---
tag: BYJHP6A906
kind: concept
name: Positive current
msc: [32U40, 32C30]
specializes: [360H7MD9RD]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_current
same_as:
  wikidata: Q7233260
  wikipedia: Positive current
---

## Summary

A positive current is a [[current|360H7MD9RD]] of bidegree $(p, p)$ on a [[complex manifold|JSGWAN2RJV]] with a positivity property.
Its wedge product with products of the forms $i \alpha \wedge \bar\alpha$, for $(1, 0)$-forms $\alpha$, is a nonnegative measure.
Pierre Lelong introduced closed positive currents, and they are now a central tool of complex geometry.
The two basic examples are the current of integration over a complex analytic subvariety and the current $dd^c u$ of a [[plurisubharmonic function|FF2P7ER3QT]] $u$.
The coefficients of a positive current are complex measures, and closed positive currents carry [[Lelong numbers|WVB60RKC3Z]] that measure their singularities.

## Explanations

### L1 human

A current is a generalised differential form whose coefficients may be measures or even distributions.
In complex geometry there is a natural notion of positivity for forms of type $(p, p)$, which extends the idea of a nonnegative function or of a positive semidefinite Hermitian matrix.
A positive current is a current that is positive in this sense.
The simplest example is integration over a complex curve $C$ in $\C^2$: it sends a $2$-form to its integral over $C$, and complex curves always carry a positive orientation.
Another example is $i \partial\bar\partial \abs{z}^2$, the standard Kähler form of $\C^n$.