  Using currents with minimal singularities, we construct pointwise minimal
multiplicities for a real pseudo-effective $(1,1)$-class $\alpha$ on a compact
complex $n$-fold
  $X$, which are the local obstructions to the numerical effectivity of
$\alpha$. The negative part of $\alpha$ is then defined as the real effective
divisor $N(\alpha)$ whose multiplicity along a prime divisor $D$ is just the
generic multiplicity of $\alpha$ along $D$, and we get in that way a divisorial
Zariski decomposition of $\alpha$ into the sum of a class $Z(\alpha)$ which is
nef in codimension 1 and the class of its negative part $N(\alpha)$, which is
exceptional in the sense that it is very rigidly embedded in $X$. The positive
parts $Z(\alpha)$ generate a modified nef cone, and the pseudo-effective cone
is shown to be locally polyhedral away from the modified nef cone, with
extremal rays generated by exceptional divisors. We then treat the case of a
surface and a hyper-K\"ahler manifold in some detail: under the intersection
form (resp. the
  Beauville-Bogomolov form), we characterize the modified nef cone and the
exceptional divisors; our divisorial Zariski decomposition is orthogonal, and
is thus a rational decomposition, which fact accounts for the usual existence
statement of a Zariski decomposition on a projective surface, which is thus
extended to the hyper-K\"ahler case. Finally, we explain how the divisorial
Zariski decomposition of (the first Chern class of) a big line bundle on a
projective manifold can be characterized in terms of the asymptotics of the
linear series $|kL|$ as $k\to\infty$.
