Powers of ideals and convergence of Green functions with colliding poles

TL;DR

This paper investigates the convergence behavior of Green functions associated with families of ideals of holomorphic functions vanishing at points converging to a single point, establishing conditions under which the Green functions converge to a well-defined limit.

Contribution

It proves that if all powers of the ideals converge, then the Green functions also converge to an upper envelope of scaled Green functions, clarifying a previously uncertain aspect.

Findings

Green functions converge locally uniformly away from the pole

Limit Green function is an upper envelope of scaled Green functions

Results explain asymptotics in 3-point models

Abstract

Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green functions to the Green function of the limit ideal; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers of the ideals converge to some ideals, we prove that the Green functions converge, locally uniformly away from the limit pole, to a function which is essentially the upper envelope of the scaled Green functions of the limits of the powers. As examples, we consider ideals generated by hyperplane sections of a holomorphic curve near its singular point. In particular, our result explains recently obtained asymptotics for 3-point models.