Limits of multipole pluricomplex Green functions

TL;DR

This paper investigates the limits of multipole pluricomplex Green functions in complex analysis, establishing conditions under which these functions converge or diverge based on the algebraic properties of associated ideals.

Contribution

It provides a detailed analysis of the convergence behavior of pluricomplex Green functions related to vanishing ideals, linking convergence to the Hilbert-Samuel multiplicity and ideal structure.

Findings

Convergence of Green functions occurs when the ideal's Hilbert-Samuel multiplicity equals the number of poles.

If the multiplicity exceeds the number of poles, the Green functions do not converge to the ideal's Green function.

In the case of three poles, convergence depends on the limiting directions of the points, with specific conditions leading to strict inequalities.

Abstract

Let $S_\epsilon$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$\epsilon$ tends to 0. To each set $S_\epsilon$ we associate its vanishing ideal $I_\epsilon$ and the pluricomplex Green function $G_\epsilon$ with poles on the set. Suppose that, as $\epsilon$ tends to 0, the vanishing ideals converge to $I$ (local uniform convergence, or equivalently convergence in the Douady space), and that $G_\epsilon$ converges to $G$, locally uniformly away from the origin; then the length (i.e. codimension) of $I$ is equal to $N$ and $G \ge G_I$. If the Hilbert-Samuel multiplicity of $I$ is strictly larger than $N$, then $G_\epsilon$ cannot converge to $G_I$. Conversely, if the Hilbert-Samuel multiplicity of $I$ is equal to $N$, (we say that $I$ is a complete intersection ideal), then $G_\epsilon$ does converge to $G_I$. We work out the case of three poles; when the directions defined by any two of the three points converge to limits which don't all coincide, there is convergence, but $G > G_I$.