Residue Currents and Analytic Continuation

Abstract

From the computational point of view, the theory of multidimensional residues arises as follows. Given a complex n-dimensional manifold Χ, equipped with an orientation, and a closed subset S,we consider a d-closed p-form φ, smooth on Χ\S, singular on S, and we wish to compute ∫ _γ φ where γ is an element in Z _p (Χ\S). Such an integral depends only on the homology class of γ in H ^p (Χ\S) and on the cohomology class of ϕ in H _P (X\S).The two main cases studied by Leray are the case where S is a n −1 dimensional submanifold of X and the case where S = S _l U…U S _M , where S _j , j = 1,…, M are n − 1 dimensional submanifolds of X in general position, this means that, for any set of distinct indices j ₁,…, j _k , where 1 ≤ k ≤ M, for any point a in ${S_{{j_1}}} \cap \ldots \cap {S_{{j_k}}}$, there is a system of local coordinates about a, such that in these coordinates,${S_{{j_l}}} = \left\{ {{s_{{j_l}}}\left( \zeta \right) = 0} \right\},l = 1, \ldots,k$, and the s _jl are holomorphic functions such that

$$ grad\left( {{s_{j1}}} \right)\left( a \right) \wedge \ldots \wedge grad\left( {{s_{{j_k}}}} \right)\left( a \right) \ne 0.$$

In the first case, the result of Leray [28] can be stated as follows.