# (Pluri)potential compactifications

## TL;DR

This paper introduces a new pluripotential compactification of complex manifolds using pluricomplex Green functions, providing an invariant and compactification method akin to the Martin compactification in potential theory.

## Contribution

It establishes the existence of a norming volume form ensuring all negative plurisubharmonic functions are integrable, leading to a novel compactification of complex manifolds.

## Key findings

- Existence of a norming volume form V on M
- Compactness of the set of bounded negative plurisubharmonic functions
- Embedding of M into a compact set via pluricomplex Green functions

## Abstract

Using pluricomplex Green functions we introduce a compactification of a complex manifold $M$ invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form $V$ on $M$ such that all negative plurisubharmonic functions on $M$ are in $L^1(M,V)$. Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point $w\in M$ with the normalized pluricomplex Green function with pole at $w$ we get an imbedding of $M$ into a compact set and the closure of $M$ in this set is the pluripotential compactification.