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, C n ∖ { 0 } , n ≥ 2 \mathbb {C}^n \setminus \{0\}, n \ge 2 and B N ∖ A \mathbb {B}^N \setminus A , N ≥ 2 N \ge 2 and A ⊂ B N 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 C ∗ \mathbb {C}^{\ast } and show that it behaves very differently in this case.