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

Abstract

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 $\mathbb{R}$-divisor on $X$. Given an expression $(\ast )D{\sim }_{\mathbb{R}}{t}_1{H}_1+\dots +t_s{H}_s$ with $t_i\in \mathbb{R}$ and $H_i$ very ample, we define the $(\ast )$-restricted volume of $D$ to $Z$ and we show that it coincides with the usual restricted volume when $Z⊈{\mathbf{B}}_{+}(D)$. 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, Mustaţă, 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.

10 pages

References in corpus (1)

• Boundedness of pluricanonical maps of varieties of general type

Cited by in corpus (1)

• Restricted volumes on Kähler manifolds