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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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