Continuity properties of finely plurisubharmonic functions and pluripolarity

Abstract

We prove that every bounded finely plurisubharmonic function can be locally (in the pluri-fine topology) written as the difference of two usual plurisubharmonic functions. As a consequence, finely plurisubharmonic functions are continuous with respect to the pluri-fine topology. Moreover, we show that − ∞ sets of finely plurisubharmonic functions are pluripolar, hence graphs of finely holomorphic functions are pluripolar.