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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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