Characteristic classes and multidimensional reciprocity laws

Abstract

In this note, we formulate and prove a non-commutative generalization of the reciprocity laws of Parshin and Kato in higher dimensional class field theory [Kat][P1][P4]. This can be regarded as a geometric realization of the reciprocity laws implicit in the Gersten-Quillen complex in algebraic K-theory [Q]. We expect this work to be related to the “higher dimensional Langlands program”, which has yet to be formulated precisely (although see [Kap]). Let X be an k-dimensional complex analytic space (possibly singular). Let F := (C0 ⊂ C1 ⊂ · · · ⊂ Ck = X) be a flag of irreducible subspaces in X, with dim Ci = i (we often omit Ck from the notation). For any analytic open set V ⊂ X, we constructed in [Br-M1] a homology class κF,V ∈ Hk(V ;Z) and showed that it satisfies the homological reciprocity law described in Theorem 2.1. Let ĉp denote the characteristic class of Deligne-Beilinson [Be] which refines the usual pth Chern class cp. In section 3, we define the non-commutative symbol F,V at the “place” F , by pulling back ĉk+1 via the evaluation map BGL(n,O(V )) × V → BGL(n,C) and taking the “slant-product” of the resulting class with κF,V . Here O(V ) denotes the algebra of holomorphic functions on V and we are viewing GL(n,O(V )) as a discrete group. It is important to note that the non-commutative symbol is not just a number, rather it is the cohomology class in Hk+1(BGL(n,O(V )),C∗) of a degree k + 1 group cocycle on GL(n,O(V )). Since the homology class κF,V satisfies a reciprocity law, we obtain