Dual class of a subvariety
Let \(E\) be a holomorphic vector bundle of rank \(k\) over an \(n\)-dimensional complex manifold \(M\) and \(X\) a pure \(k\)-codimensional analytic subspace of \(M\) defined by the zero-set of a regular holomorphic section \(s\) of \(E\). With respect to \(s\) the author defines and describes for the top Chern class \(c_{k}(E)\in H^{2k}(M,\mathbb C)\) a canonical localisation \(c_{k}(E,s)\in H^{2k}(M,M\backslash X;\mathbb C)\). He proves that for \(X\) compact this class corresponds to the class \([X]\in H_{2n-2k}(X,\mathbb C)\) via the Alexander duality \(H^{2k}(M,M\backslash X;\mathbb C)\to H_{2n-2k}(X,\mathbb C)\). For \(M\) compact it gives the known result that \(c_{k}(E)\) corresponds via Poincaré duality to the homology class of \(X\) in \(H_{2n-2k}(M,\mathbb C)\). In the situation \(k=n\) the author calculates the residue \(\text{Res}_{c_{n}}(s,E;p)\in H^{0}(\{p\};\mathbb C)=\mathbb C\) at an isolated zero \(p\) of \(s\) and shows that it can be expressed in terms of the Grothendieck residue symbol. This result is a main tool for the proof of the above theorem. Moreover it leads to the known fact that the global intersection number of \(n\) divisors on \(M\) with common intersection a finite set is the sum of the local intersection numbers. Finally the author studies the situation where \(X\) may not be compact but where the singularity set of \(X\) has a compact connected component \(S\). Considering \(X\) as a relative cycle in \(M\) modulo \(M\backslash S\) he proves a duality theorem analogous to the one above with \(E\) restricted to compact \(\mathcal C^{\infty}\)-submanifolds of \(M\).
scientific article; zbMATH DE number 1523369