effective ${\mathbf{R}}$-divisors whose support $D_1,...,D_r$ have negative definite Gram matrix $(D_i\cdot D_j)$.\
The difference $Z(\alpha):=\alpha-\{N(\alpha)\}$ is a real $(1,1)$-class on $X$ which we call the Zariski projection of $\alpha$. It is not a nef class, but is somehow nef in codimension 1. More precisely, we define the modified nef cone of a Kähler $n$-fold to be the closed convex cone generated by the classes in ${H^{1,1}(X,{\mathbf{R}})}$ which can be written as the push-forward of a Kähler class by a modification. We then show that the Zariski projection $Z(\alpha)$ of a pseudo-effective class $\alpha$ belongs to this modified nef cone. The decomposition $\alpha=Z(\alpha)+\{N(\alpha)\}$ we call the divisorial Zariski decomposition, and it is just induced by the Siu decomposition of a positive current with minimal singularities in $\alpha$ when the latter is big. For such a big class, we give a criterion to recognize a decomposition $\alpha=p+\{N\}$ into a modified nef and big class and the class of an effective real divisor as the divisorial Zariski decomposition of $\alpha$, in terms of the non-Kähler locus of $p$ (cf. section 3.5)\
The geometric picture is now as follows: the pseudo-effective cone of a compact complex $n$-fold $X$ is locally polyhedral away from the 