-Kähler iff \(Q = R\). As a consequence, \((M,g)\) is locally conformally Hermitian-flat and semi-Kähler iff \((M,g)\) is Hermitian-flat.