Remarks on the higher dimensional Suita conjecture

TL;DR

This paper investigates the behavior of a higher-dimensional analogue of Suita's conjecture, focusing on strongly pseudoconvex domains and specific egg domains in complex space, providing explicit limit computations.

Contribution

It introduces and analyzes the invariant $F^k_D(z)$ for higher-dimensional domains, extending Suita's conjecture to complex spaces of dimension two or more.

Findings

Behavior of $F^k_D(z)$ on strongly pseudoconvex domains analyzed.

Explicit limiting behavior computed for certain egg domains in $C^2$.

Abstract

To study the analog of Suita's conjecture for domains $D \subset \mathbb{C}^n$, $n \ge 2$, B\l ocki introduced the invariant $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, 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 $z$. In this note, we study the behaviour of $F^k_D(z)$ (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in $\mathbb{C}^2$.