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A theory of characteristic currents associated with a singular connection
F. Reese Harvey, H. Blaine Lawson, Jr.
Astérisque 213 (1993), 160 pages
This monograph presents a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil theory for smooth bundle maps which, for smooth connections on E and F, establishes formulas of the type
Here is a standard characteristic form, Res is an associated smooth ``residue" form computed canonically in terms of curvature is a rectifiable current depending only on the singular structure of , and T is a canonical, functorial transgression form with coefficients in L1loc. The theory encompasses such classical topics as: Poincaré-Lelong Theory, Bott-Chern Theory, Chern-Weil Theory, and formulas of Hopf. Applications include: A new proof of the Riemann-Roch Theorem for vector bundles over algebraic curves; A -generalization of the Poincaré-Lelong Formula; Universal formulas for the Thom class as an equivariant characteristic form (i. e. , canonical formulas for a de Rham representative of the Thom class of a bundle with connection); A Differentiable Riemann-Roch-Grothendieck Theorem at the level of forms and currents. A variety of formulas relating geometry and characteristic classes are deduced as direct consequences of the theory.
Class. math. : 53C07, 49Q15 (principale), 58H99, 57R20 (secondaire)