Restricted volumes and divisorial Zariski decompositions (Q2851613)

Let \(D\) be a big divisor on a complex smooth projective variety \(X\), and let \(V\) be an irreducible subvariety of \(X\). The notion of restricted volume \(\text{vol}_{X|V}(D)\) of \(D\) along \(V\) has been introduced by \textit{H. Tsuji} [Osaka J. Math. 43, No. 4, 967--995 (2006; Zbl 1142.14012)] and applies in various situations. Essentially, as a function of \(D\), \(\text{vol}_{X|V}(D)\) depends only on the numerical class of \(D\). The author relates the existence of a Zariski decomposition for \(D\) to the fact that, as a function of \(V\), \(\text{vol}_{X|V}(D)\) depends only on the numerical class of \(V\). Moreover, he describes \(\text{vol}_{X|V}(D)\) analytically, in terms of positive \((1,1)\)-currents and integrals over the regular locus of \(V\), generalizing \textit{S. Boucksom}'s description of the usual volume [Int. J. Math. 13, No. 10, 1043--1063 (2002; Zbl 1101.14008)]. This enables him to extend the notion of restricted volume to transcendental classes in \(H^{1,1}(M, \mathbb R)\) on a compact Kähler manifold \(M\) and to revisit the above mentioned relationship in this analytic setting, for an appropriate notion of Zariski decomposition for transcendental classes.