Positive current

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, $i \partial\bar\partial u$ 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.