Positive line bundles on arithmetic surfaces. (English) Zbl 0788.14017

Zhang, Shouwu

Ann. Math. (2) 136, No. 3, 569-587 (1992).

The paper, which is part of the author’s doctoral thesis, contains two main results: a Nakai-Moishezon theorem on an arithmetic surface, result which was conjectured by Szpiro, and an analogue of a conjecture of Bogomolov about the discreteness of algebraic points on an algebraic curve.
More precisely, let \(X\) be an arithmetic surface and \(\overline L=(L,\| \|)\) a Hermitian line bundle on \(X\) \((L\) invertible sheaf on \(X\), \(\| \|\) a continuous Hermitian metric on \(L_ \mathbb C\), invariant under the complex conjugation of \(C_ \mathbb C)\). A nonzero section \(\ell\) of \(L\) on \(X\) is “strictly effective” if \(\|\ell\|(x)<1\) for all \(x\in X_ \mathbb C\). One says that \(\overline L\) is “ample” if \(L\) is ample, the curvature form \(\omega(\overline L)\) of \(\overline L\) is semipositive, and there is a basis of \(\Gamma(L^ n)\) over \(\mathbb Z\) consisting of strictly effective sections for all sufficiently large \(n\). One says that \(\overline L\) is “positive” if \(\omega(\overline L)\) is semipositive, \(\overline L\cdot\overline L\) (the intersection number of Deligne) is positive, and \(\deg(\overline L| D):=\deg[\text{ div}\;\ell]-\sum_{x\in D_ \mathbb C}\log\|\ell\|(x)\) \((\ell\) a section of \(L| D)\) is positive for any integral divisor \(D\) on \(X\). The Nakai-Moishezon arithmetic theorem proved by the author says that: \(\overline L\) is ample if and only if \(\overline L\) is positive. The analogue of Bogomolov’s conjecture has the following statement:
Let \(C\to \mathbb G^ n_ m\) \((\mathbb G_ m=\) the multiplicative group) be an embedding of a curve defined over a number field \(K\). Assume that \(C\) is not a translate of a subgroup of \(G^ n_ m\). Then \(C(\overline K)\) is discrete under a certain semipositive distance function \(d_ \infty\) on \(\mathbb G^ n_ m(\overline{\mathbb Q})\) defined in terms of a canonical height function on the projective line \(\mathbb P^ 1(\overline{\mathbb Q})\).
The proofs use results of G. Faltings [Ann. Math. (2) 119, 387–424 (1984; Zbl 0559.14005)] and a theorem of G. Tian [J. Differ. Geom. 32, No. 1, 99–130 (1990; Zbl 0706.53036)] on Fubini-Study metrics.

Reviewer: I.Coandă (Bucureşti)

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14G05	Rational points
11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G50	Heights
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60	Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14H25	Arithmetic ground fields for curves

Keywords:

Nakai-Moishezon theorem; arithmetic surface; discreteness of algebraic points on an algebraic curve; hermitian line bundle; canonical height

Citations:

Zbl 0559.14005; Zbl 0706.53036