×

On separating subgroups and local residues. (Russian) Zbl 0671.32005

Let X be a complex manifold of complex dimension n, and \(T_ 1,...,T_ m\) be analytic subsets of X whose pure codimension equals 1 (i.e. positive divisors), \(T=T_ 1\cup...\cup T_ m\). In this paper the case \(m>n\) is considered. The separating subgroup \(H^*_ n(X\setminus T)\) of the homology group \(H_ n(X\setminus T)\) is introduced in which each cycle, called separating cycle, has the property that the integral of a meromorphic form along it can be represented as the sum of local residues. As a topological characterization, the author gives a necessary condition for a cycle \(\gamma\) to be separating as follows: \(\gamma\) \(\sim 0\) in \(X\setminus (T_{j_ 1}\cup...\cup T_{j_{n-1}})\) for any choice of n-1 indices \(\{j_ 1,...,j_{n-1}\}\subset \{1,2,...,m\}\). Having added some supplementary conditions on X and \(T_ i\) (especially X is Stein), the above condition turns to be also sufficient. A basis of the separatig subgroup and the coefficient formula of expansion of a separating cycle with respect to this basis are given. The author also obtains a complete set of independent relations among separating cycles and, correspondingly, the set of relations among local residues which is just the local multidimensional analog of the theorem on complete sum of residues.
Reviewer: J.Na

MSC:

32A27 Residues for several complex variables
PDFBibTeX XMLCite
Full Text: EuDML