On the residues of Baum-Bott
Consider a compact complex manifold \(V\) of complex dimension \(n\), a holomorphic vector field \(v\) on \(V\), a vector bundle \(E\) of rank \(r\) on \(V\), and a \(\mathbb{C}\)-action \(\theta_v\) on \(E\). If \(v\) has no singularities, the Chern numbers \(c_I(E)\cap [V]\) are zero. If \(v\) has singularities Bott has shown that the Chern numbers localize near these singularities. The author obtains some generalizations of the results of Bott and Baum-Bott. After presenting some definitions and known results, the author presents a proof for the existence of residues. Next he obtains a generalization for the case where \(v\) is degenerate along a holomorphic submanifold \(W\). First he obtains the residues by using a special covering (Theorem A). Then the covering is suitably modified in order to obtain Theorems B,C, and D. Some applications of the obtained results are presented.
scientific article; zbMATH DE number 5799095