Restricted volumes of effective divisors

TL;DR

This paper explores the properties of restricted volume of effective divisors, linking it with reduced volume and asymptotic intersection numbers, extending key geometric concepts to broader divisor classes and establishing new approximation results.

Contribution

It extends the theory of restricted volume to effective divisors, generalizing previous results and establishing a Fujita-type approximation in this broader context.

Findings

Established a relation between restricted volume and reduced volume.

Proved a Fujita-type approximation for effective divisors.

Analyzed boundedness of asymptotic multiplier ideals.

Abstract

We study the restricted volume of effective divisors, its properties and the relationship with the related notion of reduced volume, defined via multiplier ideals, and with the asymptotic intersection number. We build upon the fundamental work of Lazarsfeld and Mustata relating the restricted volume of big divisors to the volume of the associated Okounkov body. We extend their constructions and results to the case of effective divisors, recovering some results of Kaveh and Khovanskii, proving a Fujita-type approximation in this larger setting and studying the restricted volume function. In order to relate the reduced volume and the asymptotic intersection number we investigate a boundedness property of asymptotic multiplier ideals and prove it holds, for instance, for finitely generated divisors. In this way we obtain also a complete picture for the canonical divisor of an arbitrary smooth projective variety and for nef divisors on varieties of dimension at most 3.