$H^0(kP)\to H^0(kL)$ is necessarily an isomorphism in that case, relating the decomposition to the original problem.\
The proof of the uniqueness in this decomposition shows that the negative part $N$ only depends on the class $\{D\}$ of $D$ in the Néron-Severi group $NS(X)$, so that $\{D\}\mapsto\{P\}$ yields a map from part of the pseudo-effective cone to the nef cone, which we want to study geometrically.\
Building upon the construction by J.-P.Demailly of metrics with minimal singularities on a pseudo-effective line bundle $L$ over a compact complex $n$-fold, we define the minimal multiplicity $\nu(\alpha,x)$ of an arbitrary real pseudo-effective $(1,1)$-class $\alpha$ on a compact complex $n$-fold $X$ at some point $x\in X$. This multiplicity $\nu(\alpha,x)$ is the local obstruction at $x$ to the numerical effectivity of $\alpha$, and we then get the negative part of such a class $\alpha$ by setting $N(\alpha)=\sum\nu(\alpha,D)D$, where $D$ ranges over the prime divisors of $X$ and $\nu(\alpha,D)$ is the generic multiplicity of $\alpha$ along $D$ (cf. section 3). This negative part $N(\alpha)$ is an effective ${\mathbf{R}}$-divisor which is exceptional in the sense that it is very rigidly imbedded in $X$. When $X$ is a surface, the divisors we obtain in that way are exactly the 