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The purpose of this paper is to study holomorphic approximation and approximation of $$\overline{\partial }$$ -closed forms in complex manifolds of complex dimension $$n\ge 1$$ . We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the $${{\mathcal {C}}}^\infty $$ -smooth and the $$L^2$$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

In this paper, we consider the problem of solving the Cauchy–Riemann equation with prescribed support in a domain of a complex manifold for forms or currents. We are especially interested in the case when the domain is a Hartogs triangle in C 2 \mathbb {C}^2 or C P 2 \mathbb {C}\mathbb {P}^2 . In particular, we show that the strong L 2 L^2 Dolbeault cohomology group on the Hartogs triangle in C P 2 \mathbb {C}\mathbb {P}^2 is infinitely dimensional.

Let \(\Omega ={\widetilde{\Omega }}{\setminus } \overline{D}\) where \({\widetilde{\Omega }}\) is a bounded domain with connected complement in \({\mathbb {C}}^n\) (or more generally in a Stein manifold) and D is relatively compact open subset of \({\widetilde{\Omega }}\) with connected complement in \(\widetilde{\Omega }\). We obtain characterizations of pseudoconvexity of \({\widetilde{\Omega }}\) and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups of \(\Omega \) on various function spaces. In particular, we show that if the boundaries of \({\widetilde{\Omega }}\) and D are Lipschitz and \(C^2\)-smooth respectively, then both \({\widetilde{\Omega }}\) and D are pseudoconvex if and only if 0 is not in the spectrum of the \(\overline{\partial }\)-Neumann Laplacian of \(\Omega \) on (0, q)-forms for \(1\le q\le n-2\) when \(n\ge 3\); or 0 is not a limit point of the spectrum of the \(\overline{\partial }\)-Neumann Laplacian on (0, 1)-forms when \(n=2\).

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli., Small typos corrected; Final Version; To appear in Indiana University Mathematics Journal

In this paper we study the non-closed range of the Cauchy-Riemann operator for relatively compact domains in \(\mathbb C^{n}\) or in a complex manifold. We give necessary and sufficient conditions for the \(L^{2}\) closed range property for \(\overline{\partial }\) on bounded Lipschitz domains in \(\mathbb C^{2}\) with connected complement. It is proved for the Hartogs triangle that \(\overline{\partial }\) does not have closed range for (0, 1)-forms smooth up to the boundary, even though it has closed range in the weak \(L^{2}\) sense. An example is given to show that \(\overline{\partial }\) might not have closed range in \(L^{2}\) on a Stein domain in complex manifold.

Let M be a C∞-submanifold of codimension k in Cn , 1 ≤ k ≤ n − 1. For a ∈ M , a collection ρ = (U ; ρ1, . . . , ρk) will be called a defining collection of M at a if U ⊆ Cn is open, a ∈ U , ρ1, . . . , ρk are real C∞-functions on U , U ∩ M = { ζ ∈ U ∣ ρ1(ζ ) = . . . ρk(ζ ) = 0 } = ∅ and dρ1(ζ ) ∧ . . . ∧ dρk(ζ ) = 0 for all ζ ∈ U ∩ M . For a ∈ M , let T a (M) be the complex tangent space of M at a. We assume that M is a generic C R-submanifold of Cn which means, by definition, that

We search a cohomological and a geometrical characterization of the open subsets of a strictly pseudoconvex boundary in a Stein manifold on which one can solve the tangential Cauchy–Riemann equation in all bidegrees.On cherche une caractérisation cohomologique et géométrique des ouverts du bord d'un domaine strictement pseudoconvexe relativement compact d'une variété de Stein sur lesquels on peut résoudre l'équation de Cauchy–Riemann tangentielle en tout bidegré.

Nous etudions la resolubilite locale de l'operateur de Cauchy-Riemann tangentiel sur un voisinage w d'un point z 0 d'une sous-variete CR generique q-concave M de codimension quelconque de C n . Nous construisons une formule d'homotopie pour le ∂ b sur ω, lorsque ω est l'intersection de M et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un operateur de resolution pour toute forme ∂ b -fermee sur ω. Nous en deduisons des estimations L P et des estimations holderiennes jusqu'au bord pour la solution de l'equation de Cauchy-Riemann tangentielle sur ω.

Soit D ⊂⊂ C n , n ≥ 2, un domaine a bord C 2 et K ⊂ ∂D un compact tel que ∂D\K soit connexe. On etudie l'extension holomorphe des fonctions CR definies sur ∂D\K a des sous-ensembles de D. On dit que K est CR-convexe si son enveloppe A(D)-convexe, A(D) - hull(K), verifie K = ∂D∩A(D) - hull(K) (A(D) designe l'espace des fonctions holomorphes sur D et continues sur D). Le theoreme principal de cet article prouve l'extension holomorphe a ∂D\A(D) - hull(K), si K est CR-convexe.

Using Serre duality in CR manifolds and integral operators for the solution of the tangential Cauchy–Riemann equation with compact support, we prove a separation theorem of Andreotti–Vesentini type for the \(\)-cohomology in q-concave real hypersurfaces.

We prove a separation criterion for the compactly supported Dolbeault cohomology. As an application we give a simple proof of the following result: If X is an ( n − q )-convex complex manifold of dimension n, 1≤q≤n−1 , and K is a compact subset of X which admits a basis of q-convex neighborhoods, then H p , n − q ( X \ K , E ) is separated for all p and each holomorphic vector bundle E over X.

We prove a vanishing theorem for the -cohomology in top degree on 1-concave CR generic manifolds.

Resume Si M est une sous-variete CR generique, q-concave de classe C∞, de codimension reelle k d'une variete analytique complexe X de dimension n, les complexes de Cauchy-Riemann tangentiels tronques τ≤q−1[ɛp,*](M), τ≥n−k−q+2[ɛp, *](M) des formes differentielles de classe C∞ et τ≤q−1 [D′p.*](M), τ≥n−k−q+2[D′p,*](M) des courants sur M sont quasi-isomorphes pour 0 ≤ p ≤ n.

Applying the methods of Serre duality in the setting of CR manifolds we prove approximation theorems and we study the Hartogs-Bochner phenomenon in 1-concave CR generic manifolds.

Notation. We denote byC0 s,r(X,E) the space of continuousE-valued (s, r)forms on X (we omit E, when E is the trivial line bundle), by Z0 s,r(X,E) the subspace of ∂-closed forms, and byE0 s,r(X,E) the subspace of ∂-exact forms (E0 s,0(X,E) := {0}). As usual, H(X,E) : = Z0 s,r(X,E)/E 0 s,r(X,E). 0.1. Definition. X will be called q-concave-q∗-convex where q, q∗ are integers with 1 ≤ q ≤ n− 1 and 0 ≤ q∗ ≤ n− 1 if X is connected and there exists a real C2 function ρ on X such that if

Let X be a complex manifold. The study of the closed-range property of the Cauchy–Riemann equations is of fundamental importance both from the sheaf theoretic point of view and the PDE point of view. In terms of the associated cohomology, it means that the corresponding cohomology is Hausdorff, hence separated. There are many known results for the Hausdorff property of such cohomologies in complex manifolds (see in particular [16–18]). For instance, it is well-known that for a bounded pseudoconvex domain D inCn , the Dolbeault cohomology H p,q(D) in the Frechet space C∞p,q(D) vanishes for all q > 0. It is also known that the L2 cohomology also vanishes. Much less is known about the cohomologies whose topology does not have the Hausdorff property, even for domains in Cn (an example is given in [13], section 14). In this paper, we study duality of the Cauchy–Riemann complex in various function spaces and the Hausdorff property of the corresponding cohomologies. Such duality is classical if

We study the stability of the embeddability of compact 2 2 -concave CR manifolds in complex manifolds under small horizontal perturbations of the CR structure.

We start this chapter by giving various characterisations of the compact sets K in the boundary of a strictly pseudoconvex domain D in a Stein manifold of dimension n which have the following property: any continuous CR function on \(\partial D\backslash K\) can be extended holomorphically to the whole of D. We will obtain a geometric characterisation of such sets for n = 2 and a cohomological characterisation of such sets for n ⩾ 3. Amongst other things, we prove that the suffcient cohomological condition given in Theorem 5.1 of Chapter V is necessary if the ambient manifold is Stein and the domain D is assumed strictly pseudoconvex. We end the section with a geometric characterisation of the compact sets K such that any continuous CR function defined on \(\partial D\backslash K\)which is orthogonal to the set of \(\overline{\partial}\) -closed (n; n−1)-forms whose support does not meet K can be extended holomorphically to the whole of D. When K is empty this condition is just the hypothesis of Theorem 3.2 of Chapter IV.

In this chapter we define one of the fundamental tools of integral representation theory in complex analysis, namely the Bochner–Martinelli–Koppelman kernel. This kernel generalises the Cauchy kernel on ℂ to ℂn. It enables us to prove an integral representation formula, the Bochner–Martinelli–Koppelman formula, which extends Cauchy’s formula to (p, q) differential forms in ℂn. This formula plays an important role in the study of the operator \(\overline{\partial}\): in particular, we prove using this formula our first results on the existence of solutions to the Cauchy–Riemann equation in ℂn by considering the case where the data has compact support. Hartog’s phenomenon, a special case of which was studied in Chapter I, follows from the existence of a compactly supported solution to the Cauchy–Riemann equations for n ⩾ 2 when the right-hand side of the equation is a compactly supported form of bidegree (0, 1). The links between the vanishing of compactly supported Dolbeault cohomology groups in bidegree (0, 1) and Hartog's phenomenon will be explored in greater detail in Chapter V. We will also use the Bochner–Martinelli–Koppelman formula to study the regularity of the operator \(\bar{\partial}\) by proving a Holder hypoellipticity theorem.

In this chapter we study the local properties of holomorphic functions of several complex variables which can be deduced directly from the classical theory of holomorphic functions in one complex variable. The basis for our work is a Cauchy formula for polydiscs which generalises the classical Cauchy formula. Most of the theorems proved in this chapter extend well-known theorems for holomorphic functions in dimension 1 (such as the open mapping theorem, the maximum principle, Montel's theorem and the local inversion theorem) to multivariable analysis. However, when we try to extend holomorphic function a phenomenon which is specific to n-dimensional space with n ⩾ 2 appears, namely Hartog's phenomenon. A special case of this phenomenon, which is studied in detail in Chapter III, is discussed at the end of this chapter.

The aim of this chapter is to study the Hartogs–Bochner phenomenon on complex analytic manifolds. We start by studying the relationship between Hartogs’ phenomenon and the vanishing of the Dolbeault cohomology group with compact support in bidegree (0, 1). We then give some cohomological conditions which enable us to extend a CR function of class \(C^\infty\) defined on a subset of the boundary of a domain to a holomorphic function on the whole domain. This work generalises the geometric situation studied at the end of Chapter IV. This work generalises the geometric situation studied at the end of Chapter IV. Proving similar results for CR functions of class Ck requires two extra elements: a theorem on local resolutions of \(\partial\) and an isomorphism theorem between the various cohomology groups \(H^{p,q}_{\alpha}\)(X). This isomorphism theorem follows from the local resolution and some sheaf-theoretic results which are given in Appendix B. The existence of the resolution is proved by solving \(\overline{\partial}\) in convex domains with C2 boundary using a new integral formula, the Cauchy–Fantappie formula.

This chapter is devoted to solving the Levi problem – or in other words, to proving that any pseudoconvex open set in ℂn is a domain of holomorphy. We proceed by studying \(\overline{\partial}\) in pseudoconvex open sets using local integral representation formulas for strictly pseudoconvex domains and then applying H. Grauert’s bumping technique.

At the end of Chapter I and in Chapter III we met open sets in ℂ n on which any holomorphic function can be extended to a larger open set. The open sets which do not have this property are called domains of holomorphy: in this chapter we study such open sets. We start by giving a characterisation of domains of holomorphy in terms of holomorphic convexity (the Cartan–Thullen theorem). We then introduce the notion of pseudoconvexity in order to get a more analytic characterisation of domains of holomorphy. This requires us to define plurisubharmonic functions. We then prove that every domain of holomorphy is pseudoconvex: the converse, which is known as the Levi problem, is studied in Chapter VII.

Whilst studying Hartogs’ phenomenon in Chapter III we proved that if D is a simply connected bounded domain in ℂn,n ⩾ 2, then any holomorphic function defined on a neighbourhood of the boundary of D can be extended to a holomorphic function on D. It follows that the restriction to \(\partial\)D of a holomorphic function defined in a neighbourhood of \(\partial\)D is the boundary value of a holomorphic function on D which is continuous on \(\overline{D}\). We now try to characterise the boundary values of holomorphic functions on a bounded domain \(D \subset \mathbb{C}^n\) which are continuous on \(\overline{D}\). The main result of this chapter is Bochner’s extension theorem for CR functions defined on the boundary of a domain. Its proof uses the Bochner–Martinelli transform which is studied in Section 1. We also prove our first generalisation of Bochner’s theorem to CR functions which are only defined on part of the boundary of the domain. This generalisation is also based on the properties of the Bochner–Martinelli transform but it requires two extra ingredients: Stokes’ formula for CR functions and the integrals of the Bochner–Martinelli kernel.

We prove sharp anisotropic H\"older estimates for the local solutions of the tangential Cauchy-Riemann equation in q-concave CR manifolds and we derive the same kind of estimates for global solutions when the manifold is compact.

We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some \( \bar \partial _b \)-cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups.

International audience; Using functional analysis, we derive from local homotopy formulas for the tangential Cauchy-Riemann operator a global homotopy formula for compact CR manifolds without loss of regularity.

Using functional analysis and a Friedrichs approximation lemma for first order differential operators, we derive a global homotopy formula in large degrees for the tangential Cauchy-Riemann operator from local homotopy formulas without loss of regularity.

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface M and holomorphic convexity of compact sets in M, or bounded in part by M. Applications include extendability of Cauchy-Riemann functions, solvability of the dibar_b-equation, approximability of holomorphic and CR functions, and global regularity of the dibar-Neumann operator., Comment: Final version, Trans. Amer. Math. Soc

LetX be a complex manifold with finitely many ends such that each end is either q-concave or (n−q)-convex. If $q< \tfrac{1}{2}n$ , then we prove that Hpn−q(X) is Hausdorff for all p. This is not true in general if $q \geqslant \tfrac{1}{2}n$ (Rossi’s example with n=2 and q=1). If all ends are q-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for $q \geqslant \tfrac{1}{2}n$ ). Moreover the result was already known in the case when the q-concave ends can be ‘filled in’ (again also for $q \geqslant \tfrac{1}{2}n$ ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, $q< \tfrac{1}{2}n$ .

Let X be a complex manifold of dimension n and suppose an open subset Z of X is given such that \( \bar Z\backslash Z \) is compact.

Etant donne un domaine D relativement compact d'une variete de Stein M de dimension n, n ⩾ 2, on montre que toute fonction continue, CR, definie sur un ouvert connexe de ∂D ayant un complementaire K dont l'enveloppe holomorphiquement convexe dans M ne rencontre pas ¯ D∖K, se prolonge en une fonction holomorphe sur D.

Etant donne une forme differentielle C1 ou un courant d'ordre nul, f, a support compact sur une hypersurface reelle d'une variete de Stein, on definit, en utilisant les noyaux globaux de Henkin et Leiterer, une transformee de Bochner-Martinelli generalisee de f et on etudie son comportement au voisinage de l'hypersurface. Dans le cas ou l'hypersurface est le bord d'un domaine relativement compact et ou f est une mesure verifiant les conditions de Cauchy-Riemann, on obtient une generalisation du theoreme d'extension de Bochner.