Further remarks on the higher dimensional Suita conjecture

TL;DR

This paper investigates the boundary behavior of a biholomorphic invariant related to the Bergman kernel and Kobayashi indicatrix in higher-dimensional complex domains, focusing on h-extendible and strongly pseudoconvex polyhedral domains.

Contribution

It provides new insights into the limiting boundary behavior of the invariant $F^k_D(z)$ on specific classes of complex domains, extending previous understanding.

Findings

Boundary behavior characterized for h-extendible domains

Boundary behavior characterized for strongly pseudoconvex polyhedral domains

Enhanced understanding of biholomorphic invariants in complex analysis

Abstract

For a domain $D \subset \mathbb C^n$, $n \ge 2$, let $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda\big(I^k_D(z)\big)$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. This biholomorphic invariant was introduced by B\locki and in this note, we study its limiting boundary behaviour on two classes of domains namely, $h$-extendible and strongly pseudoconvex polyhedral domains.