Mathematics – Complex Variables
Scientific paper
2011-03-11
International Journal of Mathematics Vol. 23, No. 6 (2012)
Mathematics
Complex Variables
41 p., version 2. A section linking our notion of convergence to the topology of the Douady space has been added. Some typos h
Scientific paper
10.1142/S0129167X12500656
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 $G > G_I$.
Magnusson Jon I.
Rashkovskii Alexander
Sigurdsson Ragnar
Thomas Pascal J.
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