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Multidimensional reciprocity laws. (English) Zbl 0857.11062

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.

MSC:

11R70 \(K\)-theory of global fields
19F05 Generalized class field theory (\(K\)-theoretic aspects)
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