Continuity Properties of Finely Plurisubharmonic Functions and pluripolarity

TL;DR

This paper establishes that bounded finely plurisubharmonic functions are locally representable as differences of standard plurisubharmonic functions, ensuring their continuity in the pluri-fine topology and demonstrating the pluripolarity of their -infinity sets and graphs of finely holomorphic functions.

Contribution

It introduces a local representation of bounded finely plurisubharmonic functions as differences of classical plurisubharmonic functions, advancing the understanding of their continuity and pluripolarity.

Findings

Bounded finely plurisubharmonic functions are locally differences of plurisubharmonic functions.

Finely plurisubharmonic functions are continuous in the pluri-fine topology.

The -infinity sets of finely plurisubharmonic functions are pluripolar, as are graphs of finely holomorphic functions.

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 -infinity sets of finely plurisubharmonic functions are pluripolar, hence graphs of finely holomorphic functions are pluripolar.