### abstract

- Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms c/(?; V) T(X, Ax). Here A'x is the sheaf of Beilinson adeles and V is an adelic connection. When X is smooth WT{X, A'x) = H^R(X), the algebraic De Rham cohomology, and c,-(?) = (q(?; V)) 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. 0. Introduction. Let X be a scheme of finite type over a field k. According to Beilinson (Be), given any quasi-coherent ?x-module M and an integer q9 there is a flasque Ox-module A^dCM), called the sheaf of adeles. This is a gen eralization of the classical adeles of number theory (cf. Example 2.3). Moreover, there are homomorphisms d: Aj?ed(.M) ? A^(M) which make Ared(.M) into a complex, and M ?> Ared(.M) is quasi-isomorphism. Now let Qx, be the algebra of Kahler differential forms on X. In (HY) we proved that the sheaf