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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.

MSC:
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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