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**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.

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 |