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Positive line bundles on arithmetic varieties. (English) Zbl 0861.14018
For an arithmetic variety (maybe singular at its generic fiber) we prove several fundamental results.
We prove the Hilbert-Samuel formula. When the generic fiber is smooth, this was done by H. Gillet and C. Soulé [Invent. Math. 110, No. 3, 473-543 (1992; Zbl 0777.14008)].
We prove the Nakai-Moishezon theorem. As a consequence, we prove a successive minima theorem for the variety (not necessarily linear).
We prove the Bogomolov conjecture for the torus $$\mathbb{G}^n_m$$.

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 11G35 Varieties over global fields
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##### References:
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