Conformally K\"ahler surfaces and orthogonal holomorphic bisectional   curvature
  We show that a compact complex surface which admits a conformally K\"ahler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric. 
Mustafa Kalafat and Caner Koca
math.DG math-ph math.MP