Invariant pluricomplex Green functions. (English) Zbl 0844.31004

Jakóbczak, Piotr (ed.) et al., Topics in complex analysis. Proceedings of the semester on complex analysis, held in autumn of 1992 at the International Banach Center in Warsaw, Poland. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 31, 207-226 (1995).
This article is a survey of results leading to the construction of a plurisubharmonic Green function. For \(D \subset \mathbb{C}^n\) a domain, \(a \in D\) and \(\text{PSH} (D)\) the plurisubharmonic functions in \(D\), one defines the pluricomplex Green function by \[ g_D (z,a) = \sup \bigl\{ u(z) \mid u \in \text{PSH} (D),\;u |_D< 0,\;u(z) - \log |z - a |< C_K,\;K \text{ compact}\}. \] This function has many points in common with the classical Green function of harmonic analysis. For instance, if \((dd^c)^n\) is the complex Monge-Ampère operator, then \((dd^cg_D(z,a))^n\equiv 0\) on \(D-\{a\}\). It is also invariant with respect to biholomorphic maps. On the other hand, it presents some startling differences. For instance, it is not symmetric in \(z\) and \(a\). The author exposes the basic properties of the pluricomplex Green function and illustrates them with a wide variety of explicit computations for diverse domains.
For the entire collection see [Zbl 0816.00022].


31C10 Pluriharmonic and plurisubharmonic functions
32U05 Plurisubharmonic functions and generalizations
32F45 Invariant metrics and pseudodistances in several complex variables