---
tag: N0MAGPW8AP
kind: concept
name: Pluripolar set
msc: [32U15, 32U20]
sources:
  - url: https://en.wikipedia.org/wiki/Pluripolar_set
same_as:
  wikidata: Q7205562
  wikipedia: Pluripolar set
---

## Summary

A pluripolar set is a subset of $\C^n$ that lies, at least locally, in the set where a [[plurisubharmonic function|FF2P7ER3QT]] takes the value $-\infty$.
Pluripolar sets are the negligible sets of pluripotential theory, in the same way as [[polar sets|C95QJ7YH5T]] are the negligible sets of classical potential theory.
They have Lebesgue measure zero, and plurisubharmonic functions bounded above extend across closed pluripolar sets.
By [[Josefson's theorem|V6XSDT5C21]], a set that is pluripolar locally is pluripolar for one plurisubharmonic function on all of $\C^n$.

## Explanations

### L1 human

The function $\log\abs{f}$ of a holomorphic function $f$ equals $-\infty$ exactly where $f$ vanishes.
So the zero set of a holomorphic function, such as the complex line $z_1 = 0$ in $\C^2$, is pluripolar.
Pluripolar sets are small sets: they cannot contain an open set, and they have volume zero.
A countable union of pluripolar sets is again pluripolar, so the union of countably many complex lines is pluripolar.
The real line $\R^n$ inside $\C^n$ is not pluripolar, although it has volume zero.