The $\partial$-Operator and Real Holomorphic Vector Fields

TL;DR

This paper explores conditions under which the $ar{ abla}$-operator on weighted Bergman spaces exhibits holomorphicity properties, focusing on real holomorphic gradient fields on K"ahler and conformally K"ahler manifolds, and provides new estimates for the $ar{ abla}$-equation.

Contribution

It characterizes weight functions with real holomorphic gradient fields on specific Hermitian manifolds and analyzes their impact on the $ar{ abla}$-complex, including new estimates.

Findings

Identified all multi-radial weight functions with real holomorphic gradient fields on $\

Discovered metrics with holomorphic torsion on complex space forms, including Hopf and hyperbolic metrics.

Derived a new estimate for the $ar{ abla}$-equation on the unit ball with half hyperbolic metric.

Abstract

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 K\"ahler and a suitable density condition holds, the $\partial$-complex exhibits an interesting holomorphicity/duality property when $(\bar\partial\psi)^{\sharp}$ is holomorphic (i.e., when the real gradient field $\mathrm{grad}_h\psi$ 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 K\"ahler and conformally K\"ahler manifolds and their relationship to the $\partial$-complex on weighted Bergman spaces. For K\"ahler metrics with multi-radial potential functions on $\mathbb C^n$ we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally K\"ahler 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-K\"ahler 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.