---
author:
- Sébastien Boucksom
title: Higher dimensional Zariski decompositions
---

\[section\] \[theo\][Corollary]{} \[theo\][Definition]{} \[theo\][Proposition]{} \[theo\][Lemma]{}

$Author's$ $address$: Institut Fourier, 100 rue des Maths, BP74, 38402 Saint-Martin d’Hères Cedex, France.\
$e-mail$: sbouckso@ujf-grenoble.fr\
$Abstract$: using currents with minimal singularities, we construct 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-Kähler manifold in some detail: under the intersection form (resp. the Beauville-Bogomolov form), we characterize the 