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 strange (nonpseudoconvex) domain D^F in C^2 that cannot be filled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside D^F. However, one should point out that the standard, unrestricted disc method usually allows discs to go outsise the domain (just think of Levi pseudoconcavity). Confirming these rather ancient expectations, we show that the global Hartogs extension theorem can be established in such a natural way.
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 $X$.
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…
We discuss various known generalizations of the classical Hartogs' extension theorem on Stein spaces with arbitrary singularities and present an analytic proof based on d-bar methods.
The aim of this paper is to present an extension theorem for the functions separately holomorphic on generalized (N,k)-crosses with pluripolar 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.
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.
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.
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.
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…
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…
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.
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 $\mathbf T$.
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…
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.
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.
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,…
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…
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.
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.
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.
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.
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.
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.
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.
An $L^2$ version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the $\bar{\partial}$-operator is established. This duality is used to study the solution of the $\bar{\partial}$-equation with prescribed support. Applications are given to $\bar{\partial}$-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions.
On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions
CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院、中国科学院吴文俊数学重点实验室）
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^…
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,…
