A celebrated result of Sui-Yau says that manifolds with positive bisectional curvature are biholomorphic to complex projective space. In this talk we will introduce new curvature conditions that provide characterizations of cohomology complex projective spaces. For example, the curvature tensor of a Kaehler manifold induces an operator on symmetric holomorphic 2-tensors, called Calabi operator. This operator is the identity for complex projective space with the Fubini Study metric. We show that a compact n-dimensional Kaehler manifold with n/2-positive Calabi curvature operator has the rational cohomology of complex projective space. The complex quadric shows that this result is sharp if n is even. This talk is based on joint work with K. Broder, J. Nienhaus, P. Petersen, J. Stanfield.