Envelopes Of Holomorphy Research Articles

A Uniqueness Theorem for the Two-Dimensional Sigma Function

We prove that the sigma functions of Weierstrass (g = 1) and Klein (g = 2) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of 2g linear differential heat equations in a nonholonomic frame (for a function of 3g variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For g =1, we give a complete description of the envelopes of holomorphy of such solutions.

Effective actions of the general indefinite unitary groups on holomorphically separable manifolds

We determine all holomorphically separable complex manifolds of dimension p + q which admit smooth envelopes of holomorphy and effective general indefinite unitary group actions of size p + q. Also we give exact description of the automorphism groups of those complex manifolds. As an application we consider a characterization of those complex manifolds by their automorphism groups.

On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces

On a normal Stein variety [Formula: see text], we study the thickening problem, i.e. the problem whether the assumption that a compact set [Formula: see text] is contained in the interior of another compact set, [Formula: see text] implies that the same inclusion holds for their holomorphic hulls. An affirmative answer is given for [Formula: see text] with isolated quotient singularities. On any Stein space [Formula: see text] with isolated singularities, we prove thickening for those hulls which have analytic structure at the singular points, obtaining a limitation for possible counter-examples. In dimension [Formula: see text], we finally relate the holomorphic hulls to analytic extension from parts of strictly pseudoconvex boundaries.

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

In this paper we investigate invariant domains in $\, \Xi^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,\Xi^+$, recently introduced by Kr\otz and Opdam, contains the crown domain $\,\Xi\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,\Xi$. We prove that the envelope of holomorphy of an invariant domain in $\,\Xi^+$, which is contained neither in $\,\Xi\,$ nor in $\,S^+$, is univalent and coincides with $\,\Xi^+$. This fact, together with known results concerning $\,\Xi\,$ and $\,S^+$, proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in $\,\Xi^+\,$ and completes the classification of invariant Stein domains therein.

A Note on Envelopes of Holomorphy

Let p:X⟶M be a Riemann domain over a connected n-dimensional complex submanifold M of $\mathbb{C}^{N}$ and let $\mathcal{F}\subset \mathcal{O}(X)$ be such that $p\in\mathcal{F}^{N}$ . Our aim is to discuss relations between the $\mathcal{F}$ -envelope of holomorphy of (X,p) in the sense of Riemann domains over M and the $\mathcal{F}$ -envelope of holomorphy of X in the sense of complex manifolds.

Topological criteria for schlichtness

AbstractWe give two sufficient criteria for schlichtness of envelopes of holomorphy in terms of topology. These are weakened converses of results of Kerner and Royden. Our first criterion generalizes a result of Hammond in dimension 2. Along the way, we also prove a generalization of Royden's theorem.

Envelopes of holomorphy and extension of functions of bounded type

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not necessarily hold for unbounded sets, answering a question posed by Hirschowitz in 1972. We also consider extensions to open subsets of the bidual, present some Banach–Stone type results and show some properties of the spectrum when the domain is the unit ball of ℓ p .

Holomorphic extension of CR functions, envelopes of holomorphy, and removable singularities

Holomorphic Extension of CR Functions, Envelopes of Holomorphy, and Removable Singularities

Komplexe Analysis

The workshop Komplexe Analysis , organised by Jean-Pierre Demailly (Grenoble), Klaus Hulek (Hannover), Ngaiming Mok (Hong Kong) and Thomas Peternell (Bayreuth) was held August 24th–August 30, 2008. This meeting was well attended with 46 participants from Europe, US, and the Far East. The participants included several leaders in the field as well as many young (non-tenured) researchers. The aim of the meeting was to present recent important results in several complex variables and complex geometry with particular emphasis on topics linking different areas of the field, as well as to discuss new directions and open problems. Altogether there were nineteen talks of 60 minutes each, a programme which left sufficient time for informal discussions and joint work on research projects. One of the topics at the center of the conference was the classification theory of higher dimensional varieties. Y. Kawamata lectured on the connections between the minimal model programme and derived categories; A. Corti discussed an approach to the finite generation of the canonical ring without minimal models, but still in connection with the seminal work which was presented by J. McKernan in the last Complex Analysis meeting in Oberwolfach 2006, where the finite generation of the canonical ring of varieties of general type was announced. Extension theorems, non vanishing and positivity result for certain direct image sheaves play a role in the global classification of complex manifolds. This was largely discussed by M. Paun and B. Berndtsson. In their work analytic methods are central, whereas the talks by Kawamata and Corti were more of an algebraic nature. Also very much on the analytic side and connected to Berndtsson's talk, H. Tsuji lectured on generalised Kähler–Einstein metrics. Families of projective manifolds over higher-dimensional base spaces were considered in the talk by S. Kebekus. Direct images of coherent sheaves also play a central role in this context. About five years ago, Campana introduced new variations on the concept of “orbifolds”; they were already the suject of talks in past sessions and have turned out to be of increasing interest – in the present session, new results on the hyperbolicity of orbifolds were presented in the talk by E. Rousseau. As to varieties with special geometry, K. Oguiso spoke on non-algebraic hyperkähler manifolds and, with a rather different flavour, F. Catanese on complex and real threefolds fibered by rational curves, with a special emphasis on real algebraic geometry. J. Chen discussed the influence of terminal singularities in three-dimensional geometry, a more algebraic topic. On the analytic side, A. Teleman reported on recent progress in the classification of non-Kähler surfaces in the so called Kodaira class VII, using gauge-theoretical methods, and S. K. Yeung lectured on new results on fake projective planes. Group actions and envelopes of holomorphy were the topics of the talk by X. Zhou. S. Boucksom discussed equidistribution of Fekete points on complex manifolds, in relation with energy functionals for Monge-Ampère operators. R. Lazarsfeld presented a very interesting new approach to study properties of linear systems and line bundles via convex geometry. Overall, moduli spaces appeared to be a central theme in the workshop, and were discussed extensively in at least four talks: V. Gritsenko considered moduli spaces of K3-surfaces; S. Grushevsky spoke on intersection numbers of divisor on the moduli space of curves, and K. Ludwig and G. Farkas lectured on the moduli spaces of spin and Prym curves, their singularities, Kodaira dimension and enumerative geometry.

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.

Some envelopes of holomorphy

Some envelopes of holomorphy

On envelopes of holomorphy of model manifolds

We construct envelopes of holomorphy for model manifolds of order 4 and describe a class of such manifolds whose envelope is a cylindrical domain (with respect to certain variables) based on a Siegel domain of the second kind. This enables us to prove the holomorphic rigidity of model manifolds of this class. We also study the envelope of holomorphy of a special model manifold of type (1,4) and show it to be a domain of bounded type whose distinguished boundary coincides with the initial manifold. The holomorphic automorphism group of this domain coincides with that of the initial manifold. The envelope of holomorphy is fibred into orbits of this group. The generic orbits are 8-dimensional homogeneous non-spherical completely non-degenerate manifolds in .

Envelopes of holomorphy of Jordan curves in ℂn

A comprehensive answer is given to two related questions: (i) Can continuous functions be approximated by holomorphic ones on Jordan curves in ℂn (A.G. Vitushkin, 1964)? (ii) Is any Jordan curve in ℂn holomorphically convex (E.L. Stout, 1970)?

A schlichtness theorem for envelopes of holomorphy

Let Ω be a domain in Open image in new window. We prove the following theorem. If the envelope of holomorphy of Ω is schlicht over Ω, then the envelope is in fact schlicht. We provide examples showing that the conclusion of the theorem does not hold in Open image in new window, n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in Open image in new window whose envelopes are multiply sheeted.

On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

Dans cet article, nous associons les techniques du principe de reflexion de Lewy-Pinchuk avec celles du principe de continuite de Behnke-Sommer. Apres avoir prolonge holomorphiquement une fonction dite de reflexion a une congruence de sous-varietes de Segre, nous sommes conduits a l'etude de l'enveloppe d'holomorphie d'un domaine recouvert d'un chapeau Levi-plat lisse. D'apres notre resultat principal, tout CR-diffeomorphisme h : M → M' de classe C∞ entre deux hypersurfaces analytiques reelles globalement minimales de C n (n ≥ 2) est analytique reel en tout point de M si M' est holomorphiquement non-degeneree. Plus generalement, nous etablissons que la fonction de reflexion R' h associee a un tel diffeomorphisme CR de classe C∞ entre deux hypersurfaces analytiques reelles globalement minimales se prolonge toujours holomorphiquement a un voisinage du graphe de h dans M x M', sans aucune condition de non-degenerescence sur M'. Cet enonce fournit une nouvelle version du principe de reflexion de Schwarz en plusieurs variables complexes. Enfin, nous demontrons que toute application h: M → M' de classe C∞ et CR entre deux hypersurfaces analytiques reelles ne contenant pas de courbes holomorphes est analytique reelle en tout point de M, sans aucune condition de rang sur h.

Globalizations, fiber bundles, and envelopes of holomorphy

We describe a relationship between globalizations of local holomorphic actions on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for particular Stein Riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms. We then derive some consequences on the existence of Stein globalizations.

On Levi's problem on complex spaces and envelopes of holomorphy

On Levi's problem on complex spaces and envelopes of holomorphy

Removable singularities ofL p CR-functions on hypersurfaces

Let H be a C2 hypersurface in ℂn, n ≥ 3, and let M be a generic submanifold of H of real codimension one. We describe classes of compact removable singularities K for Lp-solutions of the tangential Cauchy-Riemann equations on H under the conditions K ⊂ M, 1 ≤ p ≤ ∞. Removability is understood here in the classical sense, but new effects occur based on compulsory analytic extension and envelopes of holomorphy. The classical theory gives results only in the case p > 1. But even for p > 1, removable singularities for Lp-solutions of the tangential Cauchy-Riemann equations may be metrically much more massive than the classical theory predicts. The results for p = 1 are close to corresponding results on removability in the sense of analytic extension justifying the name “removability” for the latter subject. No Levi-form condition on H is posed and the description is given intrinsically in terms of the CR-structure of M. This may be interesting in connection with generalizations, for example, to more general CR-manifolds instead of H.

Holomorphic functions and embedded real surfaces

The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in ℂP2 and in some other complex surfaces.

On the uniformization of Hartogs domains in $ℂ^2$ and their envelopes of holomorphy

On the uniformization of Hartogs domains in $ℂ^2$ and their envelopes of holomorphy