Soulé, C. Géométrie d’Arakelov et théorie des nombres transcendants. (Arakelov geometry and transcendental number theory). (French) Zbl 0756.14014 Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 355-371 (1991). [For the entire collection see Zbl 0743.00011.]This paper reviews some central results in Arakelov geometry. For example, an arithmetic Riemann-Roch theorem for an arithmetic variety over \(\text{Spec}(\mathbb{Z})\) is given. Also some aspects of Vojta’s proof of the Mordell conjecture are presented. A new result is a comparison of two definitions of the height of a projective variety, the definition of Faltings and one of Philippon (defined by the Chow form). Reviewer: G.van der Geer (Amsterdam) Cited in 1 ReviewCited in 6 Documents MSC: 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14C40 Riemann-Roch theorems 11J81 Transcendence (general theory) Keywords:Arakelov geometry; arithmetic Riemann-Roch theorem Citations:Zbl 0743.00011 PDFBibTeX XMLCite \textit{C. Soulé}, Astérisque 198--200, 355--371 (1991; Zbl 0756.14014)