# The \(\bar{\partial }\)-equation on a non-reduced analytic space

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## Abstract

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*.

## Mathematics Subject Classification

32A26 32A27 32B15 32C30## 1 Introduction

*X*be a smooth complex manifold of dimension

*n*and let \(\mathscr {E}_X^{0,*}\) denote the sheaf of smooth \((0,*)\)-forms. It is well-known that the Dolbeault complex

*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 \(i:X\rightarrow \Omega \), where \(\Omega \) is a pseudoconvex domain in \(\mathbb {C}^N\). If \({\mathcal Ker\,}i^*\) denotes the subsheaf of all smooth forms \(\xi \) in ambient space such that \(i^*\xi =0\) on the regular part \(X_{reg}\) of

*X*, then one defines the sheaf \(\mathscr {E}_X\) of smooth forms on

*X*simply as

*X*. Currents on

*X*are defined as the duals of smooth forms with compact support. It is readily seen that the currents \(\mu \) on

*X*so defined are in a one-to-one correspondence to the currents \(\hat{\mu }=i_*\mu \) in ambient space such that \(\hat{\mu }\) vanish on \({\mathcal Ker\,}i_*\), see, e.g., [6]. There is an induced \(\bar{\partial }\)-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 \(\mathscr {A}_X^*\) of \((0,*)\)-currents that are smooth on \(X_{reg}\) and with mild singularities at the singular part of

*X*, such that

Starting with the influential works [28, 29] by Pardon and Stern, there has been a lot of progress recently on the \(L^2\)-\(\bar{\partial }\) 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 \(\bar{\partial }\) locally in \(L^2\). For a more extensive list of references regarding the \(\bar{\partial }\)-equation on reduced singular varieties, see, e.g., [6].

In [17], a notion of the \(\bar{\partial }\)-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.

*X*. Let \(X_{reg}\) be the part of

*X*where the underlying reduced space

*Z*is smooth, and in addition \(\mathscr {O}_X\) is Cohen–Macaulay. On \(X_{reg}\) the structure sheaf \(\mathscr {O}_X\) has a structure as a free finitely generated \(\mathscr {O}_Z\)-module. More precisely, assume that we have a local embedding \(i:X\rightarrow \Omega \subset \mathbb {C}^N\) and coordinates (

*z*,

*w*) in \(\Omega \) such that \(Z=\{w=0\}\). Let \(\mathcal {J}\) be the defining ideal sheaf for

*X*on \(\Omega \). Then there are monomials \(1, w^{\alpha _1}, \ldots , w^{\alpha _{\nu -1}}\) such that each \(\phi \) in \(\mathscr {O}_\Omega /\mathcal {J}\simeq \mathscr {O}_X\) has a unique representation

*X*should admit a similar representation on \(X_{reg}\) with smooth forms \(\hat{\phi }_j\) on

*Z*. We first introduce the sheaves \(\mathscr {E}_X^{0,*}\) of smooth \((0,*)\)-forms on

*X*. By duality, we then obtain the sheaf \({\mathcal C}_X^{n,*}\) of \((n,*)\)-currents. We are mainly interested in the subsheaf \(\mathcal {PM}_X^{n,*}\) of pseudomeromorphic currents, and especially, the even more restricted sheaf \(\mathcal {W}_X^{n,*}\) 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 \(\omega _X^n\subset \mathcal {W}_X^{n,0}\) of \(\bar{\partial }\)-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 \(\omega _X^n\) in the following way. Any holomorphic function defines a morphism in \({\mathcal Hom}(\omega ^n_X,\omega ^n_X)\), 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 \(\mathcal {W}_X^{0,*}\) induces a morphism in \({\mathcal Hom}(\omega ^n_X,\mathcal {W}_X^{n,*})\), and in fact \(\mathcal {W}_X^{0,*}\rightarrow {\mathcal Hom}(\omega ^n_X,\mathcal {W}_X^{n,*})\) is an isomorphism. In the non-reduced case, we then take this as the definition of \(\mathcal {W}_X^{0,*}\). It turns out that with this definition, on \(X_{reg}\), any element of \(\mathcal {W}_X^{0,*}\) admits a unique representation (1.4), where \(\hat{\phi }_j\) are in \(\mathcal {W}_Z^{0,*}\), see Sect. 6 below for details.

Given \(v,\phi \) in \(\mathcal {W}_X^{0,*}\) we say that \(\bar{\partial }v=\phi \) if \(\bar{\partial }(v \wedge h)=\phi \wedge h\) for all *h* in \(\omega _X^n\). Following [6] we introduce semi-global integral formulas and prove that if \(\phi \) is a smooth \(\bar{\partial }\)-closed \((0,q+1)\)-form there is locally a current *v* in \(\mathcal {W}_X^{0,q}\) such that \(\bar{\partial }v=\phi \). 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 \(\mathscr {A}_X^k\subset \mathcal {W}_X^{0,k}\) and obtain our main result in this paper.

### Theorem 1.1

Let *X* be an analytic space of pure dimension *n*. There are sheaves \(\mathscr {A}_X^k\subset \mathcal {W}_X^{0,k}\) that are modules over \(\mathscr {E}_X^{0,*}\), coinciding with \(\mathscr {E}_X^{0,k}\) on \(X_{reg}\), and such that (1.2) is a resolution of the structure sheaf \(\mathscr {O}_X\).

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 non-reduced case as is described above. This is done in Sects. 4–8. The construction of integral operators to provide solutions to \(\bar{\partial }\) in Sect. 9 and the construction of the fine resolution of \(\mathscr {O}_X\) in Sect. 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 Sect. 10 however becomes significantly more involved in the non-reduced case and requires completely new ideas. In Sect. 12 we discuss the relation to the results in [17, 18] in case *X* is a local complete intersection.

## 2 Pseudomeromorphic currents

*elementary (pseudomeromorphic) current*. Let

*Z*be a reduced space of pure dimension. A current \(\tau \) 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 \(\mathcal {PM}_Z\) 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 \(\tau \) is pseudomeromorphic and has support on an analytic subset *V*, and *h* is a holomorphic function that vanishes on *V*, then \({\bar{h}} \tau =0\) and \(d{\bar{h}}\wedge \tau =0\).

*V*of some open subset \({\mathcal U}\subset Z\), the natural restriction to the open set \({\mathcal U}{\setminus } V\) of \(\tau \) has a natural extension to a pseudomeromorphic current on \({\mathcal U}\) that we denote by \(\mathbf{1}_{{\mathcal U}{\setminus } V}\tau \). Throughout this paper we let \(\chi \) denote a smooth function on \([0,\infty )\) that is 0 in a neighborhood of 0 and 1 in a neighborhood of \(\infty \). If

*h*is a holomorphic tuple whose common zero set is

*V*, then

*V*. If

*W*is another analytic set, then

*f*is a simple projection and \(\tau \) has compact support in the fiber direction. In any case we have

*dimension principle*, that states that if \(\tau \) is a pseudomeromorphic \((*,p)\)-current with support on an analytic set with codimension larger than

*p*, then \(\tau \) must vanish.

A pseudomeromorphic current \(\tau \) on *Z* has the *standard extension property*, SEP, if \(\mathbf{1}_V\tau =0\) for each germ *V* of an analytic set with positive codimension on *Z*. The set \(\mathcal {W}_Z\) of all pseudomeromorphic currents on *Z* with the SEP is a subsheaf of \(\mathcal {PM}_Z\). By (2.3), \(\mathcal {W}_Z\) is closed under multiplication by smooth forms.

*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 \(\mathbf{1}_{\{ f = 0 \}}( 1/f) = 0\). The current 1 /

*f*has the SEP and

*a*on

*Z*is

*almost semi-meromorphic*if there is a modification \(\pi :Z'\rightarrow Z\), a holomorphic section

*f*of a line bundle \(L\rightarrow Z'\) and a smooth form \(\gamma \) with values in

*L*such that \(a=\pi _*(\gamma /f)\), cf., [10, Section 4]. If

*a*is almost semi-meromorphic, then it is clearly pseudomeromorphic. Moreover, it is smooth outside an analytic set \(V\subset Z\) of positive codimension,

*a*is in \(\mathcal {W}_Z\), and in particular, \(a=\lim _{\epsilon \rightarrow 0^+}\chi (|h|/\epsilon )a\) 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 \(\mathcal {W}_X^{0,*}\), when *X* is non-reduced. Notice that if *a* is almost semi-meromorphic in *Z* then it also is in any open \({\mathcal U}\subset Z\).

### Proposition 2.1

*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 \(\tau \) is a pseudomeromorphic current in \({\mathcal U}\subset Z\), then there is a unique pseudomeromorphic current \(a\wedge \tau \) in \({\mathcal U}\) that coincides with (the naturally defined current) \(a\wedge \tau \) in \({\mathcal U}{\setminus } V\) and such that \(\mathbf{1}_V (a\wedge \tau )=0\).

- (ii)If \(W\subset {\mathcal U}\) is any analytic subset, then$$\begin{aligned} \mathbf{1}_W (a\wedge \tau )=a\wedge \mathbf{1}_W\tau . \end{aligned}$$(2.6)

*h*is a tuple that cuts out

*V*, then in view of (2.1),

### Proposition 2.2

Let *Z* be a reduced space. Then \(\mathcal {PM}_Z=\mathcal {W}_Z+\bar{\partial }\mathcal {W}_Z\).

### Proof

*Z*is smooth. Since \(\mathcal {W}_Z\) is closed under multiplication by smooth forms, so is \(\mathcal {W}_Z + \bar{\partial }\mathcal {W}_Z\). The statement that \(\mathcal {PM}_Z = \mathcal {W}_Z + \bar{\partial }\mathcal {W}_Z\) is local, and since both sides are closed under multiplication by cutoff functions, we may consider a pseudomeromorphic current \(\mu \) with compact support in \(\mathbb {C}^n\). If \(\mu \) has bidegree \((*,0)\), then it is in \(\mathcal {W}_Z\) in view of the dimension principle. Thus we assume that \(\mu \) has bidegree \((*,q)\) with \(q\ge 1\). Let

*k*is the Bochner–Martinelli kernel. Here (2.9) means that \(K\mu =p_*(k\wedge \mu \otimes 1)\), where

*p*is the projection \( \mathbb {C}^n_\zeta \times \mathbb {C}^n_z\rightarrow \mathbb {C}^n_z,\quad (\zeta ,z)\mapsto z. \) Recall that we have the Koppelman formula \( \mu =\bar{\partial }K\mu +K(\bar{\partial }\mu ) \). It is thus enough to see that \(K\mu \) is in \(\mathcal {W}_Z\) if \(\mu \) is pseudomeromorphic. Let \(\chi _\epsilon =\chi (|\zeta -z|^2/\epsilon )\). It is easy to see, by a blowup of \(\mathbb {C}^n\times \mathbb {C}^n\) along the diagonal, that

*k*is almost semi-meromorphic on \(\mathbb {C}^n\times \mathbb {C}^n\). Thus, by (2.7), \(\chi _\epsilon k\wedge (\mu \otimes 1)\rightarrow k\wedge (\mu \otimes 1)\). In view of Proposition 2.1 it follows that \(k\wedge ( \mu \otimes 1)\) is pseudomeromorphic. Finally, if

*W*is a germ of a subvariety of \(\mathbb {C}^n\) of positive codimension, then by (2.4) and (2.5),

If *Z* is not smooth, then we take a smooth modification \(\pi :Z'\rightarrow Z\). For any \(\mu \) in \(\mathcal {PM}_Z\) there is some \(\mu '\) in \(\mathcal {PM}_{Z'}\) such that \(\pi _*\mu '=\mu \), see [4, Proposition 1.2]. Since \(\mu '=\tau +\bar{\partial }u\) with \(\tau ,u\) in \(\mathcal {W}_{Z'}\), we have that \(\mu =\pi _*\tau +\bar{\partial }\pi _* u\). \(\square \)

### 2.1 Pseudomeromorphic currents with support on a subvariety

Let \(\Omega \) be an open set in \(\mathbb {C}^N\) and let *Z* be a (reduced) subvariety of pure dimension *n*. Let \(\mathcal {PM}^Z_\Omega \) denote the sheaf of pseudomeromorphic currents \(\tau \) on \(\Omega \) with support on *Z*, and let \(\mathcal {W}_\Omega ^Z\) denote the subsheaf of \(\mathcal {PM}^Z_\Omega \) of currents of bidegree \((N,*)\) with the SEP with respect to *Z*, i.e., such that \(\mathbf{1}_W \tau =0\) for all germs *W* of subvarieties of *Z* of positive codimension. The sheaf \(\mathcal {{ CH}}_\Omega ^Z\) of Coleff–Herrera currents on *Z* is the subsheaf of \(\mathcal {W}_\Omega ^Z\) of \(\bar{\partial }\)-closed (*N*, *p*)-currents, where \(p=N-n\).

### Remark 2.3

In [3, 6] \(\mathcal {{ CH}}_Z^\Omega \) 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 \(K_\Omega \) we get the sheaf \(\mathcal {{ CH}}^Z_\Omega \) in this paper. Locally these sheaves are thus isomorphic via the mapping \(\mu \mapsto \mu \wedge \alpha \), where \(\alpha \) is a non-vanishing holomorphic (*N*, 0)-form.\(\square \)

We have the following direct consequence of Proposition 2.1.

### Proposition 2.4

Let \(Z \subset \Omega \) be a subvariety of pure dimension, let *a* be almost semi-meromorphic in \(\Omega \), and assume that it is smooth generically on *Z*. If \(\tau \) is in \(\mathcal {W}_\Omega ^Z\), then \(a\wedge \tau \) is in \(\mathcal {W}_\Omega ^Z\) as well.

### Proposition 2.5

*z*,

*w*) such that \(Z=\{w=0\}\). Then \(\tau \) in \(\mathcal {W}_\Omega ^Z\) has a unique representation as a finite sum

If in addition \(\bar{\partial }\tau \) is in \(\mathcal {W}_\Omega ^Z\) then its coefficients in the expansion (2.11) are \(\bar{\partial }\tau _\gamma \), cf., (2.12). In particular, \(\bar{\partial }\tau =0\) if and only if \(\bar{\partial }\tau _\gamma =0\) for all \(\gamma \).

*Z*of smooth \((0,*)\)-forms. We assume that

*Z*is smooth and that we have coordinates (

*z*,

*w*) as before, that \(\tau \) is in \(\mathcal {W}_\Omega ^Z\), and that (2.11) holds. Moreover, we assume that \(\phi \) is a smooth \((0,*)\)-form in a neighborhood of

*Z*in \(\Omega \). For any positive integer

*M*we have the expansion

*j*. If

*M*in (2.13) is chosen so that \(\mathscr {O}(|w|^M)\tau =0\), then

### 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 \(i : Z \rightarrow Y\) is an embedding of a reduced space *Z* into a smooth manifold *Y*, then the push-forward mapping \(\tau \mapsto i_* \tau \) gives an isomorphism between currents \(\tau \) on *Z* and currents \(\mu \) on *Y* such that \(\xi \wedge \mu = 0\) for all \(\xi \) in \(\mathscr {E}_Y\) such that \(i^* \xi = 0\).

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.

### Proposition 2.6

*Z*into a smooth manifold

*Y*.

- (i)
If \(\tau \) is in \(\mathcal {PM}_Z\), then \(i_*\tau \) is in \(\mathcal {PM}_Y\).

- (ii)
If \(\tau \) is a current on

*Z*such that \(i_* \tau \) is in \(\mathcal {PM}_Y\) and \(\mathbf{1}_{Z_{sing}}(i_* \tau )=0\), then \(\tau \) is in \(\mathcal {PM}_Z\).

*Z*, we get by (2.1) that for a subvariety \(V \subset {\mathcal U}\subset Z\),

*Z*.

### Corollary 2.7

*Z*in \(\Omega \).

Notice that \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z)\) is precisely the sheaf of \(\mu \) in \(\mathcal {W}_\Omega ^Z\) such that \(\mathcal {J}\mu =0\).

### Proof

The map \(i_*\) is injective, since it is injective on any currents, and it maps into \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )\) by (2.15).

To see that \(i_*\) is surjective, we take a \(\mu \) in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )\). We assume first that we are on \(Z_\mathrm{reg}\), with local coordinates such that \(Z_\mathrm{reg} = \{ w = 0 \}\). If \(\xi \) is in \(\mathscr {E}_\Omega ^{0,*}\) and \(i^*\xi =0\), then \(\xi \) is a sum of forms with a factor \(d {\bar{w}}_j\), \(w_j\) or \(\bar{w}_j\). Since \(w_j \in \mathcal {J}\), \(w_j\) annihilates \(\mu \) by assumption, and since \(w_j\) vanishes on the support of \(\mu \), \({\bar{w}}_j\) and \(d \bar{w}_j\) annihilate \(\mu \) since \(\mu \) is pseudomeromorphic. Thus, \(\mu . \xi = 0\), so \(\mu = i_* \tau \) for some current \(\tau \) on *Z*. By Proposition 2.6 (ii), \(\tau \) is pseudomeromorphic, and by (2.15), has the SEP, i.e., \(\tau \) is in \(\mathcal {W}^{n,*}_Z\). \(\square \)

### Remark 2.8

We do not know whether \(i_* \tau \in \mathcal {PM}_\Omega ^Z\) implies that \(\tau \in \mathcal {PM}_Z\).\(\square \)

By [11, Proposition 3.12 and Theorem 3.14], we get

### Proposition 2.9

Let \(\varphi \) and \(\phi _1,\ldots ,\phi _m\) be currents in \(\mathcal {W}_Z\). If \(\varphi = 0\) on the set on \(Z_{reg}\) where \(\phi _1,\ldots ,\phi _m\) are smooth, then \(\varphi = 0\).

## 3 Local embeddings of a non-reduced analytic space

*X*be an analytic space of pure dimension

*n*with structure sheaf \(\mathscr {O}_X\) and let \(Z=X_{red}\) be the underlying reduced analytic space. For any point \(x\in X\) there is, by definition, an open set \(\Omega \subset \mathbb {C}^N\) and an ideal sheaf \(\mathcal {J}\subset \mathscr {O}_\Omega \) of pure dimension

*n*with zero set

*Z*such that \(\mathscr {O}_X\) is isomorphic to \(\mathscr {O}_\Omega /\mathcal {J}\), and all associated primes of \(\mathcal {J}\) at any point have dimension

*n*. We say that we have a local embedding \(i:X\rightarrow \Omega \subset \mathbb {C}^N\) at

*x*. There is a minimal such

*N*, called the Zariski embedding dimension \({\hat{N}}\) 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 \(i:X\rightarrow \Omega \) has locally at

*x*the form

*j*is minimal, \({\mathcal U}\) is an open subset of \(\mathbb {C}^{m}_w\), \(m=N-{\hat{N}}\), and the ideal in \(\Omega \) is \(\mathcal {J}=\widehat{\mathcal {J}}\otimes 1 +(w_1,\ldots ,w_m)\). Notice that we then also have embeddings \(Z\rightarrow \widehat{\Omega }\rightarrow \Omega \); however, the first one is in general not minimal.

*Z*is smooth, and let (

*z*,

*w*) be coordinates in \(\Omega \) such that \(Z=\{w=0\}\). We can identify \(\mathscr {O}_Z\) with holomorphic functions of

*z*, and we can define an injection

### Proposition 3.1

Assume that *Z* is smooth. Let \(\mathscr {O}_X\) have the \(\mathscr {O}_Z\)-module structure from a choice of local coordinates as above. Then \(\mathscr {O}_X\) is a coherent \(\mathscr {O}_Z\)-module, and \(\mathscr {O}_X\) is a free \(\mathscr {O}_Z\)-module at *x* if and only if \(\mathscr {O}_X\) is Cohen–Macaulay at *x*.

*regular sequence*on the

*R*-module

*M*if \(f_i\) is a non zero-divisor on \(M/(f_1,\ldots ,f_{i-1})\) for \(i=1,\ldots ,m\), and \((f_1,\ldots ,f_m) M \ne M\). If

*R*is a local ring, then \({{\mathrm{depth}}}_RM\) is the maximal length

*d*of a regular sequence \(f_1,\ldots ,f_d\) such that \(f_1,\dots ,f_d\) are contained in the maximal ideal \(\mathfrak {m}\); furthermore,

*M*is

*Cohen–Macaulay*if \({{\mathrm{depth}}}_R M = \dim _R M\), where \(\dim _R M = \dim _R (R/\mathrm{ann\,}_R M)\). If

*R*is Cohen–Macaulay, and

*M*has a finite free resolution over

*R*, then the

*Auslander–Buchsbaum*formula, [14, Theorem 19.9], gives that

*M*over

*R*. In this case,

*M*is Cohen–Macaulay as an

*R*-module if and only if

*M*has a free resolution over

*R*of length \({\text {codim}\,}M\).

### Remark 3.2

Notice that if we have a local embedding \(i:X\rightarrow \Omega \) as above, then the depth and dimension of \(\mathscr {O}_{X,x}=\mathscr {O}_{\Omega ,x}/\mathcal {J}\) as an \(\mathscr {O}_{\Omega ,x}\)-module coincide with the depth and dimension of \(\mathscr {O}_{X,x}\) as an \(\mathscr {O}_{X,x}\)-module. Thus \(\mathscr {O}_{X,x}\) is Cohen–Macaulay as an \(\mathscr {O}_{X,x}\)-module if and only if it is Cohen–Macaulay as an \(\mathscr {O}_{\Omega ,x}\)-module, and this holds in turn if and only if \(\mathscr {O}_{\Omega ,x}/\mathcal {J}\) has a free resolution of length \(N-n\).\(\square \)

### Proof of Proposition 3.1

By the Nullstellensatz there is an *M* such that \(w^\alpha \) is in \(\mathcal {J}\) in some neighborhood of *x* if \(|\alpha |=M\). Let \(\mathcal {M}\subset \mathscr {O}_\Omega \) be the ideal generated by \(\{w^\alpha ;\ |\alpha |=M\}\). Then \(\mathcal {M}'=\mathscr {O}_\Omega /\mathcal {M}\) is a free, finitely generated \(\mathscr {O}_Z\)-module. Thus, \(\mathscr {O}_\Omega /\mathcal {J}\simeq \mathcal {M}'/ \mathcal {J}\mathcal {M}'\) is a coherent \(\mathscr {O}_Z\)-module, which we note is generated by the finite set of monomials \(w^\alpha \) such that \(|\alpha | < M\).

*g*is generically non-vanishing on each irreducible component of \(Z(\mathcal {J})\). Thus \(f_1\) is a non zero-divisor if and only if \(\tilde{f}_1\) is. If it is, then \(\mathscr {O}_{X,x}/(f_1)=\mathscr {O}_{\Omega ,x}/(\mathcal {J}+(f_1))\) again has pure dimension. Thus the claim follows by induction, and the fact that \(Z(\mathcal {J}+(f_1,\ldots ,f_k)) = Z(\mathcal {J}+(\tilde{f}_1,\dots ,\tilde{f}_k))\). The claim immediately implies (3.3).

To see (3.4), we note first that \(\dim _{\mathscr {O}_{X,x}} \mathscr {O}_{X,x}\) is just the usual (geometric) dimension of *X* or *Z*, i.e., in this case, *n*. Now, \(\mathrm{ann\,}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} = \{ 0 \}\), so \(\dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} = \dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{Z,x}/(\mathrm{ann\,}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x}) = \dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{Z,x} = n\).

In the proof above, we saw that \(\mathscr {O}_X\) is generated (locally) as an \(\mathscr {O}_Z\)-module by all monomials \(w^\alpha \) with \(|\alpha |\le M\) for some *M*.

### Corollary 3.3

Assume that \(1, w^{\alpha _1},\ldots ,w^{\alpha _{\nu -1}}\) is a minimal set of generators at a given point *x* (clearly 1 must be among the generators!). Then we have a unique representation (1.4) for each \(\phi \in \mathscr {O}_{X,x}\) if and only if \(\mathscr {O}_{X,x}\) is Cohen–Macaulay.

By coherence it follows that if \(\mathscr {O}_{X,x}\) is free as an \(\mathscr {O}_{Z,x}\)-module, then \(\mathscr {O}_{Z,x'}\) is free as an \(\mathscr {O}_{Z,x'}\)-module for all \(x'\) in a neighborhood of *x*, and \(1, w^{\alpha _1},\ldots ,w^{\alpha _{\nu -1}}\) is a basis at each such \(x'\).

### Example 3.4

Let \(\mathcal {J}\) be the ideal in \(\mathbb {C}^4\) generated by \( (w_1^2, w_2^2, w_1 w_2, w_1z_2-w_2 z_1). \) It is readily checked that \(\mathscr {O}_X\) is a free \(\mathscr {O}_Z\)-module at a point on \(Z=\{w_1=w_2=0\}\) where \(z_1\) or \(z_2\) is \(\ne 0\). If, say, \(z_1\ne 0\), 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 \(\mathscr {O}_X\) is not a free \(\mathscr {O}_Z\)-module, since there is a non-trivial relation between \(w_1\) and \(w_2\).

We claim that \(\mathscr {O}_X\) 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 \(\mathcal {J}\) is primary with respect to *Z*. To see the claim, let \(\phi \) and \(\psi \) be functions such that \(\phi \psi \) is in \(\mathcal {J}\) and \(\psi \) is not in \(\sqrt{\mathcal {J}}\). The latter assumption means, in view of the Nullstellensatz, that \(\psi \) does not vanish identically on *Z*, i.e., \(\psi =a(z) + \mathscr {O}(w)\), where *a* does not vanish identically. Since in particular \(\phi \psi \) must vanish on *Z* it follows that \(\phi =\mathscr {O}(w)\). It is now easy to see that \(\phi \) is in \(\mathcal {J}\). We conclude that \(\mathcal {J}\) is primary.\(\square \)

The pure-dimensionality of \(\mathscr {O}_X\) can also be rephrased in the following way: *If*\(\phi \)*is holomorphic and is* 0 *generically, then*\(\phi =0\). If we delete the generator \(w_1w_2\) from the definition of \(\mathcal {J}\) in the example, then \(\phi =w_1w_2\) is 0 generically in \(\mathscr {O}_\Omega /\mathcal {J}\) but is not identically zero. Thus \(\mathcal {J}\) then has an embedded primary ideal at (0, 0).

### Example 3.5

*w*is a basis for \(\mathscr {O}_X=\mathscr {O}_{\mathbb {C}^2}/(w^2)\) so each function \(\phi \) in \(\mathscr {O}_X\) has a unique representation \(a_0(z)\otimes 1 +a_1(z)\otimes w\). Let us consider the new coordinates \(\zeta =z-w, \eta =w\). Then \(\mathcal {J}=(\eta ^2)\) and since

More generally, assume that, at a given point in \(X_{reg}\subset \Omega \), we have two different choices (*z*, *w*) and \((\zeta ,\eta )\) of coordinates so that \(Z=\{w=0\}=\{\eta =0\}\), and bases \(1, \ldots , w^{\alpha _{\nu -1}}\) and \(1, \ldots , \eta ^{\beta _{\nu -1}}\) for \(\mathscr {O}_X\) as a free module over \(\mathscr {O}_Z\). Then there is a \(\nu \times \nu \)-matrix *L* of holomorphic differential operators so that if \((a_j)\) is any tuple in \((\mathscr {O}_Z)^\nu \) and \((b_j)=L(a_j)\), then \( a_0\otimes 1+\cdots + a_{\nu -1}\otimes w^{\alpha _{\nu -1}}= b_0\otimes 1+\cdots + b_{\nu -1} \otimes \eta ^{\beta _{\nu -1}}+\mathcal {J}. \)

## 4 Smooth \((0,*)\)-forms on a non-reduced space *X*

Let \(i:X\rightarrow \Omega \) 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 \(\Phi \) in \(\Omega \) “vanish” on *X*, or more formally, give a meaning to \(i^*\Phi =0\). We will see, cf., Lemma 4.8 below, that the suitable requirement is that locally on \(X_{reg}\), \(\Phi \) belongs to \(\mathscr {E}_\Omega ^{0,*} \mathcal {J}+\mathscr {E}_\Omega ^{0,*}\bar{\mathcal {J}}_Z +\mathscr {E}_\Omega ^{0,*} d\bar{\mathcal {J}}_Z\), where \(\mathcal {J}_z\) is the ideal sheaf defining *Z*. However, it turns out to be more convenient to represent the sheaf \({\mathcal Ker\,}i^*\) 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.

### Proposition 4.1

### Definition 4.2

*X*as

*X*.

### Remark 4.3

It follows from Lemma 4.8 below that in case \(X=Z\) is reduced, then \(\xi \) is in \({\mathcal Ker\,}i^*\) if and only its pullback to \(X_{reg}\) vanishes. Thus our definition of \(\mathscr {E}_X^{0,*}\) is consistent with the usual one in that case.\(\square \)

### Lemma 4.4

We can realize the mapping in (4.4) as the tensor product \(\tau \mapsto \tau \wedge [w=0]\), where \([w=0]\) is the Lelong current in \(\Omega \) associated with the submanifold \(\{w=0\}\).

### Proof

To begin with, \(\iota _*\) maps pseudomeromorphic \(({\hat{N}},\hat{p}+\ell )\)-currents with support on \(Z\subset \widehat{\Omega }\) to pseudomeromorphic \((N,p+\ell )\)-currents with support on \(Z\subset \Omega \). If, in addition, \(\tau \) has the SEP with respect to *Z*, then \(\iota _*\tau \) has, as well by (2.15). Moreover, if \(\tau \) is annihilated by \(\widehat{\mathcal {J}}\), then \(\iota _*\tau \) is annihilated by \(\mathcal {J}=\widehat{\mathcal {J}}\otimes 1+(w)\). Thus the mapping (4.4) is well-defined, and it is injective since \(\iota \) is injective.

Now assume that \(\mu \) is in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )\). Arguing as in the proof of Corollary 2.7, we see that \(\mu =\iota _*\hat{\mu }\) for a current \(\hat{\mu }\) in \(\mathcal {W}^Z_{\widehat{\Omega }}\). Since \(\widehat{\mathcal {J}}=\iota ^*\mathcal {J}\) and \(\mathcal {J}\mu =0\), it follows that \(\widehat{\mathcal {J}}\hat{\mu }=0\). Thus (4.4) is surjective. \(\square \)

Since \(\iota _*\) is injective, \(\bar{\partial }\tau =0\) if and only if \(\bar{\partial }\iota _*\tau =0\), and thus we get

### Corollary 4.5

### Corollary 4.6

### Proof

It follows immediately from (4.5) that the mapping (4.6) is well-defined and injective. Given \(\widehat{\xi }\) in \(\mathscr {E}^{0,*}_{\widehat{\Omega }}\), let \(\xi =\widehat{\xi }\otimes 1\). Then \(\iota ^*\xi =\widehat{\xi }\) and so (4.6) is indeed surjective as well. \(\square \)

It follows from (4.6) and (4.3) that the sheaf \(\mathscr {E}_X^{0,*}\) is intrinsically defined on *X*. Since \(\bar{\partial }\) maps \({\mathcal Ker\,}i^*\) to \({\mathcal Ker\,}i^*\), we have a well-defined operator \(\bar{\partial }:\mathscr {E}_X^{0,*}\rightarrow \mathscr {E}_X^{0,*+1}\) such that \(\bar{\partial }^2=0\). 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 \(\mathscr {O}_X\) 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 \(1,w^{\alpha _1},\ldots ,w^{\alpha _{\nu -1}}\) of \(\mathscr {O}_X\) as a free \(\mathscr {O}_Z\)-module, which exist by Corollary 3.3, and generators \(\mu ^1,\ldots ,\mu ^m\) of \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )\).

Notice that for each smooth \((0,*)\)-form \(\Phi \) in \(\Omega \), \(\Phi \mapsto \Phi \wedge \mu ^\ell \) only depends on its class \(\phi \) in \(\mathscr {E}^{0,*}_X\), and \(\phi \) 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 \(\mathcal {W}_Z^{0,*}\). Putting all these tuples together, we get a tuple in \((\mathcal {W}_Z^{0,*})^M\), where \(M = M_1 + \cdots + M_m\) and \(M_j\) is the number of indices in (2.11) in the representation of \(\mu ^j\).

*T*is therefore generically pointwise injective.

### Lemma 4.7

Each \(\phi \) in \(\mathscr {E}_X^{0,*}\) has a unique representation (4.7) where \(\hat{\phi }_j\) are in \(\mathscr {E}_Z^{0,*}\).

### Proof

*T*, and since

*T*is generically pointwise injective we conclude that each \(\hat{\phi }_j\) vanishes. \(\square \)

By the above proof we get

### Lemma 4.8

A smooth \((0,*)\)-form \(\xi \) in \(\Omega \) is in \({\mathcal Ker\,}i^*\) if and only if \(\xi \) is in \(\mathscr {E}_\Omega ^{0,*} \mathcal {J}+\mathscr {E}_\Omega ^{0,*}\bar{\mathcal {J}}_Z +\mathscr {E}_\Omega ^{0,*} d\bar{\mathcal {J}}_Z\) on \(X_{reg}\), where \(\mathcal {J}_Z\) is the radical sheaf of *Z*.

### Remark 4.9

This is *not* the same as saying that \(\xi \) is in \(\mathscr {E}_\Omega ^{0,*} \mathcal {J}+\mathscr {E}_\Omega ^{0,*}\bar{\mathcal {J}}_Z +\mathscr {E}_\Omega ^{0,*} d\bar{\mathcal {J}}_Z\) at singular points. For a simple counterexample, consider \(\phi =x{\bar{y}}\) on the reduced space \(Z=\{xy=0\}\subset \mathbb {C}^2\).

However, this can happen also when *Z* is irreducible at a point. For example, the variety \(Z = \{ x^2 y - z^2 = 0 \} \subset \mathbb {C}^3\) 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 \(\mathscr {E}_\Omega ^{0,0} \mathcal {J}_{Z,0} + \mathscr {E}_\Omega ^{0,0} \bar{\mathcal {J}}_{Z,0}\) see [26, Proposition 9, Chapter IV], and [25, Theorem 3.10, Chapter VI].\(\square \)

### Remark 4.10

It is easy to check that if we have the setting as in the discussion at the end of Sect. 3 but \((a_j)\) is instead a tuple in \(\mathscr {E}_Z^{0,*}\), 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 \(a_0\otimes 1+\cdots + a_{\nu -1}\otimes w^{\alpha _{\nu -1}}\) and \(b_0\otimes 1+\cdots + b_{\nu -1} \otimes \eta ^{\beta _{\nu -1}}\) are equal modulo \(\mathscr {E}_\Omega ^{0,*} \mathcal {J}+\mathscr {E}_\Omega ^{0,*}\bar{\mathcal {J}}_Z +\mathscr {E}_\Omega ^{0,*} d\bar{\mathcal {J}}_Z\), and thus define the same element in \(\mathscr {E}_X^{0,*}\).\(\square \)

For future needs we prove in Sect. 6.1:

### Lemma 4.11

The morphism *T* is pointwise injective.

*A*such that

*S*and

*B*such that

## 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 \({\mathcal C}_X^{n,q}\) of (*n*, *q*)-currents on *X* is the dual sheaf of \((0,n-q)\)-test forms, i.e., forms in \(\mathscr {E}_X^{0,n-q}\) with compact support.

Here, just as in the case of reduced spaces, cf., for example [19, Section 4.2], the space of smooth forms \(\mathscr {E}_X^{0,n-q}\) is equipped with the quotient topology induced by a local embedding.

### Proposition 5.2

If \(\tau \) is in \(\mathcal {W}_\Omega ^Z\) and \(\mathcal {J}\tau =0\), then \(\xi \wedge \tau =0\) for all smooth \(\xi \) such that \(i^*\xi =0\).

### Proof

Because of the SEP it is enough to prove that \(\xi \wedge \tau =0\) on \(X_{reg}\). By assumption, \(\mathcal {J}\) annihilates \(\tau \), and by general properties of pseudomeromorphic currents, since \(\tau \) has support on *Z*, \(\bar{\mathcal {J}}_Z\) and \(d \bar{\mathcal {J}}_Z\) annihilate \(\tau \). Thus the proposition follows by Lemma 4.8. \(\square \)

### Definition 5.3

An \((n,*)\)-current \(\psi \) on *X* is in \(\mathcal {W}_X^{n,*}\) if \(i_*\psi \) is in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z)\).

### Remark 5.4

By Corollary 2.7, this definition is consistent with the previous definition of \(\mathcal {W}_X^{n,*}\) when *X* is reduced. We cannot define \(\mathcal {PM}_X^{n,*}\) in the analogous simple way, cf., Remark 2.8.\(\square \)

### Definition 5.5

If \(\psi \) is in \(\mathcal {W}^{n,*}_X\) and *a* is an almost semi-meromorphic \((0,*)\)-current on \(\Omega \) that is generically smooth on *Z*, then the product \(a \wedge \psi \) is a current in \(\mathcal {W}^{n,*}_X\) defined as follows: By definition, \(i_* \psi \) is in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )\) and by Proposition 2.4 and (2.8), one can define \(a \wedge i_* \psi \) in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )\); now \(a \wedge \psi \) is the unique current in \(\mathcal {W}^{n,*}_X\) such that \(i_*(a \wedge \psi ) = a \wedge i_* \psi \).

*h*cuts out the Zariski singular support of

*a*.

### Definition 5.6

We let \(\omega _X^n\) be the sheaf of \(\bar{\partial }\)-closed currents in \(\mathcal {W}^{n,0}_X\).

*X*is reduced \(\omega _X^n\) is the sheaf of (

*n*, 0)-forms that are \(\bar{\partial }\)-closed in the Barlet–Henkin–Passare sense. Let \(\mu ^1,\ldots ,\mu ^m\) be a set of generators for \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)\). They correspond via (5.3) to a set of generators \(h^1,\ldots , h^m\) for the \(\mathscr {O}_X\)-module \(\omega _X^n\).

We will also need a definition of \(\mathcal {PM}_X^{n,*}\). Let \(\mathcal {F}_X\) be the subsheaf of \({\mathcal C}_X^{n,*}\) of \(\tau \) such that \(i_*\tau \) is in \(\mathcal {PM}_\Omega ^Z\). If \(\tau \) is a section of \(\mathcal {F}_X\) and *W* is a subvariety of some open subset of *Z*, then \(\mathbf{1}_Wi_*\tau \) is in \(\mathcal {PM}_\Omega ^Z\), and by (2.3), \(\mathbf{1}_W i_* \tau \) is annihilated by \({\mathcal Ker\,}i^*\). Hence we can define \(\mathbf{1}_W\tau \) as the unique current in \(\mathcal {F}_X\) such that \(i_*\mathbf{1}_W\tau =\mathbf{1}_Wi_*\tau \). Clearly, \(\mathbf{1}_W\tau \) has support on *W* and it is easily checked that the computational rule (2.3) holds also in \(\mathcal {F}_X\). Moreover, \(\mathcal {F}_X\) is closed under \(\bar{\partial }\) since \(\mathcal {PM}_\Omega ^Z\) is.

### Definition 5.7

The sheaf \(\mathcal {PM}_X^{n,*}\) is the smallest subsheaf of \(\mathcal {F}_X\) that contains \(\mathcal {W}_X^{n,*}\) and is closed under \(\bar{\partial }\) and multiplication by \(\mathbf{1}_W\) 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 \(\mathcal {F}_X\), and hence it also holds for the (possibly smaller) sheaf \(\mathcal {PM}_X^{n,*}\), and in addition, (2.3) holds for forms \(\xi \) in \(\mathscr {E}_X^{0,*}\) and \(\tau \) in \(\mathcal {PM}_X^{n,*}\).

## 6 Structure form on *X*

*X*, and let \(\mathcal {J}\) be the associated ideal sheaf on \(\Omega \). In a slightly smaller set, still denoted \(\Omega \), there is a free resolution

*Z*. Let \(X_k\) be the set where \(f_k\) does not have optimal rank. Then

*pure*codimension

*p*,

*X*is Cohen–Macaulay at a point

*x*, i.e., the sheaf \(\mathscr {O}_\Omega /\mathcal {J}\) is Cohen–Macaulay at

*x*, if and only if \(x\notin X_{p+1}\). Notice that \(Z{\setminus } (X_{reg})_{red}=Z_{sing}\cup X_{p+1}\). 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*.

*Hermitian resolution*of \(\mathscr {O}_\Omega /\mathcal {J}\) in \(\Omega \). In \(\Omega {\setminus } X_k\) we have a well-defined vector bundle morphism \(\sigma _{k+1}:E_k\rightarrow E_{k+1}\), if we require that \(\sigma _{k+1}\) vanishes on \(({\text {Im}\,}f_{k+1})^\perp \), takes values in \(({\mathcal Ker\,}f_{k+1})^\perp \), and that \(f_{k+1}\sigma _{k+1}\) is the identity on \({\text {Im}\,}f_{k+1}\). Following [7, Section 2] we define smooth \(E_k\)-valued forms

*f*, \(\sigma :=\oplus \sigma _k\), and \(u:=\sum u_k\) are odd. For details, see [7]. It turns out that

*u*has a (necessarily unique) almost semi-meromorphic extension

*U*to \(\Omega \). The residue current

*R*is defined by the relation

*R*is \(\nabla _f\)-closed. In addition,

*R*has support on

*Z*and is a sum \(\sum R_k\), where \(R_k\) is a pseudomeromorphic \(E_k\)-valued current of bidegree (0,

*k*). It follows from the dimension principle that \(R=R_p+R_{p+1}+\cdots +R_N\). 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 \(\nabla _f\)-closedness implies that

*R*is \(\bar{\partial }\)-closed.

If \(\phi \) is in \(\mathcal {J}\) then \(\phi R=0\) and in fact, \(\mathcal {J}=\mathrm{ann\,}R\), see [7, Theorem 1.1].

### Remark 6.1

*f*, then we have the free resolution \(0\rightarrow \mathscr {O}_\Omega {\mathop {\rightarrow }\limits ^{f}} \mathscr {O}_\Omega \rightarrow \mathscr {O}_\Omega /(f)\rightarrow 0\); thus

*U*is just the principal value current 1 /

*f*and \(R=\bar{\partial }(1/f)\). More generally, if \(f = (f_1,\ldots ,f_p)\) is a complete intersection, then

*f*, see for example [1, Corollary 3.5].\(\square \)

*Z*.

In this section, we let \((z_1,\dots ,z_N)\) denote coordinates on \(\mathbb {C}^N\), and let \(dz := dz_1 \wedge \cdots \wedge dz_N\).

### Lemma 6.2

*b*such that

### Proof

As in [6, Section 3], see also [32, Proposition 3.2], one can prove that \(R_p=\sigma _F\mu \), where \(\mu \) is a tuple of currents in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)\) and \(\sigma _F\) is an almost semi-meromorphic current that is smooth outside \(X_{p+1}\).

Let \(b_p=\sigma _F\) and \(b_k=\alpha _k\cdots \alpha _{p+1}\sigma _F\) for \(k\ge p+1\). Then each \(b_k\) is almost semi-meromorphic, cf., [10, Section 4.1]. In view of (6.6) we have that \(R_k=b_k\mu \) 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 \(b=b_p+b_{p+1}+\cdots \). \(\square \)

By Proposition 2.4 we get

### Corollary 6.3

The current \(R\wedge dz\) is in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z)\).

From Lemma 6.2, Corollary 6.3, (5.1), and (5.3) we get the following analogue to [6, Proposition 3.3]:

### Proposition 6.4

*R*be the associated residue current. Then there exists a (unique) current \(\omega \) in \(\mathcal {W}_X^{n,*}\) such that

*b*of almost semi-meromorphic \((0,*)\)-currents in \(\Omega \), smooth outside of \(X_{p+1}\), and a tuple \(\vartheta \) of currents in \(\omega ^n_X\) such that

^{1}where \(\omega _k \in \mathcal {W}^{n,k}(X, E_{p+k})\), and if \(f^j:=f_{p+j}\), then

We will also use the short-hand notation \(\nabla _f \omega =0\). As in the reduced case, following [6], we say that \(\omega \) is a *structure form* for *X*. The products in (6.9) are defined according to Definition 5.5.

### Remark 6.5

Recall that \(X_{p+1}=\emptyset \) if *X* is Cohen–Macaulay, so in that case \(\omega =b\vartheta \), where *b* is smooth. If we take a free resolution of length *p*, then \(\omega = \omega _0\), and \(\bar{\partial }\omega _0 = f^1 \omega _1 = 0\), so \(\omega \) is in \(\omega ^n_X\).\(\square \)

### Remark 6.6

If \(X=\{f=0\}\) is a reduced hypersurface in \(\Omega \), then \(R=\bar{\partial }(1/f)\) and \(\omega \) 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 \(\omega ^n_X\) are then meromorphic, by (6.9), \(\omega \) can be represented by almost semi-meromorphic forms on *X*.

*X*is non-reduced. We recall that a differential operator is a Noetherian operator for an ideal \(\mathcal {J}\) if \(\mathcal {L}\varphi \in \sqrt{\mathcal {J}}\) for all \(\varphi \in \mathcal {J}\). It is proved by Björk, [13], see also [32, Theorem 2.2], that if \(\mu \in {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )\), then there exists a Noetherian operator \(\mathcal {L}\) for \(\mathcal {J}\) with meromorphic coefficients such that the action of \(\mu \) on \(\xi \) equals the integral of \(\mathcal {L} \xi \) over

*Z*. By (5.3), the action of

*h*in \(\omega ^n_X\) on \(\xi \) in \(\mathscr {E}^{0,*}_X\) can then be expressed as

### Proposition 6.7

### Proof

*h*is in \(\omega _X^n\), then \(i_*h\) is in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)\). We have mappings

^{2}This mapping is an intrinsic isomorphism

In particular it follows that \(\omega _X^n\) is coherent, and we have:

If \(\xi ^1, \ldots ,\xi ^m\) are generators of \({\mathcal H}^p({\mathcal Hom}(E_\bullet ^*,K_\Omega )))\), where \(\xi ^\ell = \xi ^\ell _0 \wedge dz\), then \(h^\ell :=\xi ^\ell _0 \omega _0, \ \ell =1,\ldots ,m\), generate the \(\mathscr {O}_X\)-module \(\omega _X^n\), and \(\mu ^\ell = i_* h^\ell = \xi ^\ell R_p\) generate the \(\mathscr {O}_\Omega \)-module \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)\).

### Remark 6.8

We give here an example where we can explicitly compute generators of \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )\).

### Example 6.9

### 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 \(0\in Z\) and let \(\mathcal {I}\) be the ideal generated by \(z=(z_1,\ldots ,z_n)\).

*z*; then \(\mathscr {O}(F_\bullet )\) is a free resolution of \(\mathscr {O}_\Omega /\mathcal {I}\). Since \(\mathcal {J}\) and \(\mathcal {I}\) are Cohen–Macaulay and intersect properly in \(\Omega \), the complex \(\mathscr {O}_\Omega ( (E\otimes F)_\bullet )\) is a free resolution of \(\mathscr {O}_\Omega /(\mathcal {J}+\mathcal {I})\), and the corresponding residue current is

*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 \(\mathcal {W}_X^{0,*}\) on *X*

*X*is singular we have to consider larger fine sheaves; we first define sheaves \(\mathcal {W}_X^{0,*}\supset \mathscr {E}_X^{0,*}\) of \((0,*)\)-currents. Given a local embedding \(i:X\rightarrow \Omega \) 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 \(\mathcal {W}_X^{0,*}\) shall have a unique representation

### Lemma 7.1

### Proof

*Z*where \(\widehat{\phi }_0,\ldots ,\widehat{\phi }_{\nu -1}\) are all smooth, \(\phi \wedge \tau \), as defined above, is just multiplication of the smooth form \(\phi \) by \(\tau \), and thus \(\xi \phi \wedge \tau = 0\) there. We have a unique representation

If \(\phi \) has the form (7.1) in a neighborhood of some point \(x\in X_{reg}\) and *h* is in \(\omega ^n_X\), then we get an element \(\phi \wedge h\) in \(\mathcal {W}_X^{n,*}\) defined by \(i_*(\phi \wedge h)=\phi \wedge i_*h\). It follows that \(\phi \) in this way defines an element in \({\mathcal Hom}_{\mathscr {O}_X}(\omega ^n_X, \mathcal {W}_X^{n,*})\). This sheaf is global and invariantly defined and so we can make the following global definition.

### Definition 7.2

\(\mathcal {W}_X^{0,*}={\mathcal Hom}_{\mathscr {O}_X}(\omega ^n_X, \mathcal {W}_X^{n,*})\).

If \(\phi \) is in \(\mathcal {W}^{0,*}_X\) and *h* is in \(\omega ^n_X\), we consider \(\phi (h)\) as the product of \(\phi \) and *h*, and sometimes write it as \(\phi \wedge h\).

### Theorem 7.3

The mapping (7.3) is an isomorphism in the Zariski-open subset of *X* where it is \(S_2\).

This is the subset of *X* where \({\text {codim}\,}X_k\ge k+2\), \(k\ge p+1\), cf., Sect. 6. Thus it contains all points *x* such that \(\mathscr {O}_{X,x}\) is Cohen–Macaulay. In particular, (7.3) is an isomorphism in \(X_{reg}\).

*X*has pure dimension

*p*, there is an injective mapping

*X*is reduced. Since sections of \(\omega ^n_X\) are meromorphic, see [6, Example 2.8], and thus almost semi-meromorphic and generically smooth, by Proposition 2.4 (with \(Z = X = \Omega )\) we can extend (7.3) to a morphism

### Lemma 7.4

When *X* is reduced (7.7) is an isomorphism.

Thus Definition 7.2 is consistent with the previous definition of \(\mathcal {W}^{0,*}_X\) when *X* is reduced.

### Proof

Clearly each \(\phi \) in \(\mathcal {W}^{0,*}_X\) defines an element \(\alpha \) in \({\mathcal Hom}(\omega ^n_X,\mathcal {W}^{n,*}_X)\) by \(h\mapsto \phi \wedge h\). If we apply this to a generically nonvanishing *h* we see by the SEP that (7.7) is injective.

For the surjectivity, take \(\alpha \) in \({\mathcal Hom}(\omega ^n_X,\mathcal {W}^{n,*}_X)\). If \(h'\) is nonvanishing at a point on \(X_{reg}\), then it generates \(\omega ^n_X\) and thus \(\alpha \) is determined by \(\phi := \alpha h' \) there. By [10, Theorem 3.7], \(\phi = \psi \wedge h'\) for a unique current \(\psi \) in \(\mathcal {W}^{0,*}_X\) so by \(\mathscr {O}_X\)-linearity \(\alpha h=\psi \wedge h\) for any *h*. Hence, \(\psi \) is well-defined as a current in \(\mathcal {W}^{0,*}_X\) on \(X_\mathrm{reg}\).

We must verify that \(\psi \) has an extension in \(\mathcal {W}_X^{0,*}\) across \(X_{sing}\). Since such an extension must be unique by the SEP, the statement is local on *X*. Thus we may assume that \(\alpha \) is defined on the whole of *X* and that there is a generically nonvanishing holomorphic *n*-form \(\gamma \) on *X*. Then \(\alpha \gamma \) is a section of \(\mathcal {W}^{n,*}(X)\).

Let us choose a smooth modification \(\pi :X'\rightarrow X\) that is biholomorphic outside \(X_{sing}\). Then \(\pi ^*\gamma \) is a holomorphic *n*-form on \(X'\) that is generically non-vanishing. We claim that there is a current \(\tau \) in \( \mathcal {W}^{n,0}(X')\) such that \(\pi _*\tau =\alpha \gamma \). In fact, \(\tau \) exists on \(\pi ^{-1}(X_{reg})\) since \(\pi \) is a biholomorphism there. Moreover, by [4, Proposition 1.2], \(\alpha h\) is the direct image of some pseudomeromorphic current \(\tilde{\tau }\) on \(X'\), and is therefore also the image of the (unique) current \(\tau =\mathbf{1}_{\pi ^{-1}(X_{reg})}\tilde{\tau }\) in \(\mathcal {W}^{n,*}(X')\).

By [10, Theorem 3.7] again \(\tau \) is locally of the form \(\xi \wedge ds\), where \(\xi \) is in \(\mathcal {W}^{0,*}_{X'}\) and \(ds=ds_1\wedge \cdots \wedge ds_n\) for some local coordinates *s*. Hence, \(\tau \) is a \(K_{X'}\)-valued section of \(\mathcal {W}^{0,*}(X')\), so \(\tau /\pi ^*\gamma \) is a section of \(\mathcal {W}^{0,*}(X')\). Now \(\Psi :=\pi _*(\tau /\pi ^*\gamma )\) is a section of \(\mathcal {W}^{0,*}(X)\). On \(X_{reg} \cap \{ \gamma \ne 0 \}\) we thus have that \(\Psi \wedge \gamma =\pi _*\tau =\alpha \gamma =\psi \wedge \gamma \) and so \(\Psi =\psi \) there. By the SEP it follows that \(\Psi \) coincides with \(\psi \) on \(X_{reg}\) and is thus the desired pseudomeromorphic extension to *X*. \(\square \)

### Lemma 7.5

Assume that \(X_{reg}\rightarrow \Omega \) is a local embedding and (*z*, *w*) coordinates as before. Each section \(\phi \) in \(\mathcal {W}^{0,*}_X\) has a unique representation (7.1) with \(\widehat{\phi }_j\) in \(\mathcal {W}^{0,*}_Z\).

A current with a representation (7.1) is considered as an element of \(\mathcal {W}^{0,*}_X = {\mathcal Hom}(\omega ^n_X,\mathcal {W}^{n,*}_X)\) in view of the comment after Lemma 7.1.

### Proof

*T*in (7.9) is clearly injective, and by (4.10), if \(\xi \) in \((\mathcal {W}_Z^{0,*})^M\) and \(A \xi = 0\), then \(T \eta = \xi \), if \(\eta = S \xi \).

Now take \(\phi \) in \({\mathcal Hom}(\omega ^n_X,\mathcal {W}^{n,*}_X)\). Let us choose a basis \(\mu ^1,\ldots ,\mu ^m\) for \(\omega ^n_{X}\) and let \(\tilde{\phi }\) be the element in \( (\mathcal {W}_Z^{0,*})^M\) obtained from the coefficients of \(\phi \mu ^j\) when expressed as in (2.11), cf., Sect. 4.1. We claim that \(A\tilde{\phi } = 0\). Taking this for granted, by the exactness of (7.9), \(\tilde{\phi }\) is the image of the tuple \(\hat{\phi }=S\tilde{\phi }\). Now \(\hat{\phi }\wedge \mu ^j =\phi \mu ^j\) since they are represented by the same tuple in \( (\mathcal {W}_Z^{0,*})^M\). Thus \(\hat{\phi }\) gives the desired representation of \(\phi \).

*z*, and let \(X_0\) be defined by \(\mathscr {O}_{X_0} := \mathscr {O}_\Omega /(\mathcal {J}+\mathcal {I})\), then \(\mu ^1\wedge \mu ^z,\ldots ,\mu ^m \wedge \mu ^z\) generate \(\omega ^0_{X_0}\). If we let \(\phi _0\) be the morphism in \({\mathcal Hom}(\omega ^0_{X_0},\omega ^0_{X_0})\) given by \(\phi _0(\mu ^i \wedge \mu ^z) := \phi \mu ^i \wedge \mu ^z\) (which indeed gives a well-defined such morphism), then, as in the proof of Lemma 4.11, \(\tilde{\phi }_0 = \tilde{\phi }(0)\). In addition, the sequence (4.9) for \(X_0\) is

### Example 7.6

*Z*, we say that \(\Phi \sim \Phi '\) if and only if \(\Phi -\Phi '\) is in \(\mathcal {J}\) generically on

*Z*. If \(\Phi =A/B\) and \(\Phi '=A'/B'\), where

*B*and \(B'\) are generically non-vanishing on

*Z*, the condition is precisely that \(AB'-A'B\) is in \(\mathcal {J}\). We say that such an equivalence class is a meromorphic function \(\phi \) on

*X*, i.e., \(\phi \) is in \(\mathcal {M}_X\). Clearly we have \(\mathscr {O}_X\subset \mathcal {M}_X. \) We claim that

*Z*, since it is generically holomorphic on

*Z*. As in Definition 5.5 we therefore have a current \(\Phi \wedge h\) in \(\mathcal {W}_X^{n,0}\) for

*h*in \(\omega ^n_X\). Another representative \(\Phi '\) of \(\phi \) will give rise to the same current generically and hence everywhere by the SEP. Thus \(\phi \) defines a section of \({\mathcal Hom}(\omega ^n_X,\mathcal {W}^{n,*}_X) = \mathcal {W}^{0,*}_X\).\(\square \)

By definition, a current \(\phi \) in \(\mathcal {W}^{0,*}_X\) can be multiplied by a current *h* in \(\omega ^n_X\), and the product \(\phi \wedge h\) lies in \(\mathcal {W}^{n,*}_X\). It will be crucial that we can extend to products by somewhat more general currents. Notice that \(\omega ^n_X\) is a subsheaf of \({\mathcal C}^{n,*}_X\), which is an \(\mathscr {E}^{0,*}_X\)-module. Thus, we can consider the subsheaf \(\mathscr {E}^{0,*}_X \omega ^n_X\) of \({\mathcal C}^{n,*}_X\) which consists of finite sums \(\sum \xi _i \wedge h_i\), where \(\xi _i\) are in \(\mathscr {E}^{0,*}_X\) and \(h_i\) are in \(\omega ^n_X\).

### Lemma 7.7

Each \(\phi \) in \(\mathcal {W}^{0,*}_X = {\mathcal Hom}_{\mathscr {O}_X}(\omega ^n_X,\mathcal {W}^{n,*}_X)\) has a unique extension to a morphism in \({\mathcal Hom}_{\mathscr {E}^{0,*}_X}(\mathscr {E}^{0,*}_X \omega ^n_X,\mathcal {W}^{n,*}_X)\).

### Proof

*b*. By the SEP, it is enough to prove this locally on \(X_\mathrm{reg}\), and we can then assume that \(\phi \) has a representation (7.1). By Proposition 2.9, it is then enough to prove that it is well-defined assuming that \(\widehat{\phi }_0,\dots ,\widehat{\phi }_{\nu -1}\) in (7.1) are all smooth. In this case, the right hand side of (7.10) is simply the product of \(\xi _1 \wedge h_1 + \cdots + \xi _r \wedge h_r = b\) by the smooth form \(\phi \) in \(\mathscr {E}^{0,*}_X\), and this product only depends on

*b*. \(\square \)

### Corollary 7.8

*Z*, and \(h_i\) are in \(\omega ^n_X\). Then one has a well-defined product

### Proof

The right hand side of (7.11) exists as a current in \(\mathcal {W}^{n,*}_X\), and we must prove is that it only depends on the current \(\alpha \) and not on the representation \(\sum a_i \wedge h_i\). 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 \(\phi \) and \(\alpha \) in \(\mathscr {E}^{0,*}_X \omega ^n_X\), cf., Lemma 7.7. It follows by the SEP that the right hand side only depends on \(\alpha \). \(\square \)

### Remark 7.9

Recall from (6.9) that \(\omega = b \vartheta \). If \(\phi \) is in \(\mathcal {W}^{0,*}_X\), then we can define the product \(\phi \wedge \omega \) by Corollary 7.8.

Expressed extrinsically, if \(\mu = i_* \vartheta \), and if we write \(R\wedge dz = b \mu \) as in Lemma 6.2, then we can define the product \(R\wedge dz\wedge \phi := b\mu \wedge \phi \) as a current in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z)\).\(\square \)

### Lemma 7.10

Assume that \(\phi \) is in \(\mathcal {W}^{0,*}_X\), and that \(\phi \wedge \omega = 0\) for some structure form \(\omega \), where the product is defined by Remark 7.9. Then \(\phi = 0\).

### Proof

Considering the component with values in \(E_p\), we get that \(\phi \wedge \omega _0 = 0\). By Proposition 6.7, any *h* in \(\omega ^n_X\) can be written as \(h = \xi \omega _0\), where \(\xi \) is a holomorphic section of \(E_p^*\), so by \(\mathscr {O}\)-linearity, \(\phi \wedge h = 0\), i.e., \(\phi = 0\). \(\square \)

We end this section with the following result, 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 \(\mathcal {W}_Z\) is closed under holomorphic differential operators.

### Proof

*Z*, then \(\partial \tau /\partial \zeta _1\) is in \(\mathcal {W}_Z\). Consider the current

*Z*, and it follows from (2.5) that \(\tau '\) is in \(\mathcal {W}^Z_Y\). Let

*Y*followed by a projection. It follows from (2.4) that \(p_* \tau '\) is in \(\mathcal {W}_Z\). Since

## 8 The \(\bar{\partial }\)-operator on \(\mathcal {W}_X^{0,*}\)

We already know the meaning of \(\bar{\partial }\) on \(\mathcal {W}_X^{n,*}\), and we now define \(\bar{\partial }\) on \(\mathcal {W}_X^{0,*}\).

### Definition 8.1

### Definition 8.2

We say that *v* belongs to \(\text {Dom}\,\bar{\partial }_X\) if *v* is in \(\text {Dom}\,\bar{\partial }\), i.e., \(\bar{\partial }v=\phi \) for some \(\phi \) and in addition \(\bar{\partial }(v\wedge \omega )\), a priori only in \(\mathcal {PM}_X^{n,*}\), is in \(\mathcal {W}_X^{n,*}\), for each structure form \(\omega \) from any possible embedding.

If *X* is Cohen–Macaulay, then any such \(\omega \) is of the form \(a_1 h^1+\cdots +a_m h^m\), where \(h^j\) are in \(\omega _X^n\) and \(a_j\) are smooth, see Remark 6.5, and hence \(\text {Dom}\,\bar{\partial }_X\) coincides with \(\text {Dom}\,\bar{\partial }\) in this case.

### Example 8.3

*v*is in \(\mathscr {E}_X^{0,*}\) and \(\phi =\bar{\partial }v\) in the sense in Section 4. Then clearly

*v*is in \(\text {Dom}\,\bar{\partial }_X\) and \(\bar{\partial }_Xv=\phi \).

*w*is in \(\text {Dom}\,\bar{\partial }_X\) and

*v*is in \(\mathscr {E}_X^{0,*}\), then

Let \(\chi _\delta =\chi (|h|^2/\delta )\) where *h* is a tuple of holomorphic functions that cuts out \(X_{sing}\).

### Lemma 8.4

*v*is in \(\mathcal {W}^{0,*}(X)\), and it is in \(\text {Dom}\,\bar{\partial }_X\) on \(X_\mathrm{reg}\), then

*v*is in \(\text {Dom}\,\bar{\partial }_X\) on all of

*X*if and only if

### Proof

*f*,

*v*is in \(\text {Dom}\,\bar{\partial }_X\) if and only if \(\nabla _f( v \wedge \omega )\) is in \(\mathcal {W}^{n,*}_X\) for all structure forms \(\omega \). Since

*v*is in \(\text {Dom}\,\bar{\partial }_X\) on \(X_\mathrm{reg}\), thus \(\nabla _f( v \wedge \omega )\) is in \(\mathcal {W}^{n,*}_X\) on \(X_\mathrm{reg}\). By (2.2), \(\nabla _f(v \wedge \omega )\) is then in \(\mathcal {W}^{n,*}_X\) on all of

*X*if and only if

*v*is in \(\mathcal {W}^{0,*}_X\), \(v \wedge \omega \) is in \(\mathcal {W}^{n,*}_X\), so the left hand side of (8.6) tends to \(\nabla _f(v \wedge \omega )\) when \(\delta \rightarrow 0\), whereas the second term on the right hand side of (8.6) tends to \(\mathbf{1}_{X_\mathrm{reg}}\nabla _f( v \wedge \omega )\). Thus (8.5) holds if and only if (8.3) does. Thus the first statement in the lemma is proved.

Recall, cf., (6.9), that \(\omega = b \vartheta \) where *b* is smooth on \(X_\mathrm{reg}\) and \(\vartheta \) is in \(\omega ^n_X\). By the Leibniz rule thus \(-\nabla _f( v \wedge \omega ) = \bar{\partial }v \wedge \omega \) on \(X_\mathrm{reg}\), since \(\nabla _f \omega = 0\). Therefore, (8.6) is equivalent to \( -\nabla _f(\chi _\delta v \wedge \omega ) = \bar{\partial }\chi _\delta \wedge v \wedge \omega + \chi _\delta \bar{\partial }v \wedge \omega . \) If (8.3) holds, we therefore get (8.4) when \(\delta \rightarrow 0\). \(\square \)

### Remark 8.5

In case *X* is reduced the definition of \(\bar{\partial }_X\) is precisely the same as in [6]. However, the definition of \(\bar{\partial }v=\phi \) given here, for \(v,\phi \) in \(\mathcal {W}_X^{0,*}\), does *not* coincide with the definition in, e.g., [6]. In fact, that definition means that \(\bar{\partial }(v \wedge h)=\phi \wedge h\) for all *smooth**h* in \(\omega _X^n\), which in general is a strictly weaker condition. For example, for any weakly holomorphic function *v*, we have \(\bar{\partial }(v \wedge h) = 0\) for all smooth *h* in \(\omega _X^n\), while if *X* is a reduced complete intersection, or more generally Cohen–Macaulay, then \(\bar{\partial }(v \wedge h) = 0\) for all *h* in \(\omega _X^n\) is equivalent to *v* being strongly holomorphic, see [33, p. 124] and [2].\(\square \)

We conclude this section with a lemma that shows that \(\bar{\partial }\) means what one should expect when \(\phi ,v\) are expressed with respect to a local basis \(w^{\alpha _j}\) for \(\mathscr {O}_X\) over \(\mathscr {O}_Z\) as in Lemma 7.5.

### Lemma 8.6

### Proof

Let us use the notation from the proof of Lemma 7.5. Recall that \({\hat{v}}=S\tilde{v}\). In view of (8.2) and (2.12), \(\widetilde{\bar{\partial }v}=\bar{\partial }\tilde{v}\). Since *S* is holomorphic therefore \( \widehat{\bar{\partial }v} = S \widetilde{\bar{\partial }v} = S \bar{\partial }\tilde{v} = \bar{\partial }(S\tilde{v}) = \bar{\partial }\hat{v}. \)\(\square \)

## 9 Solving \(\bar{\partial }u=\phi \) on *X*

We will find local solutions to the \(\bar{\partial }\)-equation on *X* by means of integral formulas. We use the notation and machinery from [6, Section 5]. Let \(i:X\rightarrow \Omega \subset \mathbb {C}^N\) be a local embedding such that \(\Omega \) is pseudoconvex, let \(\Omega ' \subset \subset \Omega \) be a relatively compact subdomain of \(\Omega \), and let \(X' = X \cap \Omega '\).

### Theorem 9.1

*K*and

*P*are described below; they depend on a choice of weight

*g*. Since \(\Omega \) is Stein one can find such a weight

*g*that is holomorphic in

*z*, by which we mean that it depends holomorphically on \(z\in \Omega '\) and has no components containing any \(d\bar{z}_i\), cf., Example 5.1 in [6]. In this case, \(P\phi \) is holomorphic when \(k=0\), and vanishes when \(k\ge 1\), i.e.,

We now turn to the definition of *K* and *P*. For future need, in Sect. 11, we define them acting on currents in \(\mathcal {W}^{0,*}(X)\) and not only on smooth forms. Let \(\pi : \Omega _\zeta \times \Omega _z' \rightarrow \Omega _z'\) be the natural projection. Let us choose a holomorphic Hefer form^{3}*H*, a smooth weight *g* with compact support in \(\Omega \) with respect to \(z\in \Omega '\subset \subset \Omega \), 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 \(\zeta \), i.e., the factors \(d\eta _i = d\zeta _i - dz_i\) in *H* and *B* in [6] should be replaced by just \(d\zeta _i\).

*z*, and let \(\vartheta (\gamma )\) be the current such that

*B*is smooth, we can define

### Lemma 9.2

### Proof of Lemma 9.2

In order to define the extension of (9.5) across \(\Delta \), we note first that since *B* is almost semi-meromorphic with Zariski singular support \(\Delta \), \(\vartheta (B \wedge g\wedge H)\) is an almost semi-meromorphic \((0,*)\)-current on \(\Omega _\zeta \times \Omega _z'\), which is smooth outside the diagonal. We can thus form the current \(\vartheta (B \wedge g \wedge H) \wedge R(\zeta ) \wedge d\zeta \wedge \phi (\zeta )\wedge \mu (z)\) in \(\mathcal {W}^{Z\times Z'}_{\Omega _\zeta \times \Omega _z'}\), cf., Proposition 2.4, and this is the extension of (9.5) across \(\Delta \).

From the definitions above, it is clear that (9.4) and the extension of (9.5) are \(\mathscr {O}_\Omega \)-bilinear in \(\phi \) and \(\mu \). Both these currents are annihilated by \(\mathcal {J}_z\) and \(\mathcal {J}_\zeta \), cf., (2.8), so they depend \(\mathscr {O}_\Omega /\mathcal {J}\)-bilinearly. In view of (2.4) we conclude that (9.6) and (9.7) are in \({\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {W}^{Z'}_{\Omega '})\). \(\square \)

### Proposition 9.3

If \(\phi \in \mathcal {W}^{0,k}(X)\), then \(P\phi \in \mathscr {E}^{0,k}(X')\), and if in addition *g* is holomorphic in *z*, then \(P\phi \in \mathscr {O}(X')\) if \(k=0\) and vanishes if \(k\ge 1\).

### Proof

*g*depends holomorphically on

*z*, then \(P\phi \) is holomorphic in \(\Omega '\) if \(\phi \) is a (0, 0)-current, and vanishes for degree reasons if \(\phi \) has positive degree. \(\square \)

We shall now approximate \(K\phi \) by smooth forms. Let \(B^\epsilon = \chi (|\zeta -z|^2/\epsilon ) B\).

### Proposition 9.4

### Proof

*z*. Therefore \(K^\epsilon \phi \) is smooth. As in the proof of Proposition 9.3, we obtain since \(B^\epsilon \) is smooth that

### 9.1 Proof of Theorem 9.1

Recall from Lemma 6.2 that \(R\wedge dz=b\mu \), where \(\mu \) is a tuple of currents in \({\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {{ CH}}^{Z'}_{\Omega '})\) 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 \({\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {W}^{Z'}_{\Omega '})\), the equalities hold everywhere by the SEP.

As in [6] we let \(R^\lambda =\bar{\partial }|f|^{2\lambda }\wedge U\) for \({\text {Re}\,}\lambda \gg 0\). It has an analytic continuation to \(\lambda =0\) and \(R=R^\lambda |_{\lambda =0}\). Notice that \(R(z)\wedge B\) is well-defined since it is a tensor product with respect to the coordinates \(z, \eta =\zeta -z\). Also \(R(z)\wedge R^\lambda (\zeta )\wedge B\) admits such an analytic continuation and defines a pseudomeromorphic current^{4} when \(\lambda =0\). Let \(B_{k,k-1}\) be the component of *B* of bidegree \((k,k-1)\).

### Lemma 9.5

*k*,

### Proof of Lemma 9.5

*T*be the left hand side of (9.14). In view of Proposition 2.1 it is therefore enough to check that \(\mathbf{1}_\Delta T=0\). Fix

*j*,

*k*and let

*j*,

*j*) in \(\zeta \), the current vanishes unless \(j+k\le N\). Thus the total antiholomorphic degree is \(\le N-n + N-1\). On the other hand, the current has support on \(\Delta \cap Z\times Z\simeq Z\times \{pt\}\) which has codimension \(N+N-n\). Thus it vanishes by the dimension principle.

We now prove by induction over \(\ell \ge p\) that \(\mathbf{1}_\Delta T_{\ell } = 0\). Note that by (6.6), outside of \(Z_\ell \), \(R_\ell (z) = \alpha _\ell (z) R_{\ell -1}(z)\), where \(\alpha _\ell (z)\) is smooth. Thus, outside of \(Z_\ell \times \Omega \), \(T_{\ell }\) is a smooth form times \(T_{\ell -1}\), and thus, by induction and (2.3), \(\mathbf{1}_\Delta T_{\ell }\) has its support in \(\Delta \cap (Z_\ell \times Z) \simeq Z_\ell \times \{pt\}\), which has codimension \(\ge N + \ell +1\), see (6.3). On the other hand, the total antiholomorphic degree is \(\le \ell +j+k-1 \le \ell + N -1\), so the current vanishes by the dimension principle. We conclude that (9.14) holds. \(\square \)

^{5}as for [6, (5.2)] we have the equality

*R*, where \([\Delta ]'\) denotes the part of \([\Delta ]\) where \(d\eta _i = d\zeta _i - dz_i\) has been replaced

^{6}by \(d\zeta _i\). In view of (9.14) we can put \(\lambda =0\) in (9.15), and then we get

*h*in \(\omega _X\). Thus by definition (9.1) holds.

Since \(\mathcal {W}^{0,*}_X\) is closed under multiplication by \(\mathscr {O}_X\), we get that \(\psi \) in \(\mathcal {W}^{0,*}_X\) is in \(\text {Dom}\,\bar{\partial }_X\) if and only if \(-\nabla _f(\psi \wedge \omega )\) is in \(\mathcal {W}^{n,*}_X\). Thus, we conclude from (9.17) that \(K\phi \) is in \(\text {Dom}\,\bar{\partial }_X\) since all the other terms but \(-\nabla _f(K\phi \wedge \omega )\) are in \(\mathcal {W}^{0,*}_X\).

### 9.2 Intrinsic interpretation of *K* and *P*

*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 \(X\times X'\). Given a local embedding \(i : X \rightarrow \Omega \) as before, we have an embedding \((i\times i):X\times X' \rightarrow \Omega \times \Omega '\) such that the structure sheaf is \(\mathscr {O}_{\Omega \times \Omega '}/(\mathcal {J}_X+\mathcal {J}_{X'})\). One can check that this sheaf is independent of the chosen embedding, i.e., \(\mathscr {O}_{X\times X'}\) is intrinsically defined. Thus we also have definitions of all the various sheaves on \(X\times X'\) like \(\mathscr {E}_{X\times X'}^{0,*}\). The projection \(p:X\times X'\rightarrow X'\) is determined by \(p^*\phi :\mathscr {O}_{X'}\rightarrow \mathscr {O}_{X\times X'}\), which in turn is defined so that \( p^*i^*\Phi =(i\times i)^*\pi ^*\Phi \) for \(\Phi \) in \(\mathscr {O}_{\Omega '}\), where \(\pi :\Omega \times \Omega '\rightarrow \Omega '\) as before. Again one can check that this definition is independent of the embedding, and also extends to smooth \((0,*)\)-forms \(\phi \). Therefore, we have the well-defined mapping \( p_*:{\mathcal C}_{X\times X'}^{2n, *+n}\rightarrow {\mathcal C}_{X'}^{n,*}, \) and clearly

*h*in \(\omega ^n_{X'}\) and let \(\mu =i_*h\). Then, cf., the proof of Lemma 9.2,

## 10 Regularity of solutions on \(X_{reg}\)

We have already seen, cf., Proposition 9.3, that \(P\phi \) is always a smooth form. We shall now prove that *K* preserves regularity on \(X_{reg}\). More precisely,

### Theorem 10.1

If \(\phi \) in \(\mathcal {W}^{0,*}_X\) is smooth near a point \(x \in X_\mathrm{reg}'\), then \(K\phi \) in Theorem 9.1 is smooth near *x*.

Throughout this section, let us choose local coordinates \((\zeta ,\tau )\) and (*z*, *w*) at *x* corresponding to the variables \(\zeta \) and *z* in the integral formulas, so that \(Z = \{ (\zeta ,\tau );\ \tau = 0 \}\).

### Lemma 10.2

Notice that here we cut away the diagonal \(\Delta '\) in \(Z\times Z'\) times \(\mathbb {C}_\tau \times \mathbb {C}_w\) in contrast to Proposition 9.4, where we only cut away the diagonal \(\Delta \) in \(\Omega \times \Omega '\).

### Proof

Clearly \(B^\epsilon \) is smooth so that each \(K^\epsilon \phi \) is smooth in a full neighborhood in \(\Omega '\) of *x*. Let \(T= \mu (z,w)\wedge (HR(\zeta ,\tau )\wedge B\wedge g)_N\wedge \phi \), and let \(W=\Delta '\times \mathbb {C}_\tau \times \mathbb {C}_w\). Since \(\mu (z,w)\otimes R(\zeta ,\tau )\) has support on \(\{ w=\tau =0\}\), \(T=\mathbf{1}_{\{w=\tau =0\}}T\). Therefore, \(\mathbf{1}_W T= \mathbf{1}_W\mathbf{1}_{\{w=\tau =0\}}T=0\) since \(W\cap \{ w=\tau =0\}\subset \Delta \) and \(\mathbf{1}_\Delta T=0\) by definition, cf., Proposition 2.1 (i). Now notice that \(\mathbf{1}_W T=0\) implies (9.11) and in turn (9.9) with our present choice of \(B^\epsilon \). \(\square \)

We first consider a simple but nontrivial example of Theorem 10.1.

### Example 10.3

*z*,

*w*) we can choose the Hefer form

*B*we only get a contribution from the term

### Remark 10.4

*not*converge to smooth functions in general when \(\epsilon \rightarrow 0\). For a simple example, take \(\phi =\zeta d\bar{\zeta }\otimes \tau ^m\). Then \(K^\epsilon \phi (0,w)\) tends to

*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 \(K\phi (z,w)=\lim _{\epsilon \rightarrow 0^+}K^\epsilon \phi (z,w)\) 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

*k*th term in the second sum is equal to

*M*, then the integral is at least \(C^m\). By a Taylor expansion of \(\psi (z+\zeta )\) at the point

*z*, we are thus reduced to consider

*w*if \(\beta \le m-k-1\) and a smooth function plus

### Proposition 10.5

*z*,

*w*be coordinates at a point \(x\in X_{reg}\) such that \(Z=\{w=0\}\) and \(x=(0,0)\). If \(\phi \) is smooth, and has support where the local coordinates are defined, then

*Z*.

Taking this proposition for granted we can conclude the proof of Theorem 10.1.

### Proof of Theorem 10.1

If \(\phi \equiv 0\) in a neighborhood of \(x\in X'_\mathrm{reg}\), then \(K\phi \) is smooth near *x*, cf., the proof of Proposition 9.4. Thus, it is sufficient to prove Theorem 10.1 assuming that \(\phi \) is smooth and has support near *x*.

*Z*. Let \(v={\hat{v}}_0\otimes 1+\cdots + {\hat{v}}_{\nu -1}\otimes w^{\alpha _{\nu -1}}\). In view of (2.14), \(v^\epsilon \wedge \mu \rightarrow v\wedge \mu \) for all \(\mu \) in \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )\). From Lemma 10.2 we conclude that \( v\wedge \mu = K\phi \wedge \mu \) for all such \(\mu \). Thus \(K\phi =v\) in \(\mathcal {W}_X^{0,*}\) and hence \(K\phi \) is smooth. \(\square \)

### Proof of Proposition 10.5

*X*is embedded in \(\Omega \subset \mathbb {C}^N_{\zeta ',\tau '}\). After a suitable rotation we can assume that

*Z*is the graph \(\tau '=\psi (\zeta ')\). The Bochner–Martinelli kernel in \(\Omega \) is rotation invariant, so it is

*x*. It is clear that the symbols \(\bar{{\tau }}, {\bar{w}}, d\bar{{\tau }}\) can be omitted in the expression for

*w*and set \(w = 0\).

*A*is a holomorphic \((N-n)\times n\)-matrix. Then

### Lemma 10.6

Recall that if *a* is a form, then \(L_\xi a = d(\xi \lnot a)+ \xi \lnot (d a)\), and that \(L_\xi (\beta \lnot a)=[\xi ,\beta ]\lnot a+\beta \lnot (L_\xi a)\) if \(\beta \) is another vector field.

### Proof

*e*and its dual \(e^*\) and consider the exterior algebra spanned by \(e_j, e^*_\ell ,\) and the cotangent bundle. Let

*w*, setting \(w = 0\), and evaluating the residue with respect to \(\tau \) using (2.10), we obtain a sum of integrals like

*z*.

### Lemma 10.7

*z*, and \(\Phi := \eta ^*(I+A^*A)\eta \). If \(r\ge 1\) and \(r+s \ge \ell + 1\), then we have the relation

### Proof

*L*in the integral. If a derivative with respect to \(\zeta _j\) falls on some \(\eta ^* a_i\), we get a term \(I^{r-1,s}_\ell \). If it falls on \(\mathscr {O}(|\eta |^{2s})\) we get either \(\mathscr {O}(|\eta |^{2(s-1)})\) times \(\eta ^* b\), for some tuple

*b*of smooth functions, and this gives rise to the term \(I^{r,s-1}_{\ell }\) or \(\mathscr {O}(|\eta |^{2s})\), and this gives rise to another term \(I^{r-1,s}_\ell \). If it falls on \(\phi \) or \(\tilde{\gamma }_k\) we get an additional term \(I^{r-1,s}_\ell \). The only possibility left is when the derivative falls on \(\chi _\epsilon = \chi (|\eta |^2/\epsilon )\). It remains to show that an integral of the form

*v*is smooth and strictly positive and \(\alpha \) is smooth.

*z*. \(\square \)

## 11 A fine resolution of \(\mathscr {O}_X\)

We first consider a generalization of Theorem 9.1.

### Lemma 11.1

Assume that \(\phi \in \mathcal {W}^{0,k}(X)\cap \mathscr {E}_X^{0,k}(X_{reg})\cap \text {Dom}\,\bar{\partial }_X\) and that \(K\phi \) is in \(\text {Dom}\,\bar{\partial }_X\) (or just in \(\text {Dom}\,\bar{\partial }\)). Then (9.1) holds on \(X'\).

### Proof

*z*belongs to a fixed compact subset of \(X_\mathrm{reg}'\), then

*B*is smooth in (9.5) when \(\zeta \) is in \({{\mathrm{supp}}}\bar{\partial }\chi _\delta \) for small \(\delta \). Hence it suffices to see that

### Definition 11.2

*q*)-current \(\phi \) on an open set \({\mathcal U}\subset X\) is a section of \(\mathscr {A}_X^q\) over \({\mathcal U}\), \(\phi \in \mathscr {A}^q({\mathcal U})\), if, for every \(x\in {\mathcal U}\), the germ \(\phi _x\) can be written as a finite sum of terms

*x*, defined as above, and such that \(\xi _j\) has compact support in the set where \(z\mapsto k_j(\zeta ,z)\) is defined.

Clearly \(\mathscr {A}_X^*\) is closed under multiplication by \(\xi \) in \(\mathscr {E}_X^{0,*}\). It follows from (9.8) that \(\mathscr {A}_X^*\) is a subsheaf of \(\mathcal {W}_X^{0,*}\) and from Theorem 10.1 that \(\mathscr {A}_X^k=\mathscr {E}_X^{0,*}\) on \(X_{reg}\). Clearly any operator *K* as above maps \(\mathscr {A}_X^{*+1}\rightarrow \mathscr {A}_X^*\).

### Lemma 11.3

If \(\phi \) is in \(\mathscr {A}_X\), then \(\phi \) and \(K\phi \) are in \(\text {Dom}\,\bar{\partial }_X\).

### Proof

Notice that [6, Lemma 6.4] holds in our case by verbatim the same proof, since we have access to the dimension principle for (tensor products of) pseudomeromorphic \((n,*)\)-currents, and the computation rule (2.3), cf., the comment after Definition 5.7. Since \(\mathscr {A}_X^* = \mathscr {E}^{0,*}_X\) on \(X_\mathrm{reg}\) it is enough by Lemma 8.4 to check that \( \bar{\partial }\chi _\delta \wedge \omega \wedge \phi \rightarrow 0, \) and this precisely follows from [6, Lemma 6.4]. \(\square \)

In view of Lemmas 11.1 and 11.3 we have

### Proposition 11.4

### Proof of Theorem 1.1

By definition, it is clear that \(\mathscr {A}_X^k\) are modules over \(\mathscr {E}^{0,k}_X\), and by Theorem 10.1, \(\mathscr {A}_X^k\) coincides with \(\mathscr {E}^{0,k}_X\) on \(X_\mathrm{reg}\). Since we have access to Koppelman formulas, precisely as in the proof of [6, Theorem 1.2] we can verify that \(\bar{\partial }:\mathscr {A}_X^k\rightarrow \mathscr {A}_X^{k+1}\).

It remains to prove that (1.2) is exact. We choose locally a weight *g* that is holomorphic in *z*, so the term \(P\phi \) vanishes if \(\phi \) is in \(\mathscr {A}_X^k\), \(k\ge 1\), and is holomorphic in *z* when \(k=0\). Assume that \(\phi \) is in \(\mathscr {A}_X^k\) and \(\bar{\partial }\phi =0\). If \(k\ge 1\), then \(\bar{\partial }K\phi =\phi \), and if \(k=0\), then \(\phi =P\phi \). \(\square \)

### 11.1 Global solvability

*F*defines a vector bundle \(i^*F\rightarrow X\). The sheaves \(\mathscr {A}_X^{*}(E)\) give rise to a fine resolution of the sheaf \(\mathscr {O}_X(E)\), and by standard homological algebra we have the isomorphisms

*X*is Stein. If for instance

*X*is a pure-dimensional projective variety \(i:X\rightarrow {\mathbb P}^N\), then the \(\bar{\partial }\)-equation is solvable, e.g., if

*E*is a sufficiently ample line bundle.

## 12 Locally complete intersections

*X*locally is a complete intersection, i.e., given a local embedding \(i:X\rightarrow \Omega \subset \mathbb {C}^N\) there are global sections \(f_j\) of \(\mathscr {O}(d_j)\rightarrow {\mathbb P}^N\) such that \(\mathcal {J}=(f_1,\ldots ,f_p)\), where \(p=N-n\). In particular, \(Z=\{f_1=\cdots =f_p=0\}\). In this case \({\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega )\) is generated by the single current

*q*)-form \(\phi \) in \(\mathscr {E}_X^{0,q}\) is thus represented by a current \(\Phi \wedge \mu \), where \(\Phi \) is smooth in a neighborhood of

*Z*and \(i^*\Phi =\phi \). Moreover,

*X*is Cohen–Macaulay so \(X_{reg}\) coincides with the part of

*X*where

*Z*is regular, and \(\bar{\partial }\phi =\psi \) if and only if \(\bar{\partial }(\phi \wedge \mu )=\psi \wedge \mu \).

Henkin and Polyakov introduced, see [17, Definition 1.3], the notion of *residual currents*\(\phi \)*of bidegree* (0, *q*) on a locally complete intersection \(X\subset {\mathbb P}^N\), and the \(\bar{\partial }\)-equation \(\bar{\partial }\psi =\phi \). Their currents \(\phi \) correspond to our \(\phi \) in \(\mathscr {E}_X^{0,q}\) and their \(\bar{\partial }\)-operator on such currents coincides with ours.

### Remark 12.1

In [18] Henkin and Polyakov consider a global reduced complete intersection \(X\subset {\mathbb P}^N\). They prove, by a global explicit formula, that if \(\phi \) is a global \(\bar{\partial }\)-closed smooth (0, *q*)-form with values in \(\mathscr {O}(\ell )\), \(\ell = d_1+\cdots d_p -N-1\), then there is a smooth solution to \(\bar{\partial }\psi =\phi \) at least on \(X_{reg}\), if \(1\le q\le n-1\). When \(q=n\) a necessary obstruction term occurs. However, their meaning of “\(\bar{\partial }\)-closed” is that locally there is a representative \(\Phi \) of \(\phi \) and smooth \(g_j\) such that \(\bar{\partial }\Phi =g_1f_1+\cdots +g_pf_p\). If this holds, then clearly \(\bar{\partial }\phi =0\). 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 \(\bar{\partial }\phi =0\), neither do we know whether their solution admits some intrinsic extension across \(X_{sing}\) as a current on *X*.\(\square \)

### Example 12.2

Let \(X=\{f=0\}\subset \Omega \subset \mathbb {C}^{n+1}\) be a reduced hypersurface, and assume that \(df\ne 0\) on \(X_{reg}\), so that \(\mathcal {J}=(f)\). Let \(\phi \) be a smooth (0, *q*)-form in a neighborhood of some point *x* on *X* such that \(\bar{\partial }\phi =0\). We claim that \(\bar{\partial }u=\phi \) has a smooth solution *u* if and only if \(\phi \) has a smooth representative \(\Phi \) in ambient space such that \(\bar{\partial }\Phi =fg\) for some smooth form *g*. In fact, if such a \(\Phi \) exists then \(0=f\bar{\partial }g\) and thus \(\bar{\partial }g=0\). Therefore, \(g=\bar{\partial }\gamma \) for some smooth \(\gamma \) (in a Stein neighborhood of *x* in ambient space) and hence \(\bar{\partial }(\Phi -f\gamma )=0\). Thus there is a smooth *U* such that \(\bar{\partial }U=\Phi -f\gamma \); this means that \(u=i^*U\) is a smooth solution to \(\bar{\partial }u=\phi \). Conversely, if *u* is a smooth solution, then \(u=i^* U\) for some smooth *U* in ambient space, and thus \(\Phi =\bar{\partial }U\) is a representative of \(\phi \) in ambient space. Thus \(\bar{\partial }\Phi =fg\) (with \(g=0\)).

There are examples of hypersurfaces *X* where there exist smooth \(\phi \) with \(\bar{\partial }\phi =0\) that do not admit smooth solutions to \(\bar{\partial }u=\phi \), see, e.g., [6, Example 1.1]. It follows that such a \(\phi \) cannot have a representative \(\Phi \) in ambient space as above.\(\square \)

## Footnotes

- 1.
In [6, Proposition 3.3], the sum ends with \(\omega _{n-1}\) instead of \(\omega _n\), which, as remarked above, one can indeed assume when \(n \ge 1\) and the resolution is chosen to be of length \(\le N-1\).

- 2.
- 3.
We are only concerned with the component \(H^0\) of this form, so for simplicity we write just

*H*. - 4.
- 5.
There is a sign error in [6, (5.2)] due to \(R(z) \wedge dz\) being odd with respect to the super structure. Since we here move \(R(z) \wedge dz\) to the right, we get the correct sign.

- 6.
This change is due to the fact that we do the same change of the differentials in the definition of

*H*and*B*above.

## Notes

### Acknowledgements

We thank the referee for very careful reading and many valuable remarks.

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