Vanishing theorems for Hodge numbers and the Calabi curvature operator

It is shown that a compact n-dimensional K\"ahler manifold with n2-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 n2-nonnegative Calabi curvature operator, yet bₙ =2. Furthermore, the compact K\"ahler manifolds with an n2-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.