---
tag: T19H7BJ9Y3
kind: definition
name: Positive current
msc: [32U40, 32C30]
defines: BYJHP6A906
uses: [360H7MD9RD, DC81TCR98Y, JSGWAN2RJV]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_current
---

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $0 \le p \le n$.
A [[current|360H7MD9RD]] $T$ of bidegree $(p, p)$ on $X$ is *positive* if
$$T \wedge \left(i \alpha_1 \wedge \bar\alpha_1\right) \wedge \cdots \wedge \left(i \alpha_{n-p} \wedge \bar\alpha_{n-p}\right) \ge 0$$
for every choice of smooth [[$(1, 0)$-forms|DC81TCR98Y]] $\alpha_1, \dots, \alpha_{n-p}$ on $X$.
Here an $(n, n)$-current $S$ is identified with a distribution, and $S \ge 0$ means $S(\chi) \ge 0$ for every smooth compactly supported function $\chi \ge 0$.
Then $S$ is a positive measure.
A current is *closed* if $dT = 0$ in the sense of currents.

## Remarks

The condition is the same as $T(\varphi) \ge 0$ for all compactly supported smooth strongly positive $(n - p, n - p)$-forms $\varphi$, with the current acting on forms of complementary degree.
A positive current is real, and its coefficients in local coordinates are complex measures.
Example: for $X = \C^n$ and $p = 1$, the smooth form $\omega = i \partial\bar\partial \abs{z}^2 = i \sum_j dz_j \wedge d\bar z_j$ is positive, because its coefficient matrix is the identity matrix (see the criterion for $p = 1$ below).
Example: let $Z \subseteq X$ be a complex analytic subset of pure dimension $n - p$.
By Lelong's theorem, the current of integration $[Z](\varphi) = \int_{Z_{\mathrm{reg}}} \varphi$ is a closed positive current of bidegree $(p, p)$.
Example: for $u$ [[plurisubharmonic|FF2P7ER3QT]], $i \partial\bar\partial u$ is a closed positive $(1, 1)$-current.
Non-example: on $\C^2$, the form $i \, dz_1 \wedge d\bar z_1 - i \, dz_2 \wedge d\bar z_2$ is not positive; take $\alpha_1 = dz_1$, then the product is $-i\,dz_2 \wedge d\bar z_2 \wedge i \, dz_1 \wedge d\bar z_1$, which is $-4$ times Lebesgue measure.
For $p = 1$, a $(1,1)$-form $i \sum h_{jk} \, dz_j \wedge d\bar z_k$ is positive exactly when the Hermitian matrix $(h_{jk})$ is positive semidefinite at each point.