Limits of multipole pluricomplex Green functions (Q2888830)

The authors consider a family \((S_\epsilon)_\epsilon\) of \(N\) poles \((a_\epsilon)\) in a bounded hyperconvex domain \(\Omega\) in \(\mathbb C^n\), and they study the convergence of the vanishing ideals \(\mathcal I_\epsilon\) associated to \(S_\epsilon\) and the pluricomplex Green functions \(G_\epsilon:=G_{\mathcal I_\epsilon}\).NEWLINENEWLINE The main result is that if all \(N\) points \(a_\epsilon\) go to \(0\) and if \(\lim_{\epsilon\to 0}\mathcal I_\epsilon=\mathcal I\), then \((G_\epsilon)_\epsilon\) converges to \(G_{\mathcal I}\) locally uniformly on \(\Omega\smallsetminus\{0\}\) if and only if \(\mathcal I\) is a complete intersection ideal. The proofs of the results are based essentially on some estimates for the Hilbert-Samuel multiplicity of the ideal \(\mathcal I\) and the Monge-Ampère operator associated to the plurisubharmonic function \(G_{\mathcal I}\).