Analytic residues along algebraic cycles

摘要

Let W be a q-dimensional irreducible algebraic subvariety in the affine space A_C~n, P_1, ..., P_mm elements in C[X_1, ..., X_n], and V(P) the set of common zeros of the P_j'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 ′D~(0,k)(C~n), 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_I, R∈C[X_1 ...,X_n], at the point α is the same as the residue on W (completion of W in Proj C[X_0, ..., X_n]) at the point α in the sensse of Serre (q = 1) or Kunz-Lipman (1 < q < n) of the q-differential form (R/P_1 ···P_q)dζ_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.