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- W4299800310 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 $alpha : E rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ phi = text{rm Res}_{phi}Sigma_{alpha} + dT. $$ Here $phi$ is a standard charactersitic form, $text{Res}_{phi}$ is an associated smooth ``residue'' form computed canonically in terms of curvature, $Sigma_{alpha}$ is a rectifiable current depending only on the singular structure of $alpha$, and $T$ is a canonical, functorial transgression form with coefficients in $loc$. The theory encompasses such classical topics as: Poincar'e-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 $C^{infty}$-generalization of the Poincar'e-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." @default.
- W4299800310 created "2022-10-03" @default.
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- W4299800310 date "1994-06-30" @default.
- W4299800310 modified "2023-09-24" @default.
- W4299800310 title "A theory of characteristic currents associated with a singular connection" @default.
- W4299800310 doi "https://doi.org/10.48550/arxiv.math/9407216" @default.
- W4299800310 hasPublicationYear "1994" @default.
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