Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000