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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.

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