Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X, which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of along D, and we get in that way a divisorial Zariski decomposition of into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) 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-Kahler manifold in some detail. Using the intersection form (respectively the Beauville–Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The 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?hler 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 → ∞.
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