On Grauert's examples of complete Kähler metrics

Abstract

Grauert showed that the existence of a complete Kähler 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.

10 pages