Non-negative divisors and the Grauert metric

TL;DR

This paper investigates the properties of Grauert metrics, specifically their holomorphic sectional curvatures, on the complements of principal divisors in complex Euclidean spaces, and examines how these metrics change with varying divisors.

Contribution

It provides new insights into the curvature behavior of Grauert metrics on complements of divisors and analyzes their continuous variation with respect to divisor changes.

Findings

Holomorphic sectional curvatures of Grauert metrics are characterized.

Behavior of the metrics under continuous divisor variation is analyzed.

Results contribute to understanding complex geometric structures in holomorphic domains.

Abstract

Grauert showed that it is possible to construct complete K\"{a}hler 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 $\mathbb{C}^n$, $n \ge 1$. In addition, we also study how this metric and its holomorphic sectional curvature behaves when the corresponding principal divisors vary continuously.