Atsushi Moriwaki

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 Pic(X) the group of isomorphism classes of C∞-hermitian invertible sheaves onX . For L ∈ Pic(X), the volume vol(L) of L is defined by vol(L) = lim sup m→∞ log#{s ∈ H(X,mL) | ∥s∥sup ≤ 1} md/d! . For example, if L is ample, then vol(L) = deg(ĉ(L)·d) (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 C∞hermitian invertible sheaf by the positivity of its volume (cf. Theorem 4.5). The second one is the homogeneity of the volume function, namely, vol(nL) = nvol(L) for all nonnegative integers n (cf. Proposition 4.7). By this property, it can be extended to Pic(X)⊗ Q. From viewpoint of arithmetic analogue, the most important and fundamental question is the continuity of vol : Pic(X)⊗Q → R, that is, the validity of the formula: lim e1,...,en∈Q e1→0,...,en→0 vol(L+ e1A1 + · · ·+ enAn) = vol(L) for any L,A1, . . . , An ∈ Pic(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). 1991Mathematics Subject Classification. 14G40, 11G50. 1

Bogomolov conjecture over function fields for stable curves with only irreducible fibers

Let K be a function field and C a non-isotrivial curve of genus g≥2 overK. In this paper, we will show that if C has a global stable modelwith only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.

Zariski Decompositions on Arithmetic Surfaces

In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic varieties.

Arithmetic Bogomolov-Gieseker's Inequality

Let f : X --> Spec(Z) be an arithmetic variety of dimension d >= 2 and (H, k) an arithmetically ample Hermitian line bundle on X. Let (E, h) be a rank r vector bundle on X. In this paper, we will prove that if E is semistable with respect to H on each connected component of the infinite fiber of X, then { c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 } c_1(H, k)^{d-2} >= 0. Moreover, if the equality of the above inequality holds, then E is projectively flat and h is a weakly Einstein-Hermitian metric.

Inequality of Bogomolov-Gieseker type on arithmetic surfaces

Let K be an algebraic number field, O_K the ring of integers of K, and f : X --> Spec(O_K) an arithmetic surface. Let (E, h) be a rank r Hermitian vector bundle on X such that

Torsion freeness of higher direct images of canonical bundles

E

The continuity of Deligne's pairing

is semistable on the geometric generic fiber of f. In this paper, we will prove an arithmetic analogy of Bogomolov-Giesekers inequality: c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 >= 0.

Estimation of arithmetic linear series

Let X be a non-singular projective variety and f : X ~ Y a surjective morphism of projective varieties. Koll~tr [4] proved that R~f, cox is torsion free for every i > 0, where co x is the canonical line bundle on X. The statement of his theorem is clearly local analytic with respect to Y. But his proof depends on Delignes global results (a mixed Hodge structure of an algebraic variety, semi-simplicity of a representation of a fundamental group etc.). In this paper, we extend his result to a local analytic situation under the assumption of projectivity of f. Our arguments are based on several vanishing theorems [9, 10, 13] and Zuckers result [16]. A key theorem in our paper is Theorem (2.4), which is a generalization of Kollfir [5] and Nakayama [8]. The assumption of projectivity of the morphism is crucial in our proof. But the author thinks optimistically that most of our results are valid for a K/ihler morphism. Saito proved the same results independently, who uses his theory of Hodge modules [11]. Section 1 is a preliminary section in which we prove several vanishing theorems. The author was taught Theorem (1.1) by Prof. T. Ohsawa. In Sect. 2, first we discuss the upper and lower canonical extension of a variation of Hodge structure, which are introduced in [5]. Next we prove some relationships between variations of Hodge structures and higher direct images (Theorem (2.4)). These imply the semi-positivity of Rif, tOx/r (Corollary (2.10)). In Sect. 3, we prove the torsion freeness of higher direct images of canonical bundles and its generalizations.

Toward Dirichlet’s unit theorem on arithmetic varieties

Theorem A. Let f : X → S be a flat and projective morphism of varieties over C with n = dim f . Let L0, . . . , Ln be C -hermitian line bundles on X. Then, the metric of Deligne’s pairing 〈L0, . . . , Ln〉(X/S) is continuous. Here we note that the C of hermitian line bundles in this note is slightly stronger than the sense of [11] and [8]. For simplicity, we explain it in terms of the C of functions. A function φ on X is said to be C at x ∈ X (in this note) if there are an open neighborhood U of x, a complex manifold V , and a C-function Φ on V such that U is a closed analytic subset of V and φ = Φ|U . As an application of the above theorem, we have the following result, which is, in some sense, a generalization of Kawaguchi’s Hodge index theorem [6].

Semi-ampleness of the numerically effective part of Zariski decomposition

Lazarsfeld and Mustata propose general and systematic usage of Okounkovs idea in order to study asymptotic behavior of linear series on an algebraic variety. It is a very simple way, but it yields a lot of consequences, like Fujitas approximation theorem. Yuan generalized this way to the arithmetic situation, and he established the arithmetic Fujitas approximation theorem, which was also proved by Chen independently. In this paper, we introduce arithmetic linear series and give a general way to estimate them based on Yuans idea. As an application, we consider an arithmetic analogue of the algebraic restricted volumes.