Singularities and Chern-Weil theory. I: The local MacPherson formula (Q5929536)

Let \(\alpha: E\to F\) be a smooth bundle map between vector bundles with connections on a manifold \(X\), and let \(\Phi(\Omega)\) be a Chern-Weil characteristic form of either \(E\) and \(F\). For the class of ``geometrically atomic'' maps \(\alpha\) (which are generic in all structured situations) the authors establish a canonical co-homology NEWLINE\[NEWLINE \Phi(\Omega) - \sum_{k\geq 0} \operatorname {Re}s_{\Phi,k}[\Sigma_k(\alpha)] = dT NEWLINE\]NEWLINE where \(\Sigma_k(\alpha)=\{x\in X : \dim \ker(\alpha)=k\}\), \(\operatorname {Re}s_{\Phi,k}\) is a smooth residue form along \(\Sigma_k(\alpha)\), and \(T\) is a canonical \(L^1_{\text{loc}}\)-form on \(X\). When \(\text{rank}(E)= \text{rank}(F)\) the last formula can be written NEWLINE\[NEWLINE \Phi(\Omega^F) - \Phi(\Omega^E)= \sum_{k>0} \operatorname {Re}s_{\Phi,k}[\Sigma_k(\alpha)] = dT NEWLINE\]NEWLINE which generalizes a classical formula of R.~MacPherson at the level of forms and currents. Proofs entail a direct application of methods of singular connections and of finite volume flows previously developed by the authors.