Powers of ideals and convergence of Green functions with colliding poles (Q2878706)

The main result of the paper is the following:NEWLINENEWLINELet \(\mathcal I_{\varepsilon}\) be a family of ideals of holomorphic functions on a hyperconvex domain \(\Omega\), indexed by a complex parameter \(\varepsilon\) taken from some subset of the complex plane whose closure contains \(0\). For each \(\varepsilon\), \(\mathcal I_{\varepsilon}\) consists of functions which vanish at a fixed number of distinct but varying with \(\varepsilon\) points. Under the assumption that all these varying points converge to a fixed point \(a\) and for all powers \(p\in \mathbb N\) there is the convergence of the powers of the ideals \(\mathcal I_{\varepsilon}^p\to \mathcal I_{(p)}\), where \(\mathcal I_{(p)}\) denotes just some limit ideal and the convergence is in the topology of the Douady space, when \(\varepsilon \to 0\), then one has the following convergence result on the pluricomplex (multipole or with respect to ideals) Green functions:NEWLINENEWLINENEWLINE\[NEWLINE\lim_{\varepsilon \to 0} G_{\mathcal I_{\varepsilon}}(z)=\limsup_{y\to z}\sup_{p\in\mathbb N} p^{-1}G_{\mathcal I_{(p)}}(y).NEWLINE\]NEWLINENEWLINENEWLINEThe limit is locally uniform on \(\Omega\setminus\{a\}\). It was known before that just the existence of the limit ideal \(\mathcal I_{\varepsilon}\to\mathcal I\) is not enough to guarantee the convergence of \(G_{I_{\varepsilon}}\) to \(G_{\mathcal I}\).NEWLINENEWLINEAs an application the authors show that for families of pluricomplex Green functions with fixed number of poles all reasonable notions of convergence coincide.NEWLINENEWLINEAnother result says that if in addition to the above conditions, the Hilbert-Samuel multiplicity of some limit ideal \(\mathcal I_{(p)}\) equals \(p^n\) times the number of poles, then the limit of the Green functions in the above theorem equals \(p^{-1}G_{\mathcal I_{(p)}}\).NEWLINENEWLINEThe paper ends with many illustrative examples and some open problems.