Vanishing theorems for Hodge numbers and the Calabi curvature operator

It is shown that a compact n-dimensional K\"ahler manifold with n/2-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric with its symmetric metric has n/2-nonnegative Calabi curvature operator, yet bₙ =2. Furthermore, the compact K\"ahler manifolds with an n/2-nonnegative Calabi curvature operator are classified. In addition, the previously known results for the K\"ahler curvature operator are improved when the metric is K\"ahler--Einstein.