On quadratic orthogonal bisectional curvature
  In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possible de Rham decompositions of the universal cover under this condition 
Albert Chau, Luen-Fai Tam
math.DG