## 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 H^{P}Γ(X, A^{̇}_{X}) = H^{P}_{DR}(X), 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.

Original language | English |
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Pages (from-to) | 797-839 |

Number of pages | 43 |

Journal | American Journal of Mathematics |

Volume | 121 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 1999 |

## ASJC Scopus subject areas

- Mathematics (all)