Complex product manifolds and bounds of curvature

Abstract

Let M = X × Y be the product of two complex manifolds of positive dimension. In this paper, we prove that there is no complete Kähler metric g on M such that: either (i) the holomorphic bisectional curvature of g is bounded by a negative constant and the Ricci curvature is bounded below by -C(1 + r2) where r is the distance from a fixed point; or (ii) g has nonpositive sectional curvature and the holomorphic bisectional curvature is bounded above by -B(1 +r2)-δ and the Ricci curvature is bounded below by -A(1+r2)γ where A, B, γ, δ are positive constants with γ + 2δ < 1. These are generalizations of some previous results, in particular the result of Seshadri and Zheng [8]. © 2010 International Press.