Residue currents of the Bochner-Martinelli type

Our objective is to construct residue currents from Bochner-Martinelli type kernels; the computations hold in the non complete intersection case and provide a new and more direct approach of the residue of Coleff-Herrera in the complete intersection case; computations involve crucial relations with toroidal varieties and multivariate integrals of the Mellin-Barnes type.

Of the great number of integral representation formulas for holomorphic functions in several variables, there are two that are particularly simple and use-ful, namely those given by the Cauchy kernel and by the Bochner-Martinelli kernel.It is well known, see [18], that these kernels correspond to each other via the Dolbeault isomorphism.Moreover, it is an elementary observation that the Bochner-Martinelli representation formula can be obtained by averaging the Cauchy formula over a simplex.More precisely, taking the mean value over the simplex Σ p (η) = {s ∈ R n + ; s 1 + . . .+ s p = η} of both sides in the Cauchy formula one arrives at the Bochner-Martinelli formula where c p = (−1) p(p−1)/2 (p − 1)!/(2πi) p is a constant depending only on the number of variables, and the kernel Ω is given by Ω(w) = The simplicity of the Cauchy kernel makes it a natural candidate in the definition of multidimensional residues.For instance, there is an elegant integral interpretation of the Grothendieck residue based on this kernel, see [19].In 1978 the Cauchy kernel was used by Coleff and Herrera [14] in their definition of residue currents, which goes as follows: Let f 1 , . . ., f p be a system of p holomorphic functions in some domain V ⊂ C n .For every smooth, compactly supported test form ϕ ∈ D n,n−p (V ) one considers the integral where the real-analytic chain {|f 1 | 2 = ε 1 , . . ., |f p | 2 = ε p } is oriented as the distinguished boundary of the corresponding polyhedron.It is easy to to see that, when the common zero set f −1 (0) of the system f = (f 1 , . . ., f p ) has codimension less than p (that is, when f is not a complete intersection), then the function I(ε) given by (1.1) does not have a limit as ε → 0. However, Coleff and Herrera showed that this limit does exist if one lets ε approach the origin along a special path ε(δ) = (ε 1 (δ), . . ., ε p (δ)), a so-called admissible trajectory, for which each coordinate tends to zero quicker than any power of the subsequent coordinate.In the case of a complete intersection this limit is independent of the ordering of the functions, and it seemed reasonable to expect, in this case, the existence of an unconditional limit of the function I(ε) at the origin.This turned out not to be the case, and the counterexamples of [25] and [12] show that the behaviour of the integral (1.1) near ε = 0 can be quite intricate.We have therefore found it natural to consider the residue current, associated with the mapping f : V → C p , as a limit of certain averages of the residue function I(ε).
The aim of the present paper is to study residue currents of the Bochner-Martinelli type, which may be viewed as limits of mean values of I(ε) over the simplex Σ p (η) and which can in fact be written as ( In particular, our Theorem 1.1 says that such a limit always exists and defines a (0, p)-current T f , which annihilates the integral closure of the p-th power of the ideal generated by f 1 , . . ., f p in the space of holomorphic functions in V , and which also annihilates the conjugate of any function from the radical of this ideal.In the complete intersection case T f coincides with the Coleff-Herrera current, see Theorem 4.1, and with the currents considered in the papers [23], [5], [24], so there is the natural notation As a consequence we obtain in Theorem 2.1 the alternative representation where ∂f = ∂f 1 ∧ . . .∧ ∂f p , for the current ∂(1/f 1 ) ∧ . . .∧ ∂(1/f p ).This latter limit agrees with a more classical approach to particularly simple residue currents (with measure coefficients), which was used in [3] for obtaining interpolation and division formulas.
We feel that our results are of a certain interest already in the case of a complete intersection f .Indeed, a big draw-back in the theory of residue currents has always been the difficulty (for p > 1) in giving a concise definition of them, and the above limits (1.2) and (1.3) certainly provide much more appealing definitions of T f than the previously existing ones.(We must admit though that we had to do some work in order to prove their equivalence.) We shall however not restrict ourselves to the complete intersection case.This is partly because in our existence proof we do not need this assumption, but more importantly since there is already some recent work (see for example [30], [9]), where some questions related to residue theory in the non-complete intersection case are studied.
Here is the exact formulation of our main result: Then, for any ordered subset I ⊂ {1, . . ., m} of cardinality p ≤ min(m, n), and for any test form ϕ ∈ D n,n−p (V ), the limit exists and defines the action of a (0, p) current T f, I with the following vanishing properties: (i) hT f, I = 0 for any h ∈ O(V ) which vanishes on the common zero set Moreover, T f, I depends in an alternating way on the ordering of the elements in I.
Since the currents we introduce here are similar to those introduced in [15], Section 5, it seems reasonable to expect that the constructions we propose in this paper might give some further insight regarding explicit formulations of the Ehrenpreis-Palamodov fundamental principle in the non complete intersection case (in the spirit of the formulation in [13]).
Finally, we will also explain in our paper how explicit computations involving Bochner-Martinelli currents (in the case of normal crossings, when the f j are monomials) provide interesting connections with multidimensional Mellin-Barnes integrals (see [27]).
2 Residue currents of the Bochner-Martinelli type.
In this section we give a proof of Theorem 1.1.First a piece of notation: Throughout this paper c p will denote the numerical constant (−1) defines a smooth real hypersurface in V , which inherits the standard orientation of V ⊂ C n .We denote by Γ f (η) the corresponding real analytic (2n − 1)-chain.Let further I = {i 1 , . . ., i p } be an arbitrary ordered subset of {1, . . ., m}, whose number of elements p is at most min(n, m).For any test form ϕ ∈ D n,n−p (V ) and for each η ∈ R + \ E f , we then write It follows from the co-area formula ( [16], Theorem 3.2.11,p. 248) that the almost everywhere defined map defines a compactly supported element in the weighted space L 1 (R + , t p dt).Therefore, its Mellin transform is a holomorphic function in the half-plane Re λ > p.
Lemma 2.1 For Re λ > p, the above Mellin transform may be represented as where V αβ denotes the set {ζ ∈ V ; α < f 2 < β}.The set R + \ E f (ϕ) is a countable union of such disjoint intervals ]α, β[, so it follows from Lebesgue's theorem and from the co-aerea formula that the equality (2.2) holds for all λ with Re λ > p. ♦ Our second lemma gives the existence of a meromorphic continuation of the Mellin transform M f, I which is in fact holomorphic across the imaginary axis.Its value at the origin is of particular interest to us.

Lemma 2.2
The function λ → M f, I (ϕ, λ) can be meromorphically continued to the whole complex plane, and the poles of the extended function are strictly negative rational numbers.Moreover, the map defines the action of a (0, p)-current T f, I on V such that hT f, I = 0 for any h ∈ O(V ) which vanishes on the common zero set The current T f, I is hence supported by Z(f ), and moreover, one has Proof.Clearly one can reduce the problem to the case where the support of the test form is an arbitrarily small neighborhood of a point z 0 in Z(f ), and for the sake of simplicity we will reduce ourselves, via a change of variables, to the case z 0 = 0. We will therefore assume that Supp ϕ ⊂ W , where W is a neighborhood of the origin such that there exists a desingularisation (X , Π), X being a n-dimensional complex manifold and Π a proper holomorphic map The existence of such a pair (X , Π) follows from Hironaka's theorem [20].
For Re λ sufficiently large, one can write M f, I (ϕ, λ) as a finite sum of terms where ω is a local chart on X , coming from a finite covering of the compact subset π * (Supp ϕ), and ρ is the function from the partition of unity (subordinate to the covering) which corresponds to the local chart ω.Thanks to the normal crossing condition (i), one can assume that in a system of local coordinates on ω centered at the origin, where the u j are invertible elements in O(ω) and the α jk are positive integers.If one of the vectors α j := (α j1 , . . ., α jn ), j = 1, . . ., m, is zero, the corresponding function of λ in (2.4) is entire as λ → Π * f 2λ is.So that the interesting case occurs when all the α j are nonzero.
In order to study such a term, we use an idea that has already been extensively developed in [4].Let ∆ be the closed convex hull (in R n + ) of (The brackets here stand for the usual scalar product in the affine space R n .)The set of all closures of the equivalence classes for this relation is a fan Σ(∆) (see [1] and [17].)Such a fan can be refined ( [22]) in order that all cones are simple ones, so that the corresponding toric variety X is a ndimensional complex manifold; local charts correspond to different copies of C n which are glued together via invertible monoidal transformations from the n-dimensional torus T n into itself.Since the union of the cones in this fan is R n + , the projection map Π : In each chart on X (the coordinates being τ 1 , . . ., τ n ), one can write where µ j = Π * m j is also a monomial.Moreover, since the toric variety X is associated with the Newton polyedron ∆ attached to α 1 , . . ., α m , there exists an index j ∈ {1, . . ., m} such that µ divides all monomials µ j , j = 1, . . ., m (see [1].)This implies that where ũ is a non-vanishing positive real analytic function in .Since Π * and Π * commute with ∂ and ∂, one has where the (ξ ρ, ) correspond to a smooth partition of unity subordinate to Π * (Supp ρ) and θ ,1 and θ ,2 are smooth forms of bidegree (0, p) and (0, p−1) respectively.
For any smooth functions ψ ∈ D(Ω) and υ ∈ C ∞ (Ω), where Ω ⊂ C, such that υ > 0 on Supp ψ, one can see immediately, just integrating by parts, that the maps defined for Re λ > p by extend to meromorphic maps with poles in {r ∈ Q, r < 0}.The value at λ = 0 corresponds to the action of a distribution (with support at the origin) on the test function ψ in the first case; the value at λ = 0 is 0 in the second case.Moreover, the distribution that appears in the first case is annihilated by s.
It follows from the above remark that each term in the right hand side of (2.6) can be meromorphically continued as a function of λ with poles in {r ∈ Q, r < 0}.The value at the origin of the meromorphic continuation of any function of the form (2.6) corresponds to the action of a (0, p)-current in V .Summing up all functions of λ of the form (2.6), we find that the function λ → M f, I (ϕ, λ) can be meromorphically continued to the whole plane, with strictly negative rational poles.The value at λ = 0 corresponds to the action of a (0, p)-current T f, I .
Suppose that h ∈ O(V ), h = 0 on V .It follows from the Nullstellensatz that for any ϕ ∈ D n,n−p (V ), one has h N (ϕ) ∈ (f 1 , . . ., f m ) loc for some integer N (ϕ).For any Π and Π involved in the resolutions of singularities used in the proof, and for any ρ, , ξ ρ, as before, we have the estimate In order to prove the last assertion in the statement of Lemma 2.2, assume . This implies that, given a local chart on any toric manifold such as X , the differential form Π * Π * n−p l=1 dζ i l (which has antiholomorphic functions as coefficients) vanishes on the analytic variety {µ j (τ ) = 0}, where µ j is the distinguished monomial corresponding to the local chart .Every conjugate coordinate τ k , such that τ k is involved in µ j then divides each coefficient of Π * Π * n−p l=1 dζ i l , which does not contain dτ k .This implies that for any local chart , the integrand in (2.5) does not contain antiholomorphic singularities (such singularities come from logarithmic derivatives and therefore are cancelled by the corresponding term Π * Π * ϕ.)The proof of our Lemma 2.2 is complete.♦ Let us recall the definition of the integral closure of an ideal A in the ring n O z 0 of germs of holomorphic functions of n variables at a point z 0 ∈ C n .A germ h at z 0 is in the integral closure of A if and only if it satisfies a relation of integral dependency where and only if, at any point z 0 ∈ V , the germ h z 0 belongs to the integral closure (in n O z 0 ) of the ideal A z 0 generated by the germs at z 0 of all elements in A.
Lemma 2.3 Let T f, I be the current occurring in the preceeding lemma.For any h ∈ O(V ) which is locally in the integral closure of the ideal (f 1 , . . ., f m ) p we then have Proof.Replacing ϕ by hϕ and arguing as in the proof of Lemma 2.2, we decompose the function λ → M f, I (hϕ, λ) into a finite sum of expressions of the type (2.5) (modulo an entire function which vanishes at the origin).The only thing we have to show is that, for any h which locally belongs to the integral closure of (f 1 , . . ., f m ) p in O(V ), the value at λ = 0 of the analytic continuation of This implies that the monomial µ j divides Π * Π * h.We now make the following observation: for any domain Ω ⊂ C, and any smooth functions ψ ∈ D(Ω) and υ ∈ C ∞ (Ω), such that υ > 0 on Supp ψ, explicit integration by parts provides meromorphic continuations of the maps defined for Re λ > p by Their poles are again strictly negative rational numbers, and they both vanish for λ = 0.This shows that the value at λ = 0 of the analytic continuation of any of the functions is also equal to zero.Since our original function of λ (as in (2.2), but with hϕ instead of ϕ) is a combination of such expressions, its analytic continuation also vanishes at the origin.This proves our result.♦ The last lemma that we will need in order to conclude the proof of our Theorem 1.1 is concerned with rapid decrease in imaginary directions.
Then, for any natural number k and any real numbers α, β, such that λ θ,j+1 < α < β < λ θ,j , j ∈ N * or λ θ,0 < α < β, there is a constant γ(k, α, β) such that (2.9) (In other words, the function F θ is rapidly decreasing at infinity in any closed vertical strip which is free of poles.) Proof.Our proof was inpired by an argument used in [2].Let G be the holomorphic function in V × V defined as Consider a point z 0 in V where f j (z 0 ) = 0, j = 1, . . ., m.By Corollary 9.10 in Chapter 5 of [10], there exists a neighborhood V(z 0 ) of (z As was proved in [21], Section 6, there is then an operator where D (z 0 ,z 0 ) denotes the ring of holomorphic differential operators in 2n complex variables with coefficients in the ring 2n O (z 0 ,z 0 ) of germs of holomorphic functions at the point (z 0 , w 0 ), such that and (2.11) If we now make the substitution w = z, and use the fact that the operators ∂/∂z l and ∂/∂z l commute with multiplication by z k and z k , respectively, we find that in a neighborhood V (z 0 ) of z 0 , there holds the identity (in the sense of distributions) This functional equation (2.12), used in the form and then iterated (as in an argument quoted from [2]), provides the rapid decrease of F θ on closed vertical strips in the λ-plane which are pole-free.
Notice that the fact that the meromorphic continuation of F θ exists (with poles organized as a decreasing sequence of strictly negative rational numbers) follows (as in our proof of Lemma 2.1) from Hironaka's theorem on resolution of singularities.The proof of Lemma 2.3. is thereby complete.♦ Proof of Theorem 1.1.We have now collected all elements for the proof of our Theorem 1.1.Recall that the Mellin transform of the function η → J f, I (ϕ, η) defined as in (2.1), is equal to the function λ → M f, I (ϕ, λ) described in Lemma 2.1.The Fourier-Laplace inversion formula then tells us that, for γ 0 > 0 large enough, the identity holds for every posistive η.We know from Lemma 2.2 that there is a positive number ε 0 such that the only pole of in the closed vertical strip Γ := Re λ ∈ [−ε 0 , γ 0 ] is the origin, and that the residue is M f, I (ϕ, 0).It follows from Lemma 2.4 that the function (2.13) is rapidly decreasing at infinity on the strip Γ.We can apply the residue formula and get that J f, I (ϕ, η) is equal to We conclude that the limit (1.4) exists and equals M f, I (ϕ, 0).We get the conclusions (i) and (iii) of Theorem 1.1 from Lemma 2.2, and conclusion (ii) from Lemma 2.3.Our main Theorem 1.1 is thus proved.♦ One can also realize the action of all the currents T f, I in Theorem 1.1 as limits of solid volume integrals.More precisely, we have the following theorem: Theorem 2.1 Let f 1 , . . ., f m be holomorphic functions in some open set V ∈ C n .For any ordered subset I ⊂ {1, . . ., m} of cardinality p ≤ min(m, n), let T f, I be the current defined in (1.4).Then one has the representation (2.14) In particular, if m ≤ n and I = {1, . . ., m}, we have (2.15) Proof.Notice first that the integral in (2.14) is absolutely convergent, for the differential form inside the integral has bounded coefficients.Let us fix τ > 0. If [α, β] denotes any interval in R + which does not contain a critical value for the mapping Supp ϕ ζ → f (ζ) 2 , it follows from Fubini's theorem that Since the critical values for f 2 which are attained on Supp ϕ form a negligible closed subset of R + , we get from Lebesgue's theorem and from the continuity at η = 0 of η → J f, I (ϕ, η) that for any τ > 0 ( We just note that for τ > 0, where lim τ →0 ρ(τ ) = 0. Using the short-hand notation J(η) := J f, I (ϕ, η) we can now rewrite the right hand side of (2.17) as for ε < A and ε arbitrary small, with lim τ →0 + ρε,A (τ ) = 0. Since we can choose ε arbitrarily small, we have This concludes the proof of Theorem 2.1.♦ When m ≤ n, it is well known (see [5] or [24]) that for any test form ϕ ∈ D n,n−m (V ), the function df k ∧ ϕ can be continued from the cone Re λ j > 1, 1 ≤ j ≤ n, to a meromorphic function in the entire space C m , with polar set Sing Γ f included in a union of hyperplanes β 0 + β 1 λ 1 + • • • + β m λ m = 0, where β 0 ∈ N, (β 1 , . . ., β m ) ∈ N m \ {0}.This is obtained immediately using Hironaka's theorem [20].It seems interesting to relate this meromorphic continuation λ → Γ f (λ, ϕ) to the computation of the residue currents we introduced in Theorem 1.1.We have the following result in this direction. ) , then for any fixed τ > 0, and for for any γ ∈ C, the integral is absolutely convergent.
Proof.This result was proved in [29] in the complete intersection case.In fact, this hypothesis is not necessary and the whole proof goes through as follows.Fix τ > 0. For any ζ in V such that f 1 . . .f m (ζ) = 0, one has, if γ1 , . . .γm are strictly positive numbers with sum strictly less than m, This is just a standard iteration of formula 6.422 (3), p. 657 in [18].Notice that this idea has been extensively used in [6].If we change s k into 1 − s k and let γ k = 1 − γk , k = 1, . . ., m, formula (2.22) can be rewritten as If we assume that the γ j are all very close to 1, we can apply Lebesgue's and Fubini's theorems in order to get, for such τ , Using Bernstein-Sato functional identities (see [28]) or resolution of singularities (which leads us to the normal crossing case) together with integration by parts, one can see that the function can be estimated by in any vertical strip Re s ∈ K, K ⊂⊂ R m , which does not intersect the polar set of this function (in particular when K ⊂]0, ∞[ m ), the constants C(Re s) and N (Re s) being uniform in Re s in this strip.Similar estimates hold for the function Therefore, because of the rapid decrease of the Gamma function on vertical lines, we get the uniform rapid decrease at infinity for the function in any vertical strip Re s ∈ K, K ⊂⊂ R n which is pole-free (in particular when K ⊂]0, ∞[ m .)Thus, one can apply Cauchy's formula and replace (γ 1 , . . ., γ m ) in (2.23) by any element in ]0, 1[ m .The first assertion of Theorem 2.2 follows from these computations, together with Theorem 2.1.The second assertion in the theorem is a consequence (in view of Cauchy's theorem) of the uniform rapid decrease at infinity in vertical strips (which are pole-free) for the function Using the second part of this statement, it would be interesting to analyze how S(C, τ ) changes when one moves from the original cell ]0, 1[ m into the contiguous ones.The difference between S(C 1 , τ ) and S(C 2 , τ ) should appear (at least formally) as a (finite or infinite) sum of iterated residues for the function relatively to collections of m independent affine polar divisors.We will elaborate this idea somewhat in our computations in Section 3. Our assumption is motivated by the fact that for any τ > 0, for γ > 0 very close to 0 where (this is proved as formula (2.23), just using formula 6.422 (3), p. 657 in [18] this time without iterating it.)Thanks to Lemma 2.4, we have the rapid decrease of F (as a function of s) on vertical strips which are pole free.Using Cauchy's formula and moving the integration path in (2.24) to the left, we deduce from the fact that the poles of F are in ] − ∞, −m[ the existence of an asymptotic development for the function along the basis (1, τ α (log τ ) µ ) α∈Q + , µ∈N .It seems reasonable to think that the coefficients in this asymptotic development should be expressible as (infinite or finite) sums of residues corresponding to the meromorphic differential form This is precisely the point we will emphazise in the examples detailed in the next section.
In this section, we will compute the action of some of the currents T f, I .Our approach is to deal only with the normal crossing case (even, in order to make things more simple, assume that the f j are all momomials) and profit (as we already did when we stated Theorem 2.2) from some combinatorial basic identities which correspond to multivariate analogs for the integral representation of the beta function as an inverse Mellin transform.
3.1 A simple example when m = 2 and f 1 divides f 2 .
Assume that n > 1 and f 1 and f 2 are defined in a neighborhood V of the origin in C n .Let We conclude in this case that This corresponds to the action of a current whose support is the zero set of the ideal (f 1 , f 2 ).Note that, in this case, the essential intersection (in the sense of [14]) of the divisors {f 1 = 0} and {f 2 = 0} (in this order) is empty, so that the Coleff-Herrera current associated to the sequence (f 1 , f 2 ) in this order would be zero.On the other hand, the Coleff-Herrera current associated to the sequence ( where h is the product of irreducible factors in h which are coprime with f 1 .In any case, the Coleff-Herrera current for this example is either 0, either a residual current supported by the origin and therefore differs from our current T f, {1,2} .

The normal crossing case m = n ≥ 2, and relations
with Mellin-Barnes integrals.
In the space C n we consider a system of monomials and the Bochner-Martinelli type current (1.3) corresponding to this system.According to Theorem 2.2, this current may be represented as the limit where the integrand is given by the n-form In the monomial case under consideration the possible poles of the function s → Γ f (s; ϕ) consist of the n families of hyperplanes where α j denotes the j'th column vector in the matrix, whose row vectors are α 1 , . . ., α n .In other words, if α i = (α i1 , . . ., α in ), then α j = (α 1j , . . ., α nj ).
In the real subspace R n with variables x j = Re s j , j = 1, . . ., n, we now introduce the cone and we let K 0 denote the intersection of K with the closed halfspace Let us write q for the codimension of K 0 .It is clear that if q = 0, then K 0 will contain interior points of Π − , whereas in the case q ≥ 1 the intersection K 0 = K ∩ Π − consists of a (n − q)-dimensional face of the cone K, contained in the hyperplane x 1 + • • • + x n = 0, i.e the boundary of Π − .Up to a mere re-numbering of the faces, we may suppose that We have the Proposition 3.1 If q = 0, then the current T f , defined by (3.2), is equal to zero.In case q ≥ 1, it admits the representation where |α j | denotes the sum of the components of the vector α j , and F is a certain hypergeometric function (whose representation as a Mellin-Barnes integral is given in formula (3.9) below).In particular, if q = n, then Remark 3.1.The complex codimension of the support of T f is equal to q, which is the real codimension of the cone K 0 .
Proof of the proposition.First of all we observe that by (the second statement of) Theorem 2.2 we may enlarge the cube ]0, 1[ n , consisting of admissible values for γ in the integral (3.2), to the convex polyhedron M obtained by intersecting the interior of the cone K with the open cone {x ∈ R n ; x 1 < 1, . . ., x n < 1}.
• If q = 0, i.e dim R n K 0 = n, then we can choose the point γ in M so that |γ| < 0. Therefore, in view of the factor τ −|s| , the restriction of the form (3.3) to γ + i R n tends to zero as τ → 0 + .It then follows that the limit (3.2) is equal to zero, and hence T f = 0.
• Assume now that q ≥ 1. Letting M q denote the relative interior of the intersection K 0 ∩ M , we have the following formula which decreases the number of integrations in (3.2).Lemma 3.1 The limit of the n-fold integral (3.2) may be written as the where γ q ∈ M q , and Res L q is the q-fold Poincaré-Leray residue class of the meromorphic form Γ f ds, taken with respect to the intersection L q = L 1 ∩ . . .∩ L q of the hyperplanes To prove the lemma we establish first the following asymptotic (as τ → 0 + ) formula: Here γ 1 is a point in the (n − 1)-dimensional polyhedron M 1 , which is the relative interior of the intersection ) To this end we consider the ray emanating from the point γ and parallel to one of the edges of the cone K. Now, what matters to us is the fact that this ray intersects the face α 1 , x = 0 no later than the hyperplane |x| = 0 (the intersections occur simultaneously when q = 1), and that among the polar hyperplanes of the form s → ω τ (s), the ray intersects only L 1 .Letting γ 1 denote the point of intersection between and {x ∈ R n ; α 1 , x = 0} = Re L 1 , we thus see that, for any two points γ and κ of lying on different sides of γ 1 , the Cauchy formula yields Res L 1 ω τ (s). (3.5) Choosing κ lying inside Π − , i.e with the property |κ| < 0, we find, in view of the presence of the factor τ −|s| in the form ω τ (s), that the integral over κ + i R n in (3.6) tends to zero (is o(τ )) as τ → 0 + .In this way we obtain (3.5).
In order to prove formula (3.4) we observe, that the integral in the right hand side of (3.5) has the same structure, but in the (n−1)-dimensional space L 1 , so repeating q − 1 times the residue theorem we arrive at the identity lim Since we assumed the face K 0 is contained in {|x| = 0}, we have L q ⊂ {s; |s| = 0}, which means that the restriction of s → τ −|s| Γ(1 + |s|) to L q is identically equal to 1. Recalling the expression (3.3) for the form ω τ , we may thus conclude that It follows from this that the right hand integral in (3.7) is actually independent of τ .Hence there is no need to take a limit, and we have completed the proof of our lemma.
An easy computation now leads to the following expresion: where ζ = (ζ , ζ ), θ is a holomorphic function in a neighborhood of the origin, belonging to the ideal λ 1 , . . ., λ q , and

Thus we get
Now, applying (3.2), (3.4) together with Fubini's theorem, we find that the action of the current T f may be expressed as where F is a function of the hypergeometric type, representable as the Mellin-Barnes integral where j (λ ) is the j'th component of the vector (λ ) = A −1 (0 , λ ).The proposition is thereby proved.♦ Remark 3.2.In case q = 1 it is not hard to actually compute the integral (3.9) by using the methods of [27] and [26].The result of this computation then gives a rational function.4 The complete intersection case.
In this section, we consider m ≤ n holomorphic functions f 1 , . . ., f m defining a complete intersection in a domain V ⊂ C n .It follows from Theorem 1.1 (iii) that {1, . . ., m} is the only subset I that can give a non-zero current T f, I .We use the simpler notation T f for the corresponding current T f, {1,...,m} .We shall now prove that T f in fact coincides with the residue current in the sense of .) Proof.As mentioned in the introduction (see formula (1.2)), the actions of the two currents T f and m k=1 ∂ 1 f k on test forms which are ∂-closed in a neighborhood of {f 1 = • • • = f m = 0} coincide.The problem is to show this remains true for any test form.For this, we will need two preparatory lemmas.
Lemma 4.1 ([4], [24]) Let p ≥ 2 and g 1 , . . ., g p be p holomorphic functions of n variables defining a complete intersection in an open subset V of C n .Then, for any test form ϕ ∈ D n,n−p+1 (V ), the function of two complex variables can be continued from {Re λ 1 > 0, Re λ 2 > 1} as a meromorphic function in two complex variables.Moreover, this meromorphic continuation M g (λ; ϕ) can be written near the origin in C 2 as where k 0 , . . ., k N are holomorphic near the origin and the ρ jl ( resp.σ jl ) are constants in N (resp.in N * .) Proof.The fact that the function (4.1) can be meromorphically continued to C 2 and that its continuation has the form (4.2) near the origin is proved in details in [4], p. 70-72, from formula (3.34) up to formula (3.40).To be more precise, in the mentioned reference, only the meromorphic continuation of Then, for any test form ϕ ∈ D n,n−p+1 (V ), the function of two complex variables can be continued from {Re λ 1 > 0, Re λ 2 > 1} as a meromorphic function in two complex variables.Moreover, this meromorphic continuation Proof.Although implicitly given in [6] and [4], Section 5, the proof of this lemma is there more suggested than detailed, so we will write it out here completely following the basic ideas one can find for example in [7], proof of Proposition 9 or in Section 2 above.We first localize the problem and use a resolution of singularities (X , Π) for the hypersurface where the u k are invertible holomorphic functions on the local chart ω and the α kl are positive integers.Then, as in section 2, our analytic function λ → N (λ, ϕ) appears as the sum of terms where ρ = ρ ω is the function associated to the local chart in some partition of unity subordonned to Supp Π * ϕ.In order to decompose an integral of the form (4.4), we use the toric variety X (together with the projection proper map Π : (see the proof of lemma 2.2 in section 2 above.)This introduces a new decomposition of (4.4), with expressions of the form where µ is the distinguished monomial among the µ j = Π * m j , j = 2, . . ., p, the υ k are the holomorphic functions defined as Π * (u k m k ) = υ k µ j (note that υ j is invertible in ) and ξ ρ, comes from a partition of unity related to a covering of Π * Supp ρ.Such an expression (4.5) can be written as where θ ,1,λ 2 and θ ,2,λ 2 are smooth forms with respective types (0, p) and (0, p − 1) depending holomorphically on the parameter λ 2 .It is now immediate that the meromorphic continuation of λ → N (λ, ϕ) exists and that its polar set is included in a collection of hyperplanes β 0 + β 1 λ 1 + β 2 λ 2 = 0, where β 0 ∈ N and (β 1 , β 2 ) ∈ N 2 \ {0}.In order to see that there are no polar hyperplanes with β = 0 (and then we will be done), we need to look more carefully at the analytic continuation of expressions of the form where τ is among the coordinates that divide the distinguished monomial µ j .If τ does not appear in the decomposition of Π * m 1 , then the integration by parts which is necessary in order to raise the singularity τ implies just a division of the expression by λ 2 (instead of a combination of λ 1 and λ 2 as it should be if the hypothesis was not fulfilled.)Since λ 2 was in the numerator, the new expression (after integration by parts with respect to τ ) is holomorphic near the origin in C 2 .If τ appears in the decomposition of Π * m 1 , it means that Π • Π{τ = 0} is included in the n − p dimensional analytic set {g 1 = • • • = g p = 0}.This implies (for dimension reasons) that any antiholomorphic differential form Π * Π * j∈I dζ j , when I ⊂ {1, . . ., n}, #I = n − p + 1, vanishes identically on τ = 0, which means that all its coefficients have τ as a factor.In such a case, has only holomorphic singularities and therefore defines a holomorphic function of λ at the origin.This completes the proof of Lemma 4.2.♦ λ 2 and up to Re λ 2 > 0) and integration with respect to ζ 1 commute when Re λ 0 1 >> 0. Then we will get that for Re λ 0 1 >> 0, we have (p − 1)!N g ((λ 0 1 , 0), ϕ) = N g ((λ 0 1 , 0), ϕ) .Following the analytic continuation, this time with respect to λ 1 , we get that (p − 1)!k 0 (0, 0) = k0 (0, 0) which means, if we refer to Lemma 2.2 and to [23] or [12], Theorem 6.2.1, p. 107, that T g, {1,...,p−1} (ϕ) = and concludes the proof of our inductive assumption when n − p = k.
It remains to explain why that analytic continuation (with respect to λ 2 up to Re λ 2 > −η) and integration with respect to ζ 1 commute when Re λ 0 1 >> 0 in (4.8) and (4.9).This was already explained in [4].We will use here a different approach, based on the use of Bernstein-Sato relations instead of resolution of singularities.Such an approach seems more natural.It follows from Proposition 3 in [8] that there exist analytic functions h 1 and h 2 in one complex variable u, defined in D(0, r 1 ), r