Connectedness in the Pluri-fine Topology

TL;DR

This paper investigates the properties of connectedness in the pluri-fine topology on complex Euclidean spaces, demonstrating how removing pluripolar sets preserves connectedness and exploring the behavior of finely plurisubharmonic functions.

Contribution

It provides new insights into the structure of open sets in the pluri-fine topology and establishes results on the invariance of -infinity sets for finely plurisubharmonic functions.

Findings

Removing pluripolar sets from connected sets preserves connectedness.

The union of finely connected components along complex lines forms a pluri-finely connected neighborhood.

If a finely plurisubharmonic function is -infinity on a pluri-finely open set, it is identically -infinity.

Abstract

We study connectedness in the pluri-fine topology on $\CC^n$ and obtain the following results. If $\Omega$ is a pluri-finely open and pluri-finely connected set in $\CC^n$ and $E\subset\CC^n$ is pluripolar, then $\Omega\setminus E$ is pluri-finely connected. The proof hinges on precise information about the structure of open sets in the pluri-fine topology: Let $\Omega$ be a pluri-finely open subset of $\CC^{n}$. If $z$ is any point in $\Omega$, and $L$ is a complex line passing through $z$, then obviously $\Omega \cap L$ is a finely open neighborhood of $z$ in $L$. Now let $C_L$ denote the finely connected component of $z$ in $\Omega\cap L$. Then $\cup_{L\ni z} C_L$ is a pluri-finely connected neighborhood of $z$. As a consequence we find that if $v$ is a finely plurisubharmonic function defined on a pluri-finely connected pluri-finely open set, then $v= -\infty$ on a pluri-finely open subset implies $v\equiv -\infty$.