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Computing the pluricomplex Green function with two poles. (English) Zbl 1078.32021
Let \(\Omega\) be a domain in \(\mathbb{C}^n\). For a plurisubharmonic (plsh) function \(u\) on \(\Omega\) we denote by \(\nu_u(w)\) the Lelong number of \(u\) at a point \(w \in \Omega\). By an admissible function we mean a nonnegative function \(\nu\) on \(\Omega\) with a finite support \(\text{supp} (\nu)\). We write \(\nu= \nu_1 w_1 +\cdots+ \nu_N w_N\), if \(\text{supp} (\nu)=\{ w_1,\dots,w_N\}\) and \(\nu (w_j)=\nu_j\), \(1\leq j \leq N\). Given an admissible function \(\nu\) on \(\Omega\), we define the pluricomplex Green function with pole defined by \(\nu\) as \[ g(z;\nu) := \sup \{ u(z)\mid u\leq 0,\;u\text{ is plsh on }\Omega,\;\nu_u(w)\geq \nu (w),\forall w \in \Omega \}. \] The subject of the paper is the relationship between \(g(\cdot; \nu)\) and the Lempert function \(\delta (\cdot;\nu)\). This function is defined as follows: An analytic disc \(\phi:\mathbb{D} \rightarrow \Omega\) is called \(\nu\)-admissible, if \(\text{supp} (\nu) \subset \phi (\mathbb{D})\). We associated to such an analytic disc the quantity \[ d(\phi) = \sum_{w \in \text{supp}(\nu)} \inf \{ \nu (w)\log | \zeta| \mid \zeta \in \phi^{-1}(w)\}. \] Then the Lempert function is defined by \[ \delta (z;\nu) := \inf \{d(\phi)\mid \phi \text{a \(\nu\)-admissible anlytic disc},\;\phi (0)=z\}. \] In general we always have \(g (\cdot ; \nu ) \leq \delta (\cdot;\nu)\) and equality holds, iff \(\delta (\cdot;\nu)\) is plsh on \(\Omega\). By Lempert’s work, this is known to be true if \(\Omega\) is a convex domain. There is a conjecture which is due to D. Coman: If \(\Omega\) is a bounded convex domain, then \(g (\cdot ; \nu ) = \delta (\cdot;\nu)\). But this conjecture does not hold in full generality. For example, Carlehed-Wiegerinck considered the case that \(\Omega\) is the bidisc and \(\nu = \nu_1w_1+\nu_2w_2\), where \(\nu_1\neq \nu_2\) and \(w_1,w_2 \in \mathbb{D} \times \{0\}\), where they proved, that \(g (\cdot ; \nu ) < \delta (\cdot;\nu)\).
The author considers the case \(\nu = w_1+w_2\), with \(w_1 = (p_1,0)\) and \(w_2=(0,p_2)\) and proves \(g (\cdot ; \nu )= \delta (\cdot;\nu)\).
Furthermore, he studies the case where \(\Omega\) is the unit ball in \(\mathbb{C}^2\). In the case when \(\nu=w_1+w_2\), Coman’s conjecture is true (as was shown by Coman himself). The author now treats the case \(\nu= w + 2 w'\), where \(w=0\) and \(w'=(1/2,\,0)\). Here he shows that \(\delta (\cdot;\nu)\) is not plsh, in particular Coman’s conjecture is false in this situation.

MSC:
32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
32F45 Invariant metrics and pseudodistances in several complex variables
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