Residues of holomorphic maps between manifolds
Let \(f: V\to W\) be a holomorphic map between complex analytic manifolds \(V\) and \(W\) respectively of dimensions \(n\) and \(q\), \(n\geq q.\) Assume that the singular set \(\Sigma\) of \(f\) is not empty. Then the Chern classes \(c_{n-q+i} [TV -f^{-1}(TW)]\) of the virtual bundle \([TV -f^{-1}(TW)]\) localize near \(\Sigma\) for \(i = 1,\dots,q.\) If \(\Sigma\) is compact, then the residue of \(c_{n-q+i} [TV -f^{-1}(TW)]\) is defined in the homology group \(H_{2q-2i}(\Sigma)\). The author computes these residues assuming the singular fibres of \(f\) are locally isolated complete intersections and the singular set \(\Sigma\) is smooth. If \(q = 1\) and \(V\) is compact, the Fulton formula computes the cohomology class \([c_n(V) - (c_{n-1}(V) \cup f^\ast_{c_1}(W))]\cap[V]\) using the Milnor number. The main result of the paper is a generalization of this formula for all codimension \(q.\) A generalization of the formulas of Riemann-Hurwitz type is deduced from it. Two examples, for which the residues are not equal to zero, are considered.
scientific article; zbMATH DE number 1448535