Complex product manifolds and bounds of curvature

TL;DR

This paper proves that certain complete Kähler metrics with specific curvature bounds cannot exist on products of complex manifolds, extending previous results by Seshadri and Zheng.

Contribution

It establishes new nonexistence results for complete Kähler metrics with prescribed curvature bounds on product complex manifolds.

Findings

No such metrics exist under the given curvature conditions.

Generalizes previous nonexistence results.

Extends bounds to broader classes of curvature conditions.

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.