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.
Categories: math.CV