Christine Laurent-Thiébaut

Stein compacts in Levi-flat hypersurfaces

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 ∂ b -equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the ∂-Neumann operator.

On serre duality

Abstract 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.

Some applications of Serre duality in {CR} manifolds

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. In this paper we study Serre duality in CR manifolds to get some approximation theorem and a better understanding of the Hartogs-Bochner phenomenon in 1-concave CR manifolds. Because of the lack of the Dolbeault isomorphism for the ∂b-cohomology in CR manifolds, Serre duality in this setting is not an exact copy of Serre duality in complex manifolds. The main point is that one has to be particularly careful and always to differentiate cohomology with smooth coefficients and current coefficients. Our main theorem in CR-manifolds (Theorem 3.2) is a consequence of rather classical abstract results on duality for complexes of topological vector spaces which we recall in section 1 (cf. [3], [9], [21]). Note that in the case of compact CR manifolds, Serre duality was already studied by Hill and Nacinovich in [13]. Associating Serre duality with Malgrange’s theorem on the vanishing of the ∂b-cohomology in top degree [19] and regularity for the tangential CauchyRiemann operator in bidegree 0, 1 [5], we give several applications for 1-concave CR manifolds in section 4. One of the most interesting is perhaps the study of the Hartogs-Bochner phenomenon. More precisely, we consider a connected non compact -smooth 1-concave CR manifold M and a relatively compact open subset D with -smooth boundary in M such that M D is connected. The question is : does any smooth CR function f on ∂D extend to a CR function F in D. It turns out that, for the Hartogs-Bochner phenomenon, 1-concave CR manifolds are rather similar to non compact complex manifolds. In both cases the Hartogs-Bochner phenomenon holds if D is sufficiently small, by a result of Henkin [11] in the CR case and since it holds in n for the complex case; but it may be false in general, it is sufficient to consider a domain D which contains the zero set of a CR or a holomorphic function (some example is given in [12] for the CR case and in [16] for the complex case). Moreover, if we remove the connectedness of M D and if we strengthen the assumption on f by some orthogonality condition to ∂-closed forms due to Weinstock [23], the HartogsBochner phenomenon holds in complex manifolds. In this paper we prove (Theorem 4.3) that Partially supported by HCM Research Network CHRX CT94 0468 1991 Mathematics Subject Classification. 32F40-32F10

The Andreotti-Vesentini separation theorem and global homotopy representation

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

A separation theorem and Serre duality for the Dolbeault cohomology

AbstractLetX be a complex manifold with finitely many ends such that each end is eitherq-concave or (n−q)-convex. If

Stability of the embeddability under perturbations of the CR structure for compact CR manifolds

q< \tfrac{1}{2}n

Uniform estimates for the Cauchy-Riemann equation on q-convex wedges

, then we prove thatHpn−q(X) is Hausdorff for allp. This is not true in general if