Alberto Camara, Notes on Arakelov theory, 2011 (pdf)
The theory originates in
Suren Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR Izv. 8 (6): 1167–1180, 1974, doi; Theory of intersections on an arithmetic surface, Proc. ICM Vancouver 1975, vol. 1, 405–408, Amer. Math. Soc. 1975, djvu, pdf
After Arakelov there were main improvements by Faltings and Gillet and Soulé.
Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424, MR86e:14009, doi; Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983), no. 3, 337–347, MR85m:14061, doi
a completely algebraic replacement (using generalized schemes whose local models are spectra of commutative algebraic monads) for the original mixed approach is proposed; it is not known if that approach can be closely and precisely compared with the traditional.