The Closed and $\partial \overline{\partial} -$ Closed Current

Question

Let $M$ be a Complex Manifold of complex dimension $m$. Let $T$ be a Current on $M$.

What does Directed Current mean?

The current $T$ is said to be Closed if $dT=0$.

What does The Current $T$ is $\partial \overline{\partial} -$ Closed mean?

Answer 1

The operators $d, \partial, \overline \partial, \partial \overline \partial$ act on currents by duality, for instance $\langle dT, \alpha\rangle = \langle T, d \alpha \rangle$ (up to some sign) where $\alpha$ is a test form, and analogously for the other operators. Then, $\partial \overline \partial$- closed means that $\partial \overline \partial T = 0$.

A directed current is a current that is "tangent" to some foliation or lamination. There are different ways of defining them.

Positive closed directed currents are given locally by expressions of the form $\int [L_a] d \mu(\alpha)$, where $[L_a]$ is the current of integration along a plaque of the lamination/foliation directing the current and $\mu$ is a measure on a transverse section.

For positive $\partial \overline \partial$- closed directed currents what you get is something of the form $\int h_a [L_a] d \mu(\alpha)$ for some harmonic functions $h_a$.

A good reference is the recent survey

V. A. Nguyen - Ergodic theorems for laminations and foliations: recent results and perspectives. Acta Math. Vietnam., 46 (2021), no. 1, 9–101. link.