Adelic Chern Forms and Applications

الأصل · EN

Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms cᵢ(E;∇) in Γ(X, A²ⁱₓ). Here A.ₓ is the sheaf Beilinson adeles and ∇ is an adelic connection. When X is smooth these adeles calculate the algebraic De Rham cohomology, and cᵢ(E) = [cᵢ(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.