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.

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

In this paper, we study some properties of the [Formula: see text]th-order weighted reduced Bergman kernels for planar domains, [Formula: see text]. Specifically, we look at Ramadanov type theorems, localization, and boundary behavior of the weighted reduced Bergman kernel and its higher-order counterparts. We also give a transformation formula for these kernels under biholomorphisms.

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in C n , n ≥ 1.In addition, we also study how this metric and its holomorphic sectional curvature behave when the corresponding principal divisors vary continuously.