On the $L^2$ volume of Bergman spaces

TL;DR

This paper investigates the volume properties of Bergman spaces on certain complex manifolds, demonstrating that specific volume measures are infinite, thus providing insights into their geometric structure.

Contribution

It proves that the Calabi and Mabuchi volumes of Bergman spaces on products of projective manifolds and projective spaces are infinite, addressing a conjecture by Shiffman-Zelditch.

Findings

Calabi volume of Bergman spaces is infinite

Mabuchi volume of Bergman spaces is infinite

Supports conjecture by Shiffman-Zelditch

Abstract

In this paper, we show that the Calabi volume and Mabuchi volume of Bergman spaces on the product of a projective manifold and a projective space is infinite. Our result is inspired by a conjecture of Shiffman-Zelditch in [arXiv:2303.11559].