Adelic Chern forms and applications

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; ∇) ∈ Γ(X, A²ⁱ_X). Here A^̇_X is the sheaf of Beilinson adeles and ∇ is an adelic connection. When X is smooth H^PΓ(X, A^̇_X) = H^P_DR(X), the algebraic De Rham cohomology, and C_i(E) = [c_i(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.