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

TL;DR

This paper extends the theory of restricted volumes and augmented base loci to real divisors on normal projective varieties, providing new characterizations and generalizations using recent advances in algebraic geometry.

Contribution

It introduces a definition of restricted volume for real divisors and generalizes key results relating base loci and volume vanishing to this broader setting.

Findings

Restricted volume coincides with the usual when the subvariety is outside the augmented base locus.

Characterization of the augmented base locus as the union of subvarieties where the restricted volume vanishes.

The largest open subset where the Kodaira map is an isomorphism is identified with the complement of the augmented base locus.

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 $(\ast) \ D \sim_{\mathbb R} t_1 H_1 + \ldots + t_s H_s$ 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 $Z \not\subseteq \mathbf{B}_+(D)$. 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.