Arakelov geometry studies a scheme X over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X. This extra Hermitian structure is applied as a substitute, for the failure of the scheme Spec(Z) to be a complete variety.
In (Arakelov 1974, 1975) Arakelov defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields. In Gerd Faltings (1984), Faltings extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context.
Arakelov's theory was generalized by Gillet and Soulé to higher dimensions. That is, Gillet and Soulé defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soulé is the arithmetic Riemann–Roch theorem of Gillet & Soulé (1992), an extension of the Grothendieck–Riemann–Roch theorem to arithmetic varieties. For this one defines arithmetic Chow groups CHp(X) of an arithmetic variety X, and defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups. The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete proof of this theorem was only published recently by Gillet, Rössler and Soulé.
Arakelov's intersection theory for arithmetic surfaces was developed further by Bost (1999). The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space L21. In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.
Arithmetic Chow groups
An arithmetic cycle of codimension p is a pair (Z, g) where Z ∈ Zp(X) is a p-cycle on X and g is a Green current for Z, a higher dimensional generalization of a Green function. The arithmetic Chow group of codimension p is the quotient of this group by the subgroup generated by certain "trivial" cycles.
The arithmetic Riemann–Roch theorem
The usual Grothendieck–Riemann–Roch theorem describes how the Chern character ch behaves under pushforward of sheaves, and states that ch(f*(E))= f*(ch(E)TdX/Y), where f is a proper morphism from X to Y and E is a vector bundle over f. The arithmetic Riemann–Roch theorem is similar except that the Todd class gets multiplied by a certain power series. The arithmetic Riemann–Roch theorem states
- X and Y are regular projective arithmetic schemes.
- f is a smooth proper map from X to Y
- E is an arithmetic vector bundle over X.
- is the arithmetic Chern character.
- TX/Y is the relative tangent bundle
- is the arithmetic Todd class
- R(X) is the additive characteristic class associated to the formal power series
- Manin & Panchishkin (2008) pp.400–401
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