$T_1\wedge T_2$ lies in the product cohomology class $\{T_1\}\cdot\{T_2\}\in H^{2,2}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})$.\
We will only need the following two special cases: if $T_1$ is a closed positive $(1,1)$-current with analytic singularities along a subscheme of codimension at least $2$, then $T_1\wedge T_2$ exists for every closed positive $(1,1)$-current $T_2$.\
If $D_1$ and $D_2$ are two distinct prime divisors, then $[D_1]\wedge
[D_2]$ is a well defined closed positive $(2,2)$-current. Since its support is clearly contained in the set-theoretic intersection $D_1\cap D_2$ (whose codimension is at least $2$), we have $[D_1]\wedge [D_2]=\sum a_j[Y_j]$, where the $Y_j$’s are the components of $D_1\cap D_2$. In fact, it can be shown that $\sum
a_jY_j$ is just the $2$-cycle associated to the scheme-theoretic intersection $D_1\cap D_2$, thus $[D_1]\wedge[D_2]$ is just the integration current associated to the cycle $D_1\cdot D_2$.

The modified nef cone
---------------------

For our purposes, we need to introduce a new cone in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$, which is somehow the cone of classes that are nef in codimension 1. Let $X$ be a compact complex $n$-fold, and $\omega$ be some reference Hermitian form.

Let $\alpha$ be a class in ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$.

\(i) $\alpha$ is said to be a modified Kähler class iff it contains a Kähler current $T$ with 