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