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Regularity properties of the Azukawa metric. (English) Zbl 0986.32016
The pluricomplex Green function $$g_D(w,\cdot)$$ with pole at $$w$$ is defined in a domain $$D$$ of $$\mathbb{C}^n$$ as the supremum of $$u\in\text{PSH}(D)$$ such that $$u<0$$ and $$u-\log\|\cdot-w\|$$ is bounded above near $$w$$ and the Azukawa pseudometric is defined on $$D\times \mathbb{C}^n$$ by $A_D(z;X)=\limsup_{\lambda\to 0} (g_D(z,z+\lambda X)-\log|\lambda|).$ The main results of the paper are the following: If $$\varepsilon(w)=\liminf_{z\to \partial D} g_D(w,z)>-\infty$$ and $$g_D(w,\cdot)$$ is continuous, then $$\liminf$$ in the definition of $$A_D$$ may be replaced by $$\lim$$. If the assumption holds at every point $$w\in D$$, then $$A_D$$ is continuous, $\lim_{w',w''\to w, w'\neq w''}(g_D(w',w'')-g_D(w'',w'))=0$ and $A_D(w;X)=\lim_{w',w''\to w, w'\neq w'',(w'-w'')/\|w'-w''\|\to X} (g_D(w',w'')-\log\|w'-w''\|).$ The author also defines the Green function with finitely many weighted poles and compares it with weighted sums of Green functions with single poles.

##### MSC:
 32U35 Plurisubharmonic extremal functions, pluricomplex Green functions 32U05 Plurisubharmonic functions and generalizations 31C10 Pluriharmonic and plurisubharmonic functions
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