ar X iv : 0 90 9 . 52 82 v 1 [ m at h . D G ] 2 9 Se p 20 09 COMPLEX PRODUCT MANIFOLDS AND BOUNDS OF CURVATURE

Let M = X×Y be the product of two complex manifolds of positive dimensions. 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+r) 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 + r) and the Ricci curvature is bounded below by −A(1 + r) 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].