Paper detail

Limits of multipole pluricomplex Green functions

Let $S_ε$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$ε$ tends to 0. To each set $S_ε$ we associate its vanishing ideal $I_ε$ and the pluricomplex Green function $G_ε$ with poles on the set. Suppose that, as $ε$ tends to 0, the vanishing ideals converge to $I$ (local uniform convergence, or equivalently convergence in the Douady space), and that $G_ε$ 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 $G_ε$ 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 $G_ε$ 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 $G > G_I$.