In this article we study compact Kähler 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.
"On quadratic orthogonal bisectional curvature." J. Differential Geom. 92 (2) 187 - 200, October 2012. https://doi.org/10.4310/jdg/1352297805