Abstract

Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms ci(E;∇) ∈ Γ(X,A2iX). Here A·X is the sheaf of Beilinson adeles and ∇ is an adelic connection. When X is smooth HpΓ(X,A·X) = H pDR(X), the algebraic De Rham cohomology, and ci(E) = [ci(E;∇)] are the usual Chern classes. We include three applications of the construction: (1) existence of adelic secondary (Chern-Simons) characteristic classes on any smooth X and any vector bundle E; (2) proof of the Bott Residue Formula for a vector field action; and (3) proof of a Gauss-Bonnet Formula on the level of differential forms, namely in the De Rham-residue complex.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 797-839
Launched on MUSE
1999-08-01
Open Access
No
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