Complex product manifolds cannot be negatively curved

Abstract

We show that if $M = X \times Y$ is the product of two complex manifolds (of positive dimensions), then $M$ does not admit any complete K\"ahler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric.Comment: 6 Pages. To appear in The Asian Journal of Mathematic