Complex product manifolds cannot be negatively curved

TL;DR

This paper proves that complex product manifolds and certain fiber bundles cannot support complete Kähler metrics with uniformly negative bisectional curvature, highlighting restrictions on their geometric structures.

Contribution

It establishes a non-existence result for negatively curved Kähler metrics on complex products and fiber bundles, extending understanding of curvature constraints in complex geometry.

Findings

Complex products lack complete negatively curved Kähler metrics.

Holomorphic fiber-bundles also cannot admit such metrics.

Results impose curvature restrictions on complex geometric structures.

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.