# The $\partial$-complex on weighted Bergman spaces on Hermitian manifolds

## TL;DR

This paper studies the $ar{
abla}$-complex on weighted Bergman spaces over Hermitian manifolds, extending known results from $C^n$ and applying findings to specific metrics in the unit ball.

## Contribution

It generalizes the $ar{
abla}$-complex framework to Hermitian manifolds with duality conditions, providing new estimates for the $ar{
abla}$-equation solutions in this setting.

## Key findings

- Extended the $ar{
abla}$-complex to Hermitian manifolds with duality conditions.
- Derived new estimates for solutions of the $ar{
abla}$-equation on weighted Bergman spaces.
- Applied results to complex hyperbolic and conformally Kähler metrics in the unit ball.

## Abstract

In this paper, we investigate the $\partial$-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in $\mathbb{C}^n$, studied earlier by the first-named author, in which the adjoint of the differentiation is the multiplication by $z$. The results are applied to two important examples in the unit ball, namely, the complex hyperbolic metric and a conformally K\"ahler metric which are related to Bergman spaces with so-called "exponential" and "standard" weights, respectively. In particular, we obtain new estimates for the solutions of the $\partial$-equation on these weighted Bergman spaces.