Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler (Q799959)

Assuming the Hermitian holomorphic sectional curvature of a Hermitian metric on a compact complex surface to be constant as a function on the unit tangent bundle, it is proved that the metric actually has to be Kähler if that constant is negative or zero. Strong limitations are given for a complex surface to admit such a metric for which this constant is positive.