A new explicit complete Kähler metric on ℂⁿ

Abstract

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on ${\mathbb{C}}^n$ [Klembeck P. A complete Kähler metric of positive curvature on ${\mathbb{C}}^n$. 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.

Keywords:

• Kähler metric
• Bergman metric
• weighted Fock spaces
• reproducing kernel
• holomorphic sectional curvature

AMS Subject Classifications:

• 32A36
• 53C55

Acknowledgements

The second named author (Choi) would like to express his profound gratitude to Professor Hong Rae Cho for his guidance. This work was started and motivated during his visit to Center for Geometry and its Applications (GAIA), Pohang University of Science and Technology, and KIAS. He would like to thank members of GAIA for useful discussion, in particular, he is so indebted to Seungro Joo for the invitation. It is his great pleasure to thank Kang-Tae Kim, Wing-Keung To, Jong-Do Park for useful conversation and comments. Last but not least, the authors thank the referees for their careful comments that improved the readability of this paper. This paper is a part of the second named author's Ph.D. thesis at Pusan National University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research of first (Cho) and second (Choi) named authors was supported by National Research Foundation of Korea (NRF) of Korea [NRF-2020R1F1A1A01048601] and [NRF-2022R1A6A3A01087177].