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

Let $X$ be a normal projective variety defined over an algebraically closed field and let $Z$ be a subvariety. Let $D$ be an $\mathbb{R}$-Cartier $\mathbb R$-divisoron$X$\mathrm{.}Givenanexpression$(\ast) \ D \sim_{\mathbb R} t_1 H_1 + \ldots + t_s H_s$with$t_i \in \mathbb R$and$H_i$veryample,wedefinethe$(\ast)$-restrictedvolumeof$D$to$Z$ and we show that it coincides with the usual restricted volume when $Z\subseteq ̸{\mathbf{B}}_{+}(D)$. Then, using some recent results of Birkar (arXiv:1312.0239), we generalize to $\mathbb R$-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, Musta\c{t}\u{a}, Nakamaye and Popa, is the characterization of ${\mathbf{B}}_{+}(D)$ as the union of subvarieties on which the $(\ast )$-restricted volume vanishes; the second is that $X-{\mathbf{B}}_{+}(D)$ is the largest open subset on which the Kodaira map defined by large and divisible $(\ast )$-multiples of $D$ is an isomorphism.