Limit of Green functions and ideals, the case of four poles

TL;DR

This paper investigates the behavior of pluricomplex Green functions and associated ideals with four poles approaching the origin, revealing generic convergence patterns and exceptional cases in complex analysis.

Contribution

It establishes the generic limit of Green functions with four poles and analyzes the distinct limits of ideals, extending understanding beyond the three pole case.

Findings

Green functions with four poles have a unique generic limit.

Limits of ideals can differ even when Green functions converge.

Certain exceptional cases allow determination of ideal limits.

Abstract

We study the limits of pluricomplex Green functions with four poles tending to the origin in a hyperconvex domain, and the (related) limits of the ideals of holomorphic functions vanishing on those points. Taking subsequences, we always assume that the directions defined by pairs of points stabilize as they tend to $0$. We prove that in a generic case, the limit of the Green functions is always the same, while the limits of ideals are distinct (in contrast to the three point case). We also study some exceptional cases, where only the limits of ideals are determined. In order to do this, we establish a useful result linking the length of the upper or lower limits of a family of ideals, and its convergence.