$\nu(T,D)=0$ for all prime divisors $D$ in $X$.

\(ii) $\alpha$ is said to be a modified nef class iff, for every ${\varepsilon}>0$, there exists a closed $(1,1)$-current $T_{{\varepsilon}}$ in $\alpha$ with $T_{{\varepsilon}}\geq-{\varepsilon}\omega$ and $\nu(T_{{\varepsilon}},D)=0$ for every prime $D$.

This is again independent of the choice of $\omega$ by commensurability of the Hermitian forms. The set of modified Kähler classes is an open convex cone called the modified Kähler cone and denoted by ${\mathcal{MK}}$. Similarly, we get a closed convex cone ${\mathcal{MN}}$, the modified nef cone. Using the Siu decomposition, we immediately see that ${\mathcal{MK}}$ is non-empty iff the big cone ${\mathcal{B}}$ is non-empty, in which case ${\mathcal{MK}}$ is just the interior of ${\mathcal{MN}}$.\
[**Remark 1**]{}: upon regularizing the currents using (ii) of theorem 2.1, we can always assume that the currents involved in the definition have analytic singularities along a subcheme of codimension at least 2.\
[**Remark 2**]{}: the modified nef cone of a compact complex surface is just its nef cone (cf. section 4.2.1).\
[**Remark 3**]{}: just as for nef classes, one cannot simply take ${\varepsilon}=0$ in the definition of a modified nef class. We recall the example given in \[DPS94\]: there exists a ruled surface $X$ over an 