Analytic residues along algebraic cycles

Let W be a q-dimensional irreducible algebraic subvariety in the affine space AC, P1, ..., Pm m 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 non trivial 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 fulfillled 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 < q < n) of the qdifferential form (R/P1 · · ·Pq)dζI . 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.