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

TL;DR

This paper develops a theory of the $ar{ ext{d}}$-equation on possibly non-reduced analytic spaces, establishing a Dolbeault-Grothendieck lemma and constructing fine sheaves that resolve the structure sheaf using intrinsic Koppelman formulas.

Contribution

It introduces a new notion of the $ar{ ext{d}}$-equation on non-reduced spaces and constructs a resolution of the structure sheaf via fine sheaves derived from intrinsic Koppelman formulas.

Findings

Established a Dolbeault-Grothendieck lemma for non-reduced spaces

Constructed fine sheaves of currents resolving the structure sheaf

Developed intrinsic semi-global Koppelman formulas on $X$

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 $X$.

Equations

$$0\to {\mathcal{O}}_X\stackrel{i}{\to }{\mathcal{E}}_X^{0,0}\stackrel{\text{dbar}}{\to }{\mathcal{E}}_X^{0,1}\stackrel{\text{dbar}}{\to }\cdots \stackrel{\text{dbar}}{\to }{\mathcal{E}}_X^{0,n}\to 0$$

$${\mathcal{E}}_X:={\mathcal{E}}_{\Omega }/\mathcal{K}er{i}^{\ast }.$$

$$0\to {\mathcal{O}}_X\stackrel{i}{\to }{\mathcal{A}}_X^0\stackrel{\text{dbar}}{\to }{\mathcal{A}}_X^1\stackrel{\text{dbar}}{\to }\cdots \stackrel{\text{dbar}}{\to }{\mathcal{A}}_X^n\to 0$$

$$H^{q}(X,\mathscr{O}_{X})=\frac{{\text{Ker}\,}\big{(}\mathscr{A}^{0,q}(X)\stackrel{{\scriptstyle\dbar}}{{\to}}\mathscr{A}^{0,q+1}(X)\big{)}}{\text{Im}\big{(}\mathscr{A}^{0,q-1}(X)\stackrel{{\scriptstyle\dbar}}{{\to}}\mathscr{A}^{0,q}(X)\big{)}},\quad q\geq 1,$$

$$\varphi ={\hat{\varphi }}_0\otimes 1+{\hat{\varphi }}_1\otimes w^{{\alpha }_1}+\cdots +{\hat{\varphi }}_{\nu -1}\otimes w^{{\alpha }_{\nu -1}},$$

$$\text{dbar}\frac{1}{s_1^{a_1}}\wedge \cdots \wedge \text{dbar}\frac{1}{s_{\ell }^{a_{\ell }}}\wedge \frac{\alpha }{s_{\ell +1}^{a_{\ell +1}}\cdots s_r^{a_r}}$$

$$1_{\mathcal{U}\setminus V}\tau =\underset{ϵ\to 0^{+}}{\lim }\chi (|h{|}^2/ϵ)\tau .$$

$$1_V{1}_W\tau =1_{V\cap W}\tau .$$

$$1_V(\xi \wedge \tau )=\xi \wedge 1_V\tau .$$

$$1_V{f}_{\ast }\tau =f_{\ast }(1_{f^{-1}V}\tau ).$$

$$(1_V\tau )\otimes 1_{V^{\prime }}{\tau }^{\prime }=1_{V\times V^{\prime }}(\tau \otimes {\tau }^{\prime }).$$

$$\frac{1}{f}=\underset{ϵ\to 0^{+}}{\lim }\chi (|f{|}^2/ϵ)\frac{1}{f}.$$

$$1_W(a\wedge \tau )=a\wedge 1_W\tau .$$

$$a\wedge \tau =\underset{ϵ\to 0^{+}}{\lim }\chi (|h{|}^2/ϵ)a\wedge \tau .$$

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

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

$$\begin{gathered} 1_W{p}_{\ast }(k\wedge \mu \otimes 1)=\underset{ϵ\to 0^{+}}{\lim }{p}_{\ast }(1_{{\mathbb{C}}^n\times W}({\chi }_{ϵ}k\wedge (\mu \otimes 1)))= \\ \underset{ϵ\to 0^{+}}{\lim }{p}_{\ast }({\chi }_{ϵ}k\wedge (1_{{\mathbb{C}}^n\times W}\mu \otimes 1))=\underset{ϵ\to 0^{+}}{\lim }{p}_{\ast }({\chi }_{ϵ}k\wedge (1_{{\mathbb{C}}^n}\mu \otimes 1_{W}1))=0, \end{gathered}$$

$$\text{dbar}\frac{dw}{w^{\gamma +1}}:=\text{dbar}\frac{d{w}_1}{w_1^{{\gamma }_1+1}}\wedge \cdots \wedge \text{dbar}\frac{d{w}_p}{w_p^{{\gamma }_p+1}}$$

$$\frac{1}{(2\pi i{)}^p}\text{dbar}\frac{dw}{w^{\gamma +1}}.\xi =\frac{1}{\gamma !}{\int }_z\frac{{\partial }^{\gamma }\xi }{\partial w^{\gamma }}(z,0)$$

$$\tau =\sum _{\gamma }{\tau }_{\gamma }\wedge dz\otimes \text{dbar}\frac{dw}{w^{\gamma +1}},{\tau }_{\gamma }\in {\mathcal{W}}_Z^{0,\ast },$$

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

$$\varphi =\sum _{|\alpha |<M}{\varphi }_{\alpha }(z)\otimes w^{\alpha }+\mathcal{O}(|w{|}^M)+\mathcal{O}(\bar{w},d\bar{w}),$$

$${\varphi }_{\alpha }(z)=\frac{1}{\alpha !}\frac{\partial \varphi }{\partial w^{\alpha }}(z,0)$$

$$\varphi \wedge \tau =\sum _{\alpha \le \gamma }{\varphi }_{\alpha }\wedge {\tau }_{\gamma }\wedge dz\otimes \text{dbar}\frac{dw}{w^{\gamma -\alpha +1}},$$

$$\varphi \wedge \tau =\sum _{\ell \ge 0}\sum _{\gamma \ge 0}{\varphi }_{\gamma }\wedge {\tau }_{\ell +\gamma }\wedge dz\otimes \text{dbar}\frac{dw}{w^{\ell +1}}.$$

$$1_V(i_{\ast }\tau )=i_{\ast }(1_V\tau ),$$

$$i_{\ast }:{\mathcal{W}}_Z^{n,\ast }\to \mathcal{H}om({\mathcal{O}}_{\Omega }/\mathcal{J},{\mathcal{W}}_{\Omega }^Z),$$

$$X\stackrel{j}{\to }\hat{\Omega }\stackrel{\iota }{\to }\Omega :=\hat{\Omega }\times \mathcal{U},i=\iota \circ j,$$

$${\mathcal{O}}_Z\to {\mathcal{O}}_X,\varphi (z)\mapsto \tilde{\varphi }(z,w)=\varphi (z).$$

$${\mathrm{depth}}_{R}M+{\mathrm{pd}}_{R}M={\dim }_{R}R,$$

Full text

The authors were partially supported by grants from the Swedish Research Council.

1. 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

[TABLE]

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 $i\colon X\to\Omega$, where $\Omega$ is a pseudoconvex domain in $\mathbb{C}^{N}$. If ${\mathcal{K}er\,}i^{*}$ 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

[TABLE]

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 ${\mathcal{K}er\,}i_{*}$, see, e.g., [6]. There is an induced $\dbar$-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

[TABLE]

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

[TABLE]

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., [23, 5].

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

In [17], a notion of the $\dbar$-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 $i\colon X\to\Omega\subset\mathbb{C}^{N}$ 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

[TABLE]

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 {\scalebox{1.3}{\omega}}_{X}^{n}\subset\mathcal{W}_{X}^{n,0} of $\dbar$-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 {\scalebox{1.3}{\omega}}_{X}^{n} in the following way. Any holomorphic function defines a morphism in {\mathcal{H}om}({\scalebox{1.3}{\omega}}^{n}_{X},{\scalebox{1.3}{\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{H}om}({\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}_{X}^{n,*}), and in fact \mathcal{W}_{X}^{0,*}\to{\mathcal{H}om}({\scalebox{1.3}{\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 Section 6 below for details.

Given $v,\phi$ in $\mathcal{W}_{X}^{0,*}$ we say that $\dbar v=\phi$ if $\dbar(v\wedge h)=\phi\wedge h$ for all $h$ in {\scalebox{1.3}{\omega}}_{X}^{n}. Following [6] we introduce semi-global integral formulas and prove that if $\phi$ is a smooth $\dbar$-closed $(0,q+1)$-form there is locally a current $v$ in $\mathcal{W}_{X}^{0,q}$ such that $\dbar v=\phi$. 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 Sections 4-8. The construction of integral operators to provide solutions to $\dbar$ in Section 9 and the construction of the fine resolution of $\mathscr{O}_{X}$ in Section 11, which proves Theorem 1.1, are done pretty much in the same way as in [6]. The proof of the smoothness of the solutions of the regular part in Section 10 however becomes significantly more involved in the non-reduced case and requires completely new ideas. In Section 12 we discuss the relation to the results in [17, 18] in case $X$ is a local complete intersection.

Acknowledgements

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

2. 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, $0\leq\ell\leq r\leq m$. Then

[TABLE]

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 ${\bf 1}_{{\mathcal{U}}\setminus V}\tau$. Throughout this paper we let $\chi$ denote a smooth function on $[0,\infty)$ that is [math] in a neighborhood of [math] and $1$ in a neighborhood of $\infty$. If $h$ is a holomorphic tuple whose common zero set is $V$, then

[TABLE]

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

[TABLE]

This action of ${\bf 1}_{V}$ 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

[TABLE]

If $f\colon Z^{\prime}\to Z$ is a modification and $\tau$ is in $\mathcal{PM}_{Z^{\prime}}$ 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

[TABLE]

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

[TABLE]

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

[TABLE]

We say that a current $a$ on $Z$ is almost semi-meromorphic if there is a modification $\pi\colon Z^{\prime}\to Z$, a holomorphic section $f$ of a line bundle $L\to Z^{\prime}$ 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, $a=\lim_{\epsilon\to 0^{+}}\chi(|h|/\epsilon)a$ 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 ${\bf 1}_{V}(a\wedge\tau)=0$.

(ii) If $W\subset{\mathcal{U}}$ is any analytic subset, then

[TABLE]

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

[TABLE]

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

[TABLE]

For future reference we will need the following result.

Proposition 2.2**.**

Let $Z$ be a reduced space. Then $\mathcal{PM}_{Z}=\mathcal{W}_{Z}+\dbar\mathcal{W}_{Z}$.

Proof.

First assume that $Z$ is smooth. Since $\mathcal{W}_{Z}$ is closed under multiplication by smooth forms, so is $\mathcal{W}_{Z}+\dbar\mathcal{W}_{Z}$. The statement that $\mathcal{PM}_{Z}=\mathcal{W}_{Z}+\dbar\mathcal{W}_{Z}$ 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 $q\geq 1$. Let

[TABLE]

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 $\mathbb{C}^{n}_{\zeta}\times\mathbb{C}^{n}_{z}\to\mathbb{C}^{n}_{z},\quad(\zeta,z)\mapsto z.$ Recall that we have the Koppelman formula $\mu=\dbar K\mu+K(\dbar\mu)$. 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), $\chi_{\epsilon}k\wedge(\mu\otimes 1)\to k\wedge(\mu\otimes 1)$. 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),

[TABLE]

since ${\bf 1}_{W}1=0$. Thus $K\mu$ is in $\mathcal{W}_{Z}$.

If $Z$ is not smooth, then we take a smooth modification $\pi\colon Z^{\prime}\to Z$. For any $\mu$ in $\mathcal{PM}_{Z}$ there is some $\mu^{\prime}$ in $\mathcal{PM}_{Z^{\prime}}$ such that $\pi_{*}\mu^{\prime}=\mu$, see [4, Proposition 1.2]. Since $\mu^{\prime}=\tau+\dbar u$ with $\tau,u$ in $\mathcal{W}_{Z^{\prime}}$, we have that $\mu=\pi_{*}\tau+\dbar\pi_{*}u$. ∎

2.1. Pseudomeromorphic currents with support on a subvariety

Let $\Omega$ be an open set in $\mathbb{C}^{N}$ and let $Z$ be a (reduced) subvariety of pure dimension $n$. Let $\mathcal{PM}^{Z}_{\Omega}$ denote the sheaf of pseudomeromorphic currents $\tau$ on $\Omega$ with support on $Z$, and let $\mathcal{W}_{\Omega}^{Z}$ denote the subsheaf of $\mathcal{PM}^{Z}_{\Omega}$ of currents of bidegree $(N,*)$ with the SEP with respect to $Z$, i.e., such that ${\bf 1}_{W}\tau=0$ 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 $\dbar$-closed $(N,p)$-currents, where $p=N-n$.

Remark 2.3*.*

In [6, 3] $\mathcal{{CH}}_{Z}^{\Omega}$ 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. ∎

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

[TABLE]

for multiindices $\gamma=(\gamma_{1},\ldots,\gamma_{p})$ with $\gamma_{j}\geq 0$, and let $\gamma!:=\gamma_{1}!\cdots\gamma_{p}!$. Notice that that

[TABLE]

for test forms $\xi$. If $\tau$ is in $\mathcal{W}_{Z}$, then it follows by (2.5) and the fact that $\operatorname{supp}\dbar(1/w^{\gamma+{\bf 1}})=\{w=0\}$ that $\tau\otimes\dbar(1/w^{\gamma+{\bf 1}})$ 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

[TABLE]

where $dz:=dz_{1}\wedge\cdots\wedge dz_{n}$. If $\pi$ is the projection $(z,w)\mapsto z$, then

[TABLE]

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

[TABLE]

where

[TABLE]

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

[TABLE]

i.e.,

[TABLE]

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

2.2. Intrinsic pseudomeromorphic currents on a reduced subvariety

Currents on a reduced analytic space $Z$ are defined as the dual of the sheaf of test forms. If $i:Z\to Y$ 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 $i\colon Z\to Y$ 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 ${\bf 1}_{Z_{sing}}(i_{*}\tau)=0$, 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 $V\subset{\mathcal{U}}\subset Z$,

[TABLE]

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

Corollary 2.7**.**

We have the isomorphism

[TABLE]

where $\mathcal{J}$ is the ideal defining $Z$ in $\Omega$.

Notice that ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{W}_{\Omega}^{Z})$ 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 ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{W}^{Z}_{\Omega})$ by (2.15). To see that $i_{*}$ is surjective, we take a $\mu$ in ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{W}^{Z}_{\Omega})$. We assume first that we are on $Z_{\rm reg}$, with local coordinates such that $Z_{\rm reg}=\{w=0\}$. 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 $\mathcal{W}^{n,*}_{Z}$. ∎

Remark 2.8*.*

We do not know whether $i_{*}\tau\in\mathcal{PM}_{\Omega}^{Z}$ implies that $\tau\in\mathcal{PM}_{Z}$. ∎

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

Proposition 2.9**.**

Let $\varphi$ and $\phi_{1},\dots,\phi_{m}$ be currents in $\mathcal{W}_{Z}$. If $\varphi=0$ on the set on $Z_{reg}$ where $\phi_{1},\dots,\phi_{m}$ are smooth, then $\varphi=0$.

3. 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 $i\colon X\to\Omega\subset\mathbb{C}^{N}$ 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 $i\colon X\to\Omega$ has locally at $x$ the form

[TABLE]

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 $Z\to\widehat{\Omega}\to\Omega$; however, the first one is in general not minimal.

Now consider a fixed local embedding $i\colon X\to\Omega\subset\mathbb{C}^{N}$, 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

[TABLE]

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 $f_{1},\dots,f_{m}\in R$ is a regular sequence on the $R$-module $M$ if $f_{i}$ is a non zero-divisor on $M/(f_{1},\dots,f_{i-1})$ for $i=1,\dots,m$, and $(f_{1},\dots,f_{m})M\neq M$. If $R$ is a local ring, then $\operatorname{depth}_{R}M$ is the maximal length $d$ of a regular sequence $f_{1},\dots,f_{d}$ such that $f_{1},\dots,f_{d}$ are contained in the maximal ideal $\mathfrak{m}$; furthermore, $M$ is Cohen-Macaulay if $\operatorname{depth}_{R}M=\dim_{R}M$, where $\dim_{R}M=\dim_{R}(R/{\rm ann\,}_{R}M)$. 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

[TABLE]

where $\operatorname{pd}_{R}M$ 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 $i\colon X\to\Omega$ 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 $N-n$. ∎

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 $\mathcal{M}^{\prime}=\mathscr{O}_{\Omega}/\mathcal{M}$ is a free, finitely generated $\mathscr{O}_{Z}$-module. Thus, $\mathscr{O}_{\Omega}/\mathcal{J}\simeq\mathcal{M}^{\prime}/\mathcal{J}\mathcal{M}^{\prime}$ 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

[TABLE]

and

[TABLE]

We claim that a sequence $f_{1},\dots,f_{m}$ in $\mathscr{O}_{X,x}$ is regular (on $\mathscr{O}_{X,x}$) if and only if $\tilde{f}_{1},\dots,\tilde{f}_{m}\in\mathscr{O}_{Z,x}$ 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 $Z(\mathcal{J}+(f_{1},\dots,f_{k}))=Z(\mathcal{J}+(\tilde{f}_{1},\dots,\tilde{f}_{k}))$. 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, ${\rm ann\,}_{\mathscr{O}_{Z,x}}\mathscr{O}_{X,x}=\{0\}$, so $\dim_{\mathscr{O}_{Z,x}}\mathscr{O}_{X,x}=\dim_{\mathscr{O}_{Z,x}}\mathscr{O}_{Z,x}/({\rm ann\,}_{\mathscr{O}_{Z,x}}\mathscr{O}_{X,x})=\dim_{\mathscr{O}_{Z,x}}\mathscr{O}_{Z,x}=n$.

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 $M=\mathscr{O}_{X,x}$,

[TABLE]

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

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 $|\alpha|\leq M$ 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 $\mathscr{O}_{Z,x^{\prime}}$ is free as an $\mathscr{O}_{Z,x^{\prime}}$-module for all $x^{\prime}$ in a neighborhood of $x$, and $1,w^{\alpha_{1}},\ldots,w^{\alpha_{\nu-1}}$ is a basis at each such $x^{\prime}$.

Example 3.4*.*

Let $\mathcal{J}$ be the ideal in $\mathbb{C}^{4}$ generated by $(w_{1}^{2},w_{2}^{2},w_{1}w_{2},w_{1}z_{2}-w_{2}z_{1}).$ 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 $\neq 0$. If, say, $z_{1}\neq 0$, 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. ∎

The pure-dimensionality of $\mathscr{O}_{X}$ can also be rephrased in the following way: If $\phi$ is holomorphic and is [math] generically, then $\phi=0$. If we delete the generator $w_{1}w_{2}$ from the definition of $\mathcal{J}$ in the example, then $\phi=w_{1}w_{2}$ is [math] 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

[TABLE]

we have the representation $a_{0}(\zeta)\otimes 1+(a_{1}(\zeta)+\partial a_{0}/\partial\zeta)(\zeta)\otimes\eta$ with respect to $(\zeta,\eta)$. ∎

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

4. Smooth $(0,*)$-forms on a non-reduced space $X$

Let $i\colon X\to\Omega$ 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 ${\mathcal{K}er\,}i^{*}$ 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

[TABLE]

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

[TABLE]

so ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega}^{Z})$ 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 ${\mathcal{K}er\,}i^{*}$ by saying that $\xi$ is in ${\mathcal{K}er\,}i^{*}$ if

[TABLE]

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

Definition 4.2**.**

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

[TABLE]

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 ${\mathcal{K}er\,}i^{*}$, which then motivates this notation. Notice that ${\mathcal{K}er\,}i^{*}$ is a two-sided ideal in $\mathscr{E}_{\Omega}^{0,*}$, i.e., if $\phi$ is in $\mathscr{E}_{\Omega}^{0,*}$ and $\xi$ is in ${\mathcal{K}er\,}i^{*}$, then $\phi\wedge\xi$ and $\xi\wedge\phi$ are in ${\mathcal{K}er\,}i^{*}$. It follows that we have an induced wedge product on $\mathscr{E}_{X}^{0,*}$ such that

[TABLE]

Remark 4.3*.*

It follows from Lemma 4.8 below that in case $X=Z$ is reduced, then $\xi$ is in ${\mathcal{K}er\,}i^{*}$ 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. ∎

Lemma 4.4**.**

Using the notation of (3.1),

[TABLE]

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

Since $\iota_{*}$ is injective, $\dbar\tau=0$ if and only if $\dbar\iota_{*}\tau=0$, and thus we get

Corollary 4.5**.**

Using the notation of (3.1),

[TABLE]

is an isomorphism.

Corollary 4.6**.**

Using the notation of (3.1),

[TABLE]

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. ∎

It follows from (4.6) and (4.3) that the sheaf $\mathscr{E}_{X}^{0,*}$ is intrinsically defined on $X$. Since $\dbar$ maps ${\mathcal{K}er\,}i^{*}$ to ${\mathcal{K}er\,}i^{*}$, we have a well-defined operator $\dbar\colon\mathscr{E}_{X}^{0,*}\to\mathscr{E}_{X}^{0,*+1}$ such that $\dbar^{2}=0$. Unfortunately the sheaf complex so obtained is not exact in general, see, e.g., [6, Example 1.1] for a counterexample already in the reduced case.

4.1. Local representation on $X_{reg}$ of smooth forms

Recall that $X_{reg}$ is the open subset of $X$, where the underlying reduced space is smooth and $\mathscr{O}_{X}$ is Cohen-Macaulay. Let us fix some point in $X_{reg}$, and assume that we have local coordinates $(z,w)$ such that $Z=\{w=0\}$. We also choose generators $1,w^{\alpha_{1}},\dots,w^{\alpha_{\nu-1}}$ of $\mathscr{O}_{X}$ as a free $\mathscr{O}_{Z}$-module, which exist by Corollary 3.3, and generators $\mu^{1},\dots,\mu^{m}$ of ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}^{Z}_{\Omega})$.

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 $M=M_{1}+\dots+M_{m}$ 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

[TABLE]

where $\hat{\phi}_{j}$ are in $\mathscr{O}_{Z}$. We thus have an $\mathscr{O}_{Z}$-linear morphism

[TABLE]

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

[TABLE]

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 ${\mathcal{K}er\,}i^{*}$. Then the tuple $(\hat{\phi}_{j})$ is mapped to [math] by the matrix $T$, and since $T$ is generically pointwise injective we conclude that each $\hat{\phi}_{j}$ vanishes. ∎

By the above proof we get

Lemma 4.8**.**

A smooth $(0,*)$-form $\xi$ in $\Omega$ is in ${\mathcal{K}er\,}i^{*}$ 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 [math], but there exist points arbitrarily close to [math] 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]. ∎

Remark 4.10*.*

It is easy to check that if we have the setting as in the discussion at the end of Section 3 but $(a_{j})$ is instead a tuple in $\mathscr{E}_{Z}^{0,*}$, 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 $\mathscr{E}_{X}^{0,*}$. ∎

For future needs we prove in Section 6.1:

Lemma 4.11**.**

The morphism $T$ is pointwise injective.

We can thus choose a holomorphic matrix $A$ such that

[TABLE]

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

[TABLE]

5. Intrinsic $(n,*)$-currents on $X$

In analogy with the reduced case we have the following definition when $X$ is possibly non-reduced.

Definition 5.1**.**

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

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

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

[TABLE]

Clearly, we get a mapping $\dbar\colon{\mathcal{C}}_{X}^{n,q}\to{\mathcal{C}}_{X}^{n,q+1}$ such that $\dbar^{2}=0$.

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. ∎

Definition 5.3**.**

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

By definition we thus have the isomorphism

[TABLE]

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. ∎

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 ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{W}^{Z}_{\Omega})$ and by Proposition 2.4 and (2.8), one can define $a\wedge i_{*}\psi$ in ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{W}^{Z}_{\Omega})$; 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),

[TABLE]

if $h$ cuts out the Zariski singular support of $a$.

Definition 5.6**.**

We let {\scalebox{1.3}{\omega}}_{X}^{n} be the sheaf of $\dbar$-closed currents in $\mathcal{W}^{n,0}_{X}$.

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

[TABLE]

When $X$ is reduced {\scalebox{1.3}{\omega}}_{X}^{n} is the sheaf of $(n,0)$-forms that are $\dbar$-closed in the Barlet-Henkin-Passare sense. Let $\mu^{1},\ldots,\mu^{m}$ be a set of generators for the $\mathscr{O}_{\Omega}$-module ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega}^{Z})$. They correspond via (5.3) to a set of generators $h^{1},\ldots,h^{m}$ for the $\mathscr{O}_{X}$-module {\scalebox{1.3}{\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 ${\bf 1}_{W}i_{*}\tau$ is in $\mathcal{PM}_{\Omega}^{Z}$, and by (2.3), ${\bf 1}_{W}i_{*}\tau$ is annihilated by ${\mathcal{K}er\,}i^{*}$. Hence we can define ${\bf 1}_{W}\tau$ as the unique current in $\mathcal{F}_{X}$ such that $i_{*}{\bf 1}_{W}\tau={\bf 1}_{W}i_{*}\tau$. Clearly, ${\bf 1}_{W}\tau$ 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 $\dbar$ 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 $\dbar$ and multiplication by ${\bf 1}_{W}$ for all germs $W$ of subvarieties of $Z$.

In view of Proposition 2.2 this definition coincides with the usual definition in case $X$ is reduced. It is readily checked that the dimension principle holds for $\mathcal{F}_{X}$, and hence it also holds for the (possibly smaller) sheaf $\mathcal{PM}_{X}^{n,*}$, and in addition, (2.3) holds for forms $\xi$ in $\mathscr{E}_{X}^{0,*}$ and $\tau$ in $\mathcal{PM}_{X}^{n,*}$.

6. Structure form on $X$

Let $i\colon X\to\Omega\subset\mathbb{C}^{N}$ 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

[TABLE]

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

[TABLE]

that is pointwise exact outside $Z$. Let $X_{k}$ be the set where $f_{k}$ does not have optimal rank. Then

[TABLE]

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 $p$,

[TABLE]

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 $\dbar$-equation), we can thus choose the resolution so that $N_{0}\leq N-1$. 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 $\sigma_{k+1}\colon E_{k}\to E_{k+1}$, if we require that $\sigma_{k+1}$ vanishes on $({\text{Im}\,}f_{k+1})^{\perp}$, takes values in $({\mathcal{K}er\,}f_{k+1})^{\perp}$, 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

[TABLE]

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

[TABLE]

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 $\nabla_{f}:=f-\dbar$. 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

[TABLE]

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, $\mathcal{J}={\rm ann\,}R$, 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 $0\to\mathscr{O}_{\Omega}\stackrel{{\scriptstyle f}}{{\to}}\mathscr{O}_{\Omega}\to\mathscr{O}_{\Omega}/(f)\to 0$; thus $U$ is just the principal value current $1/f$ and $R=\dbar(1/f)$. More generally, if $f=(f_{1},\dots,f_{p})$ is a complete intersection, then

[TABLE]

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

There are almost semi-meromorphic $\alpha_{k}$ in $\Omega$, cf. [7, Section 2] and the proof of [6, Proposition 3.3], that are smooth outside $X_{k}$, such that

[TABLE]

outside $X_{k+1}$ for $k\geq p$. In view of (6.3) and the dimension principle, ${\bf 1}_{X_{k+1}}R_{k+1}=0$ 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

[TABLE]

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

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 ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega}^{Z})$ 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 $k\geq p+1$. 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 $b=b_{p}+b_{p+1}+\cdots$. ∎

By Proposition 2.4 we get

Corollary 6.3**.**

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

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

[TABLE]

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 {\scalebox{1.3}{\omega}}^{n}_{X} such that

[TABLE]

More precisely, $\omega=\omega_{0}+\omega_{1}+\cdots+\omega_{n}$111In [6, Proposition 3.3], the sum ends with $\omega_{n-1}$ instead of $\omega_{n}$, which, as remarked above, one can indeed assume when $n\geq 1$ and the resolution is chosen to be of length $\leq N-1$., where $\omega_{k}\in\mathcal{W}^{n,k}(X,E_{p+k})$, and if $f^{j}:=f_{p+j}$, then

[TABLE]

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 $\dbar\omega_{0}=f^{1}\omega_{1}=0$, so $\omega$ is in {\scalebox{1.3}{\omega}}^{n}_{X}. ∎

Remark 6.6*.*

If $X=\{f=0\}$ is a reduced hypersurface in $\Omega$, then $R=\dbar(1/f)$ 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 {\scalebox{1.3}{\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 $\mu\in{\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}^{Z}_{\Omega})$, 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 {\scalebox{1.3}{\omega}}^{n}_{X} on $\xi$ in $\mathscr{E}^{0,*}_{X}$ can then be expressed as

[TABLE]

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

[TABLE]

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

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

[TABLE]

Let $\xi=\xi_{0}\wedge dz$ be a holomorphic section of the sheaf

[TABLE]

such that $f^{*}_{p+1}\xi_{0}=0$. Then $\dbar(\xi_{0}\omega_{0})=\pm\xi_{0}\dbar\omega_{0}=\pm\xi_{0}f_{p+1}\omega_{1}=\pm(f^{*}_{p+1}\xi_{0})\omega_{1}=0$, so that $\xi_{0}\omega_{0}$ is in {\scalebox{1.3}{\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

[TABLE]

Proposition 6.7**.**

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

[TABLE]

Proof.

If $h$ is in {\scalebox{1.3}{\omega}}_{X}^{n}, then $i_{*}h$ is in ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega}^{Z})$. We have mappings

[TABLE]

where the first mapping is (6.12), and the second is $h\mapsto i_{*}h$. In view of (6.8), the composed mapping is $\xi=\xi_{0}\wedge dz\mapsto\xi R_{p}=\xi_{0}R_{p}\wedge dz$ 222There is a superstructure involved, with respect to which $R_{p}$ has even degree, and therefore $dz\wedge R_{p}=R_{p}\wedge dz$, explaining the lack of a sign in the last equality, see [7] or [6].. This mapping is an intrinsic isomorphism

[TABLE]

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

In particular it follows that {\scalebox{1.3}{\omega}}_{X}^{n} is coherent, and we have:

If $\xi^{1},\ldots,\xi^{m}$ are generators of ${\mathcal{H}}^{p}({\mathcal{H}om}(E_{\bullet}^{*},K_{\Omega})))$, 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 {\scalebox{1.3}{\omega}}_{X}^{n}, and $\mu^{\ell}=i_{*}h^{\ell}=\xi^{\ell}R_{p}$ generate the $\mathscr{O}_{\Omega}$-module ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega}^{Z})$.

Remark 6.8*.*

The isomorphism

[TABLE]

was well-known since long ago, the contribution in [3] was the realization $\xi\mapsto\xi R_{p}$. ∎

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

Example 6.9*.*

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

[TABLE]

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 $\geq p$, and $a:F_{\bullet}\to E_{\bullet}$ is a morphism of complexes, then there exists a ${\mathcal{H}om}(F_{0},E_{p+1})$-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 ${\mathcal{K}er\,}f_{p+1}^{*}$, we thus get that

[TABLE]

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

[TABLE]

where

[TABLE]

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 $a:F_{\bullet}\to E_{\bullet}$ given by

[TABLE]

Since the current $R^{F}_{2}$ is equal to the Coleff-Herrera product $\dbar(1/w_{1}^{2})\wedge\dbar(1/w_{2}^{2})$, cf. Remark 6.1, we thus get by (6.16) and Remark 6.8 that ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}^{Z}_{\Omega})$ is generated by

[TABLE]

A straightforward calculation gives the generators $\mu_{1}$ and $\mu_{2}$ above. ∎

6.1. Proof of Lemma 4.11

Since $T$ is generically injective, it is clearly injective if $n=0$. We are going to reduce to this case. Fix the point $0\in Z$ 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 [math] 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

[TABLE]

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

[TABLE]

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

Let $\mu^{1},\ldots,\mu^{m}$ be a minimal set of generators for the $\mathscr{O}_{\Omega}$-module ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}^{Z}_{\Omega})$ at [math]. Then $\mu^{j}=\xi^{j}R^{E}_{p}$, where $\xi^{j}$ is a minimal set of generators for

${\mathcal{H}}^{p}({\mathcal{H}om}(E_{\bullet},K_{\Omega}))$. Notice that

[TABLE]

Since ${\mathcal{H}}^{n}({\mathcal{H}om}(F_{\bullet},\mathscr{O}_{\Omega}))$ is generated by $1$, it follows that ${\mathcal{H}}^{N}({\mathcal{H}om}((E\otimes F)_{\bullet},K_{\Omega}))$ is generated by $\xi^{j}\otimes 1$. We conclude that ${\mathcal{H}om}(\mathscr{O}_{\Omega}/(\mathcal{J}+\mathcal{I}),\mathcal{{CH}}^{\{0\}}_{\Omega})$ 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 $R^{F}_{n}=\mu^{z}=\dbar(1/z^{{\bf 1}})$.

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 $\phi\dbar(1/z^{{\bf 1}})=\phi(0,\cdot)\dbar(1/z^{{\bf 1}})$ we have that

[TABLE]

The morphism constructed in (4.8) for $X_{0}$ instead of $X$ is then $T_{0}=T(0)$, where $T$ is the morphism (4.8) for $X$. Thus $T(0)$ is injective.

7. The intrinsic sheaf $\mathcal{W}_{X}^{0,*}$ 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 $i\colon X\to\Omega$ 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

[TABLE]

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 ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}^{Z}_{\Omega})$, expressed in the form (2.11), then

[TABLE]

is in ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{W}^{Z}_{\Omega})$.

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 $\widehat{\phi}_{0},\dots,\widehat{\phi}_{\nu-1}$ 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

[TABLE]

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 $\xi\phi\wedge\tau=0$. ∎

If $\phi$ has the form (7.1) in a neighborhood of some point $x\in X_{reg}$ and $h$ is in {\scalebox{1.3}{\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{H}om}_{\mathscr{O}_{X}}({\scalebox{1.3}{\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{H}om}_{\mathscr{O}_{X}}({\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}_{X}^{n,*}).

If $\phi$ is in $\mathcal{W}^{0,*}_{X}$ and $h$ is in {\scalebox{1.3}{\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,

[TABLE]

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 ${\text{codim}\,}X_{k}\geq k+2$, $k\geq p+1$, cf. Section 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

[TABLE]

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

[TABLE]

In view of (4.2) and (5.3), (7.5) corresponds to a morphism \mathscr{O}_{X}\to{\mathcal{H}om}({\scalebox{1.3}{\omega}}^{n}_{X},{\scalebox{1.3}{\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

[TABLE]

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 {\scalebox{1.3}{\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

[TABLE]

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{H}om}({\scalebox{1.3}{\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{H}om}({\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}^{n,*}_{X}). If $h^{\prime}$ is nonvanishing at a point on $X_{reg}$, then it generates {\scalebox{1.3}{\omega}}^{n}_{X} and thus $\alpha$ is determined by $\phi:=\alpha h^{\prime}$ there. By [10, Theorem 3.7], $\phi=\psi\wedge h^{\prime}$ 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 $X_{\rm reg}$.

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 $\pi\colon X^{\prime}\to X$ that is biholomorphic outside $X_{sing}$. Then $\pi^{*}\gamma$ is a holomorphic $n$-form on $X^{\prime}$ that is generically non-vanishing. We claim that there is a current $\tau$ in $\mathcal{W}^{n,0}(X^{\prime})$ 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 $X^{\prime}$, and is therefore also the image of the (unique) current $\tau={\bf 1}_{\pi^{-1}(X_{reg})}\tilde{\tau}$ in $\mathcal{W}^{n,*}(X^{\prime})$.

By [10, Theorem 3.7] again $\tau$ is locally of the form $\xi\wedge ds$, where $\xi$ is in $\mathcal{W}^{0,*}_{X^{\prime}}$ and $ds=ds_{1}\wedge\cdots\wedge ds_{n}$ for some local coordinates $s$. Hence, $\tau$ is a $K_{X^{\prime}}$-valued section of $\mathcal{W}^{0,*}(X^{\prime})$, so $\tau/\pi^{*}\gamma$ is a section of $\mathcal{W}^{0,*}(X^{\prime})$. Now $\Psi:=\pi_{*}(\tau/\pi^{*}\gamma)$ is a section of $\mathcal{W}^{0,*}(X)$. On $X_{reg}\cap\{\gamma\neq 0\}$ 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 $X$. ∎

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

[TABLE]

Lemma 7.5**.**

Assume that $X_{reg}\to\Omega$ 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{H}om}({\scalebox{1.3}{\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

[TABLE]

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{H}om}({\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}^{n,*}_{X}). Let us choose a basis $\mu^{1},\dots,\mu^{m}$ for {\scalebox{1.3}{\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. Section 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 [math], 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 $\mu^{1}\wedge\mu^{z},\dots,\mu^{m}\wedge\mu^{z}$ generate {\scalebox{1.3}{\omega}}^{0}_{X_{0}}. If we let $\phi_{0}$ be the morphism in {\mathcal{H}om}({\scalebox{1.3}{\omega}}^{0}_{X_{0}},{\scalebox{1.3}{\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

[TABLE]

Since $X_{0}$ is [math]-dimensional, the morphism \mathscr{O}_{X_{0}}\to{\mathcal{H}om}({\scalebox{1.3}{\omega}}_{X_{0}},{\scalebox{1.3}{\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 $(A\tilde{\phi})(0)=0$. ∎

Example 7.6* (Meromorphic functions).*

Assume that we have a local embedding $X\to\Omega$. Given meromorphic functions $\Phi,\Phi^{\prime}$ in $\Omega$ that are holomorphic generically on $Z$, we say that $\Phi\sim\Phi^{\prime}$ if and only if $\Phi-\Phi^{\prime}$ is in $\mathcal{J}$ generically on $Z$. If $\Phi=A/B$ and $\Phi^{\prime}=A^{\prime}/B^{\prime}$, where $B$ and $B^{\prime}$ are generically non-vanishing on $Z$, the condition is precisely that $AB^{\prime}-A^{\prime}B$ 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

[TABLE]

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 {\scalebox{1.3}{\omega}}^{n}_{X}. Another representative $\Phi^{\prime}$ of $\phi$ will give rise to the same current generically and hence everywhere by the SEP. Thus $\phi$ defines a section of {\mathcal{H}om}({\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}^{n,*}_{X})=\mathcal{W}^{0,*}_{X}. ∎

By definition, a current $\phi$ in $\mathcal{W}^{0,*}_{X}$ can be multiplied by a current $h$ in {\scalebox{1.3}{\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 {\scalebox{1.3}{\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}{\scalebox{1.3}{\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 {\scalebox{1.3}{\omega}}^{n}_{X}.

Lemma 7.7**.**

Each $\phi$ in \mathcal{W}^{0,*}_{X}={\mathcal{H}om}_{\mathscr{O}_{X}}({\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}^{n,*}_{X}) has a unique extension to a morphism in {\mathcal{H}om}_{\mathscr{E}^{0,*}_{X}}(\mathscr{E}^{0,*}_{X}{\scalebox{1.3}{\omega}}^{n}_{X},\mathcal{W}^{n,*}_{X}).

Proof.

The uniqueness follows by $\mathscr{E}^{0,*}_{X}$-linearity, i.e., if $b=\xi_{1}\wedge h_{1}+\dots+\xi_{r}\wedge h_{r}$ is in \mathscr{E}^{0,*}_{X}{\scalebox{1.3}{\omega}}^{n}_{X}, then one must have

[TABLE]

We must check that this is well-defined, i.e., that the right hand side does not depend on the representation $\xi_{1}\wedge h_{1}+\dots+\xi_{r}\wedge h_{r}$ of $b$. By the SEP, it is enough to prove this locally on $X_{\rm reg}$, 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 $\xi_{1}\wedge h_{1}+\dots+\xi_{r}\wedge h_{r}=b$ by the smooth form $\phi$ in $\mathscr{E}^{0,*}_{X}$, and this product only depends on $b$. ∎

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 {\scalebox{1.3}{\omega}}^{n}_{X}. Then one has a well-defined product

[TABLE]

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}{\scalebox{1.3}{\omega}}^{n}_{X}, cf. Lemma 7.7. It follows by the SEP that the right hand side only depends on $\alpha$. ∎

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

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 {\scalebox{1.3}{\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., $\phi=0$. ∎

We end this section with the following result, the first part of [10, Theorem 3.7]. We include here a different proof than the one in [10], since we believe the proof here is instructive.

Proposition 7.11**.**

If $Z$ is smooth, then $\mathcal{W}_{Z}$ 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

[TABLE]

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

[TABLE]

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

[TABLE]

it is readily verified that $p_{*}\tau^{\prime}=\partial\tau/\partial\zeta_{1}$, so we conclude that $\partial\tau/\partial\zeta_{1}$ is in $\mathcal{W}_{Z}$. ∎

8. The $\dbar$-operator on $\mathcal{W}_{X}^{0,*}$

We already know the meaning of $\dbar$ on $\mathcal{W}_{X}^{n,*}$, and we now define $\dbar$ on $\mathcal{W}_{X}^{0,*}$.

Definition 8.1**.**

Assume that $\phi,v$ are in $\mathcal{W}_{X}^{0,*}$, We say that $\dbar v=\phi$ if

[TABLE]

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

[TABLE]

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 $\text{Dom}\,\dbar_{X}$ if $v$ is in $\text{Dom}\,\dbar$, i.e., $\dbar v=\phi$ for some $\phi$ and in addition $\dbar(v\wedge\omega)$, 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 {\scalebox{1.3}{\omega}}_{X}^{n} and $a_{j}$ are smooth, see Remark 6.5, and hence $\text{Dom}\,\dbar_{X}$ coincides with $\text{Dom}\,\dbar$ in this case.

Example 8.3*.*

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

[TABLE]

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

If $w$ is in $\text{Dom}\,\dbar_{X}$ and $v$ is in $\mathscr{E}_{X}^{0,*}$, then

[TABLE]

Thus $v\wedge w$ is in $\text{Dom}\,\dbar_{X}$, and the Leibniz rule $\dbar(v\wedge w)=\dbar v\wedge w+(-1)^{\deg v}v\wedge\dbar w$ holds. ∎

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 $\text{Dom}\,\dbar_{X}$ on $X_{\rm reg}$, then $v$ is in $\text{Dom}\,\dbar_{X}$ on all of $X$ if and only if

[TABLE]

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

[TABLE]

Proof.

Since $\mathcal{W}^{n,*}_{X}$ is closed under multiplication by $f$, $v$ is in $\text{Dom}\,\dbar_{X}$ if and only if $\nabla_{f}(v\wedge\omega)$ is in $\mathcal{W}^{n,*}_{X}$ for all structure forms $\omega$. Since $v$ is in $\text{Dom}\,\dbar_{X}$ on $X_{\rm reg}$, thus $\nabla_{f}(v\wedge\omega)$ is in $\mathcal{W}^{n,*}_{X}$ on $X_{\rm reg}$. By (2.2), $\nabla_{f}(v\wedge\omega)$ is then in $\mathcal{W}^{n,*}_{X}$ on all of $X$ if and only if

[TABLE]

By the Leibniz rule,

[TABLE]

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 $\delta\to 0$, whereas the second term on the right hand side of (8.6) tends to ${\bf 1}_{X_{\rm reg}}\nabla_{f}(v\wedge\omega)$. 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 $X_{\rm reg}$ and $\vartheta$ is in {\scalebox{1.3}{\omega}}^{n}_{X}. By the Leibniz rule thus $-\nabla_{f}(v\wedge\omega)=\dbar v\wedge\omega$ on $X_{\rm reg}$, since $\nabla_{f}\omega=0$. Therefore, (8.6) is equivalent to $-\nabla_{f}(\chi_{\delta}v\wedge\omega)=\dbar\chi_{\delta}\wedge v\wedge\omega+\chi_{\delta}\dbar v\wedge\omega.$ If (8.3) holds, we therefore get (8.4) when $\delta\to 0$. ∎

Remark 8.5*.*

In case $X$ is reduced the definition of $\dbar_{X}$ is precisely the same as in [6]. However, the definition of $\dbar v=\phi$ 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 $\dbar(v\wedge h)=\phi\wedge h$ for all smooth $h$ in {\scalebox{1.3}{\omega}}_{X}^{n}, which in general is a strictly weaker condition. For example, for any weakly holomorphic function $v$, we have $\dbar(v\wedge h)=0$ for all smooth $h$ in {\scalebox{1.3}{\omega}}_{X}^{n}, while if $X$ is a reduced complete intersection, or more generally Cohen-Macaulay, then $\dbar(v\wedge h)=0$ for all $h$ in {\scalebox{1.3}{\omega}}_{X}^{n} is equivalent to $v$ being strongly holomorphic, see [33, p. 124] and [2]. ∎

We conclude this section with a lemma that shows that $\dbar$ 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 $X_{reg}\to\Omega$ and $\phi,v$ in $\mathcal{W}_{X}^{0,*}$ represented as in (7.1). Then $\dbar v=\phi$ if and only if

[TABLE]

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), $\widetilde{\dbar v}=\dbar\tilde{v}$. Since $S$ is holomorphic therefore $\widehat{\dbar v}=S\widetilde{\dbar v}=S\dbar\tilde{v}=\dbar(S\tilde{v})=\dbar\hat{v}.$ ∎

9. Solving $\dbar u=\phi$ on $X$

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

Theorem 9.1**.**

There are integral operators

[TABLE]

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

[TABLE]

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 $z\in\Omega^{\prime}$ 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 $k\geq 1$, i.e.,

[TABLE]

If $\dbar\phi=0$ in $\Omega$, and $k\geq 1$, then $K\phi$ is a solution to $\dbar v=\phi$. If $k=0$, then $\phi=P\phi$ is holomorphic. It follows that a smooth $\dbar$-closed function is holomorphic. In the reduced case this is a classical theorem of Malgrange, [24]. In Section 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 Section 11, we define them acting on currents in $\mathcal{W}^{0,*}(X)$ and not only on smooth forms. Let $\pi:\Omega_{\zeta}\times\Omega_{z}^{\prime}\to\Omega_{z}^{\prime}$ be the natural projection. Let us choose a holomorphic Hefer form333We are only concerned with the component $H^{0}$ of this form, so for simplicity we write just $H$. $H$, a smooth weight $g$ with compact support in $\Omega$ with respect to $z\in\Omega^{\prime}\subset\subset\Omega$, 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 $\Omega_{\zeta}\times\Omega^{\prime}_{z}$ 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

[TABLE]

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

[TABLE]

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 $\mathcal{W}^{Z\times Z^{\prime}}_{\Omega_{\zeta}\times\Omega_{z}^{\prime}}$, where $Z^{\prime}=Z\cap\Omega^{\prime}$. Finally, we multiply this with the smooth form $\vartheta(g\wedge H)$ to obtain (9.4). Similarly, outside of $\Delta$, the diagonal in $\Omega\times\Omega^{\prime}$, where $B$ is smooth, we can define

[TABLE]

as a tensor product of currents.

Lemma 9.2**.**

For $\mu$ in ${\mathcal{H}om}(\mathscr{O}_{\Omega^{\prime}}/\mathcal{J},\mathcal{W}^{Z^{\prime}}_{\Omega^{\prime}})$ and $\phi\in\mathcal{W}^{0,*}(X)$, the current (9.5), a priori defined as a current in $\mathcal{W}^{Z\times Z^{\prime}\setminus\Delta}_{\Omega_{\zeta}\times\Omega_{z}^{\prime}\setminus\Delta}$ 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

[TABLE]

and

[TABLE]

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

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

[TABLE]

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 $\Omega_{\zeta}\times\Omega_{z}^{\prime}$, 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 $\mathcal{W}^{Z\times Z^{\prime}}_{\Omega_{\zeta}\times\Omega_{z}^{\prime}}$, 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 ${\mathcal{H}om}(\mathscr{O}_{\Omega^{\prime}}/\mathcal{J},\mathcal{W}^{Z^{\prime}}_{\Omega^{\prime}})$. ∎

Proposition 9.3**.**

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

Proof.

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

[TABLE]

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 $\Omega^{\prime}$. If $g$ depends holomorphically on $z$, then $P\phi$ is holomorphic in $\Omega^{\prime}$ if $\phi$ is a $(0,0)$-current, and vanishes for degree reasons if $\phi$ has positive degree. ∎

We shall now see that we can approximate $K\phi$ by smooth forms. Let $B^{\epsilon}=\chi(|\zeta-z|^{2}/\epsilon)B$.

Proposition 9.4**.**

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

[TABLE]

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

The last statement means that

[TABLE]

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

[TABLE]

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

[TABLE]

which implies (9.9). ∎

9.1. Proof of Theorem 9.1

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

[TABLE]

and

[TABLE]

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 ${\mathcal{H}om}(\mathscr{O}_{\Omega^{\prime}}/\mathcal{J},\mathcal{W}^{Z^{\prime}}_{\Omega^{\prime}})$ by Corollary 6.3.

Recall from Lemma 6.2 that $R\wedge dz=b\mu$, where $\mu$ is a tuple of currents in ${\mathcal{H}om}(\mathscr{O}_{\Omega^{\prime}}/\mathcal{J},\mathcal{{CH}}^{Z^{\prime}}_{\Omega^{\prime}})$ and $b$ is an almost semi-meromorphic matrix that is smooth generically on $Z^{\prime}$. Therefore (9.12) and (9.13) hold where $b$ is smooth, in view of Lemma 7.7, and since both sides are in ${\mathcal{H}om}(\mathscr{O}_{\Omega^{\prime}}/\mathcal{J},\mathcal{W}^{Z^{\prime}}_{\Omega^{\prime}})$, the equalities hold everywhere by the SEP.

As in [6] we let $R^{\lambda}=\dbar|f|^{2\lambda}\wedge U$ 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 current444One can consider this current as $R(z)\wedge B$ multiplied by the residue of the almost semi-meromorphic current $U$ in (6.5), cf.  [10, Section 4.4]. when $\lambda=0$. Let $B_{k,k-1}$ be the component of $B$ of bidegree $(k,k-1)$.

Lemma 9.5**.**

For all $k$,

[TABLE]

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 ${\bf 1}_{\Delta}T=0$. Fix $j,k$ and let

[TABLE]

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 $j+k\leq N$. Thus the total antiholomorphic degree is $\leq N-n+N-1$. 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 $\ell\geq p$ that ${\bf 1}_{\Delta}T_{\ell}=0$. 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), ${\bf 1}_{\Delta}T_{\ell}$ has its support in $\Delta\cap(Z_{\ell}\times Z)\simeq Z_{\ell}\times\{pt\}$, which has codimension $\geq N+\ell+1$, see (6.3). On the other hand, the total antiholomorphic degree is $\leq\ell+j+k-1\leq\ell+N-1$, so the current vanishes by the dimension principle. We conclude that (9.14) holds. ∎

By the same argument555There is a sign error in [6, (5.2)] due to $R(z)\wedge dz$ being odd with respect to the super structure. Since we here move $R(z)\wedge dz$ to the right, we get the correct sign. as for [6, (5.2)] we have the equality

[TABLE]

also for our $R$, where $[\Delta]^{\prime}$ denotes the part of $[\Delta]$ where $d\eta_{i}=d\zeta_{i}-dz_{i}$ has been replaced666This change is due to the fact that we do the same change of the differentials in the definition of $H$ and $B$ above. by $d\zeta_{i}$. In view of (9.14) we can put $\lambda=0$ in (9.15), and then we get

[TABLE]

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

[TABLE]

or equivalently,

[TABLE]

Multiplying by suitable holomorphic $\xi_{0}$ in $E_{p}^{*}$ such that $f^{*}_{p+1}\xi_{0}=0$, cf.  Proposition 6.7, we see that $\phi\wedge h=\dbar(K\phi\wedge h)+K(\dbar\phi)\wedge h+P\phi\wedge h$ for all $h$ in {\scalebox{1.3}{\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 $\text{Dom}\,\dbar_{X}$ 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 $\text{Dom}\,\dbar_{X}$ since all the other terms but $-\nabla_{f}(K\phi\wedge\omega)$ are in $\mathcal{W}^{n,*}_{X}$.

9.2. Intrinsic interpretation of $K$ and $P$

So far we have defined $K$ and $P$ by means of currents in ambient space. We used this approach in order to avoid introducing push-forwards on a non-reduced space. However, we will sketch here how this can be done. We must first define the product space $X\times X^{\prime}$. Given a local embedding $i:X\to\Omega$ as before, we have an embedding $(i\times i)\colon X\times X^{\prime}\to\Omega\times\Omega^{\prime}$ such that the structure sheaf is $\mathscr{O}_{\Omega\times\Omega^{\prime}}/(\mathcal{J}_{X}+\mathcal{J}_{X^{\prime}})$. One can check that this sheaf is independent of the chosen embedding, i.e., $\mathscr{O}_{X\times X^{\prime}}$ is intrinsically defined. Thus we also have definitions of all the various sheaves on $X\times X^{\prime}$ like $\mathscr{E}_{X\times X^{\prime}}^{0,*}$. The projection $p\colon X\times X^{\prime}\to X^{\prime}$ is determined by $p^{*}\phi\colon\mathscr{O}_{X^{\prime}}\to\mathscr{O}_{X\times X^{\prime}}$, which in turn is defined so that $p^{*}i^{*}\Phi=(i\times i)^{*}\pi^{*}\Phi$ for $\Phi$ in $\mathscr{O}_{\Omega^{\prime}}$, where $\pi\colon\Omega\times\Omega^{\prime}\to\Omega^{\prime}$ 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 $p_{*}\colon{\mathcal{C}}_{X\times X^{\prime}}^{2n,*+n}\to{\mathcal{C}}_{X^{\prime}}^{n,*},$ and clearly

[TABLE]

As before we have the isomorphism

[TABLE]

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

[TABLE]

Now, take $h$ in {\scalebox{1.3}{\omega}}^{n}_{X^{\prime}} and let $\mu=i_{*}h$. Then, cf. the proof of Lemma 9.2,

[TABLE]

Thus we can define $K\phi$ intrinsically by

[TABLE]

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

[TABLE]

It is natural to write

[TABLE]

although the formal meaning is given by (9.19) and (9.20).

10. Regularity of solutions on $X_{reg}$

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

Theorem 10.1**.**

If $\phi$ in $\mathcal{W}^{0,*}_{X}$ is smooth near a point $x\in X_{\rm reg}^{\prime}$, 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

[TABLE]

Notice that here we cut away the diagonal $\Delta^{\prime}$ in $Z\times Z^{\prime}$ times $\mathbb{C}_{\tau}\times\mathbb{C}_{w}$ in contrast to Proposition 9.4, where we only cut away the diagonal $\Delta$ in $\Omega\times\Omega^{\prime}$.

Proof.

Clearly $B^{\epsilon}$ is smooth so that each $K^{\epsilon}\phi$ is smooth in a full neighborhood in $\Omega^{\prime}$ of $x$. Let $T=\mu(z,w)\wedge(HR(\zeta,\tau)\wedge B\wedge g)_{N}\wedge\phi$, and let $W=\Delta^{\prime}\times\mathbb{C}_{\tau}\times\mathbb{C}_{w}$. Since $\mu(z,w)\otimes R(\zeta,\tau)$ has support on $\{w=\tau=0\}$, $T={\bf 1}_{\{w=\tau=0\}}T$. Therefore, ${\bf 1}_{W}T={\bf 1}_{W}{\bf 1}_{\{w=\tau=0\}}T=0$ since $W\cap\{w=\tau=0\}\subset\Delta$ and ${\bf 1}_{\Delta}T=0$ by definition, cf. Proposition 2.1 (i). Now notice that ${\bf 1}_{W}T=0$ implies (9.11) and in turn (9.9) with our present choice of $B^{\epsilon}$. ∎

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 $R=\dbar(1/{\tau}^{m+1})$. For an arbitrary point $(z,w)$ we can choose the Hefer form

[TABLE]

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

[TABLE]

Let $\Omega^{\prime}\subset\subset\Omega$ 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 $\overline{\Omega^{\prime}}$. Then we can choose a holomorphic weight $g=\varphi+\cdots$, see, [6, Example 5.1] with respect to $\Omega^{\prime}$, 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

[TABLE]

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

[TABLE]

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 $\Omega^{\prime}$ such that

[TABLE]

It follows that

[TABLE]

Notice that

[TABLE]

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

[TABLE]

A simple computation yields that

[TABLE]

where

[TABLE]

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

[TABLE]

It is well-known that these Cauchy integrals $a_{k}(z)$ are smooth solutions to $\dbar v=\hat{\phi}_{k}d\bar{z}$ in $Z^{\prime}=Z\cap\Omega^{\prime}$. Thus $K\phi$ is smooth. ∎

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 $\epsilon\to 0$. For a simple example, take $\phi=\zeta d\bar{\zeta}\otimes\tau^{m}$. Then $K^{\epsilon}\phi(0,w)$ tends to

[TABLE]

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 $K\phi(z,w)=\lim_{\epsilon\to 0^{+}}K^{\epsilon}\phi(z,w)$ 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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

if $\beta\geq m-k$. 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. ∎

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

[TABLE]

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

[TABLE]

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 $x\in X^{\prime}_{\rm reg}$, 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

[TABLE]

from $\partial^{\ell}_{w}v^{\epsilon}|_{w=0}$, $|\ell|\leq M$ 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 $\hat{v}_{j}^{\epsilon}\to\hat{v}_{j}$ 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), $v^{\epsilon}\wedge\mu\to v\wedge\mu$ for all $\mu$ in ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}^{Z}_{\Omega})$. 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. ∎

Proof of Proposition 10.5.

Assume that $X$ is embedded in $\Omega\subset\mathbb{C}^{N}_{\zeta^{\prime},\tau^{\prime}}$. After a suitable rotation we can assume that $Z$ is the graph $\tau^{\prime}=\psi(\zeta^{\prime})$. The Bochner-Martinelli kernel in $\Omega$ is rotation invariant, so it is

[TABLE]

where

[TABLE]

We now choose the new coordinates $\zeta=\zeta^{\prime},\ {\tau}={\tau}^{\prime}-\psi(\zeta^{\prime})$ 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 ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega}^{Z})$. Thus we are to compute $\partial_{w}^{\ell}|_{w=0}$ of integrals like

[TABLE]

where $k\leq n$ 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

[TABLE]

since $\bar{\tau}$ and $d\bar{\tau}$ annihilate $\dbar(1/\tau^{\alpha+1})$, 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

[TABLE]

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

[TABLE]

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

[TABLE]

Lemma 10.6**.**

Let

[TABLE]

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

[TABLE]

If we have a modification $\pi\colon\tilde{W}\to\Omega\times\Omega$ such that locally $\pi^{*}\eta=\eta_{0}\eta^{\prime}$, where $\eta_{0}$ is a holomorphic function, then

[TABLE]

where $\beta$ is smooth.

Recall that if $a$ is a form, then $L_{\xi}a=d(\xi\neg a)+\xi\neg(da)$, and that $L_{\xi}(\beta\neg a)=[\xi,\beta]\neg a+\beta\neg(L_{\xi}a)$ 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

[TABLE]

Notice that $\gamma_{k}$ is a sum of terms like

[TABLE]

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

[TABLE]

where $\nu_{j}$ and $b_{j}$ are smooth. Since

[TABLE]

the lemma now follows. ∎

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

[TABLE]

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

Lemma 10.7**.**

Let

[TABLE]

where $a_{1},\dots,a_{r}$ 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 $i=1,\dots,\rho$, $\phi$ is a test form with support close to $z$, and $\Phi:=\eta^{*}(I+A^{*}A)\eta$. If $r\geq 1$ and $r+s\geq\ell+1$, then we have the relation

[TABLE]

when $\epsilon\to 0$.

Proof.

If

[TABLE]

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

[TABLE]

Thus

[TABLE]

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

[TABLE]

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

[TABLE]

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

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

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

[TABLE]

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 $z$. ∎

11. A fine resolution of $\mathscr{O}_{X}$

We first consider a generalization of Theorem 9.1.

Lemma 11.1**.**

*Assume that $\phi\in\mathcal{W}^{0,k}(X)\cap\mathscr{E}_{X}^{0,k}(X_{reg})\cap\text{Dom}\,\dbar_{X}$ and that $K\phi$ is in $\text{Dom}\,\dbar_{X}$ (or just in $\text{Dom}\,\dbar$). Then (9.1) holds on $X^{\prime}$. *

Proof.

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

[TABLE]

for $z\in X^{\prime}_{reg}$. Clearly the left hand side tends to $\phi\wedge h$ when $\delta\to 0$. From Lemma 9.2 it follows that $K(\chi_{\delta}\phi)\wedge h\to K\phi\wedge h$. Thus the first term on the right hand side of (11.1) tends to $\dbar(K\phi)\wedge h$. In the same way the second and third terms on the right hand side tend to $K(\dbar\phi)\wedge h$ and $P\phi\wedge h$, respectively. It remains to show that the last term tends to [math]. If $z$ belongs to a fixed compact subset of $X_{\rm reg}^{\prime}$, then $B$ is smooth in (9.5) when $\zeta$ is in $\operatorname{supp}\dbar\chi_{\delta}$ for small $\delta$. Hence it suffices to see that

[TABLE]

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

[TABLE]

or equivalently, $\omega(\zeta)\wedge\dbar\chi_{\delta}\wedge\phi(\zeta)\to 0,$ which follows by Lemma 8.4 since $\phi$ is in $\text{Dom}\,\dbar_{X}$. We have thus proved that

[TABLE]

The first term on the right hand side is equal to $\dbar(\chi_{\delta}K\phi\wedge h)-\dbar\chi_{\delta}\wedge K\phi\wedge h$, where the latter term tends to [math] if $K\phi$ is in $\text{Dom}\,\dbar_{X}$ or just in $\text{Dom}\,\dbar$, cf.  Lemma 8.4. Thus we get

[TABLE]

which precisely means that (9.1) holds. ∎

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

[TABLE]

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 $\mathscr{A}_{X}^{*+1}\to\mathscr{A}_{X}^{*}$.

Lemma 11.3**.**

If $\phi$ is in $\mathscr{A}_{X}$, then $\phi$ and $K\phi$ are in $\text{Dom}\,\dbar_{X}$.

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 $X_{\rm reg}$ it is enough by Lemma 8.4 to check that $\dbar\chi_{\delta}\wedge\omega\wedge\phi\to 0,$ and this precisely follows from [6, Lemma 6.4]. ∎

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

[TABLE]

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 $X_{\rm reg}$. Since we have access to Koppelman formulas, precisely as in the proof of [6, Theorem 1.2] we can verify that $\dbar\colon\mathscr{A}_{X}^{k}\to\mathscr{A}_{X}^{k+1}$.

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 $\mathscr{A}_{X}^{k}$, $k\geq 1$, and is holomorphic in $z$ when $k=0$. Assume that $\phi$ is in $\mathscr{A}_{X}^{k}$ and $\dbar\phi=0$. If $k\geq 1$, then $\dbar K\phi=\phi$, and if $k=0$, then $\phi=P\phi$. ∎

11.1. Global solvability

Assume that $E\to X$ 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 $i\colon X\to\Omega$ and a holomorphic vector bundle $F\to\Omega$, then $F$ defines a vector bundle $i^{*}F\to X$. 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

[TABLE]

Thus, if $\phi\in\mathscr{A}^{q+1}(X,E)$, $\dbar\phi=0$, and its canonical cohomology class vanishes, then the equation $\dbar\psi=\phi$ 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 $i\colon X\to{\mathbb{P}}^{N}$, then the $\dbar$-equation is solvable, e.g., if $E$ is a sufficiently ample line bundle.

12. Locally complete intersections

Let us consider the special case when $X$ locally is a complete intersection, i.e., given a local embedding $i\colon X\to\Omega\subset\mathbb{C}^{N}$ there are global sections $f_{j}$ of $\mathscr{O}(d_{j})\to{\mathbb{P}}^{N}$ 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 ${\mathcal{H}om}(\mathscr{O}_{\Omega}/\mathcal{J},\mathcal{{CH}}_{\Omega})$ is generated by the single current

[TABLE]

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 $\dbar\phi=\psi$ if and only if $\dbar(\phi\wedge\mu)=\psi\wedge\mu$.

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 $\dbar$-equation $\dbar\psi=\phi$. Their currents $\phi$ correspond to our $\phi$ in $\mathscr{E}_{X}^{0,q}$ and their $\dbar$-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 $\dbar$-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 $\dbar\psi=\phi$ at least on $X_{reg}$, if $1\leq q\leq n-1$. When $q=n$ a necessary obstruction term occurs. However, their meaning of “$\dbar$-closed” is that locally there is a representative $\Phi$ of $\phi$ and smooth $g_{j}$ such that $\dbar\Phi=g_{1}f_{1}+\cdots+g_{p}f_{p}$. If this holds, then clearly $\dbar\phi=0$. 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 $\dbar\phi=0$, neither do we know whether their solution admits some intrinsic extension across $X_{sing}$ as a current on $X$. ∎

Example 12.2*.*

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

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