On Grauert's examples of complete K\"{a}hler metrics

TL;DR

This paper investigates the holomorphic sectional curvatures of complete Kähler metrics constructed by Grauert on complements of complex analytic sets, revealing diverse behaviors in different geometric contexts.

Contribution

It analyzes the curvature properties of Grauert's metrics in specific examples, providing new insights into their geometric characteristics.

Findings

Curvature computations on punctured complex spaces.

Different behavior of metrics on punctured plane.

Insights into the geometry of Grauert's examples.

Abstract

Grauert showed that the existence of a complete K\"{a}hler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb{C}^n \setminus \{0\}, n \ge 2$ and $\mathbb{B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb{B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of its restriction to the leaves of a suitable holomorphic foliation of these two examples. We also examine this metric on the punctured plane $\mathbb{C}^{\ast}$ and show that it behaves very differently in this case.