Adelic Chern Forms and Applications

1998/11/18 by Reinhold Huebl, Huebl, Reinhold, Amnon Yekutieli +1
Mathematics · #11R56 #14C17 #18G30 #53C05 #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #Primary: 14F40 #Secondary: 14F10 #math.AG #math.CT #msc:11R56 #msc:14C17 #msc:14F10 #msc:14F40 #msc:18G30 #msc:53C05

paper · pdf · doi:10.48550/arxiv.math/9811109

34 pages, AMSLaTeX, to appear in Amer. J. Math

arxiv created 1998/11/18 · arxiv updated 2009/11/30

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;∇) in Γ(X, \calA²ⁱ_X). Here \calA^._X is the sheaf Beilinson adeles and ∇ is an adelic connection. When X is smooth these adeles calculate 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.

Related

• Bott-Chern Forms and Arithmetic Intersections
• Chern Classes via Derived Determinant
• Cech - de Rham Chern character on the stack of holomorphic vector bundles
• Higher Bott Chern forms and Beilinson's regulator
• Intersection theory and Chern classes in Bott-Chern cohomology