A compactification of the Chern-Weil theory for bundle maps developed by Harvey and Lawson is described. For each smooth section ν of the compactification \IP(\doubc―⊕F)→X of a rank n complex vector bundle F→X with connection, and for each Ad-invariant polynomial ϕ on gl\sbn, 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(ν) and Div\sb∞(ν) are integrally flat currents supported on the zero and pole sets of ν, where Res\sb0(ϕ) and Res\sb∞(ϕ) are smooth residue forms which can be calculated in terms of the curvature Ω\sbF of F, where T is a canonical transgression form with coefficients in L\sbsploc1, and where ϕ(Ω\sb0) is an L\sbsploc1 form canonically defined in terms of a singular connection naturally associated to ν. These results hold for C\sp∞-meromorphic sections ν 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∞-meromorphic sections of a bundle of arbitrary rank. Analogous results hold for real vector bundles and for quaternionic line bundles.