A THEORY OF CHARACTERISTIC CURRENTS ASSOCIATED WITH A SINGULAR CONNECTION

This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particula r, a Chem-Weil-Simons theory for smooth bundle maps alpha : E --> F wh ich, for smooth connections on E and F, establishes formulas of the ty pe phi = Res(phi)SIGMA(alpha) + dT. Here phi is a standard charactersi tic form, Res(phi) is an associated smooth ''residue'' form computed c anonically in terms of curvature, SIGMA(alpha) is a rectifiable curren t depending only on the singular structure of alpha, and T is a canoni cal, functorial transgression form with coefficients in L(loc)1. The t heory encompasses such classical topics as: Poincare-Lelong Theory, Bo tt-Chem Theory, Chem-Weil Theory, and formulas of Hopf. Applications i nclude: a new proof of the Riemann-Roch Theorem for vector bundles ove r algebraic curves, a C(infinity)-generalization of the Poincare-Lelon g Formula, universal formulas for the Thom class as an equivariant cha racteristic form (i.e., canonical formulas for a de Rham representativ e 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 ar e deduced as direct consequences of the theory.