A note on meromorphic convexity

We give examples of Stein domains D in C ² such that C ² ? D is either completely pluripolar or union of germs of (principal) hypersurfaces not intersecting D such that D fails to be meromorphically convex in C ². © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Some open problems concerning q-convexity and Stein spaces

We discuss some open problems in the theory of analytic pseudoconvexity. We focus our attention especially on q-complete spaces and Stein spaces.     

C k -estimates for the -equation on convex domains of finite type

For a bounded convex domain with C^∞ smooth boundary of finite type m and q=1, . . . ,n−1, we construct a -solving integral operator T*_q such that for all k ∈ ℕ and the usual C^k and -norms the operator is continuous.     

Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions, the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148(2): 619–693, 1998; J Symplectic Geom 3:565–587, 2005).      

Extension of C R Structures on three dimensional compact pseudoconvex C R manifolds

Let M be a smooth compact orientable pseudoconvex C R manifold of real dimension three and assume that there is a smooth function λ which is strictly subharmonic in any direction where the Levi-form vanishes on M. Then we extend the given C R structure on M to an integrable almost complex structure on the concave side of M. As an application, if M is a non-compact pseudoconvex C R manifold of real dimension three, we prove that the given C R structure on M can be locally extended to an integrable almost complex structure on the concave side of M. Partially supported by Korea research foundation Grant (KRF-2001-015-DP0016) and by R14-2002-044-01000-0 from KRF     

An interpolation theorem for proper holomorphic embeddings

Given a Stein manifold x of dimension n > 1, a discrete sequence , and a discrete sequence where , there exists a proper holomorphic embedding satisfying f(a _j ) = b _j for every j = 1,2,... Forstnerič and Prezelj supported by grants P1-0291 and J1-6173, Republic of Slovenia. Kutzschebauch supported by Schweizerische National fonds grant 200021-107477/1. Ivarsson supported by The Wenner-Gren Foundations.     

Characterisation of homogeneous polynomials which are constant on complex-tangential curves in the boundary of the unit ball of

We prove that a complex-tangential curve γ in the boundary of the unit ball of having the property that there exists a homogeneous polynomial P such that P=1 on γ has constant curvature. This implies that a homogeneous polynomial P having the property that there exists a closed complex-tangential curve γ (respectively a totally real 2-dimensional submanifold) in the boundary of the unit ball of such that P=1 on γ (respectively |P|=1 on γ) reduces to a monomial by a unitary chage of variables. These results represent a positive answer to conjectures of H. O. Kim.     

The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.     

The Bremermann–Dirichlet problem for unbounded domains of $${\mathbb{C}}^n$$

Given an unbounded strongly pseudoconvex domain Ω and a continuous real valued function h defined on bΩ, we study the existence of a (maximal) plurisubharmonic function Φ on Ω such that Φ|_b Ω = h. Supported by the MURST project “Geometric Properties of Real and Complex Manifolds”.     

Problème du bord dans les variétés $q$-convexes et phénomène de Hartogs-Bochner

Résumé.   Soient E un fibré vectoriel holomorphe au dessus d'une variétéq-complète X et M une sous-variété réelle de dimension 2p-1 () de E. Nous montrons que le problème du bord pour M est résoluble si et seulement si M est maximalement complexe. Dans le cas où nous retrouvons le théorème de Harvey et Lawson [17]. Comme conséquence nous obtenons une généralisation du théorème de Hartogs-Bochner pour les applications CR-méromorphesà valeurs dans une variétéq-convexe et une généralisation au cas CR-méromorphe du théorème d'extension de Chazal [5].

---

Let E be a holomorphic vector bundle over a -complete manifold X and M be a real submanifold of dimension 2p-1 () of E. We prove that M has a solution to the boundary problem in X if and only if M is maximally complex. In the case , this is a result of Harvey and Lawson [17]. As a consequence, we obtain a generalization of the Hartogs-Bochner theorem for CR-meromorphic maps taking their values in -convex manifolds and a generalization to the CR-meromorphic case of a theorem of Chazal [5].

Received: 27 April 2000 / Published online: 23 July 2001     

On holomorphic Banach vector bundles over Banach spaces

Let X be a Banach space with a countable unconditional basis (e.g., X = ℓ ₂ Hilbert space), Ω ⊂ X pseudoconvex open, E → Ω a locally trivial holomorphic vector bundle with a Banach space Z for fiber type, the sheaf of germs of holomorphic sections of E → Ω, and Z ₁ the Banach space . Then and Ω × Z ₁ are holomorphically isomorphic, is acyclic and E is so to speak stably trivial over Ω in a generalized sense. We also show that if E is continuously trivial over Ω, then E is holomorphically trivial over Ω. In particular, if Z = ℓ ₂ or Ω is contractible, then E is holomorphically trivial over Ω. Some applications are also given. To my dear little daughter, Sári Mangala, on her third birthday. Supported in part by a Research Initiation Grant from Georgia State University.     

Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ ₁(w), . . . , Δ _r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ _k (w)/Δ _k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.     

Logarithmic moduli spaces for surfaces of class VII

In this paper we describe logarithmic moduli spaces of pairs (S, D) consisting of a minimal surface S of class VII with second Betti number b ₂ > 0 together with a reduced maximal divisor D of b ₂ rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of S\D and for the associated holomorphic contraction . Part of this work was done while the first author visited the University of Osnabrück under the program “Globale Methoden in der komplexen Geometrie” of the DFG and while the second author visited the Max-Planck-Institut für Mathematik in Bonn and the LATP, Université de Provence. We thank these institutions for their hospitality and for financial support. Furthermore the authors wish to thank Georges Dloussky for numerous discussions on surfaces of class VII.     

Multipliers in Hardy-Sobolev Spaces

For 1 < p < ∞ and 0 ≤ n − sp < 1, we give characterizations of the space of pointwise multipliers of the holomorphic Hardy-Sobolev spaces H_s^p on the unit ball B of As an application of these results we obtain a corona theorem for these spaces.     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.      

A $\overline{\partial}$ -Theoretical Proof of Hartogs’ Extension Theorem on Stein Spaces with Isolated Singularities

Let X be a connected normal Stein space of pure dimension d≥2 with finitely many isolated singularities. By solving a weighted -equation with compact support on a desingularization of X, we derive Hartogs’ Extension Theorem on X by the -idea due to Ehrenpreis.      

On semianalytic subsets of rigid analytic varieties

We show some topological properties of semianalytic subsets of rigid analytic varieties: curve selection lemma, the closure of a semianalytic subsetS is semianalytic, for every quasi-compact morphismf. As an application we show that a morphismf: X Y of rigid analytic varieties is open at a pointx X if and only if SpecO _X,x SpecO _Y,f(x) is surjective.     

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)² ? 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.     

Levi problem and semistable quotients

A complex space X is in class 𝒬 _G if it is a semistable quotient of the complement to an analytic subset of a Stein manifold by a holomorphic action of a reductive complex Lie group G. It is shown that every pseudoconvex unramified domain over X is also in 𝒬 _G .     

Cohomology of locally q-complete sets in Stein manifolds

We prove that if X is a Stein complex manifold of dimension n and Ω???X a locally q-complete open set in X with q?≤?n?2, then the cohomology groups H ^p (Ω , OΩ) vanish if p?≥?q and OΩ is the sheaf of germs of holomorphic functions on Ω.