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Complex product manifolds cannot be negatively curved. (English) Zbl 1147.53317

Summary: 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 Kähler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions