---
tag: FGSKR6ZS3Q
kind: concept
name: Plurisubharmonic exhaustion function
msc: [32U10, 32T35]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
---

## Summary

A plurisubharmonic exhaustion function on a complex manifold is a [[plurisubharmonic function|FF2P7ER3QT]] whose sublevel sets are relatively compact.
It tends to $+\infty$ at the "boundary" or "infinity" of the manifold.
Domains in $\C^n$ with such a function are exactly the pseudoconvex domains.
Complex manifolds with a smooth strictly plurisubharmonic one are exactly the Stein manifolds.

## Explanations

### L1 human

An exhaustion function is like a height function on a bowl.
Each region below a given height is a compact piece, and the pieces fill up the whole space as the height grows.
Asking the function to be plurisubharmonic means that it is subharmonic on every complex line, a complex form of convexity.
On $\C^n$ the function $\abs{z}^2$ is a plurisubharmonic exhaustion function: its sublevel sets are balls.
On the unit disc, $-\log(1 - \abs{z}^2)$ is one: it is subharmonic and blows up at the circle.