A theory of characteristic currents associated with a singular connection

Authors:
Reese Harvey and H. Blaine Lawson

Journal:
Bull. Amer. Math. Soc. **31** (1994), 54-63

MSC:
Primary 58A25; Secondary 32C30, 32L10, 53C65, 57R20

DOI:
https://doi.org/10.1090/S0273-0979-1994-00497-5

MathSciNet review:
1254076

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Abstract: This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps which, for smooth connections on *E* and *F*, establishes formulas of the type

*T*is a canonical, functorial transgression form with coefficients in . 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), and 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.

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DOI:
https://doi.org/10.1090/S0273-0979-1994-00497-5

Article copyright:
© Copyright 1994
American Mathematical Society