In this paper, a generalization of the "sector property" theorem first pioneered by Baouendi, Rothschild and Treves is given. The main contribution consists in showing that if a submanifold of $\C^n$ with higher codimension is locally presented in a weighted normal form, similar to that described by Bloom and Graham, then a characterization is given as to when a vector in the tangent space belongs to the analytic wave front set for the set of locally defined CR functions on the submanifold. This characterization is described in terms of the sign of the inner product of this vector with the local graphing functions for the submanifold on sectors of suitable size along complex lines in its tangent space. Examples are given to show that under certain circumstances, the results are sharp. Previous results by Baouendi et. al. contained a semi-rigidity assumption which is not assumed in the present paper. The hypoanalytic wave front set determines the cone of directions in which CR functions extend analytically to the ambient space and thus provides an explicit description of the local hull of holomorphy of a submanifold of $\C^n$.
Combining V. Koziarz's observation about the regularity of some modified section related to the initial extension with J. McNeal--D. Varolin's regularity argument, we generalize two theorems of McNeal--Varolin for the $L^2$ extension of $\bar\partial$-closed high-degree forms on a Stein manifold to the weakly pseudoconvex Kähler case under mixed positivity conditions.
We establish a new generalization of an $L^2$ extension theorem of Ohsawa-Takegoshi type. The improvement in the theorem is that it allows the usual curvature assumptions to be significantly weakened in certain favorable settings. The favorable settings come about, for example, when the underlying structure (e.g., the underlying manifold, or the holomorphic Hermitian line bundle whose sections are being extended) has certain potential theoretic positivity. The simplest case occurs when the underlying manifold supports a function with self-bounded gradient in the sense of McNeal, but there are …
In the present paper, we show that given a compact Kähler manifold $(X,ω)$ with a Kähler metric $ω$, and a complex submanifold $V\subset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any quasi-plurisubharmonic function $φ$ on $V$ such that $ω|_V+\sqrt{-1}\partial\bar\partialφ\geq \varepsilonω|_V$ with $\varepsilon>0$ can be extended to a quasi-plurisubharmonic function $Φ$ on $X$, such that $ω+\sqrt{-1}\partial\bar\partial Φ\geq \varepsilon'ω$ for some $\varepsilon'>0$. This is an improvement of results in \cite{WZ20}. Examples satisfying the assumption t…
For some class of mappings, there are investigated problems connected with a possibility of continuous extension to a boundary on Riemannian manifolds. In particular, for so-called ring mappings, there is proved a result related to continuous extension to an isolated boundary point. Besides that, similar theorems hold for more general boundaries. As application of developed technique, there is proves a theorem about continuous extension of mapping of Orlicz--Sobolev class to an isolated singularity.
Given a proper, open, holomorphic map of Kähler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil-Petersson metric extends as a positive current. In the projective case, the Weil-Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus,…
We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of $\C^N$ and a real-algebraic CR-submanifold of $\C^{N'}$.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.
We prove an analogue of E. Levi's Continuity Principle for meromorphic mappings with values in arbitrary compact complex manifolds in place of the Riemann sphere $\cc\pp^1$. The result is achieved by introducing a new extension method for meromorphic mappings. One of the corollaries reads as follows: If a compact complex surface $X$ is not "among the known ones" then for every domain $Ω$ in a Stein surface every meromorphic mapping $f:Ω\to X$ is in fact holomorphic and extends as a holomorphic mapping $\hat f:\hat D\to X$ of the envelope of holomorphy $\hat D$ of $D$ into $X$. In this last ver…
Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.
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.
In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.
We give a real-analytic section for the Teichmüller projection onto the VMO-Teichmüller space by using the variant of Beurling-Ahlfors extension by heat kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this result, we prove that the VMO-Teichmüller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichmüller space admits a real-analytic contraction mapping.
Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space $B(S,X)$ of Borel maps. As an application we prove two theorems about bounded holomorphic functions on $C(S,X)$, one reminiscent of the Monodromy Theorem, the other of Liouville's Theorem.
For contact manifolds $(M, η)$ a complexification $M^c$ is constructed to which the contact form $η$ extends such that the exterior derivative of the extended form is Kählerian. In the case of a proper action of an extendable Lie group this construction is realized in an equivariant way. In a simultaneous stratification of $M$ and $M^c$ according to the istropy type, it is shown that the Kählerian reduction of the complexification can be seen as the complexification of the contact reduction.
In this paper we prove an extension of a result of Gromov, Henkin and Shubin [GHS] on holomorphic L_{2} functions on coverings of strongly pseudoconvex manifolds.
We show that all the common definitions of quasiregular mappings $f\colon M\to N$ between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ are quantitatively equivalent with precise dependences of the quasiregularity constants. As an immediate consequence, we obtain that if $f$ is $1$-quasiregular according to one of the definitions, then it is also $1$-quasiregular according to any other definition. In particular, this recovers a recent theorem of Capogna et al. on the equivalence of $1$-quasiconformal mappings. Our main results answer affirmatively a few open questi…
Let $(X,ω)$ be a compact Kähler manifold with a Kähler form $ω$ of complex dimension $n$, and $V\subset X$ is a compact complex submanifold of positive dimension $k<n$. Suppose that $V$ can be embedded in $X$ as a zero section of a holomorphic vector bundle or rank $n-k$ over $V$. Let $φ$ be a strictly $ω|_V$-psh function on $V$. In this paper, we prove that there is a strictly $ω$-psh function $Φ$ on $X$, such that $Φ|_V=φ$. This result gives a partial answer to an open problem raised by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of Kähler currents. We also discuss possible extensi…
Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseu…
Let $M$ be a $CR$ submanifold of a complex manifold $X$. The main result of this article is to show that $CR$-hypoellipticity at $p_0\in{M}$ is necessary and sufficient for holomorphic extension of all germs of $CR$ functions to an ambient neighborhood in $X$. As an application, we obtain that $CR$-hypoellipticity implies the existence of generic embeddings and prove holomorphic extension for a large class of $CR$ manifolds satisfying a higher order Levi pseudoconcavity condition.
In this paper we take up the problem of describing the CR vector bundles M over compact standard CR manifolds S, which are themselves standard CR manifolds. They are associated to special graded Abelian extensions of semisimple graded CR algebras.
Given a complex manifold $X$, any Kähler class defines an affine bundle over $X$, and any Kähler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact Kähler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Szőke and to the existence question for good complex…
Higher order higher spin operators are generalizations of $kth$-powers of the Dirac operator. In this paper, we study higher order higher spin operators defined on some conformally flat manifolds, namely cylinders and Hopf manifolds. We will also construct the kernels of these operators on these manifolds.
We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension formulas, we prove several deformation invariance theorems for Hodge numbers on some certain classes of complex manifolds, without use of Frölicher inequality or the topological invariance of Betti numbers.
In this paper we will show how to construct holomorphic L^{p}-functions on unbranched coverings of strongly pseudoconvex manifolds. Also, we prove some extension and approximation theorems for such functions.
We prove an extension theorem of "Ohsawa-Takegoshi type" for Dolbeault q$-classes of cohomology ($q\geq 1$) on smooth compact hypersurfaces in a weakly pseudoconvex Kähler manifold
We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.
Let $M$ be a compact complex manifold. In this paper we give a simple proof of the bimeromorphic invariance of the higher Todd genera of $M$, a result first proved implicitly by Brasselet-Schürmann-Yokura using algebraic methods.
