Analytic residues along algebraic cycles

Let W be a q-dimensional irreducible algebraic subvariety in the affine space A(C)(n), P-1,..., P-m m elements in C[X-1,... X-n], and V(P) the set of common zeros of the P-j's in U. Assuming that \W\ is not included in V(P), one can attach to P a family of nontrivial W-restricted residual currents in 'D-0,D-k(C-n), 1less than or equal tokless than or equal tomin(m,q), with support on \W\. These currents (constructed following an analytic approach) inherit most of the properties that are fulfilled in the case q = n. When the set \W\ boolean AND V(P) is discrete and nt = q, we prove that for every point alpha is an element of \W\ boolean AND V(P) the W-restricted analytic residue of a (q, 0)-form R dzeta(1), R is an element of C [X-0,..., X-n], at the point alpha is the same as the residue on W (completion of W in Proj C[X-0,..., X-n]) at the point a in the sense of Serre (q = 1) or Kunz-Lipman (1 < q < n) of the q-differential form (R/P-1... P-q)dzeta(I). We will present a restricted affine version of Jacobi's residue formula and applications of this formula to higher dimensional analogues of Reiss (or Wood) relations, corresponding to situations where the Zariski closures of \W\ and V(P) intersect at infinity in an arbitrary way. (C) 2004 Elsevier Inc. All rights reserved.