A theory of characteristic currents associated with a singular connection

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 $\alpha :E\to F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$\varphi ={\text{Res}}_{\varphi }{\Sigma }_{\alpha }+dT.$$ Here $\varphi$ is a standard charactersitic form, ${\text{Res}}_{\varphi }$ 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: 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 $C^{\infty }$-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.

10 pages

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