2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77731Let 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 Γ(X, \cal{A}^{2i}_{X}). Here \cal{A}^{.}_{X} is the sheaf Beilinson adeles and \nabla is an adelic connection. When X is smooth these adeles calculate 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.34 pages, AMSLaTeX, to appear in Amer. J. MathAlgebraic GeometryCategory TheoryPrimary: 14F40; Secondary: 14F10, 14C17, 11R56, 18G30, 53C05text