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Restricted volumes and divisorial Zariski decompositions. (English) Zbl 1277.14006
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 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 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.

14C20 Divisors, linear systems, invertible sheaves
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32J27 Compact Kähler manifolds: generalizations, classification
32U40 Currents
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