with respect to $x\in X$.

Point (ii) enables us in particular to approximate a Kähler current $T$ inside its cohomology class by Kähler currents $T_k$ with analytic singularities, with a very good control of the singularities. A big class therefore contains plenty of Kähler currents with analytic singularities.

Intersection of currents
------------------------

Just as cycles, currents can be intersected provided their singular sets are in an acceptable mutual position. Specifically, let $T$ be a closed positive $(1,1)$-current on a complex manifold $X$. Locally, we have $T=dd^c\varphi$ with $\varphi$ a pluri-subharmonic function, which is well defined modulo a pluri-harmonic (hence smooth) function. We therefore get a globally well-defined unbounded locus $L(T)$, which is the complement of the open set of points near which $\varphi$ is locally bounded. Assume now that $T_1$, $T_2$ are two closed positive $(1,1)$-currents such that $L(T_j)$ is contained in an analytic set $A_j$ (which may be $X$); locally, we write $T_j=dd^c\varphi_j$ with $\varphi_j$ a pluri-subharmonic function. If $A_1\cap A_2$ has codimension at least $2$, then it is shown in \[Dem92\] that $\varphi_1dd^c\varphi_2$ has locally finite mass, and that $dd^c\varphi_1\wedge dd^c\varphi_2:=dd^c(\varphi_1
dd^c\varphi_2)$ yields a globally defined closed positive $(2,2)$-current, denoted by $T_1\wedge T_2$. It is also true that 