Complex immersions and Arakelov geometry. (English) Zbl 0744.14015

Bismut, Jean-Michel; Gillet, Henri; Soulé, Christophe

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 249-331 (1990).

[For the entire collection see Zbl 0717.00008.]
This paper forms part of a larger program to prove a Grothendieck- Riemann-Roch theorem in the framework of Arakelov theory. Such a theorem involves Chern classes in the arithmetic Chow groups which are much finer invariants than the usual Chow groups. The resulting Riemann-Roch theorems can be seen as a refinements of the old Riemann-Roch theorems.
In the paper at hand the authors prove a Riemann-Roch theorem for immersions of arithmetic varieties. Besides the Chern character of the hermitian vector bundle and the Todd genus of a normal bundle with a hermitian metric it involves a secondary invariant (a so-called Bott- Chern singular current) of a resolution (a complex of hermitian vector bundles) of the direct image of the vector bundle under the immersion. – - The authors also study the functorial properties of such secondary invariants.

Reviewer: G.van der Geer (Amsterdam)

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14C40	Riemann-Roch theorems
57R20	Characteristic classes and numbers in differential topology
57N35	Embeddings and immersions in topological manifolds
14C05	Parametrization (Chow and Hilbert schemes)

Keywords:

Grothendieck-Riemann-Roch theorem; Arakelov theory; Chern classes; arithmetic Chow groups; Bott-Chern singular current

Citations:

Zbl 0717.00008