Hartogs type extension theorems

The Hartogs Extension Theorem for holomorphic vector bundles and sprays

We study ellipticity properties of complements of compact subsets of Stein manifolds.

A d-bar-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 isolated singularities only. By solving a weighted d-bar-equation with compact support on a desingularization of X, we derive Hartogs' Extension Theorem on X by the d-bar-idea due to Ehrenpreis.

有效与数值有效除子补集的Hartogs型延拓定理

Hartogs type extension theorem for the complement of effective and numerically effective divisors

本文利用同调代数方法和Demailly-Peternell消失定理，将Ohsawa的Hartogs延拓现象推广到Kähler流形中具有连通支撑的有效且nef除子的补集上，并给出无基点除子补集的几何刻画。

A d-bar-theoretical proof of Hartogs' extension theorem on (n-1)-complete spaces

Let X be a connected normal complex space of dimension n>=2 which is (n-1)-complete, and let p: M -> X be a resolution of singularities. By use of Takegoshi's generalization of the Grauert-Riemenschneider vanishing theorem, we deduce H^1_{cpt}(M,O)=0, which in turn implies Hartogs' extension theorem on X by the d-bar-technique of Ehrenpreis.

Hartogs' theorem on separate holomorphicity for projective spaces

If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.

A new Hartogs type extension results for the cross-like objects

We discuss the problem of the existence of envelopes of holomorphy of the A-crosses, which leads us to the far-reaching generalizations of the famous Hartogs theorem. We also take under consideration the issue of the existence of "nice" general descriptions of envelopes of holomorphy of the cross-like objects in terms of the relative extremal function, which seems to be very natural in the light of the extension theory of separately holomorphic functions on classical crosses and (N,k)-crosses.

Hartogs Type Theorems for CR L^{2} functions on Coverings of Strongly Pseudoconvex Manifolds

We prove an analog of the classical Hartogs extension theorem for CR $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic $L_{2}$ functions on coverings of pseudoconvex manifolds.

Hartogs-Bochner type theorem in projective space

We prove the following Hartogs-Bochner type theorem: Let $M$ be a connected $C^2$ hypersurface of $P_n(\mathbb{C})$ ($n\geq 2$) which divides $P_n(\mathbb{C})$ in two connected open sets $Ω_1$ and $Ω_2$. Then there exists $i \in \{1,2\}$ such that $C^{1}$ CR functions defined on $M$ extends holomorphically to $Ω_i$.

Elementary approach to the Hartogs extension theorem

In this paper we present a proof of Hartogs' extension theorem, following T. Sobieszek's paper from 2003. Hartogs' theorem provides a large class of domains where holomorphic functions have analytic continuation to larger domains, and is "a several complex variables theorem" in nature because its conclusion is false in the complex plane. Sobieszek's proof is quite remarkable because he uses, stated in his paper without proofs, only higher-dimensional identity principle for holomorphic functions and Cauchy's integral formula for compact sets. We proved this two theorems here, making this exposi…

A metric analogue of Hartogs' theorem

In this paper we prove a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with $\mathcal{C}^2$ smooth boundary is a Kähler metric, then the universal cover of the domain is the unit ball.

Théorème de Hartogs-Bochner dans $P_2(\mathbb{C})$

The present paper have been replaced by the paper: Hartogs-Bochner type theorem in Projective Space (math.CV/0011095) in which we prove the following Hartogs-Bochner type theorem: Let $M$ be a connected $C^2$ hypersurface of $P_n(\mathbb{C})$ ($n\geq 2$) which divides $P_n(\mathbb{C})$ in two connected open sets $Ω_1$ and $Ω_2$. Then there exists $i \in \{1,2\}$ such that $C^{1}$ CR functions defined on $M$ extends holomorphically to $Ω_i$.

Sobolev映射的复流形与环路空间中的Hartogs型定理

Complex manifolds of Sobolev mappings and a Hartogs-type theorem in loop spaces

本文回顾带边界紧实流形到复流形的广义环路空间的复结构，并证明该空间中全纯映射的Hartogs型延拓定理。

On the Hartogs extension theorem for unbounded domains in $\mathbb{C}^n$

Let $Ω\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $Ω$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partialΩ$ has a holomorphic extension to $Ω$. For unbounded domains this extension property may fail, for example if $Ω$ contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of $\mathbb{C}^n\backslash\overlineΩ$ is $\mathbb{C}^n$. It seems that it is a first result in the literature which gives a geometric characterization of…

Extendability and the $\overline \partial$ Operator on the Hartogs Triangle

In this paper it is shown that the Hartogs triangle $\mathbf T$ in $\mathbf C^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $\mathbf T$.

Hartogs companions and holomorphic extensions in arbitrary dimension

We show that every holomorphic map $f\in\mathcal{H}(Ω\setminus K)$ ($K\subsetΩ\subset\mathbb{C}^n$, with $K$ compact, $Ω$ open, and $n\ge2$), has a unique "\emph{Hartogs companion}" $\tilde f\in\mathcal{H}(Ω)$ matching $f$ on an open subset $C_{K,Ω}\subsetΩ\setminus K$. Furthermore, $\tilde f$ extends $f$, \emph{if and only if} $\mathbb{C}^n\setminus K$ is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for Gâteaux holomorphic maps $f\in\mathc…

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of graph(F)\cup(\partial D\times D) -- where D is the open unit disc and graph(F) denotes the graph of a continuous D-valued function F -- to the bidisc. We extend holomorphic functions by applying the Kontinuitaetssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirka's theorem.

The $L^p$ CR Hartogs separate analyticity theorem for strictly convex domains

For a large class of convex domains in $\bf C^n$, it is shown that an $L^p$ function on the boundary is CR if there are holomorphic extensions on almost all slices of D by complex lines parallel to the coordinate axes. As an application, a new operator limit for the Szegő projection is obtained.

哈托格斯现象与开嵌入、真映射、紧化及上同调

Hartogs and open embeddings, proper maps, compactifications, cohomologies

本文研究复解析簇上全纯向量丛全纯截面的哈托格斯延拓现象，分析其与开嵌入等结构及上同调的关系，还给出复近齐次代数G-簇（G为半阿贝尔簇）哈托格斯现象的凸几何判别准则。

The Hartogs extension phenomenon in toric varieties

We study the Hartogs extension phenomenon in non-compact toric varieties and its relation to the first cohomology group with compact support. We show that a toric variety admits this phenomenon if at least one connected component of the fan complement is concave, proving by this an earlier conjecture M. Marciniak.

A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f: W:=(A\times Y) \cup (X\times B)\longrightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W}:=\left\lbrace(z,w)\in X\times Y:\ \widetildeω(z,A,X)+\widetildeω(w,B,Y)<1 \right\rbrace$ such that $\hat{f}=f$ on $W\cap \hat{W},$ where $\widetildeω(\cdot,A,…

Local and Global Hartogs-Bochner Phenomenon in Tubes

A generalization of the Hartogs theorem is proved for a class of Tubes structures. We assume that the intervening commutative Lie algebra admits at least a number of globally solvable generators greater or equal to the structure codimension. We give necessary and sufficient conditions for the first cohomological group with compact support to be trivial. A such global result was previously obtained only when the supporting manifold was the Euclidean space. Further but canonical work is need to recover the Bochner extension theorem.

Osgood-Hartogs type properties of power series and smooth functions

We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given $C^\infty $ function $f$ is proved when the restriction of $f$ on analytic curves belonging to some family is analytic. Our results generalize two known statements: a theorem of P. Lelong and the Bochnak-Siciak Theorem. The questions we study fall into the category of "Osgood-Hartogs-type" problems.

Hartog扩展的线束一般化

A generalization of Hartog's extension of line bundles

本文研究了复流形上全纯线束的扩展问题，证明在特定条件下线束可唯一扩展，并推广了已有的q-角凸复流形结果。

两个八元数变量下的八元数Hessian复形与八元数多重调和函数的Hartogs–Bochner延拓

On octonionic Hessian complex and Hartogs--Bochner extension of octonionic pluriharmonic functions in two octonionic variables

本文构造八元数Hessian复形，克服八元数非结合性困难，求解非齐次Hessian方程，建立切向八元数多重调和函数的Hartogs–Bochner延拓，解决了庞加莱问题的八元数版本。

Bott-Chern cohomology and the Hartogs extension theorem for pluriharmonic functions

Let $X$ be a cohomologically $(n-1)$-complete complex manifold of dimension $n\geq 2$. We prove a vanishing result for the Bott-Chern cohomology group of type $(1, 1)$ with compact support in $X$, which combined with the well-known technique of Ehrenpreis implies a Hartogs type extension theorem for pluriharmonic functions on $X$.

A counterexample to Hartogs' type extension of holomorphic line bundles

Consider a domain $\varOmega$ in $\mathbb{C}^n$ with $n\geqslant 2$ and a compact subset $K\subset\varOmega$ such that $\varOmega\backslash K$ is connected. We address the problem whether a holomorphic line bundle defined on $\varOmega\backslash K$ extends to $\varOmega$. In 2013, Fornæss, Sibony and Wold gave a positive answer in dimension $n\geqslant 3$, when $\varOmega$ is pseudoconvex and $K$ is a sublevel set of a strongly plurisubharmonic exhaustion function. However, for $K$ of general shape, we construct counterexamples in any dimension $n\geqslant 2$. The key is a certain gluing lemma…

Bochner-Hartogs type extension theorem for roots and logarithms of holomorphic line bundles

We prove an extension theorem for roots and logarithms of holomorphic line bundles across strictly pseudoconcave boundaries: they extend in all cases except one, when dimension and Morse index of a critical point is two. In that case we give an explicit description of obstructions to the extension.

The Hartogs extension theorem on (n-1)-complete complex spaces

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not neces…

A geometrical proof of the Hartogs extension theorem

100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend holomorphically and uniquely to the domain D. It was a long-standing open problem to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906) and E.E. Levi (1911) in some special, model cases. Quite unexpectedly, Fornaess in 1998 exhibited a topologically s…

The Hartogs phenomenon in almost homogeneous varieties

We study the Hartogs extension phenomenon in noncompact almost homogeneous algebraic varieties and we prove the cohomological and weight criteria for the Hartogs phenomenon. In the case of spherical varieties, we prove a criterion for the Hartogs phenomenon in terms of colored fans.

On Hilbert-Hartogs manifolds

We prove some extension results for holomorphic mappings with values in complex Hilbert manifolds

关于分离全纯性的Hartogs定理的有理变体

Rational Variant of Hartogs' Theorem on Separate Holomorphicity

本文在不可数代数闭域上证明了Hartogs定理的有理版本：若部分函数在Zariski开集上对每个坐标分别有理，则整体为$n$元有理函数。

多向量变量的单演函数的Hartogs-Bochner延拓与Dirac复形

The Hartogs-Bochner extension for monogenic functions of several vector variables and the Dirac complex

本文研究多向量变量的单演函数，通过刻画Dirac复形的前四个空间并显式写出前三个算子，证明其为椭圆复形，进而用PDE方法获得非齐次方程解，建立Hartogs现象及边界上切向单演函数的Hartogs-Bochner延拓。

The Hartpgs-type extension theorem for meromorphic mappings into q-complete complex spaces

We prove in this note a result on extension of meromorphic mappings, which can be considered as a direct generalisation of the Hartogs extension theorem for holomorphic functions. Namely: THEOREM. Every meromorphic mapping $f:H_n^q(r)\to Y$, where $Y$ is a $q$ - -complete complex space, extends to a meromorphic mapping from $Δ^{n+q}$ to $Y$. Here $H_n^q(r):=Δ^n\times (Δ^q\setminus \barΔ_r^q)\cup Δ_r^n\times Δ^q$ is a "q-concave" Hartogs figure in $C^{n+q}$. Remark that in the case $q=1$, i.e. when $Y$ is Stein, the statement of the Theorem is exactly the Theorem of Hartogs.

Extension of analytic covers by extension of their ramification divisor

This paper deals with extension of analytic covers. We prove topological extension theorems for analytic covers. The main result is an extension theorem which only uses the extension of the ramification divisor. We give also a Thullen-type and a Hartogs-type extensions theorems.

On sharper estimates of Ohsawa-Takegoshi $L^2$-extension theorem

We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. When the Hartogs domain defined by the weight function is strictly pseudoconvex, this estimate is strictly sharper than known optimal estimates. When the weight function is radial, we prove that our estimate provides the $L^2$-minimum extension.

Cross theorems with singularities

We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary complex manifolds and M is a set of singularities which is locally pluripolar (resp. thin) in fibers.

Problème de Plateau complexe feuilleté. Phénomènes de Hartogs-Severi et Bochner pour des feuilletages CR singuliers

The purpose of this paper is to generalize in a geometric setting theorems of Severi, Brown and Bochner about analytic continuation of real analytic functions which are holomorphic or harmonic with respect to one of its variables. We prove in particular that if N is a real analytic levi-flat annulus in an open set of R^{n}\timesC^{2}, then one can find X\subsetR^{n}\timesC^{2} such that X\cupN is a levi-flat real analytic subset and X fills N in the sense that the boundary of the integration current of X is a prescribed smooth submanifold of N foliated by real curves. Moreover, real analytic f…

完全多重极集与多重次调和函数的可去奇点

Complete pluripolar sets and removable singularities of plurisubharmonic functions

本研究受已有成果启发，证明了多重次调和函数穿过紧完全多重极集的Hartogs型延拓定理，补充了Shiffman的经典定理。

A mi-chemin entre analyse complexe et superanalyse

In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration (this condition is a generalization of the classical relation 1 + i^2 = 0 in C). Under the condition (A), we get an integral representation formula for the superdifferentiable functions.We give a result of Hartogs type of separated superdifferentiability, a continuation theorem of Hartogs-Bochner type and a Liouville theorem for the superdifferentiable functions.

Entre analyse complexe et superanalyse

In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration. Under the condition (A), we get an integral representation formula for the superdifferentiable functions.We give a result of Hartogs type of separated superdifferentiability and a continuation theorem of Hartogs-Bochner type for the superdifferentiable functions.

Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds II

We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr's holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for $H^\infty$). In particular, in this context we obtain results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the Peter-Weyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets,…

Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I

We develop complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds. This, in particular, includes the results on holomorphic extension from complex submanifolds, corona type theorems, properties of divisors, holomorphic analogs of the Peter-Weyl approximation theorem, Hartogs type theorems, characterization of uniqueness sets. The model examples of these algebras are: (1) Bohr's algebra of holomorphic almost periodic functions on tube domains; (2) algebra of all fibrewise bounded holomorphic functions (e.g., arising in the corona problem for $H^…

关于 $p$-Bergman 空间维度的研究

On the dimension of the $p$-Bergman spaces

本文研究了伪凸域中 $p$-Bergman 空间维度问题，通过构造 $L^p$ 版本的延伸定理，建立了确保空间无限维的几何和势理论准则，并给出了特定域的充分条件。

Some applications of the Kohn-Rossi extension theorems

We prove extension results for meromorphic functions by combining the Kohn-Rossi extension theorems with Andreotti's theory on the algebraic and analytic dependence of meromorphic functions on pseudoconcave manifolds. Versions of Kohn-Rossi theorems for pseudoconvex domains are included.

A simplified proof of optimal $L^2$-extension theorem and extensions from non-reduced subvarieties

We give a simplified proof of an optimal version of the Ohsawa-Takegoshi $L^2$-extension theorem. We follow the variational proof by Berndtsson-Lempert and use the method in the paper of McNeal-Varolin. As an application, we give an optimal estimate for extensions from possibly non-reduced subvarieties.

Ohsawa-Takegoshi type theorem and extension of plurisubharmonic functions

We prove a Thullen type extension theorem of plurisubharmonic functions across a closed complete pluripolar set, which generalizes a theorem of Siu. Our approach depends on an Ohsawa-Takegoshi type extension theorem for a single point in a bounded complete Kähler domain, which is of independent interest.

Extension theorems of Sakai type for separately holomorphic and meromorphic functions

We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.

关于Ohsawa-Takegoshi $L^2$延拓定理与多次调和函数的可去奇点

On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions

本文推广了Ohsawa-Takegoshi $L^2$延拓定理至Stein流形上扭曲典范丛的全纯截面，并建立了多次调和函数在紧完备多极集上的Hartogs型延拓定理，填补了文献空白。

Hardy-Sobolev type inequalities and their applications

This paper is devoted to various applications of Hardy-Sobolev type inequalities. We derive a new $L^2$ estimate for the $\bar{\partial}-$equation on ${\mathbb C}^n$ which yields a quantitative generalization of the Hartogs extension theorem to the case when the singularity set is not necessary compact. We show that for any negative subharmonic function $ψ$ on ${\mathbb R}^n$, $n>2$, the BMO norm of $\log |ψ|$ is bounded above by $2\sqrt{n-2}$ and $|ψ|^γ$ satisfies a reverse Hölder inequality for every $0<γ<1$. We also show that every plurisubharmonic function is locally BMO. Several Liouville…

An extension theorem for hermitian line bundles

We prove a general extension theorem for holomorphic line bundles on reduced complex spaces, equipped with singular hermitian metrics, whose curvature currents can be extended as positive, closed currents. The result has applications to various moduli theoretic situations.

A Characterization of regular points by $L^2$ Extension Theorem

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related $L^2$ extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

An extension theorem for separately holomorphic functions with singularities, II

This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089.

Extension theorems for analytic objects associated to foliations

In this paper we will establish a structure theorem concerning the extension of analytic objects associated to germs of dimension one foliations on surfaces, through one-dimensional barriers. As an application, an extension theorem for projective transverse structures is obtained.