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

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.

### 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

### Citations:

Zbl 0559.14005; Zbl 0706.53036
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