Volumes of arithmetic Okounkov bodies

Abstract

This paper is an improvement and simplification of Yuan [Yu]. In particular,by a simpler method, it proves the identity in [Yu, Theorem A] withouttaking the limit p → ∞. The method also simplifies the proofs of Moriwaki[Mo], where more general arithmetic linear series were treated.Recall that [Yu] explored a way to construct arithmetic Okounkov bodiesfrom an arithmetic line bundle, inspired by the idea of Okounkov [Ok1, Ok2]and Lazarsfeld–Musta¸ta [LM] in the geometric case. Theorem A of [Yu]asserts that the volumes of the Okounkov bodies approximate the volume ofthe arithmetic line bundle. The main result of this paper asserts an exactidentity before taking the limit.On the other hand, Boucksom–Chen [BC] initiated a different way toconstruct Okounkov bodies in the arithmetic setting. From the proof ofthe main result in this paper, it is easy to recognize the similarity of theconstructions in [BC] and in [Yu]. We will make a comparison at the end ofthis paper.In the following, we recall the construction of [Yu] and state our mainresult. We use exactly the same notations as in [Yu] throughout this paper.1