Analytic residues along algebraic cycles

Abstract

Let W be a q-dimensional irreducible algebraic subvariety in the affine space A C n, P1,..., Pm m elements in C[X1,...,Xn], and V(P) the set of common zeros of the Pj's in C n. Assuming that W is not included in V(P), one can attach to P a family of nontrivial W-restricted residual currents in ′D0,k (Cn), 1≤k≤min(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 ∩ V(P) is discrete and m=q, we prove that for every point α∈ W ∩ V(P) the W-restricted analytic residue of a (q,0)-form R dζ 1, R∈C[X1,...,Xn], at the point α is the same as the residue on W (completion of W in Proj C [X0,...,Xn]) at the point α in the sense of Serre (q = 1) or Kunz-Lipman (1<q<n) of the q-differential form (R/P1⋯Pq)dζ1. 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. © 2004 Elsevier Inc. All rights reserved.