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LIMITS OF MULTIPOLE PLURICOMPLEX GREEN FUNCTIONS

2012; World Scientific; Volume: 23; Issue: 06 Linguagem: Inglês

10.1142/s0129167x12500656

1793-6519

Jón Magnússon, Alexander Rashkovskii, Ragnar Sigurðsson, Pascal J. Thomas,

Algebraic Geometry and Number Theory

Let Ω be a bounded hyperconvex domain in ℂ n , 0 ∈ Ω, and S ε a family of N poles in Ω, all tending to 0 as ε tends to 0. To each S ε we associate its vanishing ideal [Formula: see text] and pluricomplex Green function [Formula: see text]. Suppose that, as ε tends to 0, [Formula: see text] converges to [Formula: see text] (local uniform convergence), and that (G ε ) ε converges to G, locally uniformly away from 0; then [Formula: see text]. If the Hilbert–Samuel multiplicity of [Formula: see text] is strictly larger than its length (codimension, equal to N here), then (G ε ) ε cannot converge to [Formula: see text]. Conversely, if [Formula: see text] is a complete intersection ideal, then (G ε ) ε converges to [Formula: see text]. We work out the case of three poles.