Curvature Operators on K\"{a}hler Manifolds

Abstract

We prove that there exist K\"{a}hler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that K\"{a}hler manifolds do not satisfy the same type of rigidity with respect to the curvature operator as quaternionic hyperbolic and Cayley hyperbolic manifolds and are thus more similar to real hyperbolic manifolds in this setting. Along the way we also calculate explicit values for the eigenvalues of the curvature operator with respect to the standard complex hyperbolic metric.