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Complex immersions and Arakelov geometry. (English) Zbl 0744.14015
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.

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)
Biographic References:
Grothendieck, Alexander