**Title:
**Adelic Chern Forms and Applications

**Publication
status:** Amer. J. Math. **121** (1999) 797-839.

**Abstract:**

Let X be a variety over a field of characteristic 0. Given a
vector bundle E on X we

construct Chern forms c_{i}(E; \nabla)
\in \Gamma(X, \cal{A}^{2i}_{X}).

Here \cal{A}^{.}_{X} is the
sheaf of Beilinson adeles and \nabla is an adelic

connection.
When X is smooth H^{p} \Gamma(X, \cal{A}^{.}_{X}) =

H^{p}_{DR}(X),
the algebraic De Rham cohomology, and

c_{i}(E) = [c_{i}(E;
\nabla)] 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.

**Electronic
Preprint:**

compressed postscript file (.zip, 192K)

AMSLaTeX file (137K)

updated: 31 Aug 2008