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 the restriction of these metrics to the leaves of a suitable holomorphic foliation in 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.