The ∂-Operator and Real Holomorphic Vector Fields

Author(s)
Friedrich Haslinger, Duong Ngoc Son
Abstract

Let (M, h) be a Hermitian manifold and ψ a smooth weight function on M. The ∂-complex on weighted Bergman spaces A

2

(p,0)(M, h, e

−ψ) of holomorphic (p, 0)-forms was recently studied in [10] and [9]. It was shown that if h is Kähler and a suitable density condition holds, the ∂-complex exhibits an interesting holo-morphicity/duality property when (

¯∂ψ)

♯ is holomorphic (i.e., when the real gradient field grad

h ψ is a real holomorphic vector field.) For general Hermitian metrics, this property does not hold without the holomorphicity of the torsion tensor T

rs

p . In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kähler and con-formally Kähler manifolds and their relationship to the ∂-complex on weighted Bergman spaces. For Kähler metrics with multi-radial potential functions on C

n, we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally Kähler metrics on complex space forms, we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold S

2n−1 × S

1 and the “half” hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions ψ with real holomorphic gradient fields. They provide a wealth of triples (M, h, e

−ψ ) of Hermitian non-Kähler manifolds with weights for which the ∂-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the ∂-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the ∂-equation.

Organisation(s)
Department of Mathematics
Journal
Pure and Applied Mathematics Quarterly
Volume
18
Pages
793–833
No. of pages
41
ISSN
1558-8599
Publication date
2020
Peer reviewed
Yes
Austrian Fields of Science 2012
101008 Complex analysis
Keywords
ASJC Scopus subject areas
General Mathematics
Portal url
https://ucrisportal.univie.ac.at/en/publications/de7c54b2-cbd0-443b-826b-66fa189c3afa