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.

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updated: 31 Aug 2008