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

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.

Introduction

Let 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

$$\begin{aligned} 0\rightarrow \mathscr {O}_X{\mathop {\rightarrow }\limits ^{i}} \mathscr {E}_X^{0,0}{\mathop {\rightarrow }\limits ^{\bar{\partial }}} \mathscr {E}_X^{0,1}{\mathop {\rightarrow }\limits ^{\bar{\partial }}} \cdots {\mathop {\rightarrow }\limits ^{\bar{\partial }}} \mathscr {E}_X^{0,n}\rightarrow 0 \end{aligned}$$

(1.1)

is exact, and hence provides a fine resolution of the structure sheaf $\mathscr {O}_X$. 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 $i:X\rightarrow \Omega$, 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

$$\begin{aligned} \mathscr {E}_X:=\mathscr {E}_\Omega /{\mathcal Ker\,}i^*. \end{aligned}$$

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 $\mu$ 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

$$\begin{aligned} 0\rightarrow \mathscr {O}_X{\mathop {\rightarrow }\limits ^{i}} \mathscr {A}_X^{0}{\mathop {\rightarrow }\limits ^{\bar{\partial }}} \mathscr {A}_X^{1}{\mathop {\rightarrow }\limits ^{\bar{\partial }}} \cdots {\mathop {\rightarrow }\limits ^{\bar{\partial }}} \mathscr {A}_X^{n}\rightarrow 0 \end{aligned}$$

(1.2)

is exact, and thus a fine resolution of the structure sheaf $\mathscr {O}_X$. An immediate consequence is the representation

$$\begin{aligned} H^{q}(X,\mathscr {O}_X)=\frac{{\text {Ker}\,}\big (\mathscr {A}^{0,q}(X){\mathop {\rightarrow }\limits ^{\bar{\partial }}}\mathscr {A}^{0,q+1}(X)\big )}{\text {Im}\big (\mathscr {A}^{0,q-1}(X){\mathop {\rightarrow }\limits ^{\bar{\partial }}}\mathscr {A}^{0,q}(X)\big )}, \quad q\ge 1, \end{aligned}$$

(1.3)

of sheaf cohomology, and so (1.3) is a generalization of the classical Dolbeault isomorphism. In special cases more qualitative information of the sheaves $\mathscr {A}_X^q$ are known, see, e.g., [5, 23].

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.

The aim of this paper is to extend the construction in [6] to a non-reduced pure-dimensional 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 $\mathscr {O}_X$ 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

$$\begin{aligned} \phi =\hat{\phi }_0\otimes 1+\hat{\phi }_1\otimes w^{\alpha _1} +\cdots + \hat{\phi }_{\nu -1}\otimes w^{\alpha _{\nu -1}}, \end{aligned}$$

(1.4)

where $\hat{\phi }_j$ are in $\mathscr {O}_Z$. A reasonable notion of a smooth form on 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.

Pseudomeromorphic currents

Let $s_1,\ldots , s_m$ be coordinates in $\mathbb {C}^m$, let $\alpha$ be a smooth form with compact support, and let $a_1,\ldots , a_r$ be positive integers, $0\le \ell \le r\le m$. Then

$$\begin{aligned} \bar{\partial }\frac{1}{s_1^{a_1}}\wedge \cdots \wedge \bar{\partial }\frac{1}{s_\ell ^{a_\ell }}\wedge \frac{\alpha }{s_{\ell +1}^{a_{\ell +1}}\cdots s_r^{a_r}} \end{aligned}$$

is a well-defined current that we call an 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 $d{\bar{h}}\wedge \tau =0$.

Given a pseudomeromorphic current $\tau$ and a subvariety 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

$$\begin{aligned} \mathbf{1}_{{\mathcal U}{\setminus } V}\tau =\lim _{\epsilon \rightarrow 0^+}\chi (|h|^2/\epsilon )\tau . \end{aligned}$$

(2.1)

Notice that $\mathbf{1}_V\tau :=(1-\mathbf{1}_{{\mathcal U}{\setminus } V})\tau$ is also pseudomeromorphic and has support on V. If W is another analytic set, then

$$\begin{aligned} \mathbf{1}_V\mathbf{1}_W\tau =\mathbf{1}_{V\cap W}\tau . \end{aligned}$$

(2.2)

This action of $\mathbf{1}_V$ 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 $\xi$ is a smooth form, then

$$\begin{aligned} \mathbf{1}_V(\xi \wedge \tau )=\xi \wedge \mathbf{1}_V\tau . \end{aligned}$$

(2.3)

If $f:Z'\rightarrow Z$ is a modification and $\tau$ is in $\mathcal {PM}_{Z'}$ then $f_*\tau$ is in $\mathcal {PM}_Z$. The same holds if f is a simple projection and $\tau$ has compact support in the fiber direction. In any case we have

$$\begin{aligned} \mathbf{1}_V f_*\tau =f_*(\mathbf{1}_{f^{-1}V}\tau ). \end{aligned}$$

(2.4)

It is not hard to check that if $\tau$ is in $\mathcal {PM}_Z$ and $\tau '$ is in $\mathcal {PM}_{Z'}$, then $\tau \otimes \tau '$ is in $\mathcal {PM}_{Z\times Z'}$, see, e.g., [4, Lemma 3.3]. If $V\subset {\mathcal U}\subset Z$ and $V'\subset {\mathcal U}'\subset Z'$, then

$$\begin{aligned} (\mathbf{1}_V\tau )\otimes \mathbf{1}_{V'}\tau '=\mathbf{1}_{V\times V'}(\tau \otimes \tau '). \end{aligned}$$

(2.5)

Another basic tool is the 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.

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 $\mathbf{1}_{\{ f = 0 \}}( 1/f) = 0$. The current 1 / f has the SEP and

$$\begin{aligned} \frac{1}{f} = \lim _{\epsilon \rightarrow 0^+} \chi (|f|^{2}/\epsilon ) \frac{1}{f}. \end{aligned}$$

We say that a current 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

([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 $\tau$ 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)

Notice that if h is a tuple that cuts out V, then in view of (2.1),

$$\begin{aligned} a\wedge \tau = \lim _{\epsilon \rightarrow 0^+} \chi (|h|^2/\epsilon ) a \wedge \tau . \end{aligned}$$

(2.7)

It follows that if $\xi$ is a smooth form, then

$$\begin{aligned} \xi \wedge (a\wedge \tau )=(-1)^{\deg \xi \deg a}a\wedge (\xi \wedge \tau ). \end{aligned}$$

(2.8)

For future reference we will need the following result.

Proposition 2.2

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

Proof

First assume that 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

$$\begin{aligned} K\mu (z)=\int _\zeta k(\zeta ,z) \wedge \mu (\zeta ), \end{aligned}$$

(2.9)

where 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),

$$\begin{aligned} \mathbf{1}_W p_*(k\wedge \mu \otimes 1)= & {} \lim _{\epsilon \rightarrow 0^+}p_*\left( \mathbf{1}_{\mathbb {C}^n\times W}(\chi _\epsilon k\wedge (\mu \otimes 1))\right) \\= & {} \lim _{\epsilon \rightarrow 0^+}p_*\left( \chi _\epsilon k\wedge (\mathbf{1}_{\mathbb {C}^n\times W}\mu \otimes 1)\right) \\= & {} \lim _{\epsilon \rightarrow 0^+} p_*\left( \chi _\epsilon k\wedge (\mathbf{1}_{\mathbb {C}^n}\mu \otimes \mathbf{1}_W 1)\right) =0, \end{aligned}$$

since $\mathbf{1}_W 1=0$. Thus $K\mu$ is in $\mathcal {W}_Z$.

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 $\mu =\pi _*\tau +\bar{\partial }\pi _* u$. $\square$

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.

Assume that we have local coordinates $(z,w) \in \mathbb {C}^n \times \mathbb {C}^p$ in $\Omega$ such that $Z=\{w=0\}$. We will use the short-hand notation

$$\begin{aligned} \bar{\partial }\frac{dw}{w^{\gamma +\mathbf{1}}}:=\bar{\partial }\frac{dw_1}{w_1^{\gamma _1+1}}\wedge \cdots \wedge \bar{\partial }\frac{dw_p}{w_p^{\gamma _p+1}} \end{aligned}$$

for multiindices $\gamma =(\gamma _1,\ldots ,\gamma _p)$ with $\gamma _j\ge 0$, and let $\gamma ! := \gamma _1! \cdots \gamma _p!$. Notice that

$$\begin{aligned} \frac{1}{(2\pi i)^p}\bar{\partial }\frac{dw}{w^{\gamma +\mathbf{1}}}.\xi =\frac{1}{\gamma !} \int _z \frac{\partial ^\gamma \xi }{\partial w^\gamma }(z,0) \end{aligned}$$

(2.10)

for test forms $\xi$. If $\tau$ is in $\mathcal {W}_Z$, then it follows by (2.5) and the fact that ${{\mathrm{supp}}}\bar{\partial }(1/w^{\gamma +\mathbf{1}}) = \{ w = 0 \}$ that $\tau \otimes \bar{\partial }(1/w^{\gamma +\mathbf{1}})$ is in $\mathcal {W}_\Omega ^Z$. We have the following local structure result, see [11, Proposition 4.1 and (4.3)] and [10, Theorem 3.5].

Proposition 2.5

Assume that we have local coordinates (z, w) such that $Z=\{w=0\}$. Then $\tau$ in $\mathcal {W}_\Omega ^Z$ has a unique representation as a finite sum

$$\begin{aligned} \tau =\sum _\gamma \tau _\gamma \wedge dz \otimes \bar{\partial }\frac{dw}{w^{\gamma +\mathbf{1}}},\quad \tau _\gamma \in \mathcal {W}_Z^{0,*}, \end{aligned}$$

(2.11)

where $dz := dz_1 \wedge \cdots \wedge dz_n$. If $\pi$ is the projection $(z,w)\mapsto z$, then

$$\begin{aligned} \tau _\gamma \wedge dz=(2\pi i)^{-p} \pi _*(w^\gamma \tau ). \end{aligned}$$

(2.12)

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 $\gamma$.

Let us now consider the pairing between $\mathcal {W}_\Omega ^Z$ and germs $\phi$ at 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

$$\begin{aligned} \phi =\sum _{|\alpha | < M} \phi _\alpha (z)\otimes w^\alpha +\mathscr {O}\left( |w|^M\right) +\mathscr {O}({\bar{w}},d{\bar{w}}), \end{aligned}$$

(2.13)

where

$$\begin{aligned} \phi _\alpha (z)=\frac{1}{\alpha !}\frac{\partial \phi }{\partial w^\alpha }(z,0) \end{aligned}$$

and $\mathscr {O}({\bar{w}},d{\bar{w}})$ denotes a sum of terms, each of which contains a factor ${\bar{w}}_j$ or $d{\bar{w}}_j$ for some j. If M in (2.13) is chosen so that $\mathscr {O}(|w|^M)\tau =0$, then

$$\begin{aligned} \phi \wedge \tau =\sum _{\alpha \le \gamma }\phi _\alpha \wedge \tau _\gamma \wedge dz \otimes \bar{\partial }\frac{dw}{w^{\gamma -\alpha +\mathbf{1}}}, \end{aligned}$$

i.e.,

$$\begin{aligned} \phi \wedge \tau =\sum _{\ell \ge 0}\sum _{\gamma \ge 0} \phi _{\gamma }\wedge \tau _{\ell +\gamma } \wedge dz \otimes \bar{\partial }\frac{dw}{w^{\ell +\mathbf{1}}}. \end{aligned}$$

(2.14)

Thus $\phi \wedge \tau =0$ if and only if $\sum _{\gamma \ge 0} \phi _{\gamma }\wedge \tau _{\ell +\gamma } =0$ for all $\ell$ (which is a finite number of conditions!).

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

Assume that we have an embedding $i:Z\rightarrow Y$ of a reduced space 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 $\mathcal {PM}_Z$.

Since $i_*( i^* \chi (|h|^2/\epsilon ) \tau ) = \chi (|h|^2/\epsilon ) i_* \tau$ for any current $\tau$ on Z, we get by (2.1) that for a subvariety $V \subset {\mathcal U}\subset Z$,

$$\begin{aligned} \mathbf{1}_V (i_* \tau ) = i_*( \mathbf{1}_V \tau ), \end{aligned}$$

(2.15)

i.e., (2.4) holds also for an embedding $i : Z \rightarrow Y$. The condition $\mathbf{1}_{Z_{sing}}(i_* \tau ) = 0$ in (ii) is fulfilled if $i_* \tau$ has the SEP with respect to Z.

Corollary 2.7

We have the isomorphism

$$\begin{aligned} i_* : \mathcal {W}^{n,*}_Z \rightarrow {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega ), \end{aligned}$$

where $\mathcal {J}$ is the ideal defining 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 $\varphi = 0$.

Local embeddings of a non-reduced analytic space

Let 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

$$\begin{aligned} X{\mathop {\rightarrow }\limits ^{j}}\widehat{\Omega }{\mathop {\rightarrow }\limits ^{\iota }} \Omega :=\widehat{\Omega }\times {\mathcal U}, \quad i=\iota \circ j, \end{aligned}$$

(3.1)

where 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.

Now consider a fixed local embedding $i:X\rightarrow \Omega \subset \mathbb {C}^N$, assume that 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

$$\begin{aligned} \mathscr {O}_Z\rightarrow \mathscr {O}_X,\quad \phi (z)\mapsto \tilde{\phi }(z,w)=\phi (z). \end{aligned}$$

In this way $\mathscr {O}_X$ becomes an $\mathscr {O}_Z$-module, which however depends on the choice of coordinates.

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.

Recall that $f_1,\ldots ,f_m \in R$ is a 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

$$\begin{aligned} {{\mathrm{depth}}}_R M + {{\mathrm{pd}}}_R M = \dim _R R, \end{aligned}$$

(3.2)

where ${{\mathrm{pd}}}_R M$ is the length of a minimal free resolution of 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 $|\alpha | < M$.

We shall now show that

$$\begin{aligned} {{\mathrm{depth}}}_{\mathscr {O}_{X,x}} \mathscr {O}_{X,x} = {{\mathrm{depth}}}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} \end{aligned}$$

(3.3)

and

$$\begin{aligned} \dim _{\mathscr {O}_{X,x}} \mathscr {O}_{X,x} = \dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x}. \end{aligned}$$

(3.4)

We claim that a sequence $f_1,\ldots ,f_m$ in $\mathscr {O}_{X,x}$ is regular (on $\mathscr {O}_{X,x}$) if and only if $\tilde{f}_1,\ldots ,\tilde{f}_m\in \mathscr {O}_{Z,x}$ is regular on $\mathscr {O}_{X,x}$, where $\tilde{f}_j(z)=f_j(z,0)$. In fact, since $\mathscr {O}_{X,x}$ has pure dimension, a function $g\in \mathscr {O}_{X,x}=\mathscr {O}_{\Omega ,x}/\mathcal {J}$ is a non zero-divisor if and only if 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 $\dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} = \dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{Z,x}/(\mathrm{ann\,}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x}) = \dim _{\mathscr {O}_{Z,x}} \mathscr {O}_{Z,x} = n$.

From (3.3) and (3.4) we conclude that $\mathscr {O}_{X,x}$ is Cohen–Macaulay as an $\mathscr {O}_{Z,x}$-module if and only if it is Cohen–Macaulay (as an $\mathscr {O}_{X,x}$-module). Hence, by (3.2), with $R = \mathscr {O}_{Z,x}$ and $M = \mathscr {O}_{X,x}$,

$$\begin{aligned} {{\mathrm{depth}}}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} + {{\mathrm{pd}}}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} = n, \end{aligned}$$

so $\mathscr {O}_{X,x}$ is Cohen–Macaulay as an $\mathscr {O}_{Z,x}$-module if and only if ${{\mathrm{pd}}}_{\mathscr {O}_{Z,x}} \mathscr {O}_{X,x} = 0$, that is, if and only if $\mathscr {O}_{X,x}$ is a free $\mathscr {O}_{Z,x}$-module. $\square$

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

Let $\Omega =\mathbb {C}^2_{z,w}$ and $\mathcal {J}=(w^2)$ so that $Z=\{w=0\}$. Then 1, 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

$$\begin{aligned} a_0(z) +a_1(z)w=a_0(\zeta +\eta )+a_1(\zeta +\eta )\eta = a_0(\zeta )+(\partial a_0/\partial \zeta )(\zeta ) \eta +a_1(\zeta )\eta +\mathcal {J}\end{aligned}$$

we have the representation $a_0(\zeta )\otimes 1+(a_1(\zeta )+\partial a_0/\partial \zeta )(\zeta )\otimes \eta$ with respect to $(\zeta ,\eta )$.$\square$

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 $a_0\otimes 1+\cdots + a_{\nu -1}\otimes w^{\alpha _{\nu -1}}= b_0\otimes 1+\cdots + b_{\nu -1} \otimes \eta ^{\beta _{\nu -1}}+\mathcal {J}.$

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

If $\mathcal {J}$ has pure dimension, then

$$\begin{aligned} \mathcal {J}= \mathrm{ann\,}_{\mathscr {O}_\Omega } {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z). \end{aligned}$$

(4.1)

That is, $\phi$ is in $\mathcal {J}$ if and only if $\phi \mu =0$ for all $\mu$ in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)$. It is also well-known, see, e.g., [3, Theorem 1.5], that

$$\begin{aligned} {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)\simeq {\mathcal Ext\,}^p (\mathscr {O}_\Omega /\mathcal {J},K_\Omega ), \end{aligned}$$

(4.2)

so ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)$ is a coherent analytic sheaf. Locally we thus have a finite number of generators $\mu ^1,\ldots ,\mu ^m$. In Example 6.9, we compute explicitly such generators for the ideal $\mathcal {J}$ in Example 3.4.

Let $\xi$ be a smooth $(0,*)$-form in $\Omega$. Without first giving meaning to $i^*$, we define the sheaf ${\mathcal Ker\,}i^*$ by saying that $\xi$ is in ${\mathcal Ker\,}i^*$ if

$$\begin{aligned} \xi \wedge \mu =0, \quad \mu \in {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega ). \end{aligned}$$

Notice that if $\xi$ is holomorphic, then, in view of the duality (4.1), $\xi$ is in ${\mathcal Ker\,}i^*$ if and only if $\xi$ is in $\mathcal {J}$.

Definition 4.2

We define the sheaf of smooth $(0,*)$-forms on X as

$$\begin{aligned} \mathscr {E}_X^{0,*}:=\mathscr {E}_\Omega ^{0,*}/{\mathcal Ker\,}i^*. \end{aligned}$$

(4.3)

We will prove below that this sheaf is independent of the choice of embedding and thus intrinsic on X.

Given $\phi$ in $\mathscr {E}_\Omega ^{0,*}$, let $i^*\phi$ be its image in $\mathscr {E}_X^{0,*}$. In particular, $i^* \xi = 0$ means that $\xi$ belongs to ${\mathcal Ker\,}i^*$, which then motivates this notation. Notice that ${\mathcal Ker\,}i^*$ is a two-sided ideal in $\mathscr {E}_\Omega ^{0,*}$, i.e., if $\phi$ is in $\mathscr {E}_\Omega ^{0,*}$ and $\xi$ is in ${\mathcal Ker\,}i^*$, then $\phi \wedge \xi$ and $\xi \wedge \phi$ are in ${\mathcal Ker\,}i^*$. It follows that we have an induced wedge product on $\mathscr {E}_X^{0,*}$ such that

$$\begin{aligned} i^*(\phi \wedge \xi )=i^*\phi \wedge i^*\xi . \end{aligned}$$

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

Using the notation of (3.1),

$$\begin{aligned} \iota _*:{\mathcal Hom}_{\mathscr {O}_{\widehat{\Omega }}}(\mathscr {O}_{\widehat{\Omega }}/\widehat{\mathcal {J}},\mathcal {W}^Z_{\widehat{\Omega }}) \rightarrow {\mathcal Hom}_{\mathscr {O}_\Omega }(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z) \end{aligned}$$

(4.4)

is an isomorphism.

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

Using the notation of (3.1),

$$\begin{aligned} \iota _*:{\mathcal Hom}_{\mathscr {O}_{\widehat{\Omega }}}(\mathscr {O}_{\widehat{\Omega }}/\widehat{\mathcal {J}},\mathcal {{ CH}}^Z_{\widehat{\Omega }}) \rightarrow {\mathcal Hom}_{\mathscr {O}_\Omega }(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z) \end{aligned}$$

(4.5)

is an isomorphism.

Corollary 4.6

Using the notation in (3.1),

$$\begin{aligned} \iota ^*:\mathscr {E}_{\Omega }^{0,*}/{\mathcal Ker\,}i^*\rightarrow \mathscr {E}_{\widehat{\Omega }}^{0,*}/{\mathcal Ker\,}j^*, \end{aligned}$$

(4.6)

is an isomorphism.

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.

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 $\mu ^j$.

Recall from Corollary 3.3 that $\phi$ in $\mathscr {O}_X$ has a unique representative

$$\begin{aligned} \hat{\phi }=\hat{\phi }_0+\hat{\phi }_1\otimes w^{\alpha _1}+\cdots +\hat{\phi }_{\nu -1}\otimes w^{\alpha _{\nu -1}}, \end{aligned}$$

(4.7)

where $\hat{\phi }_j$ are in $\mathscr {O}_Z$. We thus have an $\mathscr {O}_Z$-linear morphism

$$\begin{aligned} T:(\mathscr {O}_Z)^\nu \rightarrow (\mathscr {O}_Z)^M. \end{aligned}$$

(4.8)

The morphism is injective by Proposition 4.1, and the holomorphic matrix 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

To begin with notice that a given smooth $\phi$ 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 $\mu ^j$, and the terms $\bar{w}$ and $d\bar{w}$ annihilate all the $\mu ^j$ as well since they are pseudomeromorphic with support on $\{ w = 0 \}$. On the other hand, each $w^\alpha$ not in the set of generators must be of the form

$$\begin{aligned} w^\alpha =a_0+a_1\otimes w^{\alpha _1}+\cdots +a_{\nu -1}\otimes w^{\alpha _{\nu -1}} +\mathcal {J}, \end{aligned}$$

and hence $\phi _\alpha \otimes w^\alpha$ is of the form (4.7). Thus the representation exists. To show uniqueness of the representation, we assume that $\hat{\phi }$ is in ${\mathcal Ker\,}i^*$. Then the tuple $(\hat{\phi }_j)$ is mapped to 0 by the matrix 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.

We can thus choose a holomorphic matrix A such that

$$\begin{aligned} 0\rightarrow \mathscr {O}_Z^\nu {\mathop {\rightarrow }\limits ^{T}} \mathscr {O}_Z^M{\mathop {\rightarrow }\limits ^{A}} \mathscr {O}_Z^{M'} \end{aligned}$$

(4.9)

is pointwise exact, and we can also find holomorphic matrices S and B such that

$$\begin{aligned} I = TS + BA. \end{aligned}$$

(4.10)

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.

More concretely, this means that given an embedding $i:X\rightarrow \Omega$, currents $\psi$ in ${\mathcal C}_X^{n,q}$ precisely correspond to the $(N,N-n+q)$-currents $\tau$ on $\Omega$ that vanish on ${\mathcal Ker\,}i^*$. Since ${\mathcal Ker\,}i^*$ is a two-sided ideal in $\mathscr {E}_\Omega ^{0,*}$ this holds if and only if $\xi \wedge \tau =0$ for all $\xi$ in ${\mathcal Ker\,}i^*$. It is natural to write $\tau =i_*\psi$ so that

$$\begin{aligned} i_*\psi .\xi =\psi .i^*\xi . \end{aligned}$$

Clearly, we get a mapping $\bar{\partial }:{\mathcal C}_X^{n,q}\rightarrow {\mathcal C}_X^{n,q+1}$ such that $\bar{\partial }^2=0$.

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 ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z)$.

By definition we thus have the isomorphism

$$\begin{aligned} i_*:\mathcal {W}_X^{n,*}\simeq {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z). \end{aligned}$$

(5.1)

It follows from Lemma 4.4 that $\mathcal {W}_X^{n,*}$ is intrinsically defined.

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 $i_*(a \wedge \psi ) = a \wedge i_* \psi$.

By (2.7),

$$\begin{aligned} a \wedge \psi = \lim _{\epsilon \rightarrow 0^+} \chi (|h|^2/\epsilon ) a \wedge \psi \end{aligned}$$

(5.2)

if 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 $\mathcal {W}^{n,0}_X$.

This sheaf corresponds via $i_*$ to $\bar{\partial }$-closed currents in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z)$ so we have the isomorphism

$$\begin{aligned} i_*:\omega _X^n \simeq {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z). \end{aligned}$$

(5.3)

When 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 $\mathcal {PM}_X^{n,*}$.

Structure form on X

Let $i:X\rightarrow \Omega \subset \mathbb {C}^N$ be a local embedding as before, let $p = N-n$ be the codimension of 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

$$\begin{aligned} 0 \rightarrow \mathscr {O}(E_{N_0}) {\mathop {\longrightarrow }\limits ^{f_{N_0}}} \cdots {\mathop {\longrightarrow }\limits ^{f_3}} \mathscr {O}(E_2) {\mathop {\longrightarrow }\limits ^{f_2}} \mathscr {O}(E_1) {\mathop {\longrightarrow }\limits ^{f_1}} \mathscr {O}(E_0) \end{aligned}$$

(6.1)

of $\mathscr {O}_{\Omega }/\mathcal {J}$; here $E_k$ are trivial vector bundles over $\Omega$ and $E_0$ is the trivial line bundle. This resolution induces a complex of vector bundles

$$\begin{aligned} 0 \rightarrow E_{N_0} {\mathop {\longrightarrow }\limits ^{f_{N_0}}} \cdots {\mathop {\longrightarrow }\limits ^{f_3}} E_2 {\mathop {\longrightarrow }\limits ^{f_2}} E_1 {\mathop {\longrightarrow }\limits ^{f_1}} E_0 \end{aligned}$$

(6.2)

that is pointwise exact outside Z. Let $X_k$ be the set where $f_k$ does not have optimal rank. Then

$$\begin{aligned} \cdots \subset X_{k+1} \subset X_k \subset \cdots \subset X_{p+1}\subset X_p =\cdots =X_1= Z; \end{aligned}$$

these sets are independent of the choice of resolution and thus invariants of $\mathscr {O}_{\Omega }/\mathcal {J}$. Since $\mathscr {O}_\Omega /\mathcal {J}$ has pure codimension p,

$$\begin{aligned} {\text {codim}\,}X_k \ge k+1, \quad \text{ for }\, k\ge p+1, \end{aligned}$$

(6.3)

see [14, Corollary 20.14]. Thus there is a free resolution (6.1) if and only if $X_k=\emptyset$ for $k>N_0$. Unless $n = 0$ (which is not interesting in relation to the $\bar{\partial }$-equation), we can thus choose the resolution so that $N_0\le N-1$. The variety 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.

Let us now choose Hermitian metrics on the bundles $E_k$. We then refer to (6.1) as a 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

$$\begin{aligned} u_k=(\bar{\partial }\sigma _k)\cdots (\bar{\partial }\sigma _2)\sigma _1=\sigma _k(\bar{\partial }\sigma _{k-1})\cdots (\bar{\partial }\sigma _1) \end{aligned}$$

(6.4)

in $\Omega {\setminus } X$; for the second equality, see [7, (2.3)]. We have that

$$\begin{aligned} f_1u_1=1, \quad f_{k+1}u_{k+1}-\bar{\partial }u_k=0,\quad k\ge 1, \end{aligned}$$

in $\Omega {\setminus } X$. If $f:=\oplus f_k$ and $u:=\sum u_k$, then these relations can be written economically as $\nabla _f u=1$, where $\nabla _f:= f-\bar{\partial }$. To make the algebraic machinery work properly one has to introduce a superstructure on the bundle $E=:\oplus E_k$ so that vectors in $E_{2k}$ are even and vectors in $E_{2k+1}$ are odd; hence 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

$$\begin{aligned} \nabla _f U=1-R. \end{aligned}$$

(6.5)

It follows directly that 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

In case $\mathcal {J}$ is generated by the single non-trivial function 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

$$\begin{aligned} R = \bar{\partial }\frac{1}{f_p} \wedge \cdots \wedge \bar{\partial }\frac{1}{f_1}, \end{aligned}$$

where the right hand side is the so-called Coleff–Herrera product of f, see for example [1, Corollary 3.5].$\square$

There are almost semi-meromorphic $\alpha _k$ in $\Omega$, cf., [7, Section 2] and the proof of [6, Proposition 3.3], that are smooth outside $X_k$, such that

$$\begin{aligned} R_{k+1}=\alpha _{k+1} R_k \end{aligned}$$

(6.6)

outside $X_{k+1}$ for $k\ge p$. In view of (6.3) and the dimension principle, $\mathbf{1}_{X_{k+1}}R_{k+1}=0$ and hence (6.6) holds across $X_{k+1}$, i.e., $R_{k+1}$ is indeed equal to the product $\alpha _{k+1} R_k$ 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 $(z_1,\dots ,z_N)$ denote coordinates on $\mathbb {C}^N$, and let $dz := dz_1 \wedge \cdots \wedge dz_N$.

Lemma 6.2

There is a matrix of almost semi-meromorphic currents b such that

$$\begin{aligned} R\wedge dz=b\mu , \end{aligned}$$

(6.7)

where $\mu$ is a tuple of currents in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)$.

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

Let (6.1) be a Hermitian resolution of $\mathscr {O}_\Omega /\mathcal {J}$ in $\Omega$, and let R be the associated residue current. Then there exists a (unique) current $\omega$ in $\mathcal {W}_X^{n,*}$ such that

$$\begin{aligned} i_*\omega =R\wedge dz. \end{aligned}$$

(6.8)

There is a matrix 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

$$\begin{aligned} \omega = b \vartheta . \end{aligned}$$

(6.9)

More precisely, $\omega =\omega _0+\omega _1+\cdots +\omega _n$,¹ where $\omega _k \in \mathcal {W}^{n,k}(X, E_{p+k})$, and if $f^j:=f_{p+j}$, then

$$\begin{aligned} f^0\omega _0=0, \quad f^{j+1}\omega _{j+1}-\bar{\partial }\omega _j=0 \text {, for } j\ge 0. \end{aligned}$$

(6.10)

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.

We now consider the case when 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

$$\begin{aligned}h . \xi = \int _Z \mathcal {L} \xi .\end{aligned}$$

One can then verify using this formula and (6.9) that the action of the structure form $\omega$ on a test form $\xi$ in $\mathscr {E}^{0,*}_X$ equals

$$\begin{aligned} \omega . \xi = \int _Z \tilde{\mathcal {L}} \xi , \end{aligned}$$

where $\tilde{\mathcal {L}}$ is now a tuple of Noetherian operators for $\mathcal {J}$ with almost semi-meromorphic coefficients, cf., [32, Section 4].$\square$

Notice that (6.1) gives rise to the dual Hermitian complex

$$\begin{aligned} 0\rightarrow \mathscr {O}(E_0^*) {\mathop {\rightarrow }\limits ^{f^*_1}}\cdots \rightarrow \mathscr {O}(E^*_{p-1}){\mathop {\rightarrow }\limits ^{f^*_p}} \mathscr {O}(E_p^*){\mathop {\longrightarrow }\limits ^{f^*_{p+1}}}\cdots . \end{aligned}$$

(6.11)

Let $\xi = \xi _0 \wedge dz$ be a holomorphic section of the sheaf

$$\begin{aligned} {\mathcal Hom}(E_p,K_\Omega )\simeq \mathscr {O}(E^*_p)\otimes \mathscr {O}(K_\Omega ) \end{aligned}$$

such that $f^*_{p+1}\xi _0=0$. Then $\bar{\partial }(\xi _0 \omega _0)=\pm \xi _0\bar{\partial }\omega _0=\pm \xi _0 f_{p+1}\omega _1= \pm (f^*_{p+1}\xi _0)\omega _1=0$, so that $\xi _0\omega _0$ is in $\omega _X^n$. Moreover, if $\xi _0=f^*_p\eta$ for $\eta$ in $\mathscr {O}(E^*_{p-1})$, then $\xi _0\omega _0= f^*_p\eta \omega _0=\pm \eta f_p\omega _0=0$. We thus have a sheaf mapping

$$\begin{aligned} {\mathcal H}^p({\mathcal Hom}(E_\bullet ,K_\Omega ))\rightarrow \omega _X^n, \quad \xi _0 \wedge dz \mapsto \xi _0\omega _0. \end{aligned}$$

(6.12)

Proposition 6.7

The mapping (6.12) is an isomorphism, which establishes an intrinsic isomorphism

$$\begin{aligned} {\mathcal Ext\,}^p(\mathscr {O}_\Omega /\mathcal {J},K_\Omega )\simeq \omega _X^n. \end{aligned}$$

(6.13)

Proof

If 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

$$\begin{aligned} {\mathcal H}^p({\mathcal Hom}(E_\bullet ,K_\Omega )) \rightarrow \omega _X^n {\mathop {\rightarrow }\limits ^{\simeq }} {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega ), \end{aligned}$$

(6.14)

where the first mapping is (6.12), and the second is $h \mapsto i_*h$. In view of (6.8), the composed mapping is $\xi = \xi _0 \wedge dz \mapsto \xi R_p = \xi _0 R_p \wedge dz$.² This mapping is an intrinsic isomorphism

$$\begin{aligned}{\mathcal Ext\,}^p(\mathscr {O}_\Omega /\mathcal {J},K_\Omega )\simeq {\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )\end{aligned}$$

according to [3, Theorem 1.5]. It follows that (6.12) also establishes an intrinsic isomorphism. $\square$

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

The isomorphism

$$\begin{aligned} {\mathcal H}^p({\mathcal Hom}(E_\bullet ,K_\Omega )) {\mathop {\rightarrow }\limits ^{\simeq }}{\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega ) \end{aligned}$$

(6.15)

was well-known since long ago, the contribution in [3] was the realization $\xi \mapsto \xi R_p$.$\square$

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

Let $\mathcal {J}$ be as in Example 3.4. We claim that ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )$ is generated by

$$\begin{aligned} \mu _1 := \bar{\partial }\frac{1}{w_1} \wedge \bar{\partial }\frac{1}{w_2} \wedge dz \wedge dw \text { and } \mu _2 := \left( z_1 \bar{\partial }\frac{1}{w_1^2} \wedge \bar{\partial }\frac{1}{w_2} + z_2 \bar{\partial }\frac{1}{w_1} \wedge \bar{\partial }\frac{1}{w_2^2}\right) \wedge dz \wedge dw. \end{aligned}$$

In order to prove this claim, we use the comparison formula for residue currents from [21], which states that if $\mathscr {O}(F_\bullet )$ and $\mathscr {O}(E_\bullet )$ are free resolutions of $\mathscr {O}_\Omega /\mathcal {I}$ and $\mathscr {O}_\Omega /\mathcal {J}$, respectively, where $\mathcal {I}$ and $\mathcal {J}$ have codimension $\ge p$, and $a : F_\bullet \rightarrow E_\bullet$ is a morphism of complexes, then there exists a ${\mathcal Hom}(F_0,E_{p+1})$-valued current $M_{p+1}$ such that $R^E_p a_0 = a_p R^F_p + f_{p+1} M_{p+1}$. If $\xi$ is in ${\mathcal Ker\,}f_{p+1}^*$, we thus get that

$$\begin{aligned} \xi R^E_p a_0 = \xi a_p R^F_p. \end{aligned}$$

(6.16)

We will apply this with $\mathscr {O}_\Omega (E_\bullet )$ as the free resolution

$$\begin{aligned} 0 \rightarrow \mathscr {O}_\Omega {\mathop {\longrightarrow }\limits ^{f_3}} \mathscr {O}_\Omega ^4 {\mathop {\longrightarrow }\limits ^{f_2}} \mathscr {O}_\Omega ^4 {\mathop {\longrightarrow }\limits ^{f_1}} \mathscr {O}_\Omega \rightarrow \mathscr {O}_\Omega /\mathcal {J}\rightarrow 0, \end{aligned}$$

where

$$\begin{aligned} f_3= & {} \left[ \begin{array}{c} w_2 \\ -w_1 \\ z_2 \\ -z_1 \end{array} \right] \text {, } f_2 = \left[ \begin{array}{cccc} z_2 &{} 0 &{} -w_2 &{} 0 \\ -z_1 &{} z_2 &{} w_1 &{} -w_2 \\ 0 &{} -z_1 &{} 0 &{} w_1 \\ -w_1 &{} -w_2 &{} 0 &{} 0 \end{array} \right] \quad \text { and }\\ f_1= & {} \left[ \begin{array}{cccc} w_1^2&w_1 w_2&w_2^2&z_2 w_1 - z_1 w_2 \end{array} \right] , \end{aligned}$$

and the Koszul complex $(F,\delta _{\mathbf {w}^2})$ generated by $\mathbf {w}^2 := (w_1^2,w_2^2)$, which is a free resolution of $\mathscr {O}/(w_1^2,w_2^2)$. We then take the morphism of complexes $a : F_\bullet \rightarrow E_\bullet$ given by

$$\begin{aligned} a_2 = \left[ \begin{array}{c} 0 \\ 0 \\ w_2 \\ w_1 \end{array} \right] ,\quad a_1 = \left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} 0 \\ 0 &{} 1 \\ 0 &{} 0 \end{array} \right] \text { and } a_0 = \left[ \begin{array}{c} 1 \end{array} \right] . \end{aligned}$$

Since the current $R^F_2$ is equal to the Coleff–Herrera product $\bar{\partial }(1/w_1^2) \wedge \bar{\partial }(1/w_2^2)$, cf., Remark 6.1, we thus get by (6.16) and Remark 6.8 that ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )$ is generated by

$$\begin{aligned} ({\mathcal Ker\,}f_3^*) a_2 \bar{\partial }\frac{1}{w_1^2} \wedge \bar{\partial }\frac{1}{w_2^2}. \end{aligned}$$

A straightforward calculation gives the generators $\mu _1$ and $\mu _2$ above.$\square$

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 $z=(z_1,\ldots ,z_n)$.

Let $\mathscr {O}(E_\bullet )$ be a free Hermitian resolution of $\mathscr {O}_\Omega /\mathcal {J}$ 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 $\xi \mapsto \xi R_p^E$. Let $F_\bullet$ be the Koszul complex generated by 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

$$\begin{aligned} R^{E\otimes F}_N = R^E_p \wedge R^F_n \end{aligned}$$

according to [2, Theorem 4.2]. From [3, Theorem 1.5] again it follows that the canonical isomorphism

$$\begin{aligned} {\mathcal H}^N({\mathcal Hom}((E\otimes F)_\bullet ,K_\Omega ))\rightarrow {\mathcal Hom}(\mathscr {O}_\Omega /(\mathcal {J}+\mathcal {I}),\mathcal {{ CH}}^{\{0\}}_\Omega ) \end{aligned}$$

is given by $\eta \mapsto \eta R^{E\otimes F}_N$.

Let $\mu ^1,\ldots ,\mu ^m$ be a minimal set of generators for ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )$ at 0. Then $\mu ^j=\xi ^jR^E_p$, where $\xi ^j$ is a minimal set of generators for ${\mathcal H}^p({\mathcal Hom}(E_\bullet ,K_\Omega ))$. Notice that

$$\begin{aligned} {\mathcal H}^N({\mathcal Hom}((E\otimes F)_\bullet ,K_\Omega ))={\mathcal H}^p({\mathcal Hom}(E_\bullet ,K_\Omega )) \otimes _{\mathscr {O}} {\mathcal H}^n({\mathcal Hom}(F_\bullet ,\mathscr {O}_\Omega )). \end{aligned}$$

Since ${\mathcal H}^n({\mathcal Hom}(F_\bullet ,\mathscr {O}_\Omega ))$ is generated by 1, it follows that ${\mathcal H}^N({\mathcal Hom}( (E\otimes F)_\bullet ,K_\Omega ))$ is generated by $\xi ^j\otimes 1$. We conclude that ${\mathcal Hom}(\mathscr {O}_\Omega /(\mathcal {J}+\mathcal {I}),\mathcal {{ CH}}^{\{0\}}_\Omega )$ is generated by $\xi ^j\otimes 1\cdot R^E_p\wedge R^F_n=\mu ^j\wedge \mu ^z$, $j=1,\ldots ,m$, where $R^F_n = \mu ^z = \bar{\partial }(1/z^{\mathbf{1}})$.

If $1, \ldots , w^{\alpha _{\nu -1}}$ is a basis for $\mathscr {O}_\Omega /\mathcal {J}$ as an $\mathscr {O}_Z$-module, then it is also a basis for $\mathscr {O}_{X_0} := \mathscr {O}_\Omega /(\mathcal {J}+\mathcal {I})$ as a module over $\mathscr {O}_{\{0\}}\simeq \mathbb {C}$. Since $\phi \bar{\partial }(1/z^{\mathbf{1}}) = \phi (0,\cdot )\bar{\partial }(1/z^{\mathbf{1}})$ we have that

$$\begin{aligned} \phi (z,w)\mu ^j \wedge \mu ^z= & {} \phi (z,w)\sum a^j_\ell (z)\bar{\partial }\frac{1}{w^{\ell +\mathbf{1}}}\wedge \bar{\partial }\frac{1}{z^{\mathbf{1}}}\nonumber \\= & {} \phi (0,w)\sum a^j_\ell (0)\bar{\partial }\frac{1}{w^{\ell +\mathbf{1}}}\wedge \bar{\partial }\frac{1}{z^{\mathbf{1}}}. \end{aligned}$$

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.

The intrinsic sheaf $\mathcal {W}_X^{0,*}$ on X

Our aim is to find a fine resolution of $\mathscr {O}_X$ 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 $\mathcal {W}_X^{0,*}\supset \mathscr {E}_X^{0,*}$ 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

$$\begin{aligned} \phi = \widehat{\phi }_0 \otimes 1 + \widehat{\phi }_1 \otimes w^{\alpha _1} + \cdots + \widehat{\phi }_{\nu -1} \otimes w^{\alpha _{\nu -1}}, \end{aligned}$$

(7.1)

where $\widehat{\phi }_j$ are in $\mathcal {W}^{0,*}_Z$. 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 $\mathcal {W}_Z^{0.*}$ 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.

Lemma 7.1

If $\phi$ has the form (7.1) and $\tau$ is in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )$, expressed in the form (2.11), then

$$\begin{aligned} \phi \wedge \tau := \sum _i \sum _{\gamma \ge \alpha _i} \widehat{\phi }_i \wedge \tau _\gamma \wedge dz \otimes \bar{\partial }\frac{dw}{w^{\gamma -\alpha _i+\mathbf{1}}} \end{aligned}$$

(7.2)

is in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )$.

Proof

The right hand side defines a current in $\mathcal {W}_\Omega ^Z$ since $\widehat{\phi }_i$ are in $\mathcal {W}^{0,*}_Z$ and $\tau _{\gamma }$ are in $\mathscr {O}_Z$. We have to prove that it is annihilated by $\mathcal {J}$. Take $\xi$ in $\mathcal {J}$. On the subset of 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

$$\begin{aligned} \xi \phi \wedge \tau =\sum _{\ell \ge 0} a_\ell (z)\wedge dz\otimes \bar{\partial }\frac{dw}{w^{\ell +\mathbf{1}}}, \end{aligned}$$

with $a_{\ell }$ in $\mathcal {W}_Z^{0,*}$. Since $a_\ell$ vanish on the set where all $\widehat{\phi }_j$ are smooth, we conclude from Proposition 2.9 that $a_\ell$ vanish identically. It follows that $\xi \phi \wedge \tau = 0$. $\square$

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 $\phi \wedge h$.

Since $\mathcal {W}_X^{n,*}$ are $\mathscr {E}^{0,*}_X$-modules, $\mathcal {W}_X^{0,*}$ 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,

$$\begin{aligned} \mathscr {O}_X \rightarrow {\mathcal Hom}\left( \omega ^n_X,\omega ^n_X\right) , \quad \phi \mapsto (h\mapsto \phi h). \end{aligned}$$

(7.3)

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 $X_{reg}$.

Theorem 7.3 is a consequence of the results in [22]. If X has pure dimension p, there is an injective mapping

$$\begin{aligned} \mathscr {O}_X \rightarrow {\mathcal Hom}\left( {\mathcal Ext\,}^p(\mathscr {O}_X,K_\Omega ),\mathcal {{ CH}}^Z_\Omega \right) , \end{aligned}$$

(7.4)

which by [22, Theorem 1.2 and Remark 6.11] is an isomorphism if and only if $\mathscr {O}_X$ is $S_2$. Since the image of such a morphism must be annihilated by $\mathcal {J}$ by linearity, it is indeed a morphism

$$\begin{aligned} \mathscr {O}_X \rightarrow {\mathcal Hom}\left( {\mathcal Ext\,}^p(\mathscr {O}_X,K_\Omega ),{\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}^Z_\Omega )\right) . \end{aligned}$$

(7.5)

In view of (4.2) and (5.3), (7.5) corresponds to a morphism $\mathscr {O}_X \rightarrow {\mathcal Hom}(\omega ^n_X,\omega ^n_X)$, 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

$$\begin{aligned} \mathscr {O}_\Omega /\mathcal {J}\rightarrow {\mathcal Ext\,}^p\left( {\mathcal Ext\,}^p(\mathscr {O}_\Omega /\mathcal {J},K_\Omega ),K_\Omega \right) ; \end{aligned}$$

(7.6)

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 $\omega ^n_X$ 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

$$\begin{aligned} \mathcal {W}^{0,*}_X \rightarrow {\mathcal Hom}\left( \omega ^n_X,\mathcal {W}^{n,*}_X\right) . \end{aligned}$$

(7.7)

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. $\square$

In view of (5.1) and (5.3) we have, given a local embedding $i:X\rightarrow \Omega$, the extrinsic representation

$$\begin{aligned} \mathcal {W}_X^{0,*}\simeq {\mathcal Hom}\left( {\mathcal Hom}\left( \mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z\right) , {\mathcal Hom}\left( \mathscr {O}_\Omega /\mathcal {J},\mathcal {W}_\Omega ^Z\right) \right) , \phi \mapsto (i_* h\mapsto i_*(\phi \wedge h)).\nonumber \\ \end{aligned}$$

(7.8)

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

From (4.9) we get an induced sequence

$$\begin{aligned} 0\rightarrow \left( \mathcal {W}_Z^{0,*}\right) ^\nu \xrightarrow {T}\left( \mathcal {W}_Z^{0,*}\right) ^M\xrightarrow {A}\left( \mathcal {W}_Z^{0,*}\right) ^{M'}, \end{aligned}$$

(7.9)

which is also exact. In fact, 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 $\phi$.

In view of Proposition 2.9 it is enough to prove the claim where $\tilde{\phi }$ is smooth. Let us therefore fix such a point, say 0, and show that $(A\tilde{\phi })(0) = 0$. From the proof of Lemma 4.11, if we let $\mathcal {I}$ be the ideal generated by 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

$$\begin{aligned} 0 \rightarrow \mathbb {C}^\nu {\mathop {\rightarrow }\limits ^{T(0)}} \mathbb {C}^M {\mathop {\rightarrow }\limits ^{A(0)}} \mathbb {C}^{M'}. \end{aligned}$$

Since $X_0$ is 0-dimensional, the morphism $\mathscr {O}_{X_0} \rightarrow {\mathcal Hom}(\omega _{X_0},\omega _{X_0})$ is an isomorphism by Theorem 7.3, and thus $\phi _0$ is given as multiplication by a function in $\mathscr {O}_{X_0}$, which we also denote by $\phi _0$, i.e., $\tilde{\phi }_0 = T(0) \hat{\phi }_0$. Hence, $A(0) \tilde{\phi }_0 = A(0) T(0) \hat{\phi }_0 = 0$, and thus $(A \tilde{\phi })(0) = 0$. $\square$

Example 7.6

(Meromorphic functions) Assume that we have a local embedding $X\rightarrow \Omega$. Given meromorphic functions $\Phi ,\Phi '$ in $\Omega$ that are holomorphic generically on 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

$$\begin{aligned} \mathcal {M}_X\subset \mathcal {W}_X^{0,*}. \end{aligned}$$

To see this, first notice that if we take a representative $\Phi$ in $\mathcal {M}_\Omega$ of $\phi$, then it can be considered as an almost semi-meromorphic current on $\Omega$ 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 $\Phi \wedge h$ 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

The uniqueness follows by $\mathscr {E}^{0,*}_X$-linearity, i.e., if $b = \xi _1 \wedge h_1 + \cdots + \xi _r \wedge h_r$ is in $\mathscr {E}^{0,*}_X \omega ^n_X$, then one must have

$$\begin{aligned} \phi b= \sum _i (-1)^{(\deg \xi _i)(\deg \phi )} \xi _i \wedge \phi h_i. \end{aligned}$$

(7.10)

We must check that this is well-defined, i.e., that the right hand side does not depend on the representation $\xi _1 \wedge h_1 + \cdots + \xi _r \wedge h_r$ of 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

Let $\phi$ be a current in $\mathcal {W}^{0,*}_X$ and let $\alpha$ be a current in $\mathcal {W}^{n,*}_X$ of the form $\alpha = \sum a_i \wedge h_i$, where $a_i$ are almost semi-meromorphic $(0,*)$-currents on $\Omega$ which are generically smooth on Z, and $h_i$ are in $\omega ^n_X$. Then one has a well-defined product

$$\begin{aligned} \phi \wedge \alpha = \sum (-1)^{(\deg a_i) (\deg \phi )} a_i \wedge (\phi \wedge h_i). \end{aligned}$$

(7.11)

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

Let $\tau$ be any current in $\mathcal {W}_Z$. It suffices to prove that if $\zeta$ are local coordinates on Z, then $\partial \tau /\partial \zeta _1$ is in $\mathcal {W}_Z$. Consider the current

$$\begin{aligned} \tau '=\tau \otimes \bar{\partial }\frac{dw}{2\pi i w^2} \end{aligned}$$

on the manifold $Y := Z \times \mathbb {C}_w$. Clearly $\tau '$ has support on Z, and it follows from (2.5) that $\tau '$ is in $\mathcal {W}^Z_Y$. Let

$$\begin{aligned}p : (z,w) \mapsto \zeta = (z_1+w,z_2,\ldots ,z_n),\end{aligned}$$

which is just a change of variables on Y followed by a projection. It follows from (2.4) that $p_* \tau '$ is in $\mathcal {W}_Z$. Since

$$\begin{aligned} \bar{\partial }\frac{dw}{2\pi i w^2}\cdot \xi (w)=\frac{\partial \xi }{\partial w}(0) \end{aligned}$$

it is readily verified that $p_*\tau '=\partial \tau /\partial \zeta _1$, so we conclude that $\partial \tau /\partial \zeta _1$ is in $\mathcal {W}_Z$. $\square$

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

Assume that $\phi ,v$ are in $\mathcal {W}_X^{0,*}$, We say that $\bar{\partial }v=\phi$ if

$$\begin{aligned} \bar{\partial }(v \wedge h)=\phi \wedge h,\quad h\in \omega _X^n. \end{aligned}$$

(8.1)

If we have an embedding $X\rightarrow \Omega$, (8.1) means, cf., (7.8), that

$$\begin{aligned} \bar{\partial }(v\wedge \mu )=\phi \wedge \mu ,\quad \mu \in {\mathcal Hom}\left( \mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z\right) . \end{aligned}$$

(8.2)

In view of Remark 7.9 we can define the product $\phi \wedge \omega$ for $\phi$ in $\mathcal {W}^{0,*}_X$.

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

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

$$\begin{aligned} \bar{\partial }(v\wedge \omega ) = \phi \wedge \omega + (-1)^{\deg v} v \wedge \bar{\partial }\omega . \end{aligned}$$

Since $\bar{\partial }\omega = f \omega$, and $\mathcal {W}_X^{n,*}$ is closed under multiplication with forms in $\mathscr {E}_X^{0,*}$, we get that $\bar{\partial }(v\wedge \omega )$ is in $\mathcal {W}^{n,*}_X$, so v is in $\text {Dom}\,\bar{\partial }_X$ and $\bar{\partial }_Xv=\phi$.

If w is in $\text {Dom}\,\bar{\partial }_X$ and v is in $\mathscr {E}_X^{0,*}$, then

$$\begin{aligned} \bar{\partial }( v \wedge w \wedge \omega ) = \bar{\partial }v \wedge w \wedge \omega + (-1)^{\deg v} v \wedge \bar{\partial }(w \wedge \omega ). \end{aligned}$$

Thus $v \wedge w$ is in $\text {Dom}\,\bar{\partial }_X$, and the Leibniz rule $\bar{\partial }(v \wedge w) = \bar{\partial }v \wedge w + (-1)^{\deg v} v \wedge \bar{\partial }w$ holds.$\square$

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

Lemma 8.4

If 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

$$\begin{aligned} \bar{\partial }\chi _\delta \wedge v\wedge \omega \rightarrow 0, \quad \delta \rightarrow 0, \end{aligned}$$

(8.3)

for all structure forms $\omega$. In this case,

$$\begin{aligned} -\nabla _f(v\wedge \omega ) = \bar{\partial }v \wedge \omega . \end{aligned}$$

(8.4)

Proof

Since $\mathcal {W}^{n,*}_X$ is closed under multiplication by 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

$$\begin{aligned} \mathbf{1}_{X_\mathrm{reg}} \nabla _f(v \wedge \omega ) = \nabla _f(v \wedge \omega ). \end{aligned}$$

(8.5)

By the Leibniz rule,

$$\begin{aligned} \nabla _f(\chi _\delta v \wedge \omega ) = -\bar{\partial }\chi _\delta \wedge v \wedge \omega + \chi _\delta \nabla _f (v \wedge \omega ). \end{aligned}$$

(8.6)

Since 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 smoothh 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

Assume that we have a local embedding $X_{reg}\rightarrow \Omega$ and $\phi ,v$ in $\mathcal {W}_X^{0,*}$ represented as in (7.1). Then $\bar{\partial }v=\phi$ if and only if

$$\begin{aligned} \bar{\partial }{\hat{v}}_j=\hat{\phi }_j, \quad j=0,\ldots ,\nu -1. \end{aligned}$$

(8.7)

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$

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

There are integral operators

$$\begin{aligned} K:\mathscr {E}^{0,*+1}(X)\rightarrow \mathcal {W}^{0,*}(X')\cap \text {Dom}\,\bar{\partial }_X, \quad P:\mathscr {E}^{0,*}(X)\rightarrow \mathscr {E}^{0,*}(X') \end{aligned}$$

such that, for $\phi \in \mathscr {E}^{0,k}(X)$,

$$\begin{aligned} \phi = \bar{\partial }K\phi +K(\bar{\partial }\phi ) + P\phi . \end{aligned}$$

(9.1)

The operators 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.,

$$\begin{aligned} \phi = \bar{\partial }K\phi + K(\bar{\partial }\phi ), \quad \phi \in \mathscr {E}^{0,k}(X), \quad k \ge 1. \end{aligned}$$

(9.2)

If $\bar{\partial }\phi =0$ in $\Omega$, and $k\ge 1$, then $K\phi$ is a solution to $\bar{\partial }v=\phi$. If $k=0$, then $\phi =P\phi$ is holomorphic. It follows that a smooth $\bar{\partial }$-closed function is holomorphic. In the reduced case this is a classical theorem of Malgrange [24]. In Sect. 10 we prove that $K\phi$ is smooth on $X_{reg}$.

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 form³H, 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 $d\zeta _i$.

If $\gamma$ is a current in $\Omega _\zeta \times \Omega '_z$ we let $(\gamma )_N$ be the component of bidegree $(N,*)$ in $\zeta$ and $(0,*)$ in z, and let $\vartheta (\gamma )$ be the current such that

$$\begin{aligned} \vartheta ( \gamma ) \wedge d\zeta = (\gamma )_N. \end{aligned}$$

(9.3)

Consider now $\mu$ in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {W}^Z_\Omega )$ and $\phi$ in $\mathcal {W}^{0,*}_X$. We can give meaning to

$$\begin{aligned} (g \wedge HR(\zeta ))_N\wedge \phi (\zeta ) \wedge \mu (z) \end{aligned}$$

(9.4)

as a tensor product of currents in the following way: first of all, by Remark 7.9, we can form the product $R(\zeta ) \wedge d\zeta \wedge \phi (\zeta )$ as a current in $\mathcal {W}^Z_\Omega$. In view of [11, Corollary 4.7] the tensor product $R(\zeta ) \wedge d\zeta \wedge \phi (\zeta ) \wedge \mu (z)$ is in $\mathcal {W}^{Z \times Z'}_{\Omega _\zeta \times \Omega _z'}$, where $Z' = Z \cap \Omega '$. Finally, we multiply this with the smooth form $\vartheta (g \wedge H)$ to obtain (9.4). Similarly, outside of $\Delta$, the diagonal in $\Omega \times \Omega '$, where B is smooth, we can define

$$\begin{aligned} \left( B\wedge g \wedge HR(\zeta )\right) _N\wedge \phi (\zeta ) \wedge \mu (z) \end{aligned}$$

(9.5)

as a tensor product of currents.

Lemma 9.2

For $\mu$ in ${\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {W}^{Z'}_{\Omega '})$ and $\phi \in \mathcal {W}^{0,*}(X)$, the current (9.5), a priori defined as a current in $\mathcal {W}^{Z \times Z'{\setminus } \Delta }_{\Omega _\zeta \times \Omega _z'{\setminus } \Delta }$ has an extension across $\Delta$. The current (9.4) and the extension of (9.5) depend $\mathscr {O}_\Omega /\mathcal {J}$-bilinearly on $\mu$ and $\phi$, and are such that

$$\begin{aligned} K\phi \wedge \mu := \pi _*\big ( (B\wedge g \wedge HR(\zeta ))_N\wedge \phi (\zeta ) \wedge \mu (z)\big ) \end{aligned}$$

(9.6)

and

$$\begin{aligned} P\phi \wedge \mu := \pi _*\big ( (g \wedge HR(\zeta ))_N\wedge \phi (\zeta ) \wedge \mu (z) \big ) \end{aligned}$$

(9.7)

are in ${\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {W}^{Z'}_{\Omega '})$.

It follows that $K\phi \wedge \mu$ and $P\phi \wedge \mu$ are $\mathbb {C}$-linear in $\phi$ and $\mathscr {O}_{\Omega '}/\mathcal {J}$-linear in $\mu$. In view of (7.8), by considering $\mu$ in ${\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {{ CH}}^{Z'}_{\Omega '})$, we have defined linear operators

$$\begin{aligned} K :\mathcal {W}^{0,*+1}(X) \rightarrow \mathcal {W}^{0,*}(X'), \quad P :\mathcal {W}^{0,*}(X) \rightarrow \mathcal {W}^{0,*}(X'). \end{aligned}$$

(9.8)

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

Since $\vartheta (g\wedge H)$ is smooth, we get that

$$\begin{aligned}&\pi _*\big (\vartheta (g\wedge H)\wedge R(\zeta ) \wedge d\zeta \wedge \phi \wedge \mu (z)\big )\\&\quad =\pi _*\big (\vartheta (g\wedge H)\wedge R(\zeta ) \wedge d\zeta \wedge \phi \big ) \wedge \mu (z)= \pi _*\big ((g\wedge HR)_N\wedge \phi \big ) \wedge \mu (z), \end{aligned}$$

cf., for example [20, (5.1.2)]. Thus $P\phi (z) = \pi _* \big ( (g \wedge HR(\zeta ))_N \wedge \phi \big )$ which is smooth on $\Omega '$. If 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

For any $\phi \in \mathcal {W}^{0,k}(X)$, $k\ge 1$,

$$\begin{aligned} K^\epsilon \phi := \pi _* \big ( (B^\epsilon \wedge g \wedge HR(\zeta ))_N\wedge \phi \big ) =\pi _* \big ( \vartheta (B^\epsilon \wedge g\wedge H)\wedge R(\zeta ) \wedge d\zeta \wedge \phi \big ) \end{aligned}$$

is in $\mathscr {E}^{0,k-1}(X')$ and $K^\epsilon \phi \rightarrow K\phi$ when $\epsilon \rightarrow 0$.

The last statement means that

$$\begin{aligned} K^\epsilon \phi \wedge \mu \rightarrow K\phi \wedge \mu , \quad \mu \in {\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J}, \mathcal {{ CH}}_{\Omega '}^{Z'}). \end{aligned}$$

(9.9)

Proof

Since $B^\epsilon$ is smooth, the current we push forward is $R(\zeta )\wedge \phi (\zeta )$ times a smooth form of $\zeta$ and z. Therefore $K^\epsilon \phi$ is smooth. As in the proof of Proposition 9.3, we obtain since $B^\epsilon$ is smooth that

$$\begin{aligned} K^\epsilon \phi \wedge \mu = \pi _*\big ((B^\epsilon \wedge g \wedge HR(\zeta ))_N \wedge \phi \wedge \mu (z)\big ). \end{aligned}$$

(9.10)

By (5.2) applied to $a=B$ we have that

$$\begin{aligned} (B^\epsilon \wedge g \wedge HR(\zeta ))_N \wedge \phi \wedge \mu (z) \rightarrow (B\wedge g \wedge HR(\zeta ))_N \wedge \phi \wedge \mu (z) \end{aligned}$$

(9.11)

which implies (9.9). $\square$

Proof of Theorem 9.1

By definition $K\phi$ and $P\phi$ are currents in $\mathcal {W}^{0,*}(X')$ such that (9.6) and (9.7) hold for $\mu$ in ${\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {{ CH}}^{Z'}_{\Omega '})$. We claim that

$$\begin{aligned} K\phi \wedge R\wedge dz=\pi _*\big ((B\wedge g \wedge HR(\zeta ))_N\wedge \phi \wedge R(z)\wedge dz\big ) \end{aligned}$$

(9.12)

and

$$\begin{aligned} P\phi \wedge R\wedge dz=\pi _*\big ((g \wedge HR(\zeta ))_N\wedge \phi \wedge R(z)\wedge dz\big ); \end{aligned}$$

(9.13)

here the left hand sides are defined in view of Remark 7.9, whereas the right hand sides have meaning by Lemma 9.2 and the fact that $R(z) \wedge dz$ is in ${\mathcal Hom}(\mathscr {O}_{\Omega '}/\mathcal {J},\mathcal {W}^{Z'}_{\Omega '})$ by Corollary 6.3.

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 current⁴ when $\lambda =0$. Let $B_{k,k-1}$ be the component of B of bidegree $(k,k-1)$.

Lemma 9.5

For all k,

$$\begin{aligned} B_{k,k-1} \wedge HR^\lambda (\zeta ) \wedge R(z) |_{\lambda =0}= B_{k,k-1} \wedge HR(\zeta )\wedge R(z). \end{aligned}$$

(9.14)

Proof of Lemma 9.5

Notice that the equality holds outside $\Delta$. Let 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

$$\begin{aligned} T_{\ell }=B_{k,k-1} \wedge HR_j^\lambda (\zeta )\wedge R_\ell (z)|_{\lambda =0}. \end{aligned}$$

Clearly $T_\ell =0$ if $\ell < p$ so first assume that $\ell =p$. Since $HR_j$ has bidegree (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. $\square$

By the same argument⁵ as for [6, (5.2)] we have the equality

$$\begin{aligned} \nabla _{f(z)}\big ((B \wedge g \wedge HR^\lambda (\zeta ))_N \wedge R(z)\wedge dz\big )= [\Delta ]'\wedge R(z)\wedge dz-(g \wedge HR^\lambda )_N \wedge R(z)\wedge dz , \end{aligned}$$

(9.15)

also for our R, where $[\Delta ]'$ denotes the part of $[\Delta ]$ where $d\eta _i = d\zeta _i - dz_i$ has been replaced⁶ by $d\zeta _i$. In view of (9.14) we can put $\lambda =0$ in (9.15), and then we get

$$\begin{aligned} \nabla _{f(z)}\big ((B \wedge g \wedge HR(\zeta ))_N \wedge R(z)\wedge dz\big )= [\Delta ]' \wedge R(z)\wedge dz-(H R(\zeta )\wedge g)_N \wedge R(z)\wedge dz. \end{aligned}$$

(9.16)

Multiplying (9.16) by the smooth form $\phi$, and using (9.12) and (9.13), we get

$$\begin{aligned} \phi \wedge R\wedge dz=-\nabla _f( K\phi \wedge R\wedge dz)+ K(\bar{\partial }\phi )\wedge R\wedge dz+ P\phi \wedge R\wedge dz, \end{aligned}$$

or equivalently,

$$\begin{aligned} \phi \wedge \omega =-\nabla _f(K\phi \wedge \omega )+K(\bar{\partial }\phi )\wedge \omega + P\phi \wedge \omega . \end{aligned}$$

(9.17)

Multiplying by suitable holomorphic $\xi _0$ in $E_p^*$ such that $f^*_{p+1}\xi _0=0$, cf., Proposition 6.7, we see that $\phi \wedge h=\bar{\partial }(K\phi \wedge h)+K(\bar{\partial }\phi )\wedge h+ P\phi \wedge h$ for all 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 $\mathcal {W}^{0,*}_X$.

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 $X\times X'$. 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

$$\begin{aligned} i_* p_* = \pi _* (i \times i)_*. \end{aligned}$$

(9.18)

As before we have the isomorphism

$$\begin{aligned} (i\times i)_*:\mathcal {W}^{2n,*}_{X\times X'} \simeq {\mathcal Hom}\left( \mathscr {O}_{\Omega \times \Omega '}/(\mathcal {J}_X+\mathcal {J}_{X'}), \mathcal {W}_{\Omega \times \Omega '}^{Z\times Z'}\right) . \end{aligned}$$

As in the proof of Lemma 9.2 we see that $\pi _*$ maps a current in $\mathcal {W}_{\Omega \times \Omega '}^{Z\times Z'}$ annihilated by $\mathcal {J}_{X'}$ to a current in ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J}, \mathcal {W}_{\Omega '}^{Z'})$. It follows by (9.18) that

$$\begin{aligned} p_*:\mathcal {W}_{X\times X'}^{2n, *+n}\rightarrow \mathcal {W}_{X'}^{n,*}. \end{aligned}$$

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

$$\begin{aligned} (B\wedge g \wedge HR(\zeta ))_N\wedge \phi (\zeta ) \wedge \mu (z)= (i\times i)_* \big (\vartheta ( B\wedge g \wedge H) \wedge \omega (\zeta ) \wedge \phi (\zeta ) \wedge h\big ). \end{aligned}$$

Thus we can define $K\phi$ intrinsically by

$$\begin{aligned} K\phi \wedge h= p_*\left( \vartheta (B \wedge g\wedge H) \wedge \omega (\zeta ) \wedge \phi (\zeta ) \wedge h(z)\right) . \end{aligned}$$

(9.19)

From above it follows that $K\phi \wedge h$ is in $\mathcal {W}_{X'}^{n,*}$. In the same way we can define $P\phi$ by

$$\begin{aligned} P\phi \wedge h= p_*\left( \vartheta ( g\wedge H) \wedge \omega (\zeta ) \wedge \phi (\zeta ) \wedge h(z)\right) . \end{aligned}$$

(9.20)

It is natural to write

$$\begin{aligned} K\phi (z)=\int _\zeta \vartheta (B \wedge g \wedge H) \wedge \omega (\zeta )\wedge \phi (\zeta ), \quad P\phi (z)=\int _\zeta \vartheta (g \wedge H)\wedge \omega (\zeta ) \wedge \phi (\zeta ), \end{aligned}$$

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 $P\phi$ 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

Let $B^\epsilon :=\chi (|\zeta -z|^2/\epsilon ) B$, and assume that $\phi$ has compact support in our coordinate neighborhood. Then $K\phi$ can be approximated by the smooth forms

$$\begin{aligned} K^\epsilon \phi := \pi _* \big ( (B^\epsilon \wedge g \wedge HR)_N\wedge \phi \big ). \end{aligned}$$

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

Let $X=\mathbb {C}_\zeta \subset \mathbb {C}^2_{\zeta ,{\tau }}$ and $\mathcal {J}=({\tau }^{m+1})$. Then $R=\bar{\partial }(1/{\tau }^{m+1})$. For an arbitrary point (z, w) we can choose the Hefer form

$$\begin{aligned} H=\frac{1}{2\pi i}\sum _{j=0}^m{\tau }^{m-k}w^k d{\tau }. \end{aligned}$$

From the Bochner–Martinelli form B we only get a contribution from the term

$$\begin{aligned} B_1=\frac{1}{2\pi i}\frac{(\bar{\zeta }-{\bar{z}})d\zeta +(\bar{{\tau }}-\bar{w})d{\tau }}{|\zeta -z|^2+|{\tau }-w|^2}. \end{aligned}$$

Let $\Omega '\subset \subset \Omega$ be open balls with center at the origin, and let $\varphi =\varphi (|\zeta |^2+|{\tau }|^2)$ be a smooth cutoff function with support in $\Omega$ that is $\equiv 1$ in a neighborhood of $\overline{\Omega '}$. Then we can choose a holomorphic weight $g=\varphi +\cdots$, see, [6, Example 5.1] with respect to $\Omega '$, and with support in $\Omega$. Now $1,{\tau },\ldots ,{\tau }^m$ is a set of generators for $\mathscr {O}_X$ over $\mathscr {O}_Z$. Assume that

$$\begin{aligned} \phi =(\hat{\phi }_0(\zeta )\otimes 1+\cdots +\hat{\phi }_m(\zeta )\otimes {\tau }^m)d\bar{\zeta } \end{aligned}$$

is a smooth (0, 1)-form. We want to compute $K\phi$. We know that

$$\begin{aligned} K\phi =a_0(z)\otimes 1+\cdots +a_m(z)\otimes w^m \end{aligned}$$

(10.1)

with $a_k(z)$ in $\mathcal {W}_Z^{0,0}$. By Lemma 10.2 and its proof, we have smooth $K^\epsilon \phi (z,w)$ in $\Omega '$ such that

$$\begin{aligned} K^\epsilon \phi \wedge dz \wedge dw \wedge \bar{\partial }\frac{1}{w^{m+1}}\rightarrow K\phi \wedge dz \wedge dw \wedge \bar{\partial }\frac{1}{w^{m+1}}. \end{aligned}$$

(10.2)

It follows that

$$\begin{aligned} a_k(z)=\lim _{\epsilon \rightarrow 0} \frac{1}{k!}\frac{\partial ^k}{\partial w^k} K^\epsilon \phi (z,w)\big |_{w=0}. \end{aligned}$$

Notice that

$$\begin{aligned} (B\wedge g \wedge HR({\tau }))_2= & {} B_1\wedge g_{0,0} \wedge H\wedge \bar{\partial }\frac{1}{{\tau }^{m+1}} \\= & {} - \varphi \bar{\partial }\frac{1}{{\tau }^{m+1}}\wedge \frac{1}{(2\pi i)^2}\sum _{\ell =0}^m{\tau }^{m-\ell }w^\ell d{\tau }\wedge \frac{(\bar{\zeta }-{\bar{z}})d\zeta +(\bar{{\tau }}-{\bar{w}})d{\tau }}{|\zeta -z|^2+|{\tau }-w|^2} \\= & {} - \varphi \bar{\partial }\frac{d{\tau }}{{\tau }^{m+1}}\wedge \frac{1}{(2\pi i)^2}\sum _{\ell =0}^m{\tau }^{m-\ell }w^\ell \wedge \frac{(\bar{\zeta }-\bar{z})d\zeta }{|\zeta -z|^2+|{\tau }-w|^2}. \end{aligned}$$

For each fixed $\epsilon >0$, $|\zeta -z|> 0 \text { on } {{\mathrm{supp}}}\chi _\epsilon$, cf., Lemma 10.2, so we have

$$\begin{aligned}&K^\epsilon \phi (z,w)\nonumber \\&\quad =\int _{\zeta ,{\tau }} \varphi \frac{1}{(2\pi i)^2}\sum _{\ell =0}^m \bar{\partial }\frac{d{\tau }}{{\tau }^{\ell +1}} \wedge w^\ell \chi _\epsilon \frac{(\bar{\zeta }-{\bar{z}})d\bar{\zeta }\wedge d\zeta }{|\zeta -z|^2+|{\tau }-w|^2}\wedge \sum _{k=0}^m \hat{\phi }_k(\zeta )\otimes {\tau }^k.\nonumber \\ \end{aligned}$$

(10.3)

A simple computation yields that

$$\begin{aligned} K^\epsilon \phi (z,w)=\sum _{k=0}^m a_k^\epsilon (z)\otimes w^k+\mathscr {O}({\bar{w}}), \end{aligned}$$

(10.4)

where

$$\begin{aligned} a_k^\epsilon (z)=\frac{1}{2\pi i}\int _\zeta \varphi (|\zeta |^2)\chi _\epsilon \frac{\hat{\phi }_k(\zeta )d\bar{\zeta }\wedge d\zeta }{\zeta -z}. \end{aligned}$$

Letting $\epsilon$ tend to 0 we get $K\phi$ as in (10.1), where

$$\begin{aligned} a_k(z)=\frac{1}{2\pi i}\int _\zeta \varphi (|\zeta |^2)\frac{\hat{\phi }_k(\zeta )d\bar{\zeta }\wedge d\zeta }{\zeta -z}. \end{aligned}$$

It is well-known that these Cauchy integrals $a_k(z)$ are smooth solutions to $\bar{\partial }v=\hat{\phi }_k d{\bar{z}}$ in $Z'=Z\cap \Omega '$. Thus $K\phi$ is smooth.$\square$

Remark 10.4

The terms $\mathscr {O}({\bar{w}})$ in the expansion (10.4) of $K^\epsilon \phi (z,w)$ do 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

$$\begin{aligned} w^m\int \varphi (|\zeta |^2)\frac{1}{2\pi i}\frac{|\zeta |^2d\bar{\zeta }\wedge d\zeta }{|\zeta |^2+|w|^2} \end{aligned}$$

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

$$\begin{aligned} K^\epsilon \phi (z,w)=\sum _{k=0}^m\int _\zeta \varphi (|\zeta |^2)\chi _\epsilon \frac{1}{2\pi i}\frac{(\bar{\zeta }-{\bar{z}})\hat{\phi }_k(\zeta ) d\bar{\zeta }\wedge d\zeta }{|\zeta -z|^2+|w|^2} w^k\sum _{j=0}^{m-k}\left( \frac{|w|^2}{|\zeta -z|^2+|w|^2}\right) ^j. \end{aligned}$$

It is now clear that we can let $\epsilon \rightarrow 0$. By a simple computation we then get

$$\begin{aligned} K\phi (z,w)= & {} \sum _{k=0}^m C\hat{\phi }_k(z)\otimes w^k\\&- \sum _{k=0}^m\int _\zeta \varphi (|\zeta |^2) \frac{1}{2\pi i}\frac{\hat{\phi }_k(\zeta )d\bar{\zeta }\wedge d\zeta }{\zeta -z} w^k\left( \frac{|w|^2}{|\zeta -z|^2+|w|^2}\right) ^{m-k+1}. \end{aligned}$$

Let $\psi =\varphi \hat{\phi }_k$. Then the kth term in the second sum is equal to

$$\begin{aligned} b(z,w)=\frac{1}{2\pi i}\int _\zeta \frac{\psi (z+\zeta )d\bar{\zeta }\wedge d\zeta }{\zeta } w^k \left( \frac{|w|^2}{|\zeta |^2+|w|^2}\right) ^{m-k+1}. \end{aligned}$$

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 $\psi (z+\zeta ) =\mathscr {O}(|\zeta |^M)$ for a large 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

$$\begin{aligned} \int _{|\zeta |<1}\frac{\zeta ^\alpha \bar{\zeta }^\beta }{\zeta } \left( \frac{|w|^2}{|\zeta |^2+|w|^2}\right) ^{m-k+1}. \end{aligned}$$

For symmetry reasons, they vanish, except when $\alpha =\beta +1$. Thus we are left with

$$\begin{aligned} \int _{|\zeta |<1} |\zeta |^{2\beta } \left( \frac{|w|^2}{|\zeta |^2+|w|^2}\right) ^{m-k+1} w^k= C w^k|w|^{2(m-k+1)}\int _0^1 \frac{s^\beta ds}{(s+|w|^2)^{m-k+1}} \end{aligned}$$

for non-negative integers $\beta$. The right hand side is a smooth function of w if $\beta \le m-k-1$ and a smooth function plus

$$\begin{aligned} Cw^k|w|^{2(\beta +1)}\log |w|^2 \end{aligned}$$

if $\beta \ge m-k$. The worst case therefore is when $k=m$ and $\beta =0$; then we have $w^m|w|^{2}\log |w|^2$ that we encountered above.$\square$

Proposition 10.5

Let 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

$$\begin{aligned} v^\epsilon (z,w)=\int _\zeta \chi (|\zeta -z|^2/\epsilon )(HR\wedge B\wedge g)_N\wedge \phi , \end{aligned}$$

is smooth for $\epsilon >0$, and for each multiindex $\ell$ there is a smooth form $v_\ell$ such that

$$\begin{aligned} \partial ^\ell _w v^\epsilon |_{w=0}\rightarrow v_\ell \end{aligned}$$

as currents on 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.

Recall that given a minimal generating set $1, w^{\alpha _1},\ldots , w^{\alpha _{\nu -1}}$, one gets the coefficients ${\hat{v}}^\epsilon _j$ in the representation

$$\begin{aligned} v^\epsilon ={\hat{v}}_0^\epsilon \otimes 1+\cdots +\hat{v}_{\nu -1}^\epsilon \otimes w^{\alpha _{\nu -1}} \end{aligned}$$

from $\partial ^\ell _w v^\epsilon |_{w=0}$, $|\ell |\le M$ by a holomorphic matrix, cf., the proof of Lemma 4.7. It thus follows from Proposition 10.5 that there are smooth ${\hat{v}}_j$ such that ${\hat{v}}_j^\epsilon \rightarrow {\hat{v}}_j$ as currents on 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

Assume that 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

$$\begin{aligned} B=\sigma +\sigma \wedge \bar{\partial }\sigma +\sigma \wedge (\bar{\partial }\sigma )^2+\cdots , \end{aligned}$$

where

$$\begin{aligned} \sigma =\frac{(\bar{\zeta }'-{\bar{z}}')\cdot d\zeta '+(\bar{{\tau }}'-\bar{w}')\cdot d{\tau }'}{|\zeta '-z'|^2+|{\tau }'-w'|^2}. \end{aligned}$$

We now choose the new coordinates $\zeta =\zeta ',\ {\tau }={\tau }'-\psi (\zeta ')$ in $\Omega$, so that $Z=\{(\zeta ,{\tau });\ {\tau }=0\}$.

Recall that on $X_{reg}$ we have that $R\wedge dz$ is a smooth form times $\mu =(\mu _1,\ldots ,\mu _m)$, where $\mu _j$ is a generating set for ${\mathcal Hom}(\mathscr {O}_\Omega /\mathcal {J},\mathcal {{ CH}}_\Omega ^Z)$. Thus we are to compute $\partial _w^\ell |_{w=0}$ of integrals like

$$\begin{aligned} \int _{\zeta ,{\tau }}\bar{\partial }\frac{d{\tau }}{{\tau }^{\alpha +\mathbf{1}}}\wedge B^\epsilon _k\wedge \phi (\zeta ,z,w,{\tau }), \end{aligned}$$

(10.5)

where $k\le n$ and $\phi$ is smooth with compact support near x. It is clear that the symbols $\bar{{\tau }}, {\bar{w}}, d\bar{{\tau }}$ can be omitted in the expression for

$$\begin{aligned} B^\epsilon =\chi _\epsilon B=\chi (|\zeta -z|^2/\epsilon )B, \end{aligned}$$

since $\bar{{\tau }}$ and $d \bar{{\tau }}$ annihilate $\bar{\partial }(1/\tau ^{\alpha +1})$, and since we only take holomorphic derivatives with respect to w and set $w = 0$.

Let us write $\psi (\zeta )-\psi (z)=A(\zeta ,z)\eta$, where $\eta :=\zeta -z$ is considered as a column matrix and A is a holomorphic $(N-n)\times n$-matrix. Then

$$\begin{aligned} \sigma =\frac{\eta ^*\nu }{|\zeta -z|^2+|{\tau }-w+\psi (\zeta )-\psi (z)|^2}, \end{aligned}$$

where $\nu$ is the (1, 0)-form valued column matrix

$$\begin{aligned} \nu =d\zeta +A^*d(\tau + \psi (\zeta )). \end{aligned}$$

Since $\eta ^*\nu$ is a (1, 0)-form we have that

$$\begin{aligned} B^\epsilon _k=\chi _\epsilon \frac{\eta ^*\nu \wedge ( (d\eta ^*)\nu +\eta ^*\bar{\partial }\nu )^{k-1}}{(|\zeta -z|^2+|{\tau }-w+\psi (\zeta )-\psi (z)|^2)^k}. \end{aligned}$$

Lemma 10.6

Let

$$\begin{aligned} \xi ^i=\xi ^i_1\frac{\partial }{\partial \zeta _1}+\cdots + \xi ^i_n\frac{\partial }{\partial \zeta _n} \end{aligned}$$

be smooth (1, 0)-vector fields, and let $L_i=L_{\xi ^i}$ be the associated Lie derivatives for $i=1,\ldots ,\rho$. Let

$$\begin{aligned} \gamma _k:=\eta ^*\nu \wedge ((d\eta ^*)\nu +\eta ^*\bar{\partial }\nu )^{k-1}. \end{aligned}$$

If we have a modification $\pi :\tilde{W} \rightarrow \Omega \times \Omega$ such that locally $\pi ^*\eta =\eta _0\eta '$, where $\eta _0$ is a holomorphic function, then

$$\begin{aligned} \pi ^*(L_1 \cdots L_\rho \gamma _k)=\bar{\eta }_0^k \beta , \end{aligned}$$

where $\beta$ is smooth.

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

Introduce a nonsense basis e and its dual $e^*$ and consider the exterior algebra spanned by $e_j, e^*_\ell ,$ and the cotangent bundle. Let

$$\begin{aligned} c_\ell =\eta ^*e\wedge ((d\eta ^*) e)^{\ell -1}. \end{aligned}$$

Notice that $\gamma _k$ is a sum of terms like

$$\begin{aligned} (\nu e^*\lnot )^\ell c_\ell \wedge (\eta ^*\bar{\partial }\nu )^{k-\ell }. \end{aligned}$$

Since $L_i c_\ell =0$ and $L_i(\eta ^* b)=\eta ^*L_i b$ it follows after a finite number of applications of $L_i$’s that we get

$$\begin{aligned} (\nu _1 e^*)\lnot \cdots (\nu _\ell e^*)\lnot c_\ell (\eta ^*b_1)\cdots (\eta ^*b_{k-\ell }), \end{aligned}$$

where $\nu _j$ and $b_j$ are smooth. Since

$$\begin{aligned} \pi ^* c_\ell = \bar{\eta }_0^\ell (\eta ')^* e\wedge (d(\eta ')^* e)^{\ell -1}, \end{aligned}$$

the lemma now follows.$\square$

We note that $\eta ^*(I+A^*A)\eta = |\zeta -z|^2+|\psi (\zeta )-\psi (z)|^2$. Thus, differentiating (10.5) with respect to w, setting $w = 0$, and evaluating the residue with respect to $\tau$ using (2.10), we obtain a sum of integrals like

$$\begin{aligned} \int _\zeta \chi _\epsilon \frac{(\eta ^* a_1)\cdots (\eta ^* a_{t+1})\wedge \gamma _k\wedge \phi }{(\eta ^*(I+A^*A)\eta )^{k+t+1}}, \end{aligned}$$

where $a_1,\ldots ,a_{t+1}$ are column vectors of smooth functions. We must prove that the limit of such integrals when $\epsilon \rightarrow 0$ are smooth in z.

Lemma 10.7

Let

$$\begin{aligned} I_\ell ^{r,s}=\int \chi _\epsilon \frac{(\eta ^* a_1) \cdots (\eta ^*a_r)\mathscr {O}(|\eta |^{2s}) \tilde{\gamma }_k\wedge \phi }{\Phi ^{k+\ell }}, \end{aligned}$$

where $a_1,\ldots ,a_r$ are tuples of smooth functions, $\tilde{\gamma }_k=L_1 \cdots L_\rho \gamma _k$, where $L_i =L_{\xi _i}$ are Lie derivatives with respect to smooth (1, 0)-vector fields $\xi ^i$ as above for $i=1,\ldots ,\rho$, $\phi$ is a test form with support close to z, and $\Phi := \eta ^*(I+A^*A)\eta$. If $r\ge 1$ and $r+s \ge \ell + 1$, then we have the relation

$$\begin{aligned} I^{r,s}_{\ell +1}=I^{r-1,s}_{\ell }+I^{r-1,s+1}_{\ell +1}+I^{r,s-1}_{\ell }+o(1) \end{aligned}$$

(10.6)

when $\epsilon \rightarrow 0$.

Proof

If

$$\begin{aligned} \xi = a_r^t (I+A^*A)^{-t}\frac{\partial }{\partial \zeta }, \end{aligned}$$

and $L = L_\xi$, then using that $\Phi = \eta ^t (I+A^* A)^t \bar{\eta }$, one obtains that

$$\begin{aligned} L\Phi =\eta ^* a_r +\mathscr {O}(|\eta |^2). \end{aligned}$$

(10.7)

Thus

$$\begin{aligned} I_{\ell +1}^{r,s}=\int \chi _\epsilon (\eta ^* a_1) \cdots (\eta ^* a_{r-1}) \mathscr {O}(|\eta |^{2s}) \tilde{\gamma }_k\wedge \phi L \frac{1}{\Phi ^{k+\ell }}+I_{\ell +1}^{r-1,s+1} \end{aligned}$$

in view of (10.7). We now integrate by parts by 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

$$\begin{aligned} \int _{\zeta ,z}\chi '(|\eta |^2/\epsilon )\frac{(\eta ^*a_1) \cdots (\eta ^*a_{r-1}) (\eta ^*b)}{\epsilon } \frac{\mathscr {O}(|\eta |^{2s}) \gamma _k\wedge \phi }{\Phi ^{k+\ell }} \end{aligned}$$

tends to 0, where the factor $\eta ^* b$ comes from the derivative of $|\eta |^2$. We now choose a resolution $\widetilde{V}\rightarrow \Omega \times \Omega$ such that $\eta =\eta _0\eta '$ where $\eta '$ is non-vanishing and $\eta _0$ is (locally) a monomial. Notice that $\pi ^*\Phi =|\eta _0|^2\Phi '$ where $\Phi '$ is smooth and strictly positive. In view of Lemma 10.6 we thus obtain integrals of the form

$$\begin{aligned} \int _{\widetilde{V}}\chi '(|\eta _0|^2v/\epsilon ) \frac{1}{\epsilon } \frac{\bar{\eta }_0^{r+s-\ell }}{\eta _0^{k+\ell -s}} \alpha , \end{aligned}$$

(10.8)

where v is smooth and strictly positive and $\alpha$ is smooth.

In order to see that the limit of (10.8) tends to 0, we note first that if we let

$$\begin{aligned} \tilde{\chi }(s) = s \chi '(s) + \chi (s), \end{aligned}$$

then just as $\chi$, $\tilde{\chi }$ is also a smooth function on $[0,\infty )$ that is 0 in a neighborhood of 0 and 1 in a neighborhood of $\infty$. By assumption, $r+s-\ell -1 \ge 0$. Since the principal value current $1/f^m$ acting on a test form $\beta$ can be defined as

$$\begin{aligned} \lim _{\epsilon \rightarrow 0^+} \int \chi (|f|^2 v/\epsilon ) \frac{\beta }{f^m} \end{aligned}$$

for any cut-off function as above, the principal value current $1/\eta _0^{k+\ell -s}$ acting on $\bar{\eta }_0^{r+s-\ell -1} \alpha$ equals

$$\begin{aligned} \lim _{\epsilon \rightarrow 0^+} \int _{\widetilde{V}}\chi \left( |\eta _0|^2v/\epsilon \right) \frac{\bar{\eta }_0^{r+s-\ell -1}}{\eta _0^{k+\ell -s}} \alpha = \lim _{\epsilon \rightarrow 0^+} \int _{\widetilde{V}}\tilde{\chi }\left( |\eta _0|^2v/\epsilon \right) \frac{\bar{\eta }_0^{r+s-\ell -1}}{\eta _0^{k+\ell -s}} \alpha . \end{aligned}$$

Taking the difference between the left and right hand side, we conclude that (10.8) tends to 0 when $\epsilon \rightarrow 0$. $\square$

Now we can conclude the proof of Proposition 10.5. From the beginning we have $I_{\ell }^{\ell ,0}$. After repeated applications of (10.6) we end up with

$$\begin{aligned} I_\ell ^{0,\ell }+I_{\ell -1}^{0,\ell -1}+\cdots + I_0^{0,0} +o(1). \end{aligned}$$

However, any of these integrals has an integrable kernel even when $\epsilon =0$. This means that we are back to the case in [6, Lemma 6.2], and so the limit integral is smooth in z. $\square$

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

Let $\chi _\delta$ be functions as before that cut away $X_{sing}$. From Koppelman’s formula (9.1) for smooth forms we have

$$\begin{aligned} \chi _\delta \phi \wedge h=\bar{\partial }(K (\chi _\delta \phi ))\wedge h + K(\chi _\delta \bar{\partial }\phi )\wedge h+ P(\chi _\delta \phi )\wedge h + K(\bar{\partial }\chi _\delta \wedge \phi )\wedge h, \ \ h\in \omega _X^n, \end{aligned}$$

(11.1)

for $z\in X'_{reg}$. Clearly the left hand side tends to $\phi \wedge h$ when $\delta \rightarrow 0$. From Lemma 9.2 it follows that $K(\chi _\delta \phi ) \wedge h\rightarrow K\phi \wedge h$. Thus the first term on the right hand side of (11.1) tends to $\bar{\partial }(K\phi )\wedge h$. In the same way the second and third terms on the right hand side tend to $K(\bar{\partial }\phi )\wedge h$ and $P\phi \wedge h$, respectively. It remains to show that the last term tends to 0. If 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

$$\begin{aligned} R(\zeta ) \wedge d\zeta \wedge \bar{\partial }\chi _\delta \wedge \phi (\zeta ) \wedge i_*h \rightarrow 0, \end{aligned}$$

and since this is a tensor product of currents, it suffices to see that

$$\begin{aligned} R(\zeta ) \wedge d\zeta \wedge \bar{\partial }\chi _\delta \wedge \phi (\zeta ) \rightarrow 0, \end{aligned}$$

or equivalently, $\omega (\zeta ) \wedge \bar{\partial }\chi _\delta \wedge \phi (\zeta ) \rightarrow 0,$ which follows by Lemma 8.4 since $\phi$ is in $\text {Dom}\,\bar{\partial }_X$. We have thus proved that

$$\begin{aligned} \chi _\delta \phi \wedge h =\chi _\delta \bar{\partial }(K\phi )\wedge h+\chi _\delta K(\bar{\partial }\phi )\wedge h+ \chi _\delta P\phi \wedge h. \end{aligned}$$

The first term on the right hand side is equal to $\bar{\partial }(\chi _\delta K\phi \wedge h)- \bar{\partial }\chi _\delta \wedge K\phi \wedge h$, where the latter term tends to 0 if $K\phi$ is in $\text {Dom}\,\bar{\partial }_X$ or just in $\text {Dom}\,\bar{\partial }$, cf., Lemma 8.4. Thus we get

$$\begin{aligned} \phi \wedge h= \bar{\partial }( K\phi ) \wedge h+ K(\bar{\partial }\phi )\wedge h+P\phi \wedge h, \ \ h\in \omega _X^n, \end{aligned}$$

which precisely means that (9.1) holds. $\square$

Definition 11.2

We say that a (0, 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

$$\begin{aligned} \xi _\nu \wedge K_\nu (\cdots \xi _2\wedge K_2(\xi _1\wedge K_1(\xi _0))), \end{aligned}$$

where $\xi _j$ are smooth $(0,*)$-forms and $K_j$ are integral operators with kernels $k_j(\zeta ,z)$ at 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

Let K, P be integral operators as in Theorem 9.1. Then

$$\begin{aligned} K:\mathscr {A}^{k+1}(X)\rightarrow \mathscr {A}^k(X'),\quad P:\mathscr {A}^k(X)\rightarrow \mathscr {E}^{0,k}(X'), \end{aligned}$$

and the Koppelman formula (9.1) holds.

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 $\phi =P\phi$. $\square$

Global solvability

Assume that $E\rightarrow X$ is a holomorphic vector bundle; this means that the transition matrices have entries in $\mathscr {O}_X$. For instance if we have a global embedding $i:X\rightarrow \Omega$ and a holomorphic vector bundle $F\rightarrow \Omega$, then 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

$$\begin{aligned} H^q(X,\mathscr {O}(E))=\frac{{\text {Ker}\,}(\mathscr {A}^q(X,E){\mathop {\rightarrow }\limits ^{\bar{\partial }}}\mathscr {A}^{q+1}(X,E))}{{\text {Im}\,}(\mathscr {A}^{q-1}(X,E){\mathop {\rightarrow }\limits ^{\bar{\partial }}}\mathscr {A}^{q}(X,E))},\quad q\ge 1. \end{aligned}$$

Thus, if $\phi \in \mathscr {A}^{q+1}(X,E)$, $\bar{\partial }\phi =0$, and its canonical cohomology class vanishes, then the equation $\bar{\partial }\psi =\phi$ has a global solution in $\mathscr {A}^q(X,E)$. In particular, the equation is always solvable if 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.

Locally complete intersections

Let us consider the special case when 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

$$\begin{aligned} \mu =\bar{\partial }\frac{1}{f_p}\wedge \cdots \wedge \bar{\partial }\frac{1}{f_1}\wedge dz_1\wedge \cdots \wedge dz_N, \end{aligned}$$

see, e.g., [3]. Each smooth (0, 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.$\square$