Paper detail

Higher dimensional Zariski decompositions

Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $α$ on a compact complex $n$-fold $X$, 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 $α$ along $D$, and we get in that way a divisorial Zariski decomposition of $α$ 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 exceptional 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-Kähler 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-Kähler 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 $k\to\infty$.