geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The aim of Arakelov geometry is to extend intersection theory to the case of algebraic curves over $Spec(\mathbb{Z})$, hence in arithmetic geometry.
Arakelov complemented the algebraic geometry at finite primes with a holomorphic piece at a place at infinity. Then using complex analytic geometry and Green's functions he defined the intersections numbers? using the complementary piece at infinity.
… e.g. (Durov 07)
Wikipedia: Arakelov theory
Serge Lang, Introduction to Arakelov theory Springer-Verlag, New York, 1988.
Christophe Soulé, D. Abramovich, , J.-F. Burnol, J. Kramer, Lectures on Arakelov Geometry, Cambridge University Press 1991
Robin de Jong, Explicit Arakelov geometry, PhD thesis 2004 (pdf)
Alberto Camara, Notes on Arakelov theory, 2011 (pdf)
The theory originates in
After Arakelov there were main improvements by Faltings and Gillet and Soulé.
The arithmetic Riemann-Roch theorem is due to
Henri Gillet, Christophe Soulé, An arithmetic Riemann–Roch Theorem, Invent. Math. 110: 473–543, 1992, doi
Shou-Wu Zhang, Small points and Arakelov theory, Proc. ICM 1998, vol. 2, djvu, pdf
In a recent Bonn thesis under Faltings’ supervision,
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.
See also
Jean-Benoit Bost, Klaus Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers (arXiv:math/0701343)
Jean-Benoit Bost, Klaus Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. II. The arithmetic Atiyah extension (arXiv:0807.4374)