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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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