Vanishing theorems for Hodge numbers and the Calabi curvature operator

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