Continuity of volumes on arithmetic varieties
Continuity of volumes on arithmetic varieties
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DOI：
10.1090/s1056-3911-08-00500-6
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发表时间：
2006-12
影响因子：
1.8
通讯作者：
A. Moriwaki
中科院分区：
文献类型：
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作者：
A. Moriwaki
作为几何体积函数的类比，我们在算术簇上引入了C∞-Hermite可逆层的体积函数。本文的主要结果是算术体积函数的连续性。因此，我们得到了NEF C∞-Hermite可逆层的希尔伯特-塞缪尔算术公式。引言设X是d维射影算术簇，Pic(X)是C-∞-Hermite可逆层的同构类群。对于L∈Pic(X)，L的体积VOL(L)定义为VOL(L)=LIM SUP m→∞LOG#{S∈H(X，mL)|∥S∥SUP≤1}Md/d！。例如，如果L足够大，则VOL(L)=deg(ĉ(L)·d)(cf.引理3.1)。这是域上射影簇上可逆轮的体积函数的算术模拟。几何体积函数通过大的可逆轮对二次几何起着至关重要的作用。从这个意义上说，引入它的算术模拟是非常有意义的。体积函数的第一个重要性质是用其体积的正性来刻画大的C∞厄米特可逆层。定理4.5)。第二个是体积函数的齐性，即对所有非负整数n(cf.命题4.7)。利用这个性质，可以推广到Pic(X)⊗Q。从算术模拟的观点来看，最重要和最基本的问题是VOL：PIC(X)⊗Q→R的连续性，即公式：LIM E1，…，en∈Q e1→0，…，en→0 VOL(L+e1A1+···+ENAN)=VOL(L)对任何L，A1，.。。，An∈Pic(X)⊗Q.本文的主要目的是对上述问题给出肯定的回答(参见.定理5.4)。因此，对于NEF C∞-Hermite可逆束，我们有以下算术希尔伯特-塞缪尔公式：日期：2007年1月5日，17：30(日本)，(2.0版)。1991年《数学学科分类》。14G40、11G50。1
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