Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors

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Let X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an R-Cartier R-divisor on X. Given an expression () D ᵣ t₁ H₁ + + tₛ Hₛ with tᵢ ∈ R and Hᵢ very ample, we define the ()-restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z B+(D). Then, using some recent results of Birkar (arXiv:1312.0239), we generalize to R-divisors the two main results of arXiv:1305.4284 by Boucksom, Cacciola and the author: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa, is the characterization of B+(D) as the union of subvarieties on which the ()-restricted volume vanishes; the second is that X - B+(D) is the largest open subset on which the Kodaira map defined by large and divisible ()-multiples of D is an isomorphism.