ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott's theorem and Bogomolov's conjecture, so that we can recover the original Raynaud's theorem (Manin-Mumford's conjecture). CONTENTS

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 other applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.

ABSTRACT. In this article, we generalize several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors and Zariski decompositions for arithmetic divisors on arithmetic surfaces, to the case of the adelic arithmetic divisors. CONTENTS

569License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ATSUSHI MORIWAKIAn immediate application of our inequality is a solution concerning the positivity of divisors on the moduli space of stable curves. Let g ≥ 2 be an integer, and let M g (resp. M g ) be the moduli space of stable (resp. smooth) curves of genus g over k. The boundary M g \ M g is of codimension one and has [g/2] + 1 irreducible components, say, ∆ 0 , ∆ 1 , . . . , ∆ [g/2] . The geometrical meaning of index is as follows. A general point of ∆ 0 represents an irreducible stable curve with one node, and a general point of ∆ i (i > 0) represents a stable curve consisting of a curve of genus i and a curve of genus g − i joined at one point. Let δ i be the class of ∆ i in Pic(M g ) ⊗ Q (strictly speaking, δ i = c 1 (O(∆ i )) for i = 1, and δ 1 = 1 2 c 1 (O(∆ 1 ))), and let λ be the Hodge class on M g . A fundamental problem due to Mumford [17] is to decide which Q-divisor aλ − b 0 δ 0 − b 1 δ 1 − · · · − b [g/2] δ [g/2] is positive, where a, b 0 , . . . , b [g/2] are rational numbers. Here, we can use a lot of types of positivity, namely, ampleness, numerical effectivity, effectivity, pseudoeffectivity, and so on. Besides them, we would like to introduce a new sort of positivity for our purposes. Let V be a projective variety over k and U a nonempty Zariski open set of V . A Q-Cartier divisor D on V is said to be numerically effective over U if (D · C) ≥ 0 for all irreducible curves C on V with C ∩ U = ∅. A first general result in this direction was found by Cornalba-Harris [3], Xiao [20] and Bost [2]. They proved that the Q-divisoris numerically effective over M g . As we observed in [15] and [16], it is not sharp in coefficients of δ i (i > 0). Actually, the existence of a certain refinement of the above result was predicted at the end of the paper [3]. Our solution for this problem is the following (cf. Theorem 3.2 and Proposition 1.7). Theorem B (char(k) = 0). The divisor is weakly positive over M g , i.e., if we denote the above divisor by D, then for any ample Q-Cartier divisors A on M g , there is a positive integer n such that n(D + A) is a Cartier divisor andis surjective on M g . In particular, it is pseudo-effective, and numerically effective over M g .As an application of this theorem, we can decide the cone of weakly positive divisors over M g (cf. Corollary 4.4). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use RELATIVE BOGOMOLOV'S INEQUALITY 571Moreover, using Theorem B, we can deduce a certain kind of inequality on an algebraic surface. In order to give an exact statement, we will introduce types of nodes of semistable curves. Let Z be a semistable curve over k, and P a node of Z. We can assign a number i to the node P in the following way. Let ι P : Z P → Z be the partial normalization of Z at P . If Z P is connected, then i = 0. Otherwise, i is the minimum of arithmetic genera of two connected components of Z P ....

We continue the study of the arithmetic geometry of toric varieties started by J. Burgos Gil, P. Philippon, and M. Sombra in 2011. In this text, we study the positivity properties of metrized R \mathbb {R} -divisors in the toric setting. For a toric metrized R \mathbb {R} -divisor, we give formulae for its arithmetic volume and its χ \chi -arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a higher-dimensional analogue of Dirichlet’s unit theorem for toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R \mathbb {R} -divisor, and we prove a toric arithmetic Fujita approximation theorem.