Complex product manifolds and bounds of curvature

Abstract

Let $M=X\times Y$ be the product of two complex manifolds of positive dimensions. In this paper, we prove that there is no complete K\"ahler 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 $-C(1+r^2)$ 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 $-B(1+r^2)^{-\delta}$ and the Ricci curvature is bounded below by $-A(1+r^2)^\gamma$ where $A, B, \gamma, \delta$ are positive constants with $\gamma+2\delta<1$. These are generalizations of some previous results, in particular the result of Seshadri and Zheng.