Continuity of Volumes on Arithmetic Varieties

We introduce the volume function for C-hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef C-hermitian invertible sheaf. We also give another applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker’s inequality, etc. INTRODUCTION Let X be a d-dimensional projective arithmetic variety and P̂ic(X) the group of isomorphism classes of C-hermitian invertible sheaves on X . For L ∈ P̂ic(X), the volume v̂ol(L) of L is defined by v̂ol(L) = lim sup m→∞ log#{s ∈ H(X,mL) | ‖s‖sup ≤ 1} md/d! . For example, if L is ample, then v̂ol(L) = d̂eg(ĉ(L)) (cf. Lemma 3.1). This is an arithmetic analogue of the volume function for invertible sheaves on a projective variety over a field. The geometric volume function plays a crucial role for the birational geometry via big invertible sheaves. In this sense, to introduce the arithmetic analogue of it is very significant. The first important property of the volume function is the characterization of a big Chermitian invertible sheaf by the positivity of its volume (cf. Theorem 4.5). The second one is the homogeneity of the volume function, namely, v̂ol(nL) = nv̂ol(L) for all nonnegative integers n (cf. Proposition 4.7). By this property, it can be extended to P̂ic(X)⊗ Q. From viewpoint of arithmetic analogue, the most important and fundamental question is the continuity of v̂ol : P̂ic(X)⊗ Q → R, that is, the validity of the formula: lim ǫ1,...,ǫn∈Q ǫ1→0,...,ǫn→0 v̂ol(L + ǫ1A1 + · · ·+ ǫnAn) = v̂ol(L) for any L,A1, . . . , An ∈ P̂ic(X) ⊗ Q. The main purpose of this paper is to give an affirmative answer for the above question (cf. Theorem 5.4). As a consequence, we have the following arithmetic Hilbert-Samuel formula for a nef C-hermitian invertible sheaf: Date: 5/January/2007, 17:30(JP), (Version 2.0). 1991 Mathematics Subject Classification. 14G40, 11G50. 1