The $\partial$-operator and real holomorphic vector fields

Let $(M,h)$ be a Hermitian manifold and $\psi$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)} (M,h, e^{-\psi})$ 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 $\partial$-complex exhibits an interesting holomorphicity/ duality property when $(\bar{\partial} \psi)^{\#}$ is holomorphic (i.e., when the real gradient field $\operatorname{grad}_h \psi$ is a real holomorphic vector field.) For general Hermitian metrics, this property does not hold without the holomorphicity of the torsion tensor $T_p^{rs}$.

In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds and their relationship to the $\partial$-complex on weighted Bergman spaces. For Kähler metrics with multi-radial potential functions on $\mathbb{C}^n$, 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 $\mathbb{S}^{2n-1} \times \mathbb{S}^1$ and the “half” hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions $\psi$ with real holomorphic gradient fields. They provide a wealth of triples $(M,h, e^{-\psi})$ of Hermitian non-Kähler manifolds with weights for which the $\partial$-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the $\partial$-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the $\partial$-equation.

Keywords

$\ell$-complex, weighted Bergman spaces, conformally Kähler metrics, holomorphic vector fields

2010 Mathematics Subject Classification

30H20, 32A36, 32W50, 53C55

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The first-named author was partially supported by the Austrian Science Fund, FWF-Projekt P 28 154-N35

The second-named author was supported by the Austrian Science Fund, FWFProjekt M 2472-N35.

Received 29 July 2020

Accepted 5 December 2021

Published 24 July 2022