A new explicit complete Kähler metric on ℂ n

Abstract

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on [Klembeck P. A complete Kähler metric of positive curvature on . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature.