On Volumes of Arithmetic Line Bundles II

This paper uses convex bodies to study line bundles in the setting of Arakelov theory. The treatment is parallel to [Yu2], but the content is independent. The method of constructing a convex body in Euclidean space, now called “Okounkov body”, from a given algebraic linear series was due to Okounkov [Ok1, Ok2], and was explored systematically by Kaveh–Khovanskii [KK] and Lazarsfeld–Mustaţǎ [LM]. Many important results of algebraic geometry can be derived from convex geometry through the bridge that the volume of the convex body gives the volume of the linear series. Let K be a number field, X be an arithmetic variety of relative dimension d over OK , and L be a hermitian line bundle over X . There are two important arithmetic invariants ĥ0(L) and χ(L). Their growth under tensor powers are measured respectively by vol(L) and volχ(L). In [Yu2], we have introduced the Okounkov body ∆(L) ⊂ R of L, whose volume computes vol(L). It is a natural arithmetic analogue of the construction in [LM].