Brylinski, J.-L.; McLaughlin, D. A. Multidimensional reciprocity laws. (English) Zbl 0857.11062 J. Reine Angew. Math. 481, 125-147 (1996). The authors develop a complex-analytic approach to the reciprocity laws of Parshin and Kato in higher-dimensional class field theory. On an \(n\)-dimensional complex-analytic space \(X\) (possibly singular), they define the multidimensional symbol \(\{f_1, \dots, f_{n+1}\}_F\) of \(n+1\)-meromorphic functions along a complete flag \(F\) of irreducible subspaces, by pairing a certain cohomology class associated to \(f_1, \dots, f_{n+1}\) with a homology class \(\kappa_F\) constructed from the flag. They find a new analytic formula for the symbol which they then evaluate algebraically to obtain Parshin’s formula. They also prove by direct geometric methods that this symbol satisfies various reciprocity laws. Finally they establish the analogue of all these results for varieties over a finite field. Reviewer: J.-L.Brylinski (Princeton) Cited in 2 ReviewsCited in 4 Documents MSC: 11R70 \(K\)-theory of global fields 19F05 Generalized class field theory (\(K\)-theoretic aspects) Keywords:multidimensional reciprocity laws; meromorphic functions along a complete flag of irreducible subspaces; flag-localized homology groups; higher-dimensional class field theory; Parshin’s formula PDFBibTeX XMLCite \textit{J. L. Brylinski} and \textit{D. A. McLaughlin}, J. Reine Angew. Math. 481, 125--147 (1996; Zbl 0857.11062) Full Text: Crelle EuDML