# Bergman metrics as pull-backs of the Fubini-Study metric

## TL;DR

This paper characterizes complex manifolds with constant holomorphic sectional curvature using Bergman metrics as pull-backs of the Fubini-Study metric, introduces new domains with unique curvature properties, and proposes a new conjecture.

## Contribution

It presents a novel approach to understanding Bergman metrics via pull-backs of the Fubini-Study metric and constructs new domains with distinctive curvature characteristics.

## Key findings

- Characterization of manifolds with constant holomorphic sectional curvature
- Construction of new domains with surprising curvature properties
- Formulation of a new conjecture regarding Bergman metrics

## Abstract

Domains and more generally complex manifolds whose Bergman metrics have constant holomorphic sectional curvature are characterized. Our approach is to treat the Bergman metrics as the pull-back by the Bergman-Bochner maps of the Fubini-Study metric of the complex projective space of infinite dimension. Several new domains with surprising curvature properties for their Bergman metrics are constructed. A new conjecture is also formulated at the end of the paper.