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 \[ \phi = {\operatorname {Res}}_\phi {\Sigma _\alpha } + dT.\] Here $\phi$ is a standard charactersitic form, ${\operatorname {Res}_\phi }$ 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 $L_{{\text {loc}}}^1$. 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.