Complex product manifolds cannot be negatively curved

2008/01/01 by Harish Seshadri, Seshadri, Harish, Fangyang Zheng +1 · 2 citations
Computer Science · #Artificial Intelligence in Games #Complex Variables (math.CV) #Computational Geometry and Mesh Generation #Computer Graphics and Visualization Techniques #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.0801.0284

openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if M = X × Y is the product of two complex manifolds (of positive dimensions), then M does not admit any complete Kähler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric.

Cited by

• On the negatively pinched properties of the disc bundles over negatively pinched Kähler manifolds
• Negatively curved Kähler metrics on total spaces of a class of vector bundles

Related

• Complex product manifolds and bounds of curvature
• Negative sectional curvature and the product complex structure
• Construction of negatively curved complete intersections
• Negatively curved Kähler metrics on total spaces of a class of vector bundles
• Negative holomorphic bisectional curvature of some bounded domains