# On the Berezin transforms on line bundles over the complex Hyperbolic spaces

## TL;DR

This paper introduces a generalized Berezin transform for line bundles over complex hyperbolic spaces, expressing it as a function of the G-invariant Laplacian, advancing the understanding of geometric analysis in these spaces.

## Contribution

It defines a new generalized Berezin transform on line bundles over complex hyperbolic spaces and relates it explicitly to the G-invariant Laplacian.

## Key findings

- The Berezin transform is expressed as a function of the G-invariant Laplacian.
- Provides a new framework for analyzing line bundles over complex hyperbolic spaces.
- Enhances the mathematical tools for geometric quantization in complex hyperbolic geometry.

## Abstract

We define a generalized Berezin transforms on line bundle over the complex hyperbolic space and we give it as a functions of the G-invariant laplacian on the line bundles.