  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.
