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Estimates for the \({\bar{\partial }}\)-Equation on Canonical Surfaces

  • Mats Andersson
  • Richard LärkängEmail author
  • Jean Ruppenthal
  • Håkan Samuelsson Kalm
  • Elizabeth Wulcan
Open Access
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Abstract

We study the solvability in \(L^p\) of the \({\bar{\partial }}\)-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case \(p=2\) for two natural closed extensions \({\bar{\partial }}_s\) and \({\bar{\partial }}_w\) of \({\bar{\partial }}\). For \({\bar{\partial }}_s\) we have solvability, whereas for \({\bar{\partial }}_w\) there is solvability if and only if a certain boundary condition \((*)\) is fulfilled at the singularity. Our main tool is certain integral operators for solving \({\bar{\partial }}\) introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.

Keywords

Cauchy–Riemann equations Canonical surface Koppelman formulas \(L^p\)-estimates Singular complex spaces 

Mathematics Subject Classification

32A26 32A27 32B15 32C30 32W05 

1 Introduction

The classical Dolbeault–Grothendieck lemma states that locally in \(\mathbb {C}^n\) one can solve the \({\bar{\partial }}\)-equation \({\bar{\partial }}u = \varphi \) if \(\varphi \) is a \({\bar{\partial }}\)-closed \((0,r)\)-form or current. One can obtain a solution u by a Koppelman formula; then u is obtained through multiplication of \(\varphi \) with a smooth form followed by convolution with an integrable form, the so-called Bochner–Martinelli form. Thus, one even gains some regularity; in particular, one can solve \({\bar{\partial }}\) in \(C^\infty \), \(L^p\), \(C^\alpha \), Sobolev-spaces, etc, see, e.g., [21] or [12]. On singular varieties this is not true in general. There are smooth \({\bar{\partial }}\)-closed forms which have no local smooth \({\bar{\partial }}\)-potentials, see, e.g., [22, Beispiel 1.3.4] and [4, Example 1].

Solvability of the \({\bar{\partial }}\)-equation on singular varieties has been studied in various articles, starting with among others [10, 20], and in recent years solvability in \(L^2\) has been of particular focus, see, e.g., [9, 18, 25]. There are known examples where the \({{\bar{\partial }}}\)-equation is not locally solvable in \(L^p\), for example when \(p = 1\) or \(p = 2\). On homogeneous varieties, obstructions for solvability in \(L^{p}\) have been described explicitly in [24].

In this paper, we study solvability in \(L^p\) of the \({\bar{\partial }}\)-equation in a neighborhood of a canonical singularity on a complex surface. On a surface, a singularity is canonical if and only if it is a rational double point. Such points are well-studied and have been classified a long time ago as the so-called du Val singularities, see, e.g., the survey [8]. The possible singularities are of type \(A_n\), \(n\ge 1\), \(D_n\), \(n\ge 4\), \(E_6\), \(E_7\) and \(E_8\), and can be realized as isolated hypersurface singularities in \(\mathbb {C}^3\).

Throughout the introduction, we assume that X is a surface with one isolated canonical singularity. We will further assume that \(X=\{f=0\}\subset \Omega '\), where \(\Omega ' \subset \subset \mathbb {C}^3\) is an open pseudoconvex set and f is holomorphic in a neighborhood of \(\Omega '\) and that \(\mathrm{{d}}f\ne 0\) on \(\{ f = 0 \}\) except at the singular point, which we assume is 0.

Let \({\bar{\partial }}_{sm}\) be the \({\bar{\partial }}\)-operator on smooth \((0,r)\)-forms which have support not intersecting the singularity at the origin. We will consider two extensions of \({\bar{\partial }}_{sm}\) as a closed operator on \(L^p(X)\). One of them is the minimal closed extension, i.e., the strong extension \({\bar{\partial }}_s^{(p)}\) of \({\bar{\partial }}_{sm}\), which is the graph closure of \({\bar{\partial }}_{sm}\) in \(L^p_{0,r}(X) \times L^p_{0,r+1}(X)\). That is, \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)} \subset L^p_{0,r}(X)\) if and only if there is a sequence of smooth forms \(\varphi _j \in L^p_{0,r}(X)\) with \({{\,\mathrm{supp}\,}}\varphi _j \cap \{0\} = \emptyset \) such that
$$\begin{aligned} \varphi _j \rightarrow \varphi \ \ \ \text{ in } \ \ L^p_{0,r}(X),\quad {\bar{\partial }}\varphi _j \rightarrow {\bar{\partial }}\varphi \ \ \ \text{ in } \ \ L^p_{0,r+1}(X). \end{aligned}$$
The other extension is the maximal closed extension, i.e., the weak \({\bar{\partial }}\)-operator \({\bar{\partial }}_w^{(p)}\), so that \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\subset L^p_{0,r}(X)\) if and only if \({\bar{\partial }}\varphi \in L^p(X)\).1 When it is clear from the context, we will drop the superscript (p) in \({\bar{\partial }}^{(p)}_s\) and \({\bar{\partial }}^{(p)}_w\).
Let \(\omega _X\) be the Poincaré residue of \(dz_1{\wedge }dz_2{\wedge }dz_3/f\). It is an intrinsic \({\bar{\partial }}\)-closed meromorphic (2, 0)-form on X that is holomorphic outside of 0. We will see below (Proposition 3.3 and Corollary 3.5) that there is a number \(2 < q(X) \le 4\) such that \(\omega _X \in L^q(X)\) for \(q<q(X)\). Let p(X) be the dual exponent of q(X) and let
$$\begin{aligned} {\hat{p}}(X)=\frac{4p(X)}{4-p(X)}. \end{aligned}$$
Notice that \(4/3 \le p(X)<2\) and \(2 \le {\hat{p}}(X)<4\). For precise definitions of \(L^p\)-forms and \(C^\alpha \)-forms on X, see Sect. 2.1.

In our results, we have the following condition:

If \(\varphi \) is a (0, 1)-form in \({{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\), where \(p(X) < p \le \infty \), then it is said to satisfy the condition \((*)\) if
for some sequence of cut-off functions \(\{\chi _k\}_k\), where each\(\chi _k\) is 1 in a neighborhood of 0 and the support of \(\chi _k\)approaches \(\{0\}\) when \(k \rightarrow \infty \).
This condition is independent of the sequence of cut-off functions, see Sect. 4.1, and is thus a kind of boundary condition at \(\{0\}\). If \(\varphi \) is \({\bar{\partial }}\)-closed, as in the following theorem, by Stokes’ theorem the condition \((*)\) means that
$$\begin{aligned} \int _X \omega _X{\wedge }{\bar{\partial }}\chi \wedge \varphi =0 \end{aligned}$$
(1.1)
for some smooth cutoff function \(\chi \) that is 1 in a neighborhood of 0.

Theorem 1.1

Let X be a surface as above with an isolated canonical singularity at 0.
  1. (i)

    Assume that \(p(X)<p\le 4\). If \(\varphi \) is a \({\bar{\partial }}_s\)-closed \((0,r)\)-form in \(L^p(X)\), \(r=1,2\), then there is u in the domain of \({\bar{\partial }}_s^{(p)}\) in a neighborhood of 0 such that \({\bar{\partial }}_s u=\varphi \).

     
  2. (ii)

    Assume that \({\hat{p}}(X)<p\le \infty \). If \(\varphi \) is a \({\bar{\partial }}_w\)-closed (0, 1)-form in \(L^p(X)\), then there is a solution in \(L^p\) to \({\bar{\partial }}_w u=\varphi \) in a neighborhood of 0. If \(p=\infty \), then one can choose u in \(C^\alpha \) for \(\alpha < 4/p(X)-2\). If \(\varphi \) is a (0, 2)-form the same holds for \(p(X) < p \le \infty \).

     
  3. (iii)

    Assume that \(p(X) < p\le {\hat{p}}(X)\). If \(\varphi \) is a \({\bar{\partial }}_w\)-closed (0, 1)-form in \(L^p(X)\), then there is a solution in \(L^p\) to \({\bar{\partial }}_w u=\varphi \) in a neighborhood of 0 if and only if \(\varphi \) satisfies the condition \((*)\).

     

Notice that if \({\bar{\partial }}_w u=\varphi \), then (1.1) follows from Stokes’ theorem since \(\omega _X{\wedge }{\bar{\partial }}\chi \) is a \({\bar{\partial }}\)-closed smooth form with compact support. Thus, the condition \((*)\) is necessary in the theorem. It turns out that \((*)\) is automatically fulfilled when \({\hat{p}}(X) < p \le \infty \), see the comment after the proof of Theorem 1.5. In Sect. 5 we study the condition \((*)\) explicitly for the various types of canonical singularities. Theorem 5.1 asserts that in the case of a singularity of type \(A_n\), \(n\ge 1\), any form \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w \subset L^2_{0,r}(X)\) satisfies \((*)\). For each of the other singularities, that is, of type \(D_n\), \(n\ge 4\), \(E_6\), \(E_7\) and \(E_8\), however, there is a (0, 1)-form \(\varphi \in \ker {\bar{\partial }}_w\subset L^2(X)\) such that the equation \({\bar{\partial }}_w u =\varphi \) has no solution in a neighborhood of 0, see Theorem  5.6. It follows that for these \(\varphi \) the condition \((*)\) is not satisfied.

To the best of our knowledge, the only known cases of Theorem 1.1 for general surfaces with canonical singularities are the following: Part (i) for \(p=2\) was proven in [26, Corollary  1.3]. Part (ii) for \(p=2\) and (0, 2)-forms was proven in [17, Theorem 4.3], which builds on the vanishing result from [28]. Some weaker versions of part (ii) are known as well. For \(\varphi \) with compact support, it was proven that one can find solutions in \(L^p\) (for arbitrary p) or with \(C^\alpha \)-estimates in [27]. Moreover, for continuous (0, 1)-forms \(\varphi \) with compact support, \(C^\alpha \)-estimates for solutions were obtained in [1, 2].

Various results are known for the \(A_1\)-singularity, as is detailed in the introduction of [14]. That there are obstructions to solving \({\bar{\partial }}_w\) in \(L^2\) on the \(D_4\)-singularity was proven in [19, Proposition 4.13].

As mentioned above, a large part of the study of the \({\bar{\partial }}\)-equation on singular varieties has been restricted to \(L^2\)-spaces. Integral formulas open up for new results about solvability in \(L^p\)-spaces for \(p \ne 2\), as well as other norms. For the proof of Theorem 1.1, our main tool is an integral operator introduced in [3, 4]. Keeping the notation above, let \(\Omega \subset \subset \Omega '\) be an open set containing 0 and let \(D=X\cap \Omega \). There is an operator \({\mathcal K}:C^\infty _{0,r}(X)\rightarrow C^\infty _{0,r-1}(D\setminus \{0\})\)\(r=1,2\), such that
$$\begin{aligned} \varphi ={\bar{\partial }}{\mathcal K}\varphi +{\mathcal K}{\bar{\partial }}\varphi \end{aligned}$$
(1.2)
on \(D\setminus \{0\}\). The operator is given by an intrinsic integral kernel \(K(\zeta ,z)\) on \(X\times D\setminus \{0\}\) that contains the Poincaré residue \(\omega _X\) as a factor in the first variable. In [4] it was proved that \({\mathcal K}\) and (1.2) can be extended to certain fine sheaves \({\mathcal A}_X^r\) of currents defined across 0 and coinciding with \(C^\infty _{0,r}\) outside 0, so that \({\bar{\partial }}u=\varphi \) is solvable in \({\mathcal A}_X\) as soon as \({\bar{\partial }}\varphi =0\).

To prove Theorem 1.1, we have to extend \({\mathcal K}\) and (1.2) to \(L^p\). To this end we first consider mapping properties of \({\mathcal K}\).

Theorem 1.2

The integral operator \({\mathcal K}\) extends to compact operators
$$\begin{aligned} {\mathcal K}:L^p_{0,r}(X) \rightarrow L^p_{0,r-1}(D), \ \ p(X)< p <\infty \end{aligned}$$
(1.3)
and
$$\begin{aligned} {\mathcal K}:L^\infty _{0,r}(X) \rightarrow C^\alpha _{0,r-1}(D), \ \ 0\le \alpha < 4/p(X)-2. \end{aligned}$$
(1.4)

Since the sheaves \({\mathcal A}_X^r\) are quite implicitly defined and its sections must have singularities at \(X_{sing}\) in general, it is interesting to note the following consequence of (1.4).

Corollary 1.3

For X as above we have that
$$\begin{aligned} \mathcal {A}_X^r\subset C^\alpha _{X,0,r}, \ \ \ \quad 0 \le \alpha < 4/p(X)-2. \end{aligned}$$

To obtain solutions to the \({\bar{\partial }}_s\)-equation in \(L^p\) we extend (1.2) by approximating \(\varphi \) by smooth forms with support away from 0. If \(\varphi \) is in the domain of \({\bar{\partial }}_s^{(p)}\), it follows that (1.2) holds, so if \({\bar{\partial }}\varphi =0\) we get the solution \(u={\mathcal K}\varphi \) to \({\bar{\partial }}u=\varphi \). The problem is to see that u is in the domain of \({\bar{\partial }}_s^{(p)}\). This is “harder” for large p and our upper bound is 4.

Theorem 1.4

Assume that \(p(X)<p\le 4\). If \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)} \subset L^p_{0,r}(X)\), then \(\mathcal {K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\) and
$$\begin{aligned} \varphi (z) = {\bar{\partial }}_s \mathcal {K} \varphi (z) + \mathcal {K} {\bar{\partial }}_s \varphi (z), \quad r=1,2. \end{aligned}$$
(1.5)

In case of \({\bar{\partial }}_w\) we have basically the opposite problem. Since a priori we have no approximation by smooth forms with support away from the origin it is “harder” to obtain the extension of (1.2) for small p, while it then directly follows from Theorem 1.2 that the solution is in the domain of \({\bar{\partial }}_w\).

Theorem 1.5

Assume that \(p(X)< p\le \infty \). If \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)} \subset L^p_{0,2}(X)\), then \(\mathcal {K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\) and
$$\begin{aligned} \varphi (z) = {\bar{\partial }}_w \mathcal {K} \varphi (z) + \mathcal {K} {\bar{\partial }}_w \varphi (z). \end{aligned}$$
(1.6)
The same holds for \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)} \subset L^p_{0,1}(X)\) if \({\hat{p}}(X) < p \le \infty \). If \(p(X)< p\le {\hat{p}}(X)\), and in addition \(\varphi \) satisfies the condition \((*)\), then the same conclusion holds.

Notice that Theorem 1.1 follows from Theorems 1.21.4 and 1.5 and the discussion about the necessity of the condition \((*)\) after the theorem.

Notice that if \(\varphi \) is a \({\bar{\partial }}\)-closed (0, 1)-form with compact support then it automatically satisfies \((*)\), and so we can solve \({\bar{\partial }}_w u=\varphi \) in \(L^p\) if \(p(X)<p\le \infty \). By means of a slight variation of the operator \({\mathcal K}\), introduced in [3], one can even get a solution with compact support. In case \(\varphi \) is a (0, 2)-form in \(L^p(X)\) with compact support and \({\hat{p}}(X) < p \le \infty \), then there is a solution with compact support if and only if
$$\begin{aligned} \int _X \varphi {\wedge }\ h\omega _X=0 \quad \text { for all } h\in \mathcal {O}(X), \end{aligned}$$
(1.7)
see Theorem 4.2 below.

Our interest in canonical singularities is partly motivated by the earlier works [14, 15], where similar results as above are studied for affine cones over projective complete intersections. The results about solvability in \(L^p\) obtained in these articles are in case the degree of these homogeneous varieties is small enough. Here, it is interesting to note that the degree is small if the singularities are mild in the sense of the minimal model program. It turned out that positive results about solvability in \(L^2\) hold precisely for the varieties with canonical singularities, see [15].

The results in this article overlap with results from [14, 15] only in the case of the \(A_1\)-singularity, where in [14, 15], it was shown that the \({\bar{\partial }}_w\)- and \({\bar{\partial }}_s\)-equations are locally solvable in \(L^p\) unconditionally for p in certain intervals. On a general canonical surface, as studied in this article, solvability depends on the condition \((*)\). The main novelty is the understanding of this condition and a quite sharp non-trivial estimate of the integral kernels from [4] on such a surface. The final estimate of the integral operators is done along the same lines as in [14, 15].

We now consider the case of functions. There is an integral operator \(\mathcal {P}:C^\infty _{0,0}(X)\rightarrow \mathcal {O}(D)\) in [3, 4] such that
$$\begin{aligned} \varphi ={\mathcal K}{\bar{\partial }}\varphi +\mathcal {P}\varphi \end{aligned}$$
(1.8)
on \(D\setminus \{0\}\). In order to formulate the following result about extension of (1.8) to \(L^p\) we need a condition \((*)\) for functions \(\varphi \) that is explained in Sect. 4.2 below.

Theorem 1.6

Let X be as above. Then, the operator \(\mathcal {P}\) extends to a compact operator \(\mathcal {P} : L^p_{0,0}(X) \rightarrow \mathcal {O}(D)\) for \(1 \le p \le \infty \). If \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}^{(p)}_s \subseteq L^p_{0,0}(X)\) where \(p(X) < p \le 4\), then (1.8) holds. This equality also holds if \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}^{(p)}_w \subseteq L^p_{0,0}(X)\) and either \(\hat{p}(X) <p \le \infty \) or \(p(X) < p \le \hat{p}(X)\) and \(\varphi \) satisfies the condition \((*)\).

The present paper is organised as follows. After providing some preliminaries in Sect. 2, in Sect. 3 we recall the integral formulas from [3, 4], analyse their integral kernels and prove Theorem  1.2 and its corollary. Section  4 is devoted to \({\bar{\partial }}\)-homotopy formulas and proofs of Theorems 1.4, 1.5 and 1.6 and also to a discussion of condition \((*)\). We also include a discussion of the domain of the \({\bar{\partial }}_X\)-operator from [4] and prove that \(\mathcal K\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_X\) for certain \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s\), see Theorem 4.3. In Sect. 5, we characterize the du Val singularities with respect to \((*)\). Finally, we recall some integral estimates on singular varieties from [15] in an appendix, Sect. 6.

2 Preliminaries

In this section, we specify the spaces of differential forms that we consider and explain some basic tools. Throughout the section \(i:X \hookrightarrow \Omega '\subset \mathbb {C}^N\) is an analytic variety of pure dimension n, and \(D \subset \subset X\) is an open subset of X.

2.1 \(C^\alpha \)- and \(L^p\)-Forms on an Analytic Variety

Let \(1\le p\le \infty \). Since \(D^* := D \cap X_{\mathrm{{reg}}}\) is a submanifold of some open subset of \(\mathbb {C}^N\), it inherits a Hermitian metric from \(\mathbb {C}^N\). We say that a \((0,r)\)-form \(\varphi \) is in \(L^p_{0,r}(D)\) if \(\varphi |_{D^*}\) is in \(L^p_{0,r}(D^*)\) with respect to the induced volume form \(\mathrm{{d}}V_X\). When it is clear from the context, we will drop the subscript in \(L^p_{0,r}(D)\).

It will be convenient to represent \((0,r)\)-forms on X in a certain “minimal” manner: Any \((0,r)\)-form \(\varphi \) on \(D^*\) can be written (uniquely) in the form
$$\begin{aligned} \varphi = \sum _{|I|=r} \varphi _I \mathrm{{d}}{\bar{z}}_I, \end{aligned}$$
(2.1)
where \(d{{\bar{z}}}_I=\mathrm{{d}}{\bar{z}}_{i_1}\wedge \cdots \wedge \mathrm{{d}}{\bar{z}}_{i_r}\) if \(I=\{i_1,\ldots , i_r\}\), such that
$$\begin{aligned} |\varphi |^2 (z)= {2}^r\sum |\varphi _I|^2(z) \end{aligned}$$
(2.2)
at each point \(z\in D^*\). In fact, one starts with any representation and then at each point takes the orthogonal projection of the form onto \(\Lambda ^{0,r}T^*D^*\), see, e.g., [23, Lemma 2.2.1].
In particular, then \(\varphi \in L^p_{0,r}(D)\) if and only if \(\varphi _I \in L^p(D)\) for all I. If one has an arbitrary representation of \(\varphi \) of the form (2.1), then
$$\begin{aligned} |\varphi |^2 (z) \le {2}^r\sum |\varphi _I|^2(z), \end{aligned}$$
(2.3)
and so, in general, \(\varphi \in L^p_{0,r}(D)\) if \(\varphi _I \in L^p(D)\) for all I.

Recall that a form \(\varphi \) on D is in \(C^k(D)\), \(0\le k\le \infty \), if (locally) it is the pullback of a \(C^k\)-form in ambient space; i.e., there exists a representation (2.1) such that all the coefficients (locally) admit \(C^k\)-extensions to a neighborhood of D. For \(0 \le \alpha < 1\), we say that a \((0,r)\)-form \(\varphi \) on D is \(C^\alpha \) if locally on D there is a representation (2.1) such that all the coefficients \(\varphi _I\) are \(C^\alpha \), i.e., Hölder continuous with exponent \(\alpha \), on D. It is well-known, that a function that is \(C^\alpha \) on D has a \(C^\alpha \)-extension to ambient space, see, e.g., [16]. Thus, a form \(\varphi \) on D is in \(C^\alpha \) if and only if it is the pull-back to D of a \(C^\alpha \)-form in ambient space. Notice that \(C^1(D)\subset C^\alpha (D)\) for all \(\alpha <1\). For \(\alpha = 1\), we denote the Lipschitz continuous functions by \(C^{0,1}(D)\) to avoid conflict of notation with continuously differentiable functions.

It is not hard to check that these definitions are independent of the choice of embedding of X, and hence are intrinsic notions on X. Fix an embedding \(D\rightarrow \Omega \subset \mathbb {C}^{N}\). We can then define the Hölder-norm
$$\begin{aligned} \Vert \varphi \Vert ^2_\alpha = \inf {2}^r\sum \Vert \varphi _I\Vert ^2_{\alpha }, \end{aligned}$$
(2.4)
of a form \(\varphi \) on D, where the infimum runs over all representations (2.1) of \(\varphi \) in ambient space, and the norms on the right-hand side of (2.4) are over D. This norm is, up to constants, independent of the embedding.

Remark 2.1

Regularity properties of \(\varphi \) like smoothness, Hölder continuity etc, will be reflected by the coefficients on \(D^*\) in the minimal representation (2.2) above. However, even if \(\varphi \) is smooth across the singularity, the coefficients in the minimal representation may be discontinuous there.

Using the minimal representation (2.2), and the inequality (2.3) for not necessarily minimal representations, and the analogous inequality for Hölder norms, we get the following lemma.

Lemma 2.2

If \(\mathcal {K}\) is an integral operator mapping \((0,r)\)-forms in \(\zeta \) to \((0,r-1)\)-forms in z, defined by an integral kernel
$$\begin{aligned} K(\zeta ,z) = \sum _{|L|=n,|I|=r-1,|J|={n-r}} K_{I,J,L}(\zeta ,z) \mathrm{{d}}{\overline{z}}_I \wedge {\mathrm{{d}}{\overline{\zeta }}_J} \wedge \mathrm{{d}}\zeta _L, \end{aligned}$$
then \(\mathcal {K}\) is a bounded linear map \(L_{0,r}^p(X) \rightarrow L_{0,r-1}^p(D)\) if
$$\begin{aligned} f(\zeta ) \mapsto \int _{X} K_{I,J,L}(\zeta ,z) f(\zeta ) \mathrm{{d}}V_X(\zeta ) \end{aligned}$$
is a bounded linear map \(L^p(X) \rightarrow L^p(D)\), and a bounded linear map \(L_{0,r}^\infty (X) \rightarrow C_{0,r-1}^\alpha (D)\) if
$$\begin{aligned} f(\zeta ) \mapsto \int _{X} K_{I,J,L}(\zeta ,z) f(\zeta ) \mathrm{{d}}V_X(\zeta ) \end{aligned}$$
is a bounded linear map \(L^\infty (X) \rightarrow C^\alpha (D)\).

2.2 Cut-off Functions

We will use the following cut-off functions to approximate forms by forms with support away from isolated singularities. As in the proof of Proposition 3.3 in [20], let \(\rho _k: \mathbb {R}\rightarrow [0,1]\), \(k\ge 1\), be smooth cut-off functions satisfying
$$\begin{aligned} \rho _k(x)=\left\{ \begin{array}{ll} 1,\ x\le k,\\ 0,\ x\ge k+1, \end{array}\right. \end{aligned}$$
and \(|\rho _k'|\le 2\). Moreover, let \(r: \mathbb {R}_+\rightarrow [0,1/2]\) be a smooth increasing function such that
$$\begin{aligned} r(x)=\left\{ \begin{array}{ll} x,\ &{} x\le 1/4,\\ 1/2,\ &{} x\ge 3/4, \end{array}\right. \end{aligned}$$
and \(|r'|\le 1\). As cut-off functions, we will use \(\mu _k(\zeta ):=\rho _k\big (\log (-\log r(|\zeta |))\big )\) on X if X has an isolated singularity at 0. Note that there is a constant C such that
$$\begin{aligned} \big | {\bar{\partial }}\mu _k(\zeta )\big | \le C\frac{\chi _k(|\zeta |)}{|\zeta | \big | \log |\zeta |\big |}, \end{aligned}$$
(2.5)
where \(\chi _k\) is the characteristic function of \([e^{-e^{k+1}}, e^{-e^k}]\).

Lemma 2.3

[15, Lemma 5.1] Let \(\varphi \in L^p_{0,r}(D)\) with \({\bar{\partial }}_w \varphi \in L^{p'}_{0,r+1}(D)\), where \(\frac{2n}{2n-1}\le p \le \infty \) and \(1\le p' \le \infty \). Let \(\varphi _k :=\mu _k \varphi \) and define \(1\le \lambda \le 2n\) by the relation
$$\begin{aligned} \frac{1}{\lambda } = \frac{1}{p} + \frac{1}{2n}. \end{aligned}$$
(2.6)
If \(\gamma =\min \{\lambda , p'\}\), then \( \varphi _k \rightarrow \varphi \ \text{ in } \ L^p_{0,r}(D),\quad {\bar{\partial }}\varphi _k \rightarrow {\bar{\partial }}_w \varphi \ \text{ in } \ L^\gamma _{0,r+1}(D). \)

2.3 On the Domain of the \({\bar{\partial }}_s\)-Operator

Lemma 2.4

[15, Lemma 5.2] Assume that X has an isolated singularity at \(0\in D\) and that D has smooth boundary. Let \(1\le p \le 2n\) and let \(\varphi \in L^p_{0,r}(D)\) such that \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\). Then \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\) if and only if there exists a sequence of bounded forms \(\varphi _j\in L^\infty _{0,r}(D)\), \(\varphi _j\in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\), such that
$$\begin{aligned} \varphi _j \rightarrow \varphi \ \ \hbox { in } \ L^p_{0,r}(D),\quad {\bar{\partial }}_w \varphi _j \rightarrow {\bar{\partial }}_w \varphi \ \ \text{ in } \ L^p_{0,r+1}(D). \end{aligned}$$
(2.7)

3 Integral Operators on Surfaces with Canonical Singularities

3.1 The Koppelman Integral Kernels for a Hypersurface

Let us recall the definition of the Koppelman integral operators from [4] in the situation of a hypersurface \(i:X \subset \Omega '\subset \mathbb {C}^{n+1}\) defined by \(X = \{ \zeta \in \Omega ';\ f(\zeta ) = 0 \}\), where f is a holomorphic function on \(\Omega '\) and \(\mathrm{{d}}f\) is non-vanishing on \(X_{\mathrm{{reg}}}\), where \(\Omega '\) is pseudoconvex. Let \(\Omega \subset \subset \Omega '\) be an open set and let \(D := X \cap \Omega \).

Let \(\omega _X\) be the Poincaré residue of the meromorphic form \(\mathrm{{d}}\zeta _1 \wedge \ldots \wedge \mathrm{{d}}\zeta _{n+1}/f\). This means that \(\omega _X\) is the unique meromorphic (n, 0)-form on X such that
$$\begin{aligned} \mathrm{{d}}f\wedge \omega _X =2\pi i \mathrm{{d}}\zeta _1 \wedge \cdots \wedge \mathrm{{d}}\zeta _{n+1}. \end{aligned}$$
(3.1)
In [4, Section 3] so-called structure forms were introduced as generalizations of the Poincaré residue for more general X; we will, therefore, refer to \(\omega _X\) as the structure form on X. Recall that 1 / f and \(\omega _X\) define principal value currents on \(\Omega '\) and X, respectively. Identifying these with their respective currents, \(\omega _X\) can be defined as the unique current such that
$$\begin{aligned} i_*\omega _X= {\bar{\partial }}\frac{1}{f}{\wedge }\mathrm{{d}}\zeta _1 \wedge \cdots \wedge \mathrm{{d}}\zeta _{n+1}. \end{aligned}$$
For coordinates \(\zeta =(\zeta _1, \ldots , \zeta _{n+1})\) such that \(\partial f/\partial \zeta _1\) is generically non-vanishing on \(X_{\mathrm{{reg}}}\), \(\omega _{X}\) is the pull-back of
$$\begin{aligned} 2\pi i\frac{\mathrm{{d}}\zeta _2 \wedge \cdots \wedge \mathrm{{d}}\zeta _{n+1}}{\partial f / \partial \zeta _1} \end{aligned}$$
(3.2)
to X. Alternatively, letting
$$\begin{aligned} \vartheta := 2\pi i\sum _{\ell =1}^{n+1} \frac{\overline{f'_{\ell }}}{|\partial f|^2}\frac{\partial }{\partial \zeta _{\ell }}, \end{aligned}$$
(3.3)
where \(f'_{\ell }=\partial f/\partial \zeta _{\ell }\), we have that \(\omega _X\) is realised as the pull-back to X of \(\vartheta \lrcorner \mathrm{{d}}\zeta _1\wedge \cdots \wedge \mathrm{{d}}\zeta _{n+1}\). Here, the norm \(|\partial f|\) is computed in \(\mathbb {C}^{n+1}\), i.e., \(|\partial f|^2=\sum |f_l'|^2\).
Let \(\eta _j=\zeta _j-z_j\) and let \(\delta _\eta \) be interior multiplication by \(2\pi i \sum \eta _j \partial /\partial \zeta _j\). We will consider forms with anti-holomorphic differentials of both \(\zeta \) and z but only holomorphic differentials with respect to \(\zeta \). The (full) Bochner–Martinelli form is \(B := b + b{\wedge }{\bar{\partial }}b + \dots + b{\wedge }({\bar{\partial }}b)^{n},\) where
$$\begin{aligned} b := \frac{1}{2\pi i}\frac{{{\bar{\eta }}}_1 \mathrm{{d}}\zeta _1+\cdots +{{\bar{\eta }}}_{n+1}\mathrm{{d}}\zeta _{n+1}}{|\eta |^2} = \frac{1}{2\pi i}\frac{{\bar{\eta }}\cdot \mathrm{{d}}\zeta }{|\eta |^2}. \end{aligned}$$
Notice that
$$\begin{aligned} B_k:=b{\wedge }({\bar{\partial }}b)^{k-1}=\frac{1}{(2\pi i)^k} \frac{{{\bar{\eta }}}\cdot \mathrm{{d}}\zeta {\wedge }(d{{\bar{\eta }}}{\wedge }\mathrm{{d}}\zeta )^{k-1}}{|\eta |^{2k}}=\mathcal {O}(1/|\eta |^{2k-1}), \end{aligned}$$
(3.4)
where \(d{{\bar{\eta }}}{\wedge }\mathrm{{d}}\zeta =d{{\bar{\eta }}}_1{\wedge }\mathrm{{d}}\zeta _1+\cdots + d\bar{\eta }_{n+1}{\wedge }\mathrm{{d}}\zeta _{n+1}\).

A smooth form \(g=g_{0,0}+\cdots + g_{n+1,n+1}\) in \(\Omega '\times \Omega '\), here lower indices denote bidegree, is a weight with respect to \(\Omega \) if \((\delta _\eta -{\bar{\partial }})g=0\) and \(g_{0,0}(z,z)=1\) for \(z\in {{\overline{\Omega }}}\). We say that g is holomorphic with respect to z if the coefficients are holomorphic in z and there are no anti-holomorphic differentials with respect to z.

Example 3.1

(Holomorphic weights with compact support) Let \(\chi =\chi (\zeta )\) be a cut-off function with compact support in \(\Omega '\) which is 1 in a Stein neighborhood \(\Omega ''\subset \subset \Omega '\) of \({{\overline{\Omega }}}\). Moreover, let \(s(\zeta ,z) = \sum s_i(\zeta ,z) \mathrm{{d}}\zeta _i\) be a (1, 0)-form defined for \(\zeta \in {{\,\mathrm{supp}\,}}{\bar{\partial }}\chi \) and \(z\in {{\overline{\Omega }}}\), such that \(\delta _\eta s = 1\) and s is smooth in \(\zeta \) and holomorphic in z. Such an s exists since \(\Omega ''\) is Stein in \(\Omega '\). Then
$$\begin{aligned} g := \chi - {\bar{\partial }}\chi \wedge \big (s+ s{\wedge }({\bar{\partial }}s) + \dots + s{\wedge }({\bar{\partial }}s)^{n}\big ) \end{aligned}$$
is a weight in \(\Omega '\times \Omega '\) with respect to \(\Omega \) that has compact support in \(\Omega _\zeta '\) and is holomorphic with respect to z. If \(\Omega \) is the unit ball in \(\mathbb {C}^{n+1}\) we can choose
$$\begin{aligned} s= \frac{{\overline{\zeta }} \cdot \mathrm{{d}}\zeta }{2\pi i(|\zeta |^2-{\bar{\zeta }}\cdot z)}. \end{aligned}$$
A holomorphic (1, 0)-form \(h=h_1 \mathrm{{d}}\zeta _1+\cdots + h_{n+1}\mathrm{{d}}\zeta _{n+1}\) in \(\Omega '\times \Omega '\) is a Hefer form for f if \( \delta _\eta h=f(\zeta )-f(z). \) Since \(h_j(\zeta ,\zeta )=(2\pi i)^{-1}\partial f/\partial \zeta _j\) it follows that
$$\begin{aligned} h (\zeta ,z) = (2\pi i)^{-1} \, \mathrm{{d}}f(\zeta ) + O(|\eta |), \end{aligned}$$
(3.5)
where \(O(|\eta |)\) is a holomorphic 1-form with coefficients in the ideal generated by \(\eta _1,\ldots ,\eta _{n+1}\).
Let h be such a Hefer form and let g be a weight as in Example 3.1. We can then define an integral operator \(\mathcal {K}\) that acts on forms on X and produces forms on \(D=X\cap \Omega '\) in the following way: we let
$$\begin{aligned} (\mathcal {K} \varphi )(z) = \int _{X_\zeta } K(\zeta ,z) \wedge \varphi (\zeta ), \end{aligned}$$
(3.6)
where the kernel has the form
$$\begin{aligned}&K(\zeta ,z) = \omega _X(\zeta ) \wedge {\tilde{K}}(\zeta ,z),\nonumber \\&\mathrm{{d}}\zeta _1 \wedge \cdots \wedge \mathrm{{d}}\zeta _{n+1} \wedge {\tilde{K}}(\zeta ,z) = h \wedge (g\wedge B)_n, \end{aligned}$$
(3.7)
and \((g\wedge B)_n\) denotes the components of \(g\wedge B\) of bidegree \((n,*)\), cf. [4, Section 8]. It follows that \(K(\zeta ,z)=\vartheta \lrcorner \big (h\wedge (g\wedge B)_n\big )\) and so, in view of (3.3), (3.4), and (3.5) we get that
$$\begin{aligned} K(\zeta ,z)= & {} \vartheta \lrcorner \left( \big (\mathrm{{d}}f/2\pi i + O(|\eta |)\big )\wedge \sum _ic_i(\zeta ,z)\frac{{\bar{\eta }}_i}{|\eta |^{2n}}\right) \\= & {} \vartheta \lrcorner \left( \mathrm{{d}}f/2\pi i \wedge \sum _ic_i(\zeta ,z)\frac{{\bar{\eta }}_i}{|\eta |^{2n}} + \mathrm{{d}}\zeta _1\wedge \cdots \wedge \mathrm{{d}}\zeta _{n+1}\wedge \sum _{i,j}b_{ij}(\zeta ,z)\frac{{\bar{\eta }}_i\eta _j}{|\eta |^{2n}} \right) \\= & {} \sum _{i,j,k} a_{ijk}(\zeta ,z)\frac{{\bar{\eta }}_i}{|\eta |^{2n}}\frac{f'_j\overline{f'_k}}{|\partial f(\zeta )|^2} + \omega _X(\zeta )\wedge \sum _{i,j} b_{ij}(\zeta ,z)\frac{{\bar{\eta }}_i\eta _j}{|\eta |^{2n}}, \end{aligned}$$
where the \(c_{i}\) and the \(a_{ijk}\) are smooth \((n,*)\)-forms and the \(b_{ij}\) are smooth \((0,*)\)-forms. We have thus shown

Proposition 3.2

We can write \(\mathcal {K} = \mathcal {K}_1 + \mathcal {K}_2\), where \(\mathcal {K}_1\) and \(\mathcal {K}_2\) are defined by integral kernels \(k_1\) and \(k_2\), respectively, that are sums of terms of the form
$$\begin{aligned} a(\zeta ,z)\frac{{\bar{\eta }}_i}{|\eta |^{2n}}\frac{f'_j(\zeta )\overline{f'_k}(\zeta )}{|\partial f(\zeta )|^2}, \end{aligned}$$
(3.8)
and
$$\begin{aligned} b(\zeta ,z)\wedge \omega _X(\zeta ) \frac{{{\bar{\eta }}}_i \eta _j}{|\eta |^{2n}}, \end{aligned}$$
(3.9)
respectively, where \(a(\zeta ,z)\) and \(b(\zeta ,z)\) are smooth on \(X \times D\).
We also need to consider the projection operator \(\mathcal {P}\), which is defined by
$$\begin{aligned} (\mathcal {P} \varphi )(z) = \int _{X_\zeta } P(\zeta ,z) \wedge \varphi (\zeta ), \end{aligned}$$
(3.10)
where the integral kernel \(P(\zeta ,z)\) is defined in a similar way to (3.7). Namely
$$\begin{aligned} P(\zeta ,z) = \omega _X(\zeta ) \wedge {\tilde{P}}(\zeta ,z), \end{aligned}$$
where
$$\begin{aligned} {\tilde{P}}(\zeta ,z) \wedge \mathrm{{d}}\zeta _1 \wedge \dots \wedge \mathrm{{d}}\zeta _{n+1} = h \wedge g_{n,n}, \end{aligned}$$
cf. [4, (5.5)]. Notice that since h and g are smooth, \({\tilde{P}}\) is smooth, and so \(|P(\zeta ,z)|\lesssim |\omega _X(\zeta )|\). If X has an isolated singularity in \(\Omega \) and we choose g according to Example 3.1, then for each z, \(\zeta \mapsto g_{n,n}(\zeta ,z)\) is supported away from \(X_{sing}\) and the corresponding P is then smooth in \(\zeta \) and holomorphic in z.

3.2 \(L^{2+}\)-Property of the Structure Form for a Canonical Hypersurface

Proposition 3.3

Let \(i:Y \rightarrow \Omega \subset \mathbb {C}^{n+1}\) be a hypersurface with canonical singularities and \(X\subset \subset Y\). Then there exists a real number \(q(X)>2\) such that \(\omega _Y \in L^q(X)\) for \(1\le q < q(X)\), where \(\omega _Y\) is the structure form of Y.

Proof

We denote by \(\omega _{Y}^{n}\) Grothendieck’s dualizing sheaf (sometimes also called the sheaf of Barlet–Henkin–Passare holomorphic n-forms on Y). As Y is a hypersurface, in particular Cohen–Macaulay, \(\omega _Y^n\) is a locally free \(\mathcal {O}_Y\)-module of rank one, and the structure form \(\omega _Y\) is a generator of \(\omega _Y^n\), see, e.g., [4] and (3.2).

Let \(\pi : M \rightarrow Y\) be a resolution of singularities such that the exceptional divisor has only normal crossings. Since Y has canonical singularities, \(\pi ^*\omega _Y\) extends across \(\pi ^{-1}Y_{sing}\) to a holomorphic n-form on M. Pick any Hermitian metric on M and let \(\mathrm{{d}}V_M\) be the corresponding volume form. Then \(i^{n^2}\pi ^*\omega _Y\wedge \pi ^*{\overline{\omega }}_Y=A\mathrm{{d}}V_M\) for some smooth non-negative function A on M.

Let s be local coordinates on M and let \(\mathrm{{d}}V_s=(i/2)^n ds_1{\wedge }d{\bar{s}}_1{\wedge }\ldots {\wedge }ds_n{\wedge }d{\bar{s}}_n\). Then \(\mathrm{{d}}V_s=B\mathrm{{d}}V_M\) for some smooth positive function B. Let \(\varpi =i\circ \pi \), where i is the inclusion \(Y\hookrightarrow \Omega \subset \mathbb {C}^{n+1}\). Then, on \(M\setminus \pi ^{-1}Y_{sing}\), \(s\mapsto \varpi (s)\) is a local parametrization of \(Y_{{\mathrm{{reg}}}}\subset \Omega \) and it is well-known that \(\pi ^* \mathrm{{d}}V_Y=\det H \mathrm{{d}}V_s\), where \(H=\ ^t \overline{{{\,\mathrm{Jac}\,}}\varpi } \cdot {{\,\mathrm{Jac}\,}}\varpi \ge 0\) and \({{\,\mathrm{Jac}\,}}\varpi = \big ( \partial \varpi _\nu / \partial s_\mu \big )_{\nu ,\mu }\) is the Jacobian matrix of \(\varpi \). Notice that \(\det H\) is a non-negative real-analytic function that vanishes precisely on \(\pi ^{-1}Y_{sing}\). It follows that \((\det H)^{-\epsilon /2}\) is locally integrable with respect to \(\mathrm{{d}}V_M\) for some \(\epsilon >0\). We now get
$$\begin{aligned} \pi ^*\mathrm{{d}}V_Y=\det H \mathrm{{d}}V_s=\det H B \mathrm{{d}}V_M=:C \mathrm{{d}}V_M. \end{aligned}$$
Thus, C is a globally defined function and each point in Y has a neighborhood where \(C^{-\epsilon /2}\) is integrable for some \(\epsilon >0\). Since \(\pi ^{-1}X\subset \subset M\), there is an \(\epsilon (X)>0\) such that \(C^{-\epsilon /2}\) is integrable on \(\pi ^{-1}X\) for all \(\epsilon <\epsilon (X)\).
Recall that \(|\omega _Y|^2\mathrm{{d}}V_Y=i^{n^2}\omega _Y\wedge {\overline{\omega }}_Y\). Pulling back to M we get \(\pi ^*|\omega _Y|^2C \mathrm{{d}}V_M=A \mathrm{{d}}V_M\) and thus \(\pi ^*|\omega _Y|^2=AC^{-1}\). Hence
$$\begin{aligned} \int _X |\omega _Y|^q \mathrm{{d}}V_Y=\int _{\pi ^{-1}X} A^{q/2}C^{-q/2+1} \mathrm{{d}}V_M <\infty \end{aligned}$$
(3.11)
as long as \(q-2<\epsilon (X)\), and so we may take \(q(X)=2+\epsilon (X)\). \(\square \)

Lemma 3.4

Let \(Y \subset \Omega \subset \mathbb {C}^3\) be a hypersurface with an isolated canonical singularity, and let X and q(X) be as in Proposition 3.3. Then \(q(X) \le 2+\frac{2}{m}\), where m is the maximum of the multiplicities of the divisors in the unreduced exceptional divisor in a minimal resolution of singularities of Y.

Proof

Assume that \(Y = \{ f = 0 \} \subset \Omega \subset \mathbb {C}^3\), and that Y has an isolated singularity at \(z = 0\). Then we claim that on \(Y_{{\mathrm{{reg}}}}\)
$$\begin{aligned} |\omega _Y| = c \frac{1}{|\partial f|}, \end{aligned}$$
(3.12)
for some constant c, where as above the norm \(|\omega _Y|\) is with respect to the norm on \(Y_{\mathrm{{reg}}}\) induced by the norm on \(\mathbb {C}^3\), while \(|\partial f|\) is with respect to the norm on \(\mathbb {C}^3\). Indeed, for any (2, 0)-form \(\alpha \) on \(Y_{\mathrm{{reg}}}\), one has the formula
$$\begin{aligned} |\alpha |_{Y_{{\mathrm{{reg}}}}} = \frac{|\alpha \wedge \partial f|_{\mathbb {C}^3}}{|\partial f|_{\mathbb {C}^3}}, \end{aligned}$$
and thus (3.12) follows from (3.1).

Let A and C be as in the proof of Proposition 3.3. Let \(\pi : M \rightarrow Y\) be a minimal resolution of singularities of Y. This resolution is crepant, i.e., \(\pi ^* \omega ^2_Y = \omega ^2_M\), see for example [11, Theorem 7.5.1]. Thus, the function A is strictly positive.

Since Y has an isolated singularity at 0, \(|\partial f| \lesssim |z|\), so by (3.12), \(|\omega _Y| \gtrsim 1/|z|\). Since A is strictly positive, \(\pi ^* |\omega _Y| \sim C^{-1/2}\), and it thus follows from (3.11) that for \(q \ge 2\),
$$\begin{aligned} \int _X |\omega _Y|^q \mathrm{{d}}V_Y \gtrsim \int _{\pi ^{-1} X} \frac{1}{\pi ^* |z|^{q-2}} \mathrm{{d}}V_M. \end{aligned}$$
If \(Z_i\) is an irreducible component of the unreduced exceptional divisor Z, and \(Z_i\) has multiplicity \(m_i\), then \(\pi ^* |z|^2\) vanishes to order \(2m_i\) along \(Z_i\), and thus, in order for the integral on the right-hand side to be finite, we must have that \(m_i(q-2) < 2\) for all \(m_i\). \(\square \)

In combination with a calculation of the multiplicities as in for example [11, Example 7.2.5] or [6, Proposition 3.8], we obtain the following corollary.

Corollary 3.5

If Y is a surface with an isolated \(A_n\), \(D_n\), \(E_6\), \(E_7\) or \(E_8\)-singularity, and \(X \subset \subset Y\), then q(X) is at most 4, 3, \(2+2/3\), \(2+1/2\) or \(2+1/3\), respectively.

In particular, we always have that \(q(X) \le 4\), so \(p(X) \ge 4/3\) for all surfaces with canonical singularities.

3.3 Mapping Properties of \({\mathcal K}\)

Proof of the\(L^{p}\)mapping properties in Theorem 1.2. By Proposition 3.2 we have the decomposition \(K(\zeta ,z) = k_1(\zeta ,z) + k_2(\zeta ,z)\), where
$$\begin{aligned} |k_1(\zeta ,z)| \lesssim \frac{1}{|\zeta -z|^3}, \quad \quad k_2(\zeta ,z) = \omega _X(\zeta ) \wedge \frac{b'(\zeta ,z)}{|\zeta -z|^2}, \end{aligned}$$
where \(b'(\zeta ,z)\) is bounded. By Lemma 6.3, \(k_1\) is uniformly integrable over X in \(\zeta \) as well as in z, and so \({\mathcal K}_1\) maps \(L^p(X) \rightarrow L^p(D)\) continuously for all \(1\le p\le \infty \) by the generalized Young inequality, [21, Appendix B] and Lemma 2.2.
Note that we can then decompose \({\mathcal K}_2\) into the consecutive application of two operators
$$\begin{aligned} \varphi (\zeta )\mapsto & {} \varphi (\zeta )\wedge \omega _X(\zeta ) \ \ \mapsto \ \ \int _{X} \varphi (\zeta ) \wedge \omega _X(\zeta ) \wedge \frac{b'(\zeta , z)}{|\zeta -z|^2}. \end{aligned}$$
(3.13)
To analyse this chain, choose \(2<q<q(X)\) so that \(\omega _X \in L^q(X)\). By Hölder’s inequality, the operator \(\varphi \mapsto \varphi \wedge \omega _X\) maps \(L^p(X) \rightarrow L^a(X)\) continuously for \(1\le a \le \infty \) defined by \(1/a=1/p+1/q\) (for p so that \(1/p+1/q \le 1\)).
The second operator in (3.13) can again be analysed by the generalised Young inequality. By Lemma 6.3, \(|\zeta -z|^{-2}\in L^s(X)\) in \(\zeta \) and in z for all \(s<2\), in particular for s defined by \(1/s + 1/q =1\), since \(q>2\). Then, since \(1/p=1/a-1/q=1/a+1/s-1\), it follows from the generalised Young inequality, [21, Appendix B], that
$$\begin{aligned} \varphi \mapsto \int _X\varphi (\zeta ) \wedge \frac{b'(\zeta , z)}{|\zeta -z|^2} \end{aligned}$$
maps \(L^a(X)\rightarrow L^p(D)\) continuously. Combining, we see that the composed operator (3.13) given by the kernel \(k_2\) is a bounded mapping \(L^p(X)\rightarrow L^p(D)\) for any p such that \(1/p+1/q\le 1\). Thus, \({\mathcal K}\) is a bounded mapping \(L^p(X)\rightarrow L^p(D)\) for all \(p(X)<p\le \infty \).

The kernel \(k_1\) is integrable in both variables, and by truncating it, we get a bounded kernel corresponding to a compact operator; by standard arguments, cf., for example [21, Appendix C], this converges to \({\mathcal K}_1\), and it is thus a compact operator. If we decompose the operator \({\mathcal K}_2\) as in (3.13), the same holds for the right-most operator, and thus also \({\mathcal K}_2\) is compact. \(\square \)

Proof of the \(C^\alpha \)mapping properties in Theorem 1.2. Let us first consider the operator \(\mathcal {K}\). Note that for \(\nu =1,2\), \(k_\nu (\zeta , z)\varphi (\zeta )\) is a sum of terms of the form \(k_\nu '(\zeta , z)\varphi '(\zeta ) \mathrm{{d}}\zeta _I\wedge d {{\bar{\zeta }}}_J \wedge d {\bar{z}}_K\) and \({\mathcal K}_\nu \varphi (z)\) is a sum of terms \(({\mathcal K}_\nu \varphi )'(z)\mathrm{{d}}{\bar{z}}_K:=\int k_\nu '(\zeta , z)\varphi '(\zeta ) \mathrm{{d}}\zeta _I\wedge d {{\bar{\zeta }}}_J \wedge d {\bar{z}}_K\), where \(k_\nu '(\zeta , z), \varphi '(\zeta )\), and \(({\mathcal K}_\nu \varphi )'(z)\) are functions. Using that
$$\begin{aligned} |(\mathcal {K}_\nu )'\varphi (z) - (\mathcal {K}_\nu )'\varphi (w)| \lesssim \Vert \varphi '\Vert _{L^\infty } \int |k'_\nu (\zeta ,z) - k'_\nu (\zeta ,w)|, \end{aligned}$$
it follows that \({\mathcal K}_\nu \) maps into \(C^\alpha \) if
$$\begin{aligned} \int |k'_\nu (\zeta ,z) - k'_\nu (\zeta ,w)| \lesssim |z-w|^\alpha . \end{aligned}$$
(3.14)
for each \(k_\nu '\).
For \(\nu =1\), we may assume that \(k_1\) is of the form (3.8). Then, \(k_1'\) is a sum of functions of the form (3.8) with \(a(\zeta , z)\) replaced by one of its coefficients \(a'(\zeta , z)\). We may assume that \(k_1'\) is one such function; then
$$\begin{aligned}&\int |k'_1(\zeta ,z) - k'_1(\zeta ,w)| \lesssim \int |a'(\zeta ,z) - a'(\zeta ,w)| \left| \frac{\overline{\zeta _i - z_i}}{|\zeta -z|^4} \frac{f'_j(\zeta )\overline{f'_k}(\zeta )}{|\partial f(\zeta )|^2} \right| \\&\quad + \int |a'(\zeta ,w)| \left| \left( \frac{\overline{\zeta _i - z_i}}{|\zeta -z|^4} - \frac{\overline{\zeta _i - w_i}}{|\zeta -w|^4}\right) \frac{f'_j(\zeta )\overline{f'_k}(\zeta )}{|\partial f(\zeta )|^2} \right| =: I_1(z,w) + I_2(z,w), \end{aligned}$$
Since \(a(\zeta ,z)\) depends smoothly on z, we may assume that \(|a'(\zeta ,z) - a'(\zeta ,w)| \lesssim |z-w|\), and since the remaining integrand in \(I_1(z,w)\) is integrable in \(\zeta \) by Lemma 6.3, \(I_1(z,w) \lesssim |z-w|\). The integrand in \(I_2(z,w)\) is bounded by a constant times
$$\begin{aligned} \left| \frac{\overline{\zeta _i - z_i}}{|\zeta -z|^4} - \frac{\overline{\zeta _i - w_i}}{|\zeta -w|^4}\right| , \end{aligned}$$
and by the same argument as for the Bochner–Martinelli kernel on \(\mathbb {C}^2\), see, e.g., [13, Proposition III.2.1], and using Lemma 6.3, one obtains that \(I_2(z,w) \lesssim |z-w|^\alpha \) for any \(\alpha < 1\), and thus \({\mathcal K}_1\) is \(C^\alpha \) for any \(\alpha <1\).
We next consider \(k_2\). As above it is enough to consider one of the coefficients \( b'(\zeta ,z)\omega '_X(\zeta ) {{\bar{\eta }}}_i \eta _j/|\eta |^4\) of one of the terms (3.9). In view of (2.2) we can choose the coefficient \(\omega '_X\) of \(\omega _X\) in \(L^q\) for \(1\le q<q(X)\). We divide the domain of integration X into
$$\begin{aligned} D_1 := X\cap B_{|z-w|/2}(z), \,\, D_2 := X\cap B_{|z-w|/2}(w), \,\,\text { and } \,\, D_3 := X \setminus (D_1 \cup D_2), \end{aligned}$$
where \(B_r(z)\) denotes a ball of radius r centered at z. We choose \(2<q<q(X)\) and let \(p = q/(q-1) < 2\) be the dual exponent. Since \(q(X) \le 4\) by Corollary 3.5, \(p > 4/3\). Using Hölder’s inequality and Lemma 6.4, we get
$$\begin{aligned} \int _{\zeta \in D_\nu } |k_2'(\zeta ,z)| \lesssim \left( \int _{\zeta \in D_\nu } \frac{1}{|\zeta -z|^{2p}}\right) ^{1/p} \lesssim (|z-w|^{4-2p})^{1/p} = |z-w|^{4/p-2} \end{aligned}$$
for \(\nu =1,2\). By the same argument
$$\begin{aligned} \int _{\zeta \in D_\nu } |k_2'(\zeta ,w)| \lesssim |z-w|^{4/p-2}. \end{aligned}$$
For the integral on \(D_3\), we use the following inequality,
$$\begin{aligned} \left| \frac{\overline{\zeta _i - z_i}}{|\zeta -z|^4} - \frac{\overline{\zeta _i - w_i}}{|\zeta -w|^4}\right| \lesssim |z-w| \max \left\{ \frac{1}{|\zeta -z|^4}, \frac{1}{|\zeta -w|^4} \right\} , \end{aligned}$$
see the proof of [13, Lemma III.2.2]. It follows that
$$\begin{aligned} \left| \frac{\overline{(\zeta _i - z_i)}(\zeta _j - z_j)}{|\zeta -z|^4} - \frac{\overline{(\zeta _i - w_i)}(\zeta _j - w_j)}{|\zeta -w|^4} \right| \lesssim |z-w| \max \left\{ \frac{1}{|\zeta -z|^3}, \frac{1}{|\zeta -w|^3} \right\} ,\nonumber \\ \end{aligned}$$
(3.15)
e.g., by assuming that \(|\zeta -z|\le |\zeta -w|\) and adding and subtracting \((\overline{\zeta _i-w_i})(\zeta _j-z_j)/|\zeta -w|^{4}\) inside the absolute value sign on the left-hand side. Using Hölder’s inequality as above, we get
$$\begin{aligned} \int _{\zeta \in D_3} |k_2'(\zeta ,z) - k_2'(\zeta ,w)| \lesssim \left( \int _{\zeta \in D_3} \left| \frac{\overline{(\zeta _i - z_i)}(\zeta _j - z_j)}{|\zeta -z|^4} - \frac{\overline{(\zeta _i - w_i)}(\zeta _j - w_j)}{|\zeta -w|^4} \right| ^p \right) ^{1/p}. \end{aligned}$$
By (3.15), this is bounded by
$$\begin{aligned} |z-w| \left( \int _{\zeta \in D_3} \max \left\{ \frac{1}{|\zeta -z|^{3p}}, \frac{1}{|\zeta -w|^{3p}} \right\} \right) ^{1/p}. \end{aligned}$$
Since \(p > 4/3\), it follows from Lemma 6.1 that this is bounded by a constant times
$$\begin{aligned} |z-w| \big (|z-w|^{4-3p}\big )^{1/p} = |z-w|^{4/p-2}. \end{aligned}$$
Since \(p > 4/3\), we get that \(4/p-2 < 1\). Thus, it follows that \(\mathcal {K}_2\) is \(C^\alpha \) for any \(\alpha <4/p-2\). We conclude that \(\mathcal {K}\) is \(C^\alpha \) for any \(\alpha <4/p(X)-2\).

Since (3.14) holds uniformly for \(z,w\in D\), if \(\{\varphi _j\}_j\) and thus \(\{\varphi _j'\}_j\) are bounded sequences in \(L^\infty (X)\), then \(\{(\mathcal {K} \varphi _j)'\}_j\) are equicontinuous in the \(C^\alpha ({\overline{D}})\)-norm and thus \(\mathcal {K}\) is compact by the Arzelà–Ascoli theorem. \(\square \)

Proof of Corollary 1.3

The stalk of \({\mathcal A}_X\) at the singular point is a finite sum of currents of the form
$$\begin{aligned} \xi _{\nu +1}\wedge (\mathcal {K}_\nu (\ldots \xi _3 \wedge \mathcal {K}_2(\xi _2\wedge \mathcal {K}_1\xi _1))), \end{aligned}$$
where each \(\mathcal {K}_i\) is an integral operator as in Theorem  1.2, mapping forms on \(D_i := \Omega _i \cap X\) to forms on \(D_{i+1}\), where \(\Omega = \Omega _{\nu +1} \subset \subset \Omega _\nu \subset \subset \dots \subset \subset \Omega _1 \subset \subset \mathbb {C}^3\) are pseudoconvex domains, and \(\xi _i\) are smooth forms on \(D_i\). The corollary now follows from Theorem 1.2. \(\square \)

3.4 The Operators \({\hat{{\mathcal K}}}\) and \({\hat{\mathcal {P}}}\) on Forms with Compact Support

Let \(H\subset X\) be a compact Stein subset such that D is relatively compact in the interior of H. In [3] are constructed integral operators, that we here denote by \({{\hat{{\mathcal K}}}}\) and \({{\hat{\mathcal {P}}}}\), which map smooth forms with compact support in D to smooth forms in \(X\setminus \{0\}\) that vanish outside H, such that
$$\begin{aligned} \varphi (z) = {\bar{\partial }}{{\hat{{\mathcal K}}}} \varphi (z) + {{\hat{{\mathcal K}}}} {\bar{\partial }}\varphi (z) \ \text { if } r=0,1, \quad \varphi (z) = {{\hat{\mathcal {P}}}} \varphi (z) + {{\hat{{\mathcal K}}}} {\bar{\partial }}\varphi (z) \ \text { if } r=2. \end{aligned}$$
(3.16)
In fact, \({{\hat{\mathcal {P}}}}\) maps forms with support in D to smooth forms. Moreover, \(\hat{P}\varphi =0\) unless \(r=2\). The kernels for these operators are obtained by choosing the weight g differently; with notation as in Example 3.1, we let \(\chi =\chi (z)\) and we interchange the roles of \(\zeta \) and z in the functions \(s_i(\zeta ,z)\). The resulting weight is then holomorphic in \(\zeta \) and has compact support H in z.

Since the proof of the mapping properties above essentially only uses that g is smooth, it follows that an analogue of Theorem 1.2 holds also for these operators. The subscript c denotes forms with compact support.

Theorem 3.6

In the situation of Theorem 1.2, the integral operator \({{\hat{{\mathcal K}}}}\) extends to an operator
$$\begin{aligned}&L^p_{0,r;c}(D)\rightarrow L^p_{0,r-1;c}(X),\ \ p(X)< p \le \infty , \ \ L^\infty _{0,r;c}(D) \rightarrow C^\alpha _{0,r-1;c}(X), \\&\quad \ 0\le \alpha <4/p(X)-2, \end{aligned}$$
and \({{\hat{\mathcal {P}}}}\) extends to an operator \( L^p_{0,2;c}(D)\rightarrow C^{\infty }_{0,2;c}(X),\ \ p(X) < p \le \infty . \)

Note that the operators in fact map to forms with support in the fixed compact set H.

4 Homotopy Formulas

Proof of Theorem 1.4

By [3, Theorem 1.1] the homotopy formula (1.5) holds pointwise on \(D_{\mathrm{{reg}}}\) if \(\varphi \) is smooth on X. For \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\), let \(\{\varphi _j\}_j\) be a sequence as in Lemma 2.4. We can assume that the \(\varphi _j\) are smooth and bounded and with support away from the singularity \(\{0\}\) (see the proof of Lemma 2.4 in [15]). Then the homotopy formula
$$\begin{aligned} \varphi _j = {\bar{\partial }}\mathcal {K} \varphi _j + \mathcal {K} {\bar{\partial }}\varphi _j \end{aligned}$$
(4.1)
holds on D. In fact, since \(\varphi _j\) is supported away from \(X_{sing}\) all the terms are smooth on D, see [4, Lemma 6.1]. By Theorem 1.2 we have that \(\mathcal {K} \varphi _j \rightarrow \mathcal {K}\varphi \), and \(\mathcal {K}{\bar{\partial }}\varphi _j \rightarrow \mathcal {K} {\bar{\partial }}\varphi \) in \(L^p(D)\). It only remains to show that \(\mathcal {K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\). Taking the limit \(j\rightarrow \infty \) in (4.1) implies, by Theorem  1.2, that \(\mathcal {K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\) and \( {\bar{\partial }}\mathcal {K} \varphi = \varphi - \mathcal {K} {\bar{\partial }}\varphi \) on D. As the \(\varphi _j\) are bounded, \(\{ \mathcal {K} \varphi _j\}_j\) is by Theorem 1.2 a sequence of bounded forms such that \(\mathcal {K} \varphi _j \rightarrow \mathcal {K} \varphi \) and \({\bar{\partial }}\mathcal {K} \varphi _j \rightarrow {\bar{\partial }}\mathcal {K}\varphi \) in \(L^p(D)\). Hence, \(\mathcal {K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\) by Lemma  2.4. \(\square \)

4.1 Proof of Theorem 1.5

We first remark that the condition \((*)\) from the introduction is indeed independent of the sequence of cut-off functions \(\{ \chi _k\}_k\). Indeed, if \(\{\chi _k'\}_k\) is another such sequence, then \(\chi _k-\chi _k'\) has compact support contained in \(X^*\), and on this set, \(\omega \) is \({\bar{\partial }}\)-closed. Thus,
$$\begin{aligned} \int _X \omega _X \wedge {\bar{\partial }}\chi _k \wedge \varphi - \int _X \omega _X \wedge {\bar{\partial }}\chi _k' \wedge \varphi = \int _X \omega _X \wedge (\chi _k - \chi _k') \wedge {\bar{\partial }}\varphi , \end{aligned}$$
and this tends to 0 as \(k \rightarrow \infty \) by dominated convergence since \(\omega _X \wedge {\bar{\partial }}\varphi \) is in \(L^1(X)\).

Proof of Theorem 1.5

We first note that it is enough to prove that
$$\begin{aligned} \varphi = {\bar{\partial }}{\mathcal K}\varphi + {\mathcal K}{\bar{\partial }}\varphi \end{aligned}$$
(4.2)
holds in the sense of distributions. Indeed, if it holds, then \({\bar{\partial }}\mathcal K \varphi =\mathcal K {\bar{\partial }}\varphi -\varphi \) is in \(L^p(D)\) by Theorem 1.2 since \({\bar{\partial }}\varphi \in L^p(X)\), and therefore \(\mathcal K\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\). We note that \(p > p(X) \ge 4/3\) since \(q(X) \le 4\) by Corollary 3.5.
Let \(\mu _k\) be the cut-off functions in Sect. 2.2 and let \(\varphi _k=\mu _k\varphi \). By the proof of Lemma 2.3 in [15], \(\varphi _k \rightarrow \varphi \) in \(L^p(X)\), \(\mu _k {\bar{\partial }}\varphi \rightarrow {\bar{\partial }}\varphi \) in \(L^p(X)\), and \({\bar{\partial }}\mu _k \wedge \varphi \rightarrow 0\) in \(L^{\lambda }(X)\) for \(\lambda = 4p/(p+4) > 1\) since \(p > 4/3\). Since \(\varphi _k\) has support away from the singularity, it follows as in the proof of Theorem 1.4 that the homotopy formula (4.1) holds on D. Since \(p>p(x)\), \(\mathcal K\varphi _k\) converges to \(\mathcal K\varphi \) in \(L^p(D)\) by Theorem 1.2, and it follows that \({\bar{\partial }}\mathcal K\varphi _k\) converges weakly to \({\bar{\partial }}\mathcal K\varphi \). Since \(\varphi _k = \mu _k \varphi \rightarrow \varphi \) and \(\mu _k {\bar{\partial }}\varphi \rightarrow {\bar{\partial }}\varphi \) in \(L^p(X)\), it follows from Theorem 1.2 that \(\mathcal {K}(\varphi _k) \rightarrow \mathcal {K} \varphi \) and \(\mathcal {K}(\mu _k {\bar{\partial }}\varphi ) \rightarrow \mathcal {K} {\bar{\partial }}\varphi \) in \(L^p(D)\). Thus, using that \({\bar{\partial }}\varphi _k = {\bar{\partial }}\mu _k \wedge \varphi + \mu _k {\bar{\partial }}\varphi \), it follows that (4.2) holds if and only if
$$\begin{aligned} \mathcal {K} ( {\bar{\partial }}\mu _k \wedge \varphi ) \rightarrow 0 \end{aligned}$$
(4.3)
in the sense of distributions. If \(\varphi \) is a \((0,r)\)-form, then there is nothing to prove for \(r=2\), so let us assume that \(r=1\).

We first consider the case when \(p > {\hat{p}}(X)\). Then \(\lambda > p(X)\) so (4.3) holds by Theorem 1.2, since \({\bar{\partial }}\mu _k \wedge \varphi \rightarrow 0\) in \(L^\lambda (X)\).

It remains to prove that (4.3) holds for \(p(X) < p \le {\hat{p}}(X)\) when \(\varphi \) satisfies \((*)\). To prove (4.3) we decompose \(\mathcal {K} = \mathcal {K}_1 + \mathcal {K}_2\) as in Proposition 3.2. We saw in the proof of Theorem  1.2 that \(\mathcal K_1\) is a bounded linear operator \(L^p_{0,r+1}(X)\rightarrow L^p_{0,r}(D)\) for any \(1\le p\le \infty \). It follows, in particular, that \(\mathcal {K}_1 ( {\bar{\partial }}\mu _k \wedge \varphi ) \rightarrow 0\) in the sense of distributions, since \({\bar{\partial }}\mu _k \wedge \varphi \rightarrow 0\) in \(L^{\lambda }_{0,r+1}(X)\) where \(\lambda > 1\) since \(p > 4/3\).

We next consider \(\mathcal {K}_2\). In view of Proposition  3.2 we may assume that the kernel is of the form (3.9). To prove (4.3) for \(\mathcal K_2\) we need to prove that
$$\begin{aligned} \langle \mathcal {K}_2( {\bar{\partial }}\mu _k \wedge \varphi ) , \xi \rangle = \int _z \xi (z) \wedge \int _\zeta b(\zeta ,z) \wedge \omega _X(\zeta ) \wedge \frac{\overline{\eta _i}\eta _j}{|\eta |^4} {\bar{\partial }}\mu _k(\zeta ) \wedge \varphi (\zeta ) \end{aligned}$$
(4.4)
tends to 0 as \(k\rightarrow \infty \) for each test form \(\xi \). By Fubini’s theorem, up to signs (4.4) is equal to
$$\begin{aligned} \int _\zeta \omega _X \wedge {\bar{\partial }}\mu _k \wedge \varphi \wedge \int _z b(\zeta ,z) \wedge \xi (z) \frac{\overline{\eta _i}\eta _j}{|\eta |^4}. \end{aligned}$$
(4.5)
We denote the inner integral with respect to z by \(\gamma (\zeta )\). Note that
$$\begin{aligned} \gamma = \int _z \frac{c(\zeta ,z)(\overline{\zeta _i-z_i})(\zeta _j-z_j)}{|\zeta -z|^4}, \end{aligned}$$
where \(c(\zeta , z)\) is a smooth (2, 2)-form. Now \( |\gamma (\zeta ) - \gamma (0)|\) is bounded by
$$\begin{aligned}&\int _z |c(\zeta ,z) - c(0,z)| \left| \frac{(\overline{\zeta _i - z_i})(\zeta _j-z_j)}{|\zeta -z|^4}\right| \\&\quad + |c(0,z)|\int _z \left| \frac{(\overline{\zeta _i-z_i})(\zeta _j-z_j)}{|\zeta -z|^4}- \frac{\overline{z_i}z_j}{|z|^4}\right| :=I_1+I_2. \end{aligned}$$
Since \(c(\zeta , z)\) depends smoothly on \(\zeta \), \(|c(\zeta ,z) - c(0,z)|\lesssim |\zeta |\) and thus, in view of Lemma  6.3, \(I_1 \lesssim |\zeta |\). Moreover, by (3.15) and Lemma 6.3, \(I_2\lesssim |\zeta |\int _z\max (|\zeta -z|^{-3}, |z|^{-3} )\lesssim |\zeta |\).

Since \(\varphi \) satisfies \((*)\), and this condition is independent of the choice of \(\chi _k\), we may assume that \(\chi _k = \mu _k\), and thus \(\int _\zeta \omega _X \wedge {\bar{\partial }}\mu _k \wedge \varphi \wedge \gamma (0)\) tends to 0 as \(k \rightarrow \infty \). It follows from (2.5) that \(|{\bar{\partial }}\mu _k\wedge (\gamma (\zeta )-\gamma (0))|\le C \chi _k(|\zeta |)\) when \(|\zeta |\ll 1\) and where \(\chi _k\) is as in Sect. 2.2. Since \(p>p(X)\), by Hölder’s inequality, \(\omega _X\wedge \varphi \in L^1(X)\) and therefore \(\lim _k \int _\zeta \omega _X \wedge {\bar{\partial }}\mu _k \wedge \varphi \wedge (\gamma (\zeta )-\gamma (0))=0\) by dominated convergence. Hence (4.5) tends to 0 as \(k\rightarrow \infty \). \(\square \)

It follows from the proof of Theorem 1.5 that if \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\), with \(p > {\hat{p}}(X)\), then \((*)\) is automatically fulfilled for \(\varphi \), since if \(\mu _k\) is as in Sect. 2.2, then
$$\begin{aligned} \int _X \omega _X \wedge {\bar{\partial }}\mu _k \wedge \varphi \rightarrow 0 \end{aligned}$$
by Hölder’s inequality.

It is worth noting that since the condition \((*)\) does not depend on p, we have the following consequence of Theorem 1.5:

Corollary 4.1

If the \({\bar{\partial }}_w\)-equation is locally solvable on a canonical surface for some \(p_0> p(X)\), then is is locally solvable for all \(p\ge p_0\).

Morally, this means that the number of obstructions to solving the \({\bar{\partial }}_w\)-equation in the \(L^p\)-sense is decreasing in p. Theorem 1.1 in [24] shows that the same kind of phenomenon holds for homogeneous varieties with an isolated singularity.

Let \(\varphi \in L^p_{0,1}(X)\), where \(p(X)<p\le 4\). Assume that \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s\). Then, by Theorem 1.4, \(\varphi = {\bar{\partial }}_s \mathcal {K} \varphi \) which implies particularly that \(\varphi = {\bar{\partial }}_w \mathcal {K} \varphi \). Hence, \(\varphi \) satisfies \((*)\). It would be interesting to know whether the converse is also true, i.e., if \(\varphi \) satisfies \((*)\), does it follow that \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s\)?

4.2 Proof of Theorem 1.6

As explained after Proposition 3.2, the operator \(\mathcal {P}\) is defined by an integral kernel \(P(\zeta ,z)\) that is smooth with compact support in \(\zeta \), and holomorphic in z. Therefore, \(\mathcal {P}\) extends to a compact operator \(\mathcal {P} : L^p(X) \rightarrow \mathcal {O}(D)\), cf. the proof of Theorem 1.2.

The formula (1.8) for \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\) and \(p(X) < p \le 4\) is proved in the same way as Theorem 1.4 above, using that (1.8) holds for the smooth functions \(\varphi _j\), and that \(\mathcal {P} \varphi _j \rightarrow \mathcal {P} \varphi \).

Now assume that \(\varphi \) is a function in \({{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\), where \(p(X) < p\). We say that \(\varphi \) satisfies \((*)\) if
$$\begin{aligned} \int _X \omega _X \wedge {\bar{\partial }}\chi _k \wedge \varphi \wedge \alpha \rightarrow 0 \end{aligned}$$
(4.6)
for any smooth \({\bar{\partial }}\)-closed (0, 1)-form \(\alpha \) and sequence \(\chi _k\) as in Sect. 4.1. In particular, if \(\varphi \) is \({\bar{\partial }}\)-closed, i.e., holomorphic on the regular part of X, then as X is a canonical surface, \(X_{sing}\) has codimension 2 and thus \(\varphi \) is bounded in a neighborhood of the singularity at the origin. Therefore, \(\varphi \in L^p\) for any \(p\ge 1\) and it follows as for (0, 1)-forms above that \((*)\) is satisfied.

If \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\) and \(p > {\hat{p}}(X)\), then (1.8) can be verified in the same way as Theorem 1.5 above. If instead \(p(X) < p \le {\hat{p}}(X)\) and \(\varphi \) satisfies (4.6), one just needs to make minor modifications. Namely at the point where one considers \(\gamma (\zeta )-\gamma (0)\), then \(\gamma \) is a (0, 1)-form, and one then writes \(\gamma = \sum \gamma _i \mathrm{{d}}{\bar{\zeta }}_i\), decomposes \(\gamma _i(\zeta ) = \gamma _i(0) + (\gamma _i(\zeta )-\gamma _i(0))\) and proceeds as in the proof above. The condition (4.6) is then finally applied with \(\alpha = \gamma _i(0) \mathrm{{d}}{\bar{\zeta }}_i\).

4.3 Homotopy Formulas with Compact Support

We get versions of Theorems 1.4 and 1.5 also for the operators in Theorem 3.6.

Theorem 4.2

Assume we are in the situation of Theorem 1.2.
  1. (i)
    Let \(p(X)< p \le 4\) and let \(\varphi \) be an \((0,r)\)-form in \({{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)} \subset L^p_c(D)\). Then \({\hat{K}}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\),
    $$\begin{aligned} \varphi (z) = {\bar{\partial }}_s {{\hat{{\mathcal K}}}} \varphi (z) + {{\hat{{\mathcal K}}}} {\bar{\partial }}_s\varphi (z) \ \text { if } r=0,1, \quad \varphi (z) = {\bar{\partial }}_s{{\hat{{\mathcal K}}}} \varphi (z) + {{\hat{\mathcal {P}}}} \varphi (z) \ \text { if } r=2. \end{aligned}$$
    (4.7)
     
  2. (ii)
    If \({\hat{p}}(X) < p \le \infty \) and \(\varphi \) is a \((0,r)\)-form with \(r=0,1\), in \({{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)} \subset L^p_c(D)\), then \({\hat{{\mathcal K}\varphi }} \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\) and
    $$\begin{aligned} \varphi (z) = {\bar{\partial }}_w {{\hat{{\mathcal K}}}} \varphi (z) + {{\hat{{\mathcal K}}}} {\bar{\partial }}_w \varphi (z). \end{aligned}$$
    (4.8)
    If \(p(X)< p\le {\hat{p}}(X)\), and in addition \(\varphi \) satisfies the condition \((*)\), then the same conclusion holds.
     
  3. (iii)
    If \(p(X)< p \le \infty \) and \(\varphi \) is a (0, 2)-form in \({{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)} \subset L^p_c(D)\), then \(\hat{K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)}\) and
    $$\begin{aligned} \varphi (z) = {\bar{\partial }}_w{{\hat{{\mathcal K}}}} \varphi (z) + {{\hat{\mathcal {P}}}} \varphi (z). \end{aligned}$$
    (4.9)
    The (0, 2)-form \(\varphi \) satisfies that \(\hat{P}\varphi =0\) if (1.7) holds, and if \(p > {\hat{p}}(X)\), then the converse holds.
     

These statements are proved essentially by the same arguments as in the proofs of Theorems  1.4 and 1.5. For (4.7), notice that as \(\varphi \) has compact support in D, when choosing the approximating sequences \(\{\varphi _j\}_j\) we may, in addition, assume that the \(\varphi _j\) have compact support in D as well.

For the last statement, notice that the kernel for \({{\hat{\mathcal {P}}}}\) has the form \(h\omega _X\) with respect to \(\zeta \) in a Stein neighborhood of the support of \(\varphi \). Since X is Stein, we can assume that h is holomorphic on X and so \(\hat{P}\varphi =0\) if (1.7) holds. Conversely, if \(\hat{P}\varphi =0\), then \(u=\hat{K}{\varphi }\) is a solution to \({\bar{\partial }}u=\varphi \) with support on the compact set H, see Sect. 3.4. It then follows that (1.7) holds if and only if u satisfies \((*)\), which as we saw in Sect. 4.1 is automatically satisfied for \(p > {\hat{p}}(X)\).

Note that if \(\varphi \) is a (0, 1)-form in \(L^p_c(X)\) and \({\bar{\partial }}\varphi = 0\), then it automatically satisfies \((*)\), so if \(p> p(X)\), then \(u = \hat{\mathcal {K}}\varphi \) is a solution with compact support to \({\bar{\partial }}u = \varphi \).

4.4 On the Domain of the \({\bar{\partial }}_X\)-Operator

The setting in [4] is rather different compared to this article. Here, we are mainly concerned with forms on X with coefficients in \(L^p\), while in [4], the type of forms/currents considered, denoted \(\mathcal {W}_X^r\), are “generically” smooth, see [5]. They include principal value currents \(\alpha /f\), where f is holomorphic and \(\alpha \) is smooth, and direct images of such currents, but with no growth restrictions on the singularities. For the precise definition of the sheaf \(\mathcal {W}_X^r\) we refer to [4, 5]. The \({\bar{\partial }}\)-operator considered in [4] is somewhat different from \({\bar{\partial }}_s\) and \({\bar{\partial }}_w\) considered here. For currents in \(\mathcal {W}_X^r\), one can define the product with the structure form \(\omega _X\) associated to the variety. A current \(\mu \in \mathcal {W}_X^r\) lies in \({{\,\mathrm{Dom}\,}}{\bar{\partial }}_X\) if \({\bar{\partial }}\mu \in \mathcal W_X^{r+1}\) and \({\bar{\partial }}(\mu \wedge \omega _X) ={\bar{\partial }}\mu \wedge \omega _X\). Combining our results about \(\mathcal {K}\) and the \({\bar{\partial }}_w\)- and \({\bar{\partial }}_s\)-operator with some properties about the \(\mathcal {W}_X\)-sheaves, we obtain a result for the \({\bar{\partial }}_X\)-operator, providing a partial answer to a question in [4], cf. the paragraph at the end of page 288 in [4].

Theorem 4.3

In the situation of Theorem 1.2, let \(p(X) < p \le 4\) and \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}^{(p)}_s \cap \mathcal {W}_X^r(X)\). Then \(\mathcal {K} \varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_X.\)

Note that if \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}^{(p)}_w\), where \(p > {\hat{p}}(X)\), then by Lemma 2.3, \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}^{(\lambda )}_s\), where \(\lambda > p(X)\), so also in this case, \(\mathcal {K}\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_X\).

Proof

First, by [4, Proposition 1.5], \(\mathcal K\varphi \in \mathcal W(D)\). Then, in particular \({\bar{\partial }}\mathcal K\varphi \) is a pseudomeromorphic current, see, e.g., [5]. Moreover, by Theorem 1.4\(\mathcal K\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\), and thus \({\bar{\partial }}\mathcal K\varphi \in L^p(D)\). Now a pseudomeromorphic current in \(L^p\) is in fact in \(\mathcal W\), cf. [5]. Hence \({\bar{\partial }}\mathcal K\varphi \in \mathcal W(D)\).

Since \(\mathcal K\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_s^{(p)}\), there is a sequence of smooth forms \(\psi _j\) with support away from the singularity such that \(\psi _j\rightarrow \mathcal K\varphi \) and \({\bar{\partial }}\psi _j\rightarrow {\bar{\partial }}\mathcal K\varphi \) in \(L^p(D)\). Since \(\omega _X\in L^q(X)\) for each \(q<q(X)\), by Hölder’s inequality, \(\psi _j\wedge \omega _X\) and \({\bar{\partial }}\psi _j\wedge \omega _X\) converge in \(L^1(D)\) to \(\mathcal K\varphi \wedge \omega _X\) and \({\bar{\partial }}\mathcal K\varphi \wedge \omega \), respectively. Hence
$$\begin{aligned} {\bar{\partial }}(\mathcal K\varphi \wedge \omega _X) = \lim _j {\bar{\partial }}(\psi _j\wedge \omega _X) = \lim _j {\bar{\partial }}\psi _j\wedge \omega _X = {\bar{\partial }}\mathcal K\varphi \wedge \omega _X. \end{aligned}$$
We conclude that \(\mathcal K\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_X\). \(\square \)

5 Examples and Counterexamples

In this section, we study the condition \((*)\) and \({\bar{\partial }}_w\)-Koppelman formulas for all types of canonical surface singularities: \(A_n\), \(n\ge 1\), \(D_n\), \(n\ge 4\), \(E_6\), \(E_7\) and \(E_8\). We focus on the important case of \(L^2\)-cohomology, i.e., \(p=2\). However, we also get some statements for \(p\ge 2\). All in all we obtain a complete picture about the solvability of the \({\bar{\partial }}_w\)-equation in the \(L^2\)-sense at canonical surface singularities.

5.1 The \(A_n\)-Singularities

Recall that the \(A_n\)-singularity for \(n\ge 1\) is the variety \(X = \{ f(\zeta ) = 0 \} \subset \mathbb {C}^3\), where \(f(\zeta ) = \zeta _1 \zeta _2 - \zeta _3^{n+1}\).

Theorem 5.1

Let X be (a neighborhood of the origin of) the \(A_n\) singularity \(\{\zeta _1\zeta _2=\zeta _3^{n+1}\}\subset \mathbb {C}^3\), let \(p\ge 2\), and let \(\varphi \in {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w^{(p)} \subset L^p_{0,r}(X)\). Then \(\varphi \) satisfies the condition \((*)\).

In combination with Theorem 1.1 we get the following.

Corollary 5.2

The \({\bar{\partial }}_w\)-equation is solvable at the \(A_n\)-singularity in the \(L^p\)-sense for \(p\ge 2\).

Proof of Theorem 5.1

Note that X has a branched \(n+1\) to 1 covering \(\mathbb {C}^2 \rightarrow X\) given by \(\pi (s,t) = (s^{n+1},t^{n+1},st)\). If \(\beta =\frac{i}{2}\sum \mathrm{{d}}\zeta _j\wedge \mathrm{{d}}{{\bar{\zeta }}}_j\) is the standard Kähler form on \(\mathbb {C}^3\), so that \(\beta ^2|_X = \mathrm{{d}}V_X\), then we obtain:
$$\begin{aligned} \pi ^*\beta ^2 = 2 (n+1)^2\big (|s|^{2n+2} + |t|^{2n+2} + (n+1)^2 |st|^{2n}\big ) \mathrm{{d}}V(s,t). \end{aligned}$$
(5.1)
Recall that by (3.3), we can choose a representation \(\omega _X = \sum \omega _{i,j} ~\mathrm{{d}}\zeta _i \wedge \mathrm{{d}}\zeta _j\) of the structure form such that \(|\omega _{i,j}| \lesssim 1/|\partial f|\). Since \(\pi ^*|\partial f|^2=|s|^{2n+2} + |t|^{2n+2} + (n+1)^2 |st|^{2n}\) we get from (5.1) that
$$\begin{aligned} \pi ^* |\omega _{i,j}|^2 \pi ^* \beta ^2 \lesssim \mathrm{{d}}V(s,t). \end{aligned}$$
(5.2)
Let \(\mu _k\) be cut-off functions as in Sect. 2.2 and let \(D_k = \{\zeta \in X: e^{-e^{k+1}}< |\zeta | < e^{-e^k} \}\). Then, in view of (2.5), the integral in \((*)\) is a finite sum of integrals, which are bounded by constants times integrals of the form
$$\begin{aligned} \int _{D_k} |\omega _{i,j}| \frac{1}{|\zeta |\big |\log |\zeta |\big |} |\gamma | \beta ^2\le & {} \left( \int _{D_k} \frac{|\omega _{i,j}|^2 \beta ^2}{|\zeta |^2 \log ^2|\zeta |} \right) ^{1/2} \left( \int _{D_k} |\gamma |^2 \beta ^2 \right) ^{1/2}\\=: & {} (I_{1,k})^{1/2} (I_{2,k})^{1/2} \end{aligned}$$
where \(\gamma \in L^2_{0,0}(X)\) and the inequality follows from the Cauchy–Schwarz inequality. Note that \(I_{2,k} \rightarrow 0\) for \(k\rightarrow \infty \) by dominated convergence because \(\gamma \in L^2_{0,0}(X)\) and the domain of integration shrinks to 0. Therefore, it is enough to show that \(I_{1,k}\) is uniformly bounded in  k.
From (5.2) it follows that
$$\begin{aligned} I_{1,k} \lesssim \int _{\pi ^{-1}(D_k)} \frac{\mathrm{{d}}V(s,t)}{\big ( |s|^{2n+2} + |t|^{2n+2} + |st|^2\big ) \log ^2\big ( |s|^{2n+2} + |t|^{2n+2} + |st|^2\big )}. \end{aligned}$$
(5.3)
Let us decompose
$$\begin{aligned} \pi ^{-1}(D_k) = \{ e^{-2e^{k+1}} \le |s|^{2n+2} + |t|^{2n+2} + |st|^2 \le e^{-2e^k} \}\subset \mathbb {C}^2 \end{aligned}$$
into \(E_k := \pi ^{-1}(D_k) \cap \{(1/2) e^{-2e^{k+1}} \le |st|^2 \le e^{-2e^k} \}\) and \(F_k:=\pi ^{-1}(D_k) \setminus E_k.\) Note that \(E_k\subset E_k' := \{ (s,t) \mid e^{-e^{k+2}} \le |st| \le e^{-e^k} \}\) because \(2^{-1/2} e^{-e^{k+1}} > e^{-e^{k+2}}\). Therefore we obtain by [25, Appendix B]:
$$\begin{aligned}&\int _{E_k} \frac{\mathrm{{d}}V(s,t)}{\big ( |s|^{2n+2} + |t|^{2n+2} + |st|^2\big ) \log ^2\big ( |s|^{2n+2} + |t|^{2n+2} + |st|^2\big )}\\&\quad \le \int _{E_k'} \frac{\mathrm{{d}}V(s,t)}{|st|^2 \log ^2\big ( |st|^2\big )} \le C \end{aligned}$$
uniformly in k.
Next note that on \(F_k\), \(|s|^{2n+2} + |t|^{2n+2} \ge |st|^2\). Therefore if \((s,t)\in F_k\) satisfies \(|s| \le |t|\), then \(|st|^2 \le 2|t|^{2n+2}\), and so
$$\begin{aligned} e^{-2e^{k+1}} \le&|s|^{2n+2} + |t|^{2n+2} + |st|^2&\le 4 |t|^{2n+2},\\ |t|^{2n+2} \le&|s|^{2n+2} + |t|^{2n+2} + |st|^2&\le e^{-2e^k}, \end{aligned}$$
and \(|s|^2 \le 2|t|^{2n}\). By symmetry we get that
$$\begin{aligned} F_k \subset \{A_k\le |s| \le B_k, 0 \le |t| \le \sqrt{2}|s|^n \} \cup \{A_k \le |t| \le B_k, 0 \le |s| \le \sqrt{2}|t|^n \}, \end{aligned}$$
where \(A_k =e^{-{e^{k+1}/(n+1)}}/2^{1/(n+1)}\) and \(B_k=e^{-e^{k}/(n+1)}\). Now by integration in polar coordinates
$$\begin{aligned}&\int _{F_k} \frac{\mathrm{{d}}V(s,t)}{\big ( |s|^{2n+2} + |t|^{2n+2} + |st|^2\big ) \log ^2\big ( |s|^{2n+2} + |t|^{2n+2} + |st|^2\big )} \\&\quad \lesssim \int _{A_k}^{B_k} \int _0^{\sqrt{2}r_2^n} \frac{r_1 r_2 \mathrm{{d}}r_1 \mathrm{{d}}r_2}{r_2^{2n+2} \log ^2(r_2)} = \int _{A_k}^{B_k}\frac{\mathrm{{d}}r_2}{r_2 \log ^2(r_2)} \rightarrow 0 \end{aligned}$$
when \(k \rightarrow \infty \) because the integrand is integrable over, say [0, 1 / 2]. Thus, (5.3) is uniformly bounded in k. \(\square \)

5.2 On the Euler Characteristics of the Structure Sheaf

As a preparation for the proof of the existence of obstructions for solvability of the \({\bar{\partial }}_w\)-equation at canonical singularities in the \(L^2\)-sense, we need some observations on the behaviour of the Euler characteristics of the structure sheaf under resolution of singularities.

Let \(\mathcal {F} \rightarrow X\) be a coherent analytic sheaf over a compact complex space X of pure dimension n, and let \(\chi (\mathcal F)\) be the Euler characteristic of \(\mathcal F\),
$$\begin{aligned} \chi \big (\mathcal {F}\big ) := \sum _{j=0}^n (-1)^j \dim H^j\big (X,\mathcal {F}\big ). \end{aligned}$$
If D is a divisor on X, associated to a line bundle \(L \rightarrow X\), then \(\chi \big (\mathcal {O}_X(D)\big )\) is the holomorphic Euler characteristic of L.

Proposition 5.3

Let \(\pi : M \rightarrow X\) be a resolution of singularities of a compact surface X with at most canonical singularities. Then \( \chi (\mathcal {O}_X) = \chi (\mathcal {O}_M)\).

Proof

Since X is a normal space, \(\pi _* \mathcal {O}_M=\mathcal {O}_X\). Moreover, canonical singularities are rational so that \(R^k \pi _* \mathcal {O}_M = 0\) for \( k>0\). Hence, the Leray spectral sequence gives \(H^k(X,\mathcal {O}_X) \cong H^k(M,\mathcal {O}_M)\) for \(k\ge 0\). \(\square \)

If we assume that the \({\bar{\partial }}_w\)-equation is locally solvable in the \(L^2\)-sense, then we obtain another representation of \(\chi (\mathcal {O}_X)\) for arbitrary normal complex surfaces.

Theorem 5.4

Let X be a compact normal complex surface, \(\pi : M\rightarrow X\) a resolution of singularities with only normal crossings, \(Z:=\pi ^{-1}(X_{sing})\) the unreduced exceptional divisor and \(E:=|Z|\) the exceptional divisor. If the \({\bar{\partial }}_w\)-equation is locally solvable in the \(L^2\)-sense for (0, 1)-forms, then
$$\begin{aligned} \chi \big (\mathcal {O}_X\big ) =\chi \big ( \mathcal {O}_M(Z-E) \big ). \end{aligned}$$
(5.4)

Proof

Following [25, Section 2.1], let \(\mathcal C_{0,r}\) denote the fine sheaves \(L_{0,r}^{2, loc}\cap {{\,\mathrm{Dom}\,}}{\bar{\partial }}_w\) and consider the sheaf complex
$$\begin{aligned} 0 \rightarrow \mathcal {O}_X \longrightarrow \mathcal {C}_{0,0} \overset{{\bar{\partial }}_w}{\longrightarrow } \mathcal {C}_{0,1} \overset{{\bar{\partial }}_w}{\longrightarrow } \mathcal {C}_{0,2} \rightarrow 0. \end{aligned}$$
(5.5)
It is easy to see that (5.5) is exact at \(\mathcal {C}_{0,0}\) because X is normal; a germ \(f \in \ker {\bar{\partial }}_w \subset \mathcal {C}_{0,0}\) is a holomorphic function on the regular locus of X, and so it is also strongly holomorphic by normality. Moreover, (5.5) is exact at \(\mathcal {C}_{0,2}\), see [17, Theorem 4.3]. (It is usually not difficult to solve \({\bar{\partial }}\)-equations in the highest degree, see also [28].) In general, (5.5) is not necessarily exact at \(\mathcal {C}_{0,1}\), but here, we assume that this is the case. Thus, (5.5) is a fine resolution of \(\mathcal {O}_X\); in particular \(H^k(X,\mathcal {O}_X)= H^k(\Gamma (X,\mathcal {C}_{0,\bullet }))\). By [25, Theorem 1.13] \(H^k(\Gamma (X,\mathcal {C}_{0,\bullet }))= H^k( M, \mathcal {O}_M(Z-E))\), and so
$$\begin{aligned} H^k(X,\mathcal {O}_X)=H^k( M, \mathcal {O}_M(Z-E)), \end{aligned}$$
which proves (5.4). \(\square \)

Combining Proposition 5.3 and Theorem  5.4 we get:

Corollary 5.5

Let X be a compact complex surface with at most canonical singularities. If the \({\bar{\partial }}_w\)-equation is locally solvable in the \(L^2\)-sense for (0, 1)-forms on X, then
$$\begin{aligned} \chi \big (\mathcal {O}_M) = \chi \big ( \mathcal {O}_M(Z-E)\big ) \end{aligned}$$
(5.6)
for any resolution of singularities \(\pi : M\rightarrow X\) with only normal crossings.

Thus, if we are looking for obstructions to solvability of the \({\bar{\partial }}_w\)-equation in the \(L^2\)-sense at canonical singularities, we just need to find configurations violating (5.6).

5.3 Obstructions for \({\bar{\partial }}_w\) at Canonical Singularities

Theorem 5.6

There exist obstructions to local solvability of the \({\bar{\partial }}_w\)-equation in the \(L^2\)-sense for (0, 1)-forms at singularities of type \(D_n\), \(n\ge 4\), \(E_6\), \(E_7\) and \(E_8\).

Hence, \((*)\) does not hold for all \(\varphi \in \ker {\bar{\partial }}_w \subset L^2_{0,1}\) at such singularities.

Proof

Let X be a projective variety with a single singularity of one of the types above, and \(\pi : M \rightarrow X\) a resolution of singularities with only normal crossings. In view of the discussion above it suffices to show that (5.6) does not hold. For the \(D_n\)-singularities, \(n\ge 4\), this was proved in the proof of Theorem 4.8 in [19] using the Riemann-Roch formula for regular complex surfaces
$$\begin{aligned} \chi \big (\mathcal {O}_M(Z-E)\big ) = \chi \big ( \mathcal {O}_M\big ) + \frac{1}{2} \big ( (Z-E)\cdot (Z-E) - (Z-E)\cdot K\big ), \end{aligned}$$
where K is the canonical divisor on M. Since \(\mathcal {O}(K)\) is trivial on a neighborhood of the exceptional set, \(Z_j\cdot K=0\) for any irreducible component \(Z_j\) of the exceptional set, cf. [8, page 135], and thus \((Z-E)\cdot K =0\). Pardon proved that \((Z-E)\cdot (Z-E)=-2\) so that
$$\begin{aligned} \chi \big (\mathcal {O}_M) = \chi \big ( \mathcal {O}_M(Z-E)\big ) + 1 \end{aligned}$$
(5.7)
and in particular (5.6) does not hold.
For the remaining singularities, \(E_6\), \(E_7\) and \(E_8\), we proceed analogously to [19] and show that (5.7) holds also for these singularities. Now, let \(\pi : M \rightarrow X\) be the minimal resolution of X. Then the exceptional divisor Z has normal crossings and the irreducible components \(E_j\) have self-intersection \(-2\) and pairwise intersections according to the Dynkin diagrams of \(E_6\), \(E_7\) or \(E_8\), see, e.g., [8]. The labels of the nodes in the following diagrams are the multiplicities of the corresponding divisors in the unreduced fundamental cycle Z, cf., e.g., [11, Example 7.2.5] and [6, Proposition 3.8].
This means that in the case of the \(E_6\)-singularity, we can label the irreducible components of Z so that \(Z-E=2Z_0 + Z_1 + Z_2 + Z_3\) and \(Z_\nu ^2=-2\), \(Z_0 \cdot Z_\mu =1\) if \(\mu \ge 1\), and \(Z_\nu \cdot Z_\mu =0\) if \(\mu >\nu \ge 1\). For the \(E_7\)-singularity we can label the irreducible components of Z so that \(Z-E=3Z_0 + 2Z_1 + Z_2 + Z_3+ 2Z_4+Z_5\) and \(Z_\nu ^2=-2\) for all \(\nu \), \(Z_0 \cdot Z_1 = Z_0 \cdot Z_3 = Z_0 \cdot Z_4=Z_1 \cdot Z_2=Z_4 \cdot Z_5=1\), and \(Z_\nu \cdot Z_\mu =0\) for all other combinations of \(\nu \ne \mu \). Finally, for the \(E_8\)-singularity, we have \(Z-E= 5Z_0 + 3Z_1 + Z_2 + 2Z_3+ 4Z_4+3Z_5 + 2Z_6 + Z_7\) and \(Z_\nu ^2=-2\) for all \(\nu \), \(Z_0 \cdot Z_1 = Z_0 \cdot Z_3 = Z_0 \cdot Z_4=Z_1 \cdot Z_2=Z_4 \cdot Z_5= Z_5 \cdot Z_6 = Z_6\cdot Z_7=1\), and \(Z_\nu \cdot Z_\mu =0\) for all other combinations of \(\nu \ne \mu \). In all three cases, a computation yields that \((Z-E)\cdot (Z-E)=-2\), which implies (5.7). \(\square \)

Footnotes

  1. 1.

    This is what we take as definition of \({\bar{\partial }}_w^{(p)}\) on X. However, to be precise, this definition only coincides with the maximal closed extension of \({\bar{\partial }}_{sm}\) for \(p \ge 4/3\), which is the only case of interest to us. In general, that \(\varphi \) lies in the domain of the maximal closed extension of \({\bar{\partial }}_{sm}\) means that \({\bar{\partial }}\varphi |_{X_{\mathrm{{reg}}}} \in L^p(X_{\mathrm{{reg}}})\). When \(p \ge 4/3\), it then follows that \({\bar{\partial }}\varphi \in L^p(X)\), see [23, Satz 4.3.3].

Notes

Acknowledgements

This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Grant RU 1474/2 within DFG’s Emmy Noether Programme. The first, second, and last author were partially supported by the Swedish Research Council. We would like to thank the anonymous referee for valuable comments regarding the presentation of the article.

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and the University of GothenburgGothenburgSweden
  2. 2.Department of MathematicsUniversity of WuppertalWuppertalGermany

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