Compactification problems in the theory of characteristic currents associated with a singular connection

A compactification of the Chern-Weil theory for bundle maps developed by Harvey and Lawson is described. For each smooth section $\nu$ of the compactification $\IP(\underline{\doubc}\oplus F)\to X$ of a rank n complex vector bundle $F\to X$ with connection, and for each Ad-invariant polynomial $\phi$ on ${\bf gl}\sb{n},$ there are associated current formulae generalizing those of Harvey and Lawson. These are of the form $$\eqalign{\phi(\Omega\sb{\it F}) &+\rm \nu\sp*(Res\sb\infty(\phi))\ Div\sb\infty(\nu) - \phi(\Omega\sb0)\ -\cr&\qquad\rm Res\sb0(\phi)\ Div\sb0(\nu) = {\it dT}\quad on\ {\it X},\cr}$$where Div$\sb0(\nu)$ and Div$\sb\infty(\nu)$ are integrally flat currents supported on the zero and pole sets of $\nu,$ where Res$\sb0(\phi)$ and Res$\sb\infty(\phi)$ are smooth residue forms which can be calculated in terms of the curvature $\Omega\sb{F}$ of F, where T is a canonical transgression form with coefficients in $L\sbsp{\rm loc}{1},$ and where $\phi(\Omega\sb0)$ is an $L\sbsp{\rm loc}{1}$ form canonically defined in terms of a singular connection naturally associated to $\nu.$ These results hold for $C\sp\infty$-meromorphic sections $\nu$ which are atomic. The notion of an atomic section of a vector bundle was first introduced and studied by Harvey and Semmes. The formulae obtained include a generalization of the Poincare-Lelong formula to $C\sp\infty$-meromorphic sections of a bundle of arbitrary rank. Analogous results hold for real vector bundles and for quaternionic line bundles