Analytic residues along algebraic cycles

Let W be a q-dimensional irreducible algebraic subvariety in the affine space ACn, P1,…,Pmm elements in C[X1,…,Xn], and V(P) the set of common zeros of the Pj's in Cn. 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 RdζI, R∈C[X1,…,Xn], at the point α is the same as the residue on W (completion of W in ProjC[X0,…,Xn]) at the point α in the sense of Serre (q=1) or Kunz-Lipman (1