The $\bar{\partial}$-equation on a non-reduced analytic space

Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\overline{\partial}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves $\mathcal{A}_X^q$ of $(0,q)$-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\mathscr{O}_X$. Our construction is based on intrinsic semi-global Koppelman formulas on $X$.


Introduction
Let X be a smooth complex manifold of dimension n and let E 0, * X denote the sheaf of smooth (0, * )-forms. It is well-known that the Dolbeault complex X∂ → · · ·∂ → E 0,n X → 0 is exact, and hence provides a fine resolution of the structure sheaf O X . If X is a reduced analytic space of pure dimension, then there is still a natural notion of "smooth forms". In fact, assume that X is locally embedded as i : X → Ω, where Ω is a pseudoconvex domain in C N . If Ker i * denotes the subsheaf of all smooth forms ξ in ambient space such that i * ξ = 0 on the regular part X reg of X, then one defines the sheaf E X of smooth forms on X simply as E X := E Ω /Ker i * .
It is well-known that this definition is independent of the choice of embedding of X.
Currents on X are defined as the duals of smooth forms with compact support. It is readily seen that the currents µ on X so defined are in a one-to-one correspondence to the currentsμ = i * µ in ambient space such thatμ vanish on Ker i * , see, e.g., [6]. There is an induced∂-operator on smooth forms and currents on X. In particular, (1.1) is a complex on X but in general it is not exact. In [6], Samuelsson and the first author introduced, by means of intrinsic Koppelman formulas on X, fine sheaves A * X of (0, * )-currents that are smooth on X reg and with mild singularities at the singular part of X, such that X∂ → · · ·∂ → A n X → 0 is exact, and thus a fine resolution of the structure sheaf O X . An immediate consequence is the representation of sheaf cohomology, and so (1.3) is a generalization of the classical Dolbeault isomorphism. In special cases more qualitative information of the sheaves A q X are known, see, e.g., [23,5].
Starting with the influential works [28,29] by Pardon and Stern, there has been a lot of progress recently on the L 2 -∂ theory on non-smooth (reduced) varieties; see, e.g., [15,27,31]. The point in these works, contrary to [6], is basically to determine the obstructions to solve∂ locally in L 2 . For a more extensive list of references regarding the∂-equation on reduced singular varieties, see, e.g., [6].
In [17], a notion of the∂-equation on non-reduced local complete intersections was introduced, and which was further studied in [18]. We discuss below how their work relates to ours.
The aim of this paper is to extend the construction in [6] to a non-reduced puredimensional analytic space. The first basic problem is to find appropriate definitions of forms and currents on X. Let X reg be the part of X where the underlying reduced space Z is smooth, and in addition O X is Cohen-Macaulay. On X reg the structure sheaf O X has a structure as a free finitely generated O Z -module. More precisely, assume that we have a local embedding i : X → Ω ⊂ C N and coordinates (z, w) in Ω such that Z = {w = 0}. Let J be the defining ideal sheaf for X on Ω. Then there are monomials 1, w α 1 , . . . , w α ν−1 such that each φ in O Ω /J ≃ O X has a unique representation (1.4) φ =φ 0 ⊗ 1 +φ 1 ⊗ w α 1 + · · · +φ ν−1 ⊗ w α ν−1 , whereφ j are in O Z . A reasonable notion of a smooth form on X should admit a similar representation on X reg with smooth formsφ j on Z. We first introduce the sheaves E 0, * X of smooth (0, * )-forms on X. By duality, we then obtain the sheaf C n, * X of (n, * )-currents. We are mainly interested in the subsheaf PM n, * X of pseudomeromorphic currents, and especially, the even more restricted sheaf W n, * X of such currents with the so-called standard extension property, SEP, on X. A current with the SEP is, roughly speaking, determined by its restriction to any dense Zariski-open subset.
Of special interest is the sheaf ω n X ⊂ W n,0 X of∂-closed pseudomeromorphic (n, 0)currents. In the reduced case this is precisely the sheaf of holomorphic (n, 0)-forms in the sense of Barlet-Henkin-Passare, see, e.g., [12,16].
We have no definition of "smooth (n, * )-form" on X. In order to define (0, * )currents, we use instead the sheaf ω n X in the following way. Any holomorphic function defines a morphism in Hom(ω n X , ω n X ), and it is a reformulation of a fundamental result of Roos, [30], that this morphism is indeed injective, and generically surjective. In the reduced case, multiplication by a current in W 0, * X induces a morphism in Hom(ω n X , W n, * X ), and in fact W 0, * X → Hom(ω n X , W n, * X ) is an isomorphism. In the non-reduced case, we then take this as the definition of W 0, * X . It turns out that with this definition, on X reg , any element of W 0, * X admits a unique representation (1.4), whereφ j are in W 0, * Z , see Section 6 below for details. Given v, φ in W 0, * X we say that∂v = φ if∂(v ∧ h) = φ ∧ h for all h in ω n X . Following [6] we introduce semi-global integral formulas and prove that if φ is a smooth∂-closed (0, q + 1)-form there is locally a current v in W 0,q X such that∂v = φ. A crucial problem is to verify that the integral operators preserve smoothness on X reg so that the solution v is indeed smooth on X reg . By an iteration procedure as in [6] we can define sheaves A k X ⊂ W 0,k X and obtain our main result in this paper.
Theorem 1.1. Let X be an analytic space of pure dimension n. There are sheaves A k X ⊂ W 0,k X that are modules over E 0, * X , coinciding with E 0,k X on X reg , and such that (1.2) is a resolution of the structure sheaf O X .
The main contribution in this article compared to [6] is the development of a theory for smooth (0, * )-forms and various classes of (n, * )and (0, * )-currents in the nonreduced case as is described above. This is done in Sections 4-8. The construction of integral operators to provide solutions to∂ in Section 9 and the construction of the fine resolution of O X in Section 11, which proves Theorem 1.1, are done pretty much in the same way as in [6]. The proof of the smoothness of the solutions of the regular part in Section 10 however becomes significantly more involved in the non-reduced case and requires completely new ideas. In Section 12 we discuss the relation to the results in [17,18] in case X is a local complete intersection.
Acknowledgements. We thank the referee for very careful reading and many valuable remarks.

Pseudomeromorphic currents
Let s 1 , . . . , s m be coordinates in C m , let α be a smooth form with compact support, and let a 1 , . . . , a r be positive integers, 0 ≤ ℓ ≤ r ≤ m. Then ℓ+1 · · · s ar r is a well-defined current that we call an elementary (pseudomeromorphic) current. Let Z be a reduced space of pure dimension. A current τ is pseudomeromorphic on Z if, locally, it is the push-forward of a finite sum of elementary pseudomeromorphic currents under a sequence of modifications, simple projections, and open inclusions. The pseudomeromorphic currents define an analytic sheaf PM Z on Z. This sheaf was introduced in [8] and somewhat extended in [6]. If nothing else is explicitly stated, proofs of the properties listed below can be found in, e.g., [6].
If τ is pseudomeromorphic and has support on an analytic subset V , and h is a holomorphic function that vanishes on V , thenhτ = 0 and dh ∧ τ = 0.
Given a pseudomeromorphic current τ and a subvariety V of some open subset U ⊂ Z, the natural restriction to the open set U \ V of τ has a natural extension to a pseudomeromorphic current on U that we denote by 1 U \V τ . Throughout this paper we let χ denote a smooth function on [0, ∞) that is 0 in a neighborhood of 0 and 1 in a neighborhood of ∞. If h is a holomorphic tuple whose common zero set is V , then Notice that 1 V τ := (1 − 1 U \V )τ is also pseudomeromorphic and has support on V . If W is another analytic set, then This action of 1 V on the sheaf of pseudomeromorphic currents is a basic tool. In fact one can extend this calculus to all constructible sets so that (2.2) holds, see [8].
One readily checks that if ξ is a smooth form, then The same holds if f is a simple projection and τ has compact support in the fiber direction. In any case we have Another basic tool is the dimension principle, that states that if τ is a pseudomeromorphic ( * , p)-current with support on an analytic set with codimension larger than p, then τ must vanish.
A pseudomeromorphic current τ on Z has the standard extension property, SEP, if 1 V τ = 0 for each germ V of an analytic set with positive codimension on Z. The set W Z of all pseudomeromorphic currents on Z with the SEP is a subsheaf of PM Z . By (2.3), W Z is closed under multiplication by smooth forms.
Let f be a holomorphic function (or a holomorphic section of a Hermitian line bundle), not vanishing identically on any irreducible component of Z. Then 1/f , a priori defined outside of {f = 0}, has an extension as a pseudomeromorphic current, the principal value current, still denoted by 1/f , such that 1 {f =0} (1/f ) = 0. The current 1/f has the SEP and We say that a current a on Z is almost semi-meromorphic if there is a modification π : Z ′ → Z, a holomorphic section f of a line bundle L → Z ′ and a smooth form γ with values in L such that a = π * (γ/f ), cf. [10,Section 4]. If a is almost semimeromorphic, then it is clearly pseudomeromorphic. Moreover, it is smooth outside an analytic set V ⊂ Z of positive codimension, a is in W Z , and in particular, a = lim ǫ→0 + χ(|h|/ǫ)a if h is a holomorphic tuple that cuts out (an analytic set of positive codimension that contains) V . The Zariski singular support of a is the Zariski closure of the set where a is not smooth. One can multiply pseudomeromorphic currents by almost semi-meromorphic currents; and this fact will be crucial in defining W 0, * X , when X is non-reduced. Notice that if a is almost semi-meromorphic in Z then it also is in any open U ⊂ Z.
Proposition 2.1 ([10, Theorem 4.8, Proposition 4.9]). Let Z be a reduced space, assume that a is an almost semi-meromorphic current in Z, and let V be the Zariski singular support of a.
(i) If τ is a pseudomeromorphic current in U ⊂ Z, then there is a unique pseudomeromorphic current a ∧ τ in U that coincides with (the naturally defined current) a ∧ τ in U \ V and such that 1 V (a ∧ τ ) = 0.
(ii) If W ⊂ U is any analytic subset, then Notice that if h is a tuple that cuts out V , then in view of (2.1), It follows that if ξ is a smooth form, then For future reference we will need the following result.
Proof. First assume that Z is smooth. Since W Z is closed under multiplication by smooth forms, so is W Z +∂W Z . The statement that PM Z = W Z +∂W Z is local, and since both sides are closed under multiplication by cutoff functions, we may consider a pseudomeromorphic current µ with compact support in C n . If µ has bidegree ( * , 0), then it is in W Z in view of the dimension principle. Thus we assume that µ has bidegree ( * , q) with q ≥ 1. Let where k is the Bochner-Martinelli kernel. Here (2.9) means that Kµ = p * (k ∧ µ ⊗ 1), where p is the projection C n ζ × C n z → C n z , (ζ, z) → z. Recall that we have the Koppelman formula µ =∂Kµ + K(∂µ). It is thus enough to see that Kµ is in W Z if µ is pseudomeromorphic. Let χ ǫ = χ(|ζ − z| 2 /ǫ). It is easy to see, by a blowup of C n × C n along the diagonal, that k is almost semi-meromorphic on C n × C n . Thus, by (2.7), χ ǫ k ∧ (µ ⊗ 1) → k ∧ (µ ⊗ 1). In view of Proposition 2.1 it follows that k ∧ (µ ⊗ 1) is pseudomeromorphic. Finally, if W is a germ of a subvariety of C n of positive codimension, then by (2.4) and (2.5), If Z is not smooth, then we take a smooth modification π : Z ′ → Z. For any µ in PM Z there is some µ ′ in PM Z ′ such that π * µ ′ = µ, see [4,Proposition 1.2]. Since µ ′ = τ +∂u with τ, u in W Z ′ , we have that µ = π * τ +∂π * u.

2.1.
Pseudomeromorphic currents with support on a subvariety. Let Ω be an open set in C N and let Z be a (reduced) subvariety of pure dimension n. Let PM Z Ω denote the sheaf of pseudomeromorphic currents τ on Ω with support on Z, and let W Z Ω denote the subsheaf of PM Z Ω of currents of bidegree (N, * ) with the SEP with respect to Z, i.e., such that 1 W τ = 0 for all germs W of subvarieties of Z of positive codimension. The sheaf CH Z Ω of Coleff-Herrera currents on Z is the subsheaf of W Z Ω of∂-closed (N, p)-currents, where p = N − n. Remark 2.3. In [6,3] CH Ω Z denotes the sheaf of pseudomeromorphic (0, p)-currents with support on Z and the SEP with respect to Z. If this sheaf is tensored by the canonical bundle K Ω we get the sheaf CH Z Ω in this paper. Locally these sheaves are thus isomorphic via the mapping µ → µ ∧ α, where α is a non-vanishing holomorphic (N, 0)-form.
We have the following direct consequence of Proposition 2.1.
Proposition 2.4. Let Z ⊂ Ω be a subvariety of pure dimension, let a be almost semi-meromorphic in Ω, and assume that it is smooth generically on Z. If τ is in W Z Ω , then a ∧ τ is in W Z Ω as well.
Proposition 2.5. Assume that we have local coordinates (z, w) such that Z = {w = 0}. Then τ in W Z Ω has a unique representation as a finite sum where dz := dz 1 ∧ · · · ∧ dz n . If π is the projection (z, w) → z, then If in addition∂τ is in W Z Ω then its coefficients in the expansion (2.11) are∂τ γ , cf. (2.12). In particular,∂τ = 0 if and only if∂τ γ = 0 for all γ.
Let us now consider the pairing between W Z Ω and germs φ at Z of smooth (0, * )forms. We assume that Z is smooth and that we have coordinates (z, w) as before, that τ is in W Z Ω , and that (2.11) holds. Moreover, we assume that φ is a smooth (0, * )-form in a neighborhood of Z in Ω. For any positive integer M we have the expansion where φ α (z) = 1 α! ∂φ ∂w α (z, 0) and O(w, dw) denotes a sum of terms, each of which contains a factorw j or dw j for some j. If M in (2.13) is chosen so that O(|w| M )τ = 0, then i.e., (2.14) φ Thus φ ∧ τ = 0 if and only if γ≥0 φ γ ∧ τ ℓ+γ = 0 for all ℓ (which is a finite number of conditions!).

2.2.
Intrinsic pseudomeromorphic currents on a reduced subvariety. Currents on a reduced analytic space Z are defined as the dual of the sheaf of test forms. If i : Z → Y is an embedding of a reduced space Z into a smooth manifold Y , then the push-forward mapping τ → i * τ gives an isomorphism between currents τ on Z and currents µ on Y such that ξ ∧ µ = 0 for all ξ in E Y such that i * ξ = 0. When defining pseudomeromorphic currents in the non-reduced case it is desirable that it coincides with the previous definition in case Z is reduced. From [4, Theorem 1.1] we have the following description of pseudomeromophicity from the point of view of an ambient smooth space.
Corollary 2.7. We have the isomorphism Notice that Hom(O Ω /J , W Z Ω ) is precisely the sheaf of µ in W Z Ω such that J µ = 0. Proof. The map i * is injective, since it is injective on any currents, and it maps into Hom(O Ω /J , W Z Ω ) by (2.15). To see that i * is surjective, we take a µ in Hom(O Ω /J , W Z Ω ). We assume first that we are on Z reg , with local coordinates such that Z reg = {w = 0}. If ξ is in E 0, * Ω and i * ξ = 0, then ξ is a sum of forms with a factor dw j , w j orw j . Since w j ∈ J , w j annihilates µ by assumption, and since w j vanishes on the support of µ,w j and dw j annihilate µ since µ is pseudomeromorphic. Thus, µ.ξ = 0, so µ = i * τ for some current τ on Z. By Proposition 2.6 (ii), τ is pseudomeromorphic, and by (2.15), has the SEP, i.e., τ is in W n, * Z . Remark 2.8. We do not know whether i * τ ∈ PM Z Ω implies that τ ∈ PM Z . By [11, Proposition 3.12 and Theorem 3.14], we get Proposition 2.9. Let ϕ and φ 1 , . . . , φ m be currents in W Z . If ϕ = 0 on the set on Z reg where φ 1 , . . . , φ m are smooth, then ϕ = 0.

Local embeddings of a non-reduced analytic space
Let X be an analytic space of pure dimension n with structure sheaf O X and let Z = X red be the underlying reduced analytic space. For any point x ∈ X there is, by definition, an open set Ω ⊂ C N and an ideal sheaf J ⊂ O Ω of pure dimension n with zero set Z such that O X is isomorphic to O Ω /J , and all associated primes of J at any point have dimension n. We say that we have a local embedding i : X → Ω ⊂ C N at x. There is a minimal such N , called the Zariski embedding dimensionN of X at x, and the associated embedding is said to be minimal. Any two minimal embeddings are identical up to a biholomorphism, and any embedding i : X → Ω has locally at x the form where j is minimal, U is an open subset of C m w , m = N −N , and the ideal in Ω is J = J ⊗ 1 + (w 1 , . . . , w m ). Notice that we then also have embeddings Z → Ω → Ω; however, the first one is in general not minimal. Now consider a fixed local embedding i : X → Ω ⊂ C N , assume that Z is smooth, and let (z, w) be coordinates in Ω such that Z = {w = 0}. We can identify O Z with holomorphic functions of z, and we can define an injection In this way O X becomes an O Z -module, which however depends on the choice of coordinates.  We shall now show that We claim that a sequence f 1 , . . . , f m in O X,x is regular (on O X,x ) if and only if f 1 , . . . ,f m ∈ O Z,x is regular on O X,x , wheref j (z) = f j (z, 0). In fact, since O X,x has pure dimension, a function g ∈ O X,x = O Ω,x /J is a non zero-divisor if and only if g is generically non-vanishing on each irreducible component of Z(J ). Thus f 1 is a non zero-divisor if and only iff 1 is. If it is, then O X,x /(f 1 ) = O Ω,x /(J + (f 1 )) again has pure dimension. Thus the claim follows by induction, and the fact that Z(J + (f 1 , . . . , f k )) = Z(J + (f 1 , . . . ,f k )). The claim immediately implies (3.3). To see (3.4), we note first that dim O X,x O X,x is just the usual (geometric) dimension of X or Z, i.e., in this case, n. Now, From In the proof above, we saw that O X is generated (locally) as an O Z -module by all monomials w α with |α| ≤ M for some M . Corollary 3.3. Assume that 1, w α 1 , . . . , w α ν−1 is a minimal set of generators at a given point x (clearly 1 must be among the generators!). Then we have a unique where z 1 or z 2 is = 0. If, say, z 1 = 0, then we can take 1, w 1 as generators. At the point z = (0, 0), e.g., 1, w 1 , w 2 form a minimal set of generators, and then O X is not a free O Z -module, since there is a non-trivial relation between w 1 and w 2 .
We claim that O X has pure dimension. That is, we claim that there is no embedded associated prime ideal at (0, 0); since Z is irreducible, this is the same as saying that J is primary with respect to Z. To see the claim, let φ and ψ be functions such that φψ is in J and ψ is not in √ J . The latter assumption means, in view of the Nullstellensatz, that ψ does not vanish identically on Z, i.e., ψ = a(z) + O(w), where a does not vanish identically. Since in particular φψ must vanish on Z it follows that φ = O(w). It is now easy to see that φ is in J . We conclude that J is primary.
The pure-dimensionality of O X can also be rephrased in the following way: If φ is holomorphic and is 0 generically, then φ = 0. If we delete the generator w 1 w 2 from the definition of J in the example, then φ = w 1 w 2 is 0 generically in O Ω /J but is not identically zero. Thus J then has an embedded primary ideal at (0, 0).
More generally, assume that, at a given point in X reg ⊂ Ω, we have two different choices (z, w) and (ζ, η) of coordinates so that Z = {w = 0} = {η = 0}, and bases 1, . . . , w α ν−1 and 1, . . . , η β ν−1 for O X as a free module over O Z . Then there is a ν × νmatrix L of holomorphic differential operators so that if (a j ) is any tuple in

4.
Smooth (0, * )-forms on a non-reduced space X Let i : X → Ω be a local embedding of X. In order to define the sheaf of smooth (0, * )-forms on X, in analogy with the reduced case, we have to state which smooth (0, * )-forms Φ in Ω "vanish" on X, or more formally, give a meaning to i * Φ = 0. We will see, cf. Lemma 4.8 below, that the suitable requirement is that locally on However, it turns out to be more convenient to represent the sheaf Ker i * of such forms as the annihilator of certain residue currents, and this is the path we will follow. Moreover, these currents play a central role themselves later on.
The following classical duality result is fundamental for this paper; see, e.g., [3] for a discussion.
. It is also well-known, see, e.g., [3, Theorem 1.5], that is a coherent analytic sheaf. Locally we thus have a finite number of generators µ 1 , . . . , µ m . In Example 6.9, we compute explicitly such generators for the ideal J in Example 3.4.
Let ξ be a smooth (0, * )-form in Ω. Without first giving meaning to i * , we define the sheaf Ker i * by saying that ξ is in . Notice that if ξ is holomorphic, then, in view of the duality (4.1), ξ is in Ker i * if and only if ξ is in J .

Definition 4.2.
We define the sheaf of smooth (0, * )-forms on X as E 0, * X := E 0, * Ω /Ker i * . We will prove below that this sheaf is independent of the choice of embedding and thus intrinsic on X.
Given φ in E 0, * Ω , let i * φ be its image in E 0, * X . In particular, i * ξ = 0 means that ξ belongs to Ker i * , which then motivates this notation. Notice that Ker i * is a twosided ideal in E 0, * Ω , i.e., if φ is in E 0, * Ω and ξ is in Ker i * , then φ ∧ ξ and ξ ∧ φ are in Ker i * . It follows that we have an induced wedge product on E 0, * X such that Remark 4.3. It follows from Lemma 4.8 below that in case X = Z is reduced, then ξ is in Ker i * if and only its pullback to X reg vanishes. Thus our definition of E 0, * X is consistent with the usual one in that case.
Proof. To begin with, ι * maps pseudomeromorphic (N ,p + ℓ)-currents with support on Z ⊂ Ω to pseudomeromorphic (N, p + ℓ)-currents with support on Z ⊂ Ω. If, in addition, τ has the SEP with respect to Z, then ι * τ has, as well by (2.15). Moreover, if τ is annihilated by J , then ι * τ is annihilated by J = J ⊗ 1 + (w). Thus the mapping (4.4) is well-defined, and it is injective since ι is injective. Now assume that µ is in Hom(O Ω /J , W Z Ω ). Arguing as in the proof of Corollary 2.7, we see that µ = ι * μ for a currentμ in W Z Ω . Since J = ι * J and J µ = 0, it follows that Jμ = 0. Thus (4.4) is surjective.
It follows from (4.6) and (4.3) that the sheaf E 0, * X is intrinsically defined on X. Since∂ maps Ker i * to Ker i * , we have a well-defined operator∂ : E 0, * X → E 0, * +1 X such that∂ 2 = 0. Unfortunately the sheaf complex so obtained is not exact in general, see, e.g., [6, Example 1.1] for a counterexample already in the reduced case.

4.1.
Local representation on X reg of smooth forms. Recall that X reg is the open subset of X, where the underlying reduced space is smooth and O X is Cohen-Macaulay. Let us fix some point in X reg , and assume that we have local coordinates (z, w) such that Z = {w = 0}. We also choose generators 1, w α 1 , . . . , w α ν−1 of O X as a free O Z -module, which exist by Corollary 3.3, and generators µ 1 , . . . , µ m of Hom(O Ω /J , CH Z Ω ). Notice that for each smooth (0, * )-form Φ in Ω, Φ → Φ ∧ µ ℓ only depends on its class φ in E 0, * X , and φ is in fact determined by these currents. By Proposition 2.5 each of these currents can (locally) be represented by a tuple of currents in W 0, * Z . Putting all these tuples together, we get a tuple in (W 0, * Z ) M , where M = M 1 + · · · + M m and M j is the number of indices in (2.11) in the representation of µ j .
Recall from Corollary 3.3 that φ in O X has a unique representative The morphism is injective by Proposition 4.1, and the holomorphic matrix T is therefore generically pointwise injective.
To begin with notice that a given smooth φ must have at least one such representation. In fact, taking the finite Taylor expansion (2.13) we can forget about high order terms, since they must annihilate all the µ j , and the termsw and dw annihilate all the µ j as well since they are pseudomeromorphic with support on {w = 0}. On the other hand, each w α not in the set of generators must be of the form . Thus the representation exists. To show uniqueness of the representation, we assume thatφ is in Ker i * . Then the tuple (φ j ) is mapped to 0 by the matrix T , and since T is generically pointwise injective we conclude that eachφ j vanishes.
By the above proof we get where J Z is the radical sheaf of Z. Remark 4.9. This is not the same as saying that ξ is in E 0, * Ω J + E 0, * ΩJ Z + E 0, * Ω dJ Z at singular points. For a simple counterexample, consider φ = xȳ on the reduced space However, this can happen also when Z is irreducible at a point. For example, the variety Z = {x 2 y − z 2 = 0} ⊂ C 3 is irreducible at 0, but there exist points arbitrarily close to 0 such that (Z, z) is not irreducible. In this case, the ideal of smooth functions vanishing on (Z, 0) is strictly larger than E 0,0 Ω J Z,0 + E 0,0 ΩJ Z,0 see [26, Proposition 9, Chapter IV], and [25, Theorem 3.10, Chapter VI].
Remark 4.10. It is easy to check that if we have the setting as in the discussion at the end of Section 3 but (a j ) is instead a tuple in E 0, * Z , then we can still define (b j ) = L(a j ) if we consider the derivatives in L as Lie derivatives; in fact, since a j has no holomorphic differentials, L only acts on the smooth coefficients, and it is easy to check that a 0 ⊗ 1 + · · · + a ν−1 ⊗ w α ν−1 and b 0 ⊗ 1 + · · · + b ν−1 ⊗ η β ν−1 are equal modulo E 0, * Ω J + E 0, * ΩJ Z + E 0, * Ω dJ Z , and thus define the same element in E 0, * X . For future needs we prove in Section 6.1: Lemma 4.11. The morphism T is pointwise injective.
We can thus choose a holomorphic matrix A such that is pointwise exact, and we can also find holomorphic matrices S and B such that (4.10) I = T S + BA.

5.
Intrinsic (n, * )-currents on X In analogy with the reduced case we have the following definition when X is possibly non-reduced.
Definition 5.1. The sheaf C n,q X of (n, q)-currents on X is the dual sheaf of (0, n − q)test forms, i.e., forms in E 0,n−q X with compact support.
Here, just as in the case of reduced spaces, cf. for example [19,Section 4.2], the space of smooth forms E 0,n−q X is equipped with the quotient topology induced by a local embedding.
More concretely, this means that given an embedding i : X → Ω, currents ψ in C n,q X precisely correspond to the (N, N − n + q)-currents τ on Ω that vanish on Ker i * . Since Ker i * is a two-sided ideal in E 0, * Ω this holds if and only if ξ ∧ τ = 0 for all ξ in Ker i * . It is natural to write τ = i * ψ so that Clearly, we get a mapping∂ : C n,q X → C n,q+1 X such that∂ 2 = 0.
Proof. Because of the SEP it is enough to prove that ξ ∧ τ = 0 on X reg . By assumption, J annihilates τ , and by general properties of pseudomeromorphic currents, since τ has support on Z,J Z and dJ Z annihilate τ . Thus the proposition follows by Lemma 4.8.
. By definition we thus have the isomorphism . It follows from Lemma 4.4 that W n, * X is intrinsically defined. Remark 5.4. By Corollary 2.7, this definition is consistent with the previous definition of W n, * X when X is reduced. We cannot define PM n, * X in the analogous simple way, cf. Remark 2.8.
X and a is an almost semi-meromorphic (0, * )-current on Ω that is generically smooth on Z, then the product a ∧ ψ is a current in W n, * X defined as follows: By definition, i * ψ is in Hom(O Ω /J , W Z Ω ) and by Proposition 2.4 and (2.8), one can define a∧i * ψ in if h cuts out the Zariski singular support of a.
Definition 5.6. We let ω n X be the sheaf of∂-closed currents in W n,0 X . This sheaf corresponds via i * to∂-closed currents in Hom(O Ω /J , W Z Ω ) so we have the isomorphism When X is reduced ω n X is the sheaf of (n, 0)-forms that are∂-closed in the Barlet-Henkin-Passare sense. Let µ 1 , . . . , µ m be a set of generators for the O Ω -module Hom(O Ω /J , CH Z Ω ). They correspond via (5.3) to a set of generators h 1 , . . . , h m for the O X -module ω n X . We will also need a definition of PM n, * Clearly, 1 W τ has support on W and it is easily checked that the computational rule (2.3) holds also in F X . Moreover, F X is closed under∂ since PM Z Ω is. Definition 5.7. The sheaf PM n, * X is the smallest subsheaf of F X that contains W n, * X and is closed under∂ and multiplication by 1 W for all germs W of subvarieties of Z.
In view of Proposition 2.2 this definition coincides with the usual definition in case X is reduced. It is readily checked that the dimension principle holds for F X , and hence it also holds for the (possibly smaller) sheaf PM n, * X , and in addition, (2.3) holds for forms ξ in E 0, * X and τ in PM n, * X .
6. Structure form on X Let i : X → Ω ⊂ C N be a local embedding as before, let p = N − n be the codimension of X, and let J be the associated ideal sheaf on Ω. In a slightly smaller set, still denoted Ω, there is a free resolution of O Ω /J ; here E k are trivial vector bundles over Ω and E 0 is the trivial line bundle. This resolution induces a complex of vector bundles that is pointwise exact outside Z. Let X k be the set where f k does not have optimal rank. Then these sets are independent of the choice of resolution and thus invariants of O Ω /J . Since O Ω /J has pure codimension p, see [14,Corollary 20.14]. Thus there is a free resolution (6.1) if and only if X k = ∅ for k > N 0 . Unless n = 0 (which is not interesting in relation to the∂-equation), we can thus choose the resolution so that N 0 ≤ N − 1. The variety X is Cohen-Macaulay at a point x, i.e., the sheaf O Ω /J is Cohen-Macaulay at x, if and only if x / ∈ X p+1 . Notice that Z \ (X reg ) red = Z sing ∪ X p+1 . The sets X k are independent of the choice of embedding, see [9,Lemma 4.2], and are thus intrinsic subvarieties of Z = X red , and they reflect the complexity of the singularities of X.
Let us now choose Hermitian metrics on the bundles E k . We then refer to (6.1) as a Hermitian resolution of O Ω /J in Ω. In Ω \ X k we have a well-defined vector bundle morphism σ k+1 : E k → E k+1 , if we require that σ k+1 vanishes on (Im f k+1 ) ⊥ , takes values in (Ker f k+1 ) ⊥ , and that f k+1 σ k+1 is the identity on Im f k+1 . Following [7, Section 2] we define smooth E k -valued forms in Ω \ X; for the second equality, see [7, (2.3)]. We have that in Ω\X. If f := ⊕f k and u := u k , then these relations can be written economically as To make the algebraic machinery work properly one has to introduce a superstructure on the bundle E =: ⊕E k so that vectors in E 2k are even and vectors in E 2k+1 are odd; hence f , σ := ⊕σ k , and u := u k are odd. For details, see [7]. It turns out that u has a (necessarily unique) almost semi-meromorphic extension U to Ω. The residue current R is defined by the relation It follows directly that R is ∇ f -closed. In addition, R has support on Z and is a sum R k , where R k is a pseudomeromorphic E k -valued current of bidegree (0, k). It follows from the dimension principle that R = R p + R p+1 + · · · + R N . If we choose a free resolution that ends at level N − 1, then R N = 0. If X is Cohen-Macaulay and N 0 = p in (6.1), then R = R p , and the ∇ f -closedness implies that R is∂-closed. If where the right hand side is the so-called Coleff-Herrera product of f , see for example [1,Corollary 3.5].
There are almost semi-meromorphic α k in Ω, cf. [7, Section 2] and the proof of [6, Proposition 3.3], that are smooth outside X k , such that In view of (6.3) and the dimension principle, 1 X k+1 R k+1 = 0 and hence (6.6) holds across X k+1 , i.e., R k+1 is indeed equal to the product α k+1 R k in the sense of Proposition 2.1. In particular, it follows that R k has the SEP with respect to Z.
In this section, we let (z 1 , . . . , z N ) denote coordinates on C N , and let dz := dz 1 ∧ · · · ∧ dz N . Lemma 6.2. There is a matrix of almost semi-meromorphic currents b such that where µ is a tuple of currents in Hom(O Ω /J , CH Z Ω ).
Proof. As in [6,Section 3], see also [32,Proposition 3.2], one can prove that R p = σ F µ, where µ is a tuple of currents in Hom(O Ω /J , CH Z Ω ) and σ F is an almost semimeromorphic current that is smooth outside X p+1 .
Let b p = σ F and b k = α k · · · α p+1 σ F for k ≥ p + 1. Then each b k is almost semimeromorphic, cf. [10,Section 4.1]. In view of (6.6) we have that R k = b k µ outside X p+1 since b k is smooth there. It follows by the SEP that it holds across X p+1 as well since R k has the SEP with respect to Z. We then take b = b p + b p+1 + · · · . By Proposition 2.4 we get Proposition 6.4. Let (6.1) be a Hermitian resolution of O Ω /J in Ω, and let R be the associated residue current. Then there exists a (unique) current ω in W n, * X such that There is a matrix b of almost semi-meromorphic (0, * )-currents in Ω, smooth outside of X p+1 , and a tuple ϑ of currents in ω n X such that (6.9) ω = bϑ.
We will also use the short-hand notation ∇ f ω = 0. As in the reduced case, following [6], we say that ω is a structure form for X. The products in (6.9) are defined according to Definition 5.5.
Remark 6.5. Recall that X p+1 = ∅ if X is Cohen-Macaulay, so in that case ω = bϑ, where b is smooth. If we take a free resolution of length p, then ω = ω 0 , and ∂ω 0 = f 1 ω 1 = 0, so ω is in ω n X . Remark 6.6. If X = {f = 0} is a reduced hypersurface in Ω, then R =∂(1/f ) and ω is the classical Poincaré residue form on X associated with f , which is a meromorphic form on X. More generally, if X is reduced, since forms in ω n X are then meromorphic, by (6.9), ω can be represented by almost semi-meromorphic forms on X.
We now consider the case when X is non-reduced. We recall that a differential operator is a Noetherian operator for an ideal J if Lϕ ∈ √ J for all ϕ ∈ J . It is proved by Björk, [13], see also [32,Theorem 2.2], that if µ ∈ Hom(O Ω /J , CH Z Ω ), then there exists a Noetherian operator L for J with meromorphic coefficients such that the action of µ on ξ equals the integral of Lξ over Z. By One can then verify using this formula and (6.9) that the action of the structure form ω on a test form ξ in E 0, * X equals whereL is now a tuple of Noetherian operators for J with almost semi-meromorphic coefficients, cf. [32,Section 4].
Notice that (6.1) gives rise to the dual Hermitian complex Let ξ = ξ 0 ∧ dz be a holomorphic section of the sheaf We thus have a sheaf mapping (6.12) Proposition 6.7. The mapping (6.12) is an isomorphism, which establishes an intrinsic isomorphism (6.13) . We have mappings (6.14) , where the first mapping is (6.12), and the second is h → i * h. In view of (6.8), the composed mapping is ξ = ξ 0 ∧ dz → ξR p = ξ 0 R p ∧ dz 2 . This mapping is an intrinsic isomorphism Ext p (O Ω /J , K Ω ) ≃ Hom(O Ω /J , CH Z Ω ) according to [3,Theorem 1.5]. It follows that (6.12) also establishes an intrinsic isomorphism.
We give here an example where we can explicitly compute generators of Hom(O Ω /J , CH Z Ω ).
Example 6.9. Let J be as in Example 3.4. We claim that Hom(O Ω /J , CH Z Ω ) is generated by In order to prove this claim, we use the comparison formula for residue currents from [21], which states that if O(F • ) and O(E • ) are free resolutions of O Ω /I and O Ω /J , respectively, where I and J have codimension ≥ p, and a : F • → E • is a morphism of complexes, then there exists a Hom( , we thus get that (6.16) ξR E p a 0 = ξa p R F p . We will apply this with O Ω (E • ) as the free resolution and f 1 = w 2 1 w 1 w 2 w 2 2 z 2 w 1 − z 1 w 2 , and the Koszul complex (F, δ w 2 ) generated by w 2 := (w 2 1 , w 2 2 ), which is a free resolution of O/(w 2 1 , w 2 2 ). We then take the morphism of complexes a : F • → E • given by Since the current R F 2 is equal to the Coleff-Herrera product∂(1/w 2 1 ) ∧∂(1/w 2 2 ), cf. Remark 6.1, we thus get by (6.16) and Remark 6.8 that Hom(O Ω /J , CH Z Ω ) is generated by A straightforward calculation gives the generators µ 1 and µ 2 above.
6.1. Proof of Lemma 4.11. Since T is generically injective, it is clearly injective if n = 0. We are going to reduce to this case. Fix the point 0 ∈ Z and let I be the ideal generated by z = (z 1 , . . . , z n ).
Let O(E • ) be a free Hermitian resolution of O Ω /J of minimal length p = N −n at 0 and let R E be the associated residue current. Recall that the canonical isomorphism (6.15) is realized by ξ → ξR E p . Let is given by η → ηR E⊗F N . Let µ 1 , . . . , µ m be a minimal set of generators for the O Ω -module Hom(O Ω /J , CH Z Ω ) at 0. Then µ j = ξ j R E p , where ξ j is a minimal set of generators for H p (Hom(E • , K Ω )). Notice that Since H n (Hom(F • , O Ω )) is generated by 1, it follows that H N (Hom((E ⊗ F ) • , K Ω )) is generated by ξ j ⊗ 1. We conclude that Hom(O Ω /(J + I), CH {0} Ω ) is generated by The morphism constructed in (4.8) for X 0 instead of X is then T 0 = T (0), where T is the morphism (4.8) for X. Thus T (0) is injective.
7. The intrinsic sheaf W 0, * X on X Our aim is to find a fine resolution of O X and since the complex (1.1) is not exact in general when X is singular we have to consider larger fine sheaves; we first define sheaves W 0, * X ⊃ E 0, * X of (0, * )-currents. Given a local embedding i : X → Ω at a point on X reg and local coordinates (z, w) as before, it is natural, in view of Lemma 4.7, to require that an element in W 0, * X shall have a unique representation where φ j are in W 0, * Z . In view of Remark 4.10 we should expect that the same transformation rules hold as for smooth (0, * )-forms. In particular it is then necessary that W 0. * Z is closed under the action of holomorphic differential operators, which in fact is true, see Proposition 7.11 below. We must also define a reasonable extension of these sheaves across X sing . Before we present our formal definition we make a preliminary observation.
Proof. The right hand side defines a current in W Z Ω since φ i are in W 0, * Z and τ γ are in O Z . We have to prove that it is annihilated by J . Take ξ in J . On the subset of Z where φ 0 , . . . , φ ν−1 are all smooth, φ ∧ τ , as defined above, is just multiplication of the smooth form φ by τ , and thus ξφ ∧ τ = 0 there. We have a unique representation with a ℓ in W 0, * Z . Since a ℓ vanish on the set where all φ j are smooth, we conclude from Proposition 2.9 that a ℓ vanish identically. It follows that ξφ ∧ τ = 0.
If φ has the form (7.1) in a neighborhood of some point x ∈ X reg and h is in ω n X , then we get an element φ ∧ h in W n, * X defined by i * (φ ∧ h) = φ ∧ i * h. It follows that φ in this way defines an element in Hom O X (ω n X , W n, * X ). This sheaf is global and invariantly defined and so we can make the following global definition.
Definition 7.2. W 0, * X = Hom O X (ω n X , W n, * X ). If φ is in W 0, * X and h is in ω n X , we consider φ(h) as the product of φ and h, and sometimes write it as φ ∧ h.
Since W n, * X are E 0, * X -modules, W 0, * X are as well. Before we investigate these sheaves further, we give some motivation for the definition. First notice that we have a natural injection, cf. Proposition 4.1, This is the subset of X where codim X k ≥ k + 2, k ≥ p + 1, cf. Section 6. Thus it contains all points x such that O X,x is Cohen-Macaulay. In particular, (7.3) is an isomorphism in X reg . Theorem 7.3 is a consequence of the results in [22]. If X has pure dimension p, there is an injective mapping (7.4) O X → Hom(Ext p (O X , K Ω ), CH Z Ω ), which by [22, Theorem 1.2 and Remark 6.11] is an isomorphism if and only if O X is S 2 . Since the image of such a morphism must be annihilated by J by linearity, it is indeed a morphism (7.5) O X → Hom(Ext p (O X , K Ω ), Hom(O Ω /J , CH Z Ω )). In view of (4.2) and (5.3), (7.5) corresponds to a morphism O X → Hom(ω n X , ω n X ), and the fact that it is the morphism (7.3) is a rather simple consequence of the definition of the morphism (7.4) in [22, (6.9)].
As mentioned in the introduction, Theorem 7.3 can be seen as a reformulation of a classical result of Roos, [30], which is the same statement about the injection (7.6) O Ω /J → Ext p (Ext p (O Ω /J , K Ω ), K Ω ); here we assume that the ideal has pure dimension. The equivalence of the morphisms (7.4) and (7.6) is discussed in [22,Corollary 1.4].
Let us now consider the case when X is reduced. Since sections of ω n X are meromorphic, see [6,Example 2.8], and thus almost semi-meromorphic and generically smooth, by Proposition 2.4 (with Z = X = Ω) we can extend (7.3) to a morphism (7.7) W 0, * X → Hom(ω n X , W n, * X ).
Lemma 7.4. When X is reduced (7.7) is an isomorphism.
Thus Definition 7.2 is consistent with the previous definition of W 0, * X when X is reduced.
Proof. Clearly each φ in W 0, * X defines an element α in Hom(ω n X , W n, * X ) by h → φ ∧ h. If we apply this to a generically nonvanishing h we see by the SEP that (7.7) is injective.
For the surjectivity, take α in Hom(ω n X , W n, * X ). If h ′ is nonvanishing at a point on X reg , then it generates ω n X and thus α is determined by φ := αh ′ there. By [10,Theorem 3.7], φ = ψ ∧ h ′ for a unique current ψ in W 0, * X so by O X -linearity αh = ψ ∧ h for any h. Hence, ψ is well-defined as a current in W 0, * X on X reg . We must verify that ψ has an extension in W 0, * X across X sing . Since such an extension must be unique by the SEP, the statement is local on X. Thus we may assume that α is defined on the whole of X and that there is a generically nonvanishing holomorphic n-form γ on X. Then αγ is a section of W n, * (X).
Let us choose a smooth modification π : X ′ → X that is biholomorphic outside X sing . Then π * γ is a holomorphic n-form on X ′ that is generically non-vanishing. We claim that there is a current τ in W n,0 (X ′ ) such that π * τ = αγ. In fact, τ exists on π −1 (X reg ) since π is a biholomorphism there. Moreover, by [4,Proposition 1.2], αh is the direct image of some pseudomeromorphic currentτ on X ′ , and is therefore also the image of the (unique) current τ = 1 π −1 (Xreg )τ in W n, * (X ′ ).
In view of (5.1) and (5.3) we have, given a local embedding i : X → Ω, the extrinsic representation ). Lemma 7.5. Assume that X reg → Ω is a local embedding and (z, w) coordinates as before. Each section φ in W 0, * X has a unique representation (7.1) with φ j in W 0, * Z . A current with a representation (7.1) is considered as an element of W 0, * X = Hom(ω n X , W n, * X ) in view of the comment after Lemma 7.1. Proof. From (4.9) we get an induced sequence which is also exact. In fact, T in (7.9) is clearly injective, and by (4.10), if ξ in (W 0, * Z ) M and Aξ = 0, then T η = ξ, if η = Sξ. Now take φ in Hom(ω n X , W n, * X ). Let us choose a basis µ 1 , . . . , µ m for ω n X and let φ be the element in (W 0, * Z ) M obtained from the coefficients of φµ j when expressed as in (2.11), cf. Section 4.1. We claim that Aφ = 0. Taking this for granted, by the exactness of (7.9),φ is the image of the tupleφ = Sφ. Nowφ ∧ µ j = φµ j since they are represented by the same tuple in (W 0, * Z ) M . Thusφ gives the desired representation of φ.
Example 7.6 (Meromorphic functions). Assume that we have a local embedding X → Ω. Given meromorphic functions Φ, Φ ′ in Ω that are holomorphic generically on Z, where B and B ′ are generically non-vanishing on Z, the condition is precisely that AB ′ − A ′ B is in J . We say that such an equivalence class is a meromorphic function φ on X, i.e., φ is in M X . Clearly we have O X ⊂ M X . We claim that M X ⊂ W 0, * X . To see this, first notice that if we take a representative Φ in M Ω of φ, then it can be considered as an almost semi-meromorphic current on Ω with Zariski-singular support of positive codimension on Z, since it is generically holomorphic on Z. As in Definition 5.5 we therefore have a current Φ ∧ h in W n,0 X for h in ω n X . Another representative Φ ′ of φ will give rise to the same current generically and hence everywhere by the SEP. Thus φ defines a section of Hom(ω n X , W n, * X ) = W 0, * X . By definition, a current φ in W 0, * X can be multiplied by a current h in ω n X , and the product φ ∧ h lies in W n, * X . It will be crucial that we can extend to products by somewhat more general currents. Notice that ω n X is a subsheaf of C n, * X , which is an E 0, * X -module. Thus, we can consider the subsheaf E 0, * X ω n X of C n, * X which consists of finite sums ξ i ∧ h i , where ξ i are in E 0, * X and h i are in ω n X .
Lemma 7.7. Each φ in W 0, * X = Hom O X (ω n X , W n, * X ) has a unique extension to a morphism in Hom E 0, * Proof. The uniqueness follows by E 0, * We must check that this is well-defined, i.e., that the right hand side does not depend on the representation ξ 1 ∧ h 1 + · · · + ξ r ∧ h r of b. By the SEP, it is enough to prove this locally on X reg , and we can then assume that φ has a representation (7.1). By Proposition 2.9, it is then enough to prove that it is well-defined assuming that φ 0 , . . . , φ ν−1 in (7.1) are all smooth. In this case, the right hand side of (7.10) is simply the product of ξ 1 ∧ h 1 + · · · + ξ r ∧ h r = b by the smooth form φ in E 0, * X , and this product only depends on b.
Corollary 7.8. Let φ be a current in W 0, * X and let α be a current in W n, * X of the form α = a i ∧ h i , where a i are almost semi-meromorphic (0, * )-currents on Ω which are generically smooth on Z, and h i are in ω n X . Then one has a well-defined product Proof. The right hand side of (7.11) exists as a current in W n, * X , and we must prove is that it only depends on the current α and not on the representation a i ∧ h i . Notice that all the a i are smooth outside some subvariety V of Z and there the right hand side of (7.11) is the product of φ and α in E 0, * X ω n X , cf. Lemma 7.7. It follows by the SEP that the right hand side only depends on α.
Lemma 7.10. Assume that φ is in W 0, * X , and that φ ∧ ω = 0 for some structure form ω, where the product is defined by Remark 7.9. Then φ = 0.
Proof. Considering the component with values in E p , we get that φ ∧ ω 0 = 0. By Proposition 6.7, any h in ω n X can be written as h = ξω 0 , where ξ is a holomorphic section of E * p , so by O-linearity, φ ∧ h = 0, i.e., φ = 0.
We end this section with the following result, the first part of [10, Theorem 3.7]. We include here a different proof than the one in [10], since we believe the proof here is instructive. Proposition 7.11. If Z is smooth, then W Z is closed under holomorphic differential operators.
Proof. Let τ be any current in W Z . It suffices to prove that if ζ are local coordinates on Z, then ∂τ /∂ζ 1 is in W Z . Consider the current Clearly τ ′ has support on Z, and it follows from which is just a change of variables on Y followed by a projection. It follows from it is readily verified that p * τ ′ = ∂τ /∂ζ 1 , so we conclude that ∂τ /∂ζ 1 is in W Z .
Remark 8.5. In case X is reduced the definition of∂ X is precisely the same as in [6]. However, the definition of∂v = φ given here, for v, φ in W 0, * X , does not coincide with the definition in, e.g., [6]. In fact, that definition means that∂(v ∧ h) = φ ∧ h for all smooth h in ω n X , which in general is a strictly weaker condition. For example, for any weakly holomorphic function v, we have∂(v ∧ h) = 0 for all smooth h in ω n X , while if X is a reduced complete intersection, or more generally Cohen-Macaulay, then∂(v ∧ h) = 0 for all h in ω n X is equivalent to v being strongly holomorphic, see [33, p. 124] and [2].
We conclude this section with a lemma that shows that∂ means what one should expect when φ, v are expressed with respect to a local basis w α j for O X over O Z as in Lemma 7.5. Proof. Let us use the notation from the proof of Lemma 7.5. Recall thatv = Sṽ. In view of (8.2) and (2.12), ∂ v =∂ṽ. Since S is holomorphic therefore ∂ v = S ∂ v = S∂ṽ =∂(Sṽ) =∂v. 9. Solving∂u = φ on X We will find local solutions to the∂-equation on X by means of integral formulas. We use the notation and machinery from [6, Section 5]. Let i : X → Ω ⊂ C N be a local embedding such that Ω is pseudoconvex, let Ω ′ ⊂⊂ Ω be a relatively compact subdomain of Ω, and let X ′ = X ∩ Ω ′ . Theorem 9.1. There are integral operators K : E 0, * +1 (X) → W 0, * (X ′ ) ∩ Dom∂ X , P : E 0, * (X) → E 0, * (X ′ ) such that, for φ ∈ E 0,k (X), The operators K and P are described below; they depend on a choice of weight g. Since Ω is Stein one can find such a weight g that is holomorphic in z, by which we mean that it depends holomorphically on z ∈ Ω ′ and has no components containing any dz i , cf. Example 5.1 in [6]. In this case, P φ is holomorphic when k = 0, and vanishes when k ≥ 1, i.e., (9.2) φ =∂Kφ + K(∂φ), φ ∈ E 0,k (X), k ≥ 1.
If∂φ = 0 in Ω, and k ≥ 1, then Kφ is a solution to∂v = φ. If k = 0, then φ = P φ is holomorphic. It follows that a smooth∂-closed function is holomorphic. In the reduced case this is a classical theorem of Malgrange, [24]. In Section 10 we prove that Kφ is smooth on X reg . We now turn to the definition of K and P . For future need, in Section 11, we define them acting on currents in W 0, * (X) and not only on smooth forms. Let π : Ω ζ ×Ω ′ z → Ω ′ z be the natural projection. Let us choose a holomorphic Hefer form 3 H, a smooth weight g with compact support in Ω with respect to z ∈ Ω ′ ⊂⊂ Ω, and let B be the Bochner-Martinelli form. Since we are only are concerned with (0, * )forms, we will here assume that H and B only have holomorphic differentials in ζ, i.e., the factors dη i = dζ i − dz i in H and B in [6] should be replaced by just dζ i .
If γ is a current in Ω ζ × Ω ′ z we let (γ) N be the component of bidegree (N, * ) in ζ and (0, * ) in z, and let ϑ(γ) be the current such that Consider now µ in Hom(O Ω /J , W Z Ω ) and φ in W 0, * X . We can give meaning to as a tensor product of currents in the following way: First of all, by Remark 7.9, we can form the product R(ζ)∧dζ∧φ(ζ) as a current in W Z Ω . In view of [11,Corollary 4.7] Finally, we multiply this with the smooth form ϑ(g ∧ H) to obtain (9.4). Similarly, outside of ∆, the diagonal in Ω × Ω ′ , where B is smooth, we can define as a tensor product of currents.
Proof of Lemma 9.2. In order to define the extension of (9.5) across ∆, we note first that since B is almost semi-meromorphic with Zariski singular support ∆, ϑ(B∧g∧H) is an almost semi-meromorphic (0, * )-current on Ω ζ × Ω ′ z , which is smooth outside the diagonal. We can thus form the current ϑ( , cf. Proposition 2.4, and this is the extension of (9.5) across ∆. From the definitions above, it is clear that (9.4) and the extension of (9.5) are O Ω -bilinear in φ and µ. Both these currents are annihilated by J z and J ζ , cf. (2.8), so they depend O Ω /J -bilinearly. In view of (2.4) we conclude that (9.6) and (9.7) are in Hom(O Ω ′ /J , W Z ′ Ω ′ ). Proposition 9.3. If φ ∈ W 0,k (X), then P φ ∈ E 0,k (X ′ ), and if in addition g is holomorphic in z, then P φ ∈ O(X ′ ) if k = 0 and vanishes if k ≥ 1.
Proposition 9.4. For any φ ∈ W 0,k (X), k ≥ 1, The last statement means that Since B ǫ is smooth, the current we push forward is R(ζ)∧φ(ζ) times a smooth form of ζ and z. Therefore K ǫ φ is smooth. As in the proof of Proposition 9.3, we obtain since B ǫ is smooth that By (5.2) applied to a = B we have that which implies (9.9).
Recall from Lemma 6.2 that R ∧ dz = bµ, where µ is a tuple of currents in Hom(O Ω ′ /J , CH Z ′ Ω ′ ) and b is an almost semi-meromorphic matrix that is smooth generically on Z ′ . Therefore (9.12) and (9.13) hold where b is smooth, in view of Lemma 7.7, and since both sides are in Hom(O Ω ′ /J , W Z ′ Ω ′ ), the equalities hold everywhere by the SEP.
As in [6] we let R λ =∂|f | 2λ ∧ U for Re λ ≫ 0. It has an analytic continuation to λ = 0 and R = R λ | λ=0 . Notice that R(z) ∧ B is well-defined since it is a tensor product with respect to the coordinates z, η = ζ − z. Also R(z) ∧ R λ (ζ) ∧ B admits such an analytic continuation and defines a pseudomeromorphic current 4 when λ = 0. Let B k,k−1 be the component of B of bidegree (k, k − 1). Lemma 9.5. For all k, Proof of Lemma 9.5. Notice that the equality holds outside ∆. Let T be the left hand side of (9.14). In view of Proposition 2.1 it is therefore enough to check that 1 ∆ T = 0. Fix j, k and let Clearly T ℓ = 0 if ℓ < p so first assume that ℓ = p. Since HR j has bidegree (j, j) in ζ, the current vanishes unless j + k ≤ N . Thus the total antiholomorphic degree is ≤ N −n+N −1. On the other hand, the current has support on ∆∩Z ×Z ≃ Z ×{pt} which has codimension N + N − n. Thus it vanishes by the dimension principle.
By the same argument 5 as for [6, (5.2)] we have the equality also for our R, where [∆] ′ denotes the part of [∆] where dη i = dζ i − dz i has been replaced 6 by dζ i . In view of (9.14) we can put λ = 0 in (9.15), and then we get Multiplying (9.16) by the smooth form φ, and using (9.12) and (9.13), we get or equivalently, Multiplying by suitable holomorphic ξ 0 in E * p such that f * p+1 ξ 0 = 0, cf. Proposition 6.7, we see that φ ∧ h =∂(Kφ ∧ h) + K(∂φ) ∧ h + P φ ∧ h for all h in ω X . Thus by definition (9.1) holds. Since W 0, * X is closed under multiplication by O X , we get that ψ in W 0, * X is in Dom∂ X if and only if −∇ f (ψ ∧ ω) is in W n, * X . Thus, we conclude from (9.17) that Kφ is in Dom∂ X since all the other terms but −∇ f (Kφ ∧ ω) are in W n, * X . 9.2. Intrinsic interpretation of K and P . So far we have defined K and P by means of currents in ambient space. We used this approach in order to avoid introducing push-forwards on a non-reduced space. However, we will sketch here how this can be done. We must first define the product space X × X ′ . Given a local embedding i : X → Ω as before, we have an embedding (i × i) : X × X ′ → Ω × Ω ′ such that the structure sheaf is O Ω×Ω ′ /(J X + J X ′ ). One can check that this sheaf is independent of the chosen embedding, i.e., O X×X ′ is intrinsically defined. Thus we also have definitions of all the various sheaves on X × X ′ like E 0, * X×X ′ . The projection p : X×X ′ → X ′ is determined by p * φ : O X ′ → O X×X ′ , which in turn is defined so that p * i * Φ = (i×i) * π * Φ for Φ in O Ω ′ , where π : Ω×Ω ′ → Ω ′ as before. Again one can check that this definition is independent of the embedding, and also extends to smooth (0, * )-forms φ. Therefore, we have the well-defined mapping p * : C 2n, * +n X×X ′ → C n, * X ′ , and clearly (9.18) i * p * = π * (i × i) * .
As before we have the isomorphism As in the proof of Lemma 9.2 we see that π * maps a current in W Z×Z ′ Ω×Ω ′ annihilated by J X ′ to a current in Hom(O Ω /J , W Z ′ Ω ′ ). It follows by (9.18) that Now, take h in ω n X ′ and let µ = i * h. Then, cf. the proof of Lemma 9.2, Thus we can define Kφ intrinsically by (9.19) Kφ From above it follows that Kφ ∧ h is in W n, * X ′ . In the same way we can define P φ by (9.20) It is natural to write although the formal meaning is given by (9.19) and (9.20).

Regularity of solutions on X reg
We have already seen, cf. Proposition 9.3, that P φ is always a smooth form. We shall now prove that K preserves regularity on X reg . More precisely, Throughout this section, let us choose local coordinates (ζ, τ ) and (z, w) at x corresponding to the variables ζ and z in the integral formulas, so that Z = {(ζ, τ ); τ = 0}.
Lemma 10.2. Let B ǫ := χ(|ζ − z| 2 /ǫ)B, and assume that φ has compact support in our coordinate neighborhood. Then Kφ can be approximated by the smooth forms Notice that here we cut away the diagonal ∆ ′ in Z × Z ′ times C τ × C w in contrast to Proposition 9.4, where we only cut away the diagonal ∆ in Ω × Ω ′ .
We first consider a simple but nontrivial example of Theorem 10.1.
Remark 10.4. The terms O(w) in the expansion (10.4) of K ǫ φ(z, w) do not converge to smooth functions in general when ǫ → 0. For a simple example, take φ = ζdζ ⊗τ m . Then K ǫ φ(0, w) tends to which is a smooth function of w plus (a constant times) w m |w| 2 log |w| 2 , and thus not smooth. However, it is certainly in C m . One can check that Kφ(z, w) = lim ǫ→0 + K ǫ φ(z, w) exists pointwise and defines a function in at least C m and that our solution can be computed from this limit. In fact, by a more precise computation we get from (10.3) that It is now clear that we can let ǫ → 0. By a simple computation we then get Let ψ = ϕφ k . Then the kth term in the second sum is equal to If we integrate outside the unit disk, then we certainly get a smooth function. Thus it is enough to consider the integral over the disk. Moreover, if ψ(z + ζ) = O(|ζ| M ) for a large M , then the integral is at least C m . By a Taylor expansion of ψ(z + ζ) at the point z, we are thus reduced to consider for non-negative integers β. The right hand side is a smooth function of w if β ≤ m − k − 1 and a smooth function plus The worst case therefore is when k = m and β = 0; then we have w m |w| 2 log |w| 2 that we encountered above.
Proposition 10.5. Let z, w be coordinates at a point x ∈ X reg such that Z = {w = 0} and x = (0, 0). If φ is smooth, and has support where the local coordinates are defined, then is smooth for ǫ > 0, and for each multiindex ℓ there is a smooth form v ℓ such that Taking this proposition for granted we can conclude the proof of Theorem 10.1.
Proof of Theorem 10.1. If φ ≡ 0 in a neighborhood of x ∈ X ′ reg , then Kφ is smooth near x, cf. the proof of Proposition 9.4. Thus, it is sufficient to prove Theorem 10.1 assuming that φ is smooth and has support near x.
Proof. Introduce a nonsense basis e and its dual e * and consider the exterior algebra spanned by e j , e * ℓ , and the cotangent bundle. Let c ℓ = η * e ∧ ((dη * )e) ℓ−1 .
In order to see that the limit of (10.8) tends to 0, we note first that if we let χ(s) = sχ ′ (s) + χ(s), then just as χ,χ is also a smooth function on [0, ∞) that is 0 in a neighborhood of 0 and 1 in a neighborhood of ∞. By assumption, r + s − ℓ − 1 ≥ 0. Since the principal value current 1/f m acting on a test form β can be defined as Taking the difference between the left and right hand side, we conclude that (10.8) tends to 0 when ǫ → 0. Now we can conclude the proof of Proposition 10.5. From the beginning we have I ℓ,0 ℓ . After repeated applications of (10.6) we end up with I 0,ℓ ℓ + I 0,ℓ−1 ℓ−1 + · · · + I 0,0 0 + o(1). However, any of these integrals has an integrable kernel even when ǫ = 0. This means that we are back to the case in [6, Lemma 6.2], and so the limit integral is smooth in z.

A fine resolution of O X
We first consider a generalization of Theorem 9.1.

Locally complete intersections
Let us consider the special case when X locally is a complete intersection, i.e., given a local embedding i : X → Ω ⊂ C N there are global sections f j of O(d j ) → P N such that J = (f 1 , . . . , f p ), where p = N −n. In particular, Z = {f 1 = · · · = f p = 0}. In this case Hom(O Ω /J , CH Ω ) is generated by the single current µ =∂ 1 f p ∧ · · · ∧∂ 1 f 1 ∧ dz 1 ∧ · · · ∧ dz N , see, e.g., [3]. Each smooth (0, q)-form φ in E 0,q X is thus represented by a current Φ∧µ, where Φ is smooth in a neighborhood of Z and i * Φ = φ. Moreover, X is Cohen-Macaulay so X reg coincides with the part of X where Z is regular, and∂φ = ψ if and only if∂(φ ∧ µ) = ψ ∧ µ.
Henkin and Polyakov introduced, see [17, Definition 1.3], the notion of residual currents φ of bidegree (0, q) on a locally complete intersection X ⊂ P N , and the∂-equation∂ψ = φ. Their currents φ correspond to our φ in E 0,q X and their∂-operator on such currents coincides with ours.
Remark 12.1. In [18] Henkin and Polyakov consider a global reduced complete intersection X ⊂ P N . They prove, by a global explicit formula, that if φ is a global ∂-closed smooth (0, q)-form with values in O(ℓ), ℓ = d 1 +· · · d p −N −1, then there is a smooth solution to∂ψ = φ at least on X reg , if 1 ≤ q ≤ n−1. When q = n a necessary obstruction term occurs. However, their meaning of "∂-closed" is that locally there is a representative Φ of φ and smooth g j such that∂Φ = g 1 f 1 + · · · + g p f p . If this holds, then clearly∂φ = 0. The converse implication is not true, see Example 12.2 below. It is not clear to us whether their formula gives a solution under the weaker assumption that∂φ = 0, neither do we know whether their solution admits some intrinsic extension across X sing as a current on X.
Example 12.2. Let X = {f = 0} ⊂ Ω ⊂ C n+1 be a reduced hypersurface, and assume that df = 0 on X reg , so that J = (f ). Let φ be a smooth (0, q)-form in a neighborhood of some point x on X such that∂φ = 0. We claim that∂u = φ has a smooth solution u if and only if φ has a smooth representative Φ in ambient space such that∂Φ = f g for some smooth form g. In fact, if such a Φ exists then 0 = f∂g and thus∂g = 0. Therefore, g =∂γ for some smooth γ (in a Stein neighborhood of x in ambient space) and hence∂(Φ − f γ) = 0. Thus there is a smooth U such that ∂U = Φ − f γ; this means that u = i * U is a smooth solution to∂u = φ. Conversely, if u is a smooth solution, then u = i * U for some smooth U in ambient space, and thus Φ =∂U is a representative of φ in ambient space. Thus∂Φ = f g (with g = 0).
There are examples of hypersurfaces X where there exist smooth φ with∂φ = 0 that do not admit smooth solutions to∂u = φ, see, e.g., [6, Example 1.1]. It follows that such a φ cannot have a representative Φ in ambient space as above.