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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; ∇) ∈ Γ(X, A 2i X ). Here A · X is the sheaf of Beilinson adeles and ∇ is an adelic connection. When X is smooth 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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