Divisorial Zariski Decomposition and some properties of full mass currents

Let $α$ be a big class on a compact Kähler manifold. We prove that a decomposition $α=α_1+α_2$ into the sum of a modified nef class $α_1$ and a pseudoeffective class $α_2$ is the divisorial Zariski decomposition of $α$ if and only if $\operatorname{vol}(α)=\operatorname{vol}(α_1)$. We deduce from this result some properties of full mass currents.

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