Irena Majcen
We prove that a Stein manifold admits a closed holomorphic 1-form without zeros in every class of the first cohomology group. We also prove an approximation result for closed holomorphic 1-forms without zeros defined in a neighborhood of a compact subset.
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In this article, we prove that if $Π: X\to Ω$ is a surjective holomorphic map, with $Ω$ a Stein space and $X$ a complex manifold of dimension $n\geq 3,$ and if, for every $x\in Ω$ there exists an open neighborhood $U$ such that $Π^{-1}(U)$ is Stein, then $X$ is Stein
In this paper we provide sufficient conditions for the graphs of holomorphic mappings on compact sets in complex manifolds to have Stein neighborhoods. We show that under these conditions the mappings have properties analogous to properties of holomorphic functions on compact sets in $\mathbb C^n$.
We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex analytic stratification with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result also provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is non…
We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2.
We study the stable embedding problem for a CR family of 3-dimensional strongly pseudoconvex CR manifolds with each fiber bounding a stein manifold.
We prove that a Stein manifold of dimension $d$ admits a proper holomorphic embedding into any Stein manifold of dimension at least $2d+1$ satisfying the holomorphic density property. This generalizes classical theorems of Remmert, Bishop and Narasimhan pertaining to embeddings into complex Euclidean spaces, as well as several other recent results.
We show how to extend some holomorphic bundles with fifer C^2 and base an open set in C, to bundles on the Riemann Sphere, by an extremely simple technique. In particular, it applies to examples of non-Stein bundles constructed by Skoda and Demailly. It gives an example on C, with polynomial transition automorphisms.
Suppose that Y is a complex manifold with the property that any holomorphic map from a compact convex set in a complex Euclidean space C^n (for any n) to Y is a uniform limit of entire maps from C^n to Y. We prove that a holomorphic map from a closed complex subvariety X_0 in a Stein manifold X to the manifold Y extends to a holomorphic map of X to Y provided that it extends to a continuous map. We then establish the equivalence of four Oka-type properties of a complex manifold. We also generalize a theorem of Siu and Demailly on the existence of open Stein neighborhoods of Stein subvarieties …
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,…
Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds
本文证明了Stein流形上全纯向量丛中满足Griffiths或Nakano意义下负曲率的奇异Hermitian度量可由光滑度量序列逼近，且限制在子流形上的此类光滑度量可延拓至整个丛并保持相同曲率负性。
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^…
This is a survey on the homotopy principle in complex analysis on Stein manifolds, also called the Oka principle in this context. We concentrate on the following topics: the Oka-Grauert principle (classification of holomorphic vector bundles); the homotopy principle for holomorphic mappings from Stein manifolds and, more generally, for sections of holomorphic submersions with sprays; question on removability of intersections of holomorphic mappings with complex subvarieties; embeddings and immersions of Stein manifolds in affine spaces of minimal dimension; embeddings of open Riemann surfaces …
In this paper we prove that every Stein manifold $S$ admits a proper holomorphic immersion into any Stein manifold $X$ of dimension $2\mathrm{dim}S$ enjoying the density property or the volume density property. The case $\mathrm{dim}S=1$ was proved beforehand by Andrist and Wold (Ann. Inst. Fourier (Grenoble), 64(2):681-697, 2014). This result generalizes the classical theorem of Bishop and Narasimhan for immersions to $\mathbb{C}^n$ with $n=2\mathrm{dim}S$, and it complements the proper embedding theorem proved by Andrist et al. when $\mathrm{dim}X>2\mathrm{dim}S$ (J. Anal. Math., 130:135-150…
On Point Separation of Bergman Space on Stein Manifolds with Constant Holomorphic Sectional Curvature
stein流形在特定曲率条件下，其bergman度量的定义与分离点性质密切相关，且与单位球体的biholomorphic性相关。
We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.
Given a Stein manifold with the density property, we show that under a suitable topological condition it is possible to prescribe derivatives at a finite number of points to automorphisms depending holomorphically on a Stein parameter. This is an Oka property of the manifold and is related to its holomorphic flexibility.
R. Guralnick [Linear Algebra Appl. 99, 85-96 (1988)] proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.
Finite dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them.
The paper is related to the author's talk at the Hayama Symposium in Complex Analysis in December 2000. In section 1 we survey results on the Oka principle for sections of holomorphic submersions over Stein manifolds. In section 2 we apply these results to the lifting problem for holomorphic mappings from Stein manifolds, and in section 3 we apply them to the problem of removability of intersections of holomorphic maps from Stein manifolds with closed complex subvarieties of the target space.
In this paper we prove results on the existence and homotopy classification of holomorphic submersions from Stein manifolds to other complex manifolds. We say that a complex manifold Y satisfies Property S_n for some integer n bigger or equal the dimension of Y if every holomorphic submersion from a compact convex set in C^n of a certain special type to Y can be uniformly approximated by holomorphic submersions from C^n to Y. Assuming this condition we prove the following. A continuous map f from an n-dimensional Stein manifold X to Y is homotopic to a holomorphic submersions of X to Y if and …
We apply the methods developed in [Br1] to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin [GHS] on holomorphic L^{2} functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.
The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper we construct similar formulas for cetain classes of holomorphic functions defined on coverings of such domains.
R. Guralnick (Linear Algebra Appl. 99, 85-96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, non-smooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global $\mathcal C^\infty$ similarity implies global holomorphic similarity, whereas global continuous similarity is not sufficient.
R. Guralnick (Linear Algebra Appl. 99, 85-96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, non-smooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global $\mathcal C^\infty$ similarity implies global holomorphic similarity, whereas global continuous similarity is not sufficient.
A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular tar…
Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic map from X to Y which agrees with f on A and which is homotopic to f relative to A. The analogous results in the case when the variety A is empty have been obtained in the preprint math.CV/0507212.
We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle…
Runge embeddings, approximation of biholomorphisms on Stein manifolds, and the Loewner PDE
该研究开发了适用于斯坦流形并集的全纯龙格嵌入逼近延拓原理，将其应用于多类斯坦流形相关问题，推广了现有嵌入结果并构造了洛普纳偏微分方程的解。
Complete nonsingular holomorphic foliations on Stein manifolds
本文证明了在赋予Riemannian度量的Stein流形上，特定维数下存在所有叶均拓扑闭且完备的非奇异全纯叶状结构，并证明了真全纯叶状结构可通过全纯自同构化为完备叶状结构。
It is shown that any open Riemann surface can be immersed in any Stein manifold with (volume) density property and of dimension at least 2, if the manifold possesses an exhaustion with holomorphically convex compacts such that their complement is connected. The immersion can be made into an embedding if the dimension is at least 3. As an application, it is shown that Stein manifolds with (volume) density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.
Let X be a Stein manifold and let Y be a complex manifold which admits a spray in the sense of Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, pp. 851-897 (1989)). We prove that for every closed complex subvariety X_0 of X and for every continuous map f_0 from X to Y whose restriction to X_0 is holomorphic there exists a homotopy of maps f_t from X to Y whose restrictions to X_0 agree with f_0 and such that the map f_1 is holomorphic on X. We obtain analogous results for sections of holomorphic submersions with sprays over Stein manifolds or Stein s…
We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface M and holomorphic convexity of compact sets in M, or bounded in part by M. Applications include extendability of Cauchy-Riemann functions, solvability of the dibar_b-equation, approximability of holomorphic and CR functions, and global regularity of the dibar-Neumann operator.
Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Fornæss and Sib…
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--4…
We study the classification of the pairs $(N, \,X)$ where $N$ is a Stein surface and $X$ is a complete holomorphic vector field with isolated singularities on $N$. We describe the role of transverse sections in the classification of $X$ and give necessary and sufficient conditions on $X$ in order to have $N$ biholomorphic to $\mathbb C^2$. As a sample of our results, we prove that $N$ is biholomorphic to $\mathbb C^2$ if $H^2(N,\mathbb Z)=0$, $X$ has a finite number of singularities and exhibits a non-nilpotent singularity with three separatrices or, equivalently, a singularity with first jet …
In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most results pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map $X\to Y$ between Stein manifolds is homotopic to a proper holomorphic embedding provided that $\mathrm{dim} Y > 2\mathrm{dim}\, X$ and we allow a homotopic deformation of the Stein structure on $X$.
We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit We…
We study ellipticity properties of complements of compact subsets of Stein manifolds.
Stein spaces and Stein algebras
本文证明了Stein空间与Stein代数之间的反同构关系，去除了Forster定理中关于有限维性的假设。
The Cœuré-Loeb example as an open subset of a Stein space
本文针对塞尔问题，证明了 Cœuré-Loeb 纤维化可作为 Stein 空间的开子集（即 Stein 空间中某解析集的补集），补充了该问题反例的结构性质。
We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka-Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-pre…
The Density Property for Vector Bundles
本文证明了底空间为具有稠密性质的Stein流形且全空间全纯柔性的全纯向量丛也满足稠密性质，并据此提供了一类新的具有稠密性质的Stein流形。
We prove the parametric homotopy principle for holomorphic immersions of Stein manifolds into Euclidian space and the homotopy principle with approximation on holomorphically convex sets. We write an integration by parts like formula for the solution $f$ to the problem $Lf|_Σ=g$, where $L$ is a holomorphic vector field, semi-transversal to analytic variety $Σ$.
The universal family of punctured Riemann surfaces is Stein
本文证明了任意正数n下，亏格g的n-刺紧黎曼曲面的万有Teichmüller族是Stein流形，并研究了其函数论性质，包括纤维代数函数的稠密性、到欧氏空间的纤维代数嵌入以及相对Oka原理。
We prove the following theorem: Let X be a Stein manifold of dimension at least 2 and Y a closed complete pluripolar subset of X. Given a point p in the complement of Y there is a proper holomorphic map f from the unit disc to X such that f(0)=p and the image of f avoids Y.
Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.
Holomorphic Factorization of Mappings into $ \operatorname{Sp}_{4}( \mathbb{C}) $
本文研究映入四阶复辛群的全纯映射，证明 Stein 空间上任意零伦全纯映射均可分解为含全纯元素的初等辛矩阵有限乘积。
In this paper we generalize to a certain class of Stein manifolds the Bernstein-Walsh-Siciak theorem which describes the equivalence between possible holomorphic continuation of a function $f$ defined on a compact set $K$ in $\mathbb{C}^N$ to the rapidity of the best uniform approximation of $f$ on $K$ by polynomials. We also generalize Winiarski's theorem which relates the growth rate of an entire function $f$ on $\mathbb{C}^N$ to its best uniform approximation by polynomials on a compact set.
Rosay and Rudin constructed examples of discrete subsets of C^n with remarkable properties. We generalize these constructions from C^n to arbitrary Stein manifolds. We prove: Given a Stein manifold X and a affine variety V of the same dimension there exists a discrete subset D in X such that (1) X-D is measure hyperbolic, (2) f(V) intersects D for every non-degenerate holomorphic map from V to X and (3) every automorphism of X preserving the set D is already the identity map. We also give examples which demonstrate that such discrete subsets can not be found in arbitrary non-Stein manifolds.
We characterise hypercyclic composition operators $C_φ:f\mapsto f\circφ$ on the space of functions holomorphic on $Ω$, where $Ω$ is a connected Stein manifold and $φ$ is a holomorphic self-mapping of $Ω$. In the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in $Ω$, we show that much simpler characterisation is possible (many natural classes of domains in $\CC^N$ satisfy this condition). Moreover, we show that in such a class of manifolds, and in simply connected and infinitely connected planar domains, hypercyclicity of $C_φ$ implies its hereditary …
Let D be a bounded strongly pseudoconvex domain in a Stein manifold S and let Y be a complex manifold. We prove that the graph of any continuous map from the closure of D to Y which is holomorphic in D admits a basis of open Stein neighborhoods in S x Y. Using this we show that many classical spaces of maps from the closure of D to Y which are holomorphic in D are infinite dimensional complex manifolds which are modeled on locally convex topological vector spaces (Banach, Hilbert or Frechet).
Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert proved the Oka principle for maps from Stein manifolds to complex Lie groups and homogeneous spaces, as well as for sections of fiber bundles with homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's result to sections of submersions over a Stein base which admit dominating sprays over small open sets in the base; for proof see [F. Forstn…
We consider a family E_m(D,M) of holomorphic bundles constructed as follows: to any given M in GL_n(Z), we associate a "multiplicative automorphism" f of (C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with fiber D, and monodromy f. We show that the function theory on E_m(D,M) depends nontrivially on the parameters m, M and D. Our main result is that E_m(D,M) is Stein if and only if m log(r(M)) <= 2 π^2, where r(M) denotes the max of the spectral radii of M and its inverse. As corollaries, we: -- …
The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is $\scrO(S,X)$ a deformation retract of $\scrC(S,X)$? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in…
In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic contact structure on a Stein domain $Ω\subset X$ which is diffeotopic to $X$. We also prove a parametric h-principle in this setting, analogous to Gromov's h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-princip…
Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a complex Banach manifold. When D is the unit disc in C (or any other topologically trivial strongly pseudoconvex domain in a Stein manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with n=dim Y. Analogous results hold for maps which are holomorphic in D and of class C^r up to the boundary for any positive integer r. We also establish the O…
We generalize to intersection of strictly $c$ -convex domains in Stein manifold, $ L^{r}-L^{s}$ and Lipschitz estimates for the solutions of the $ \bar \partial $ equation done by Ma and Vassiliadou for domains in $ {\mathbb{C}}^{n}.$ For this we use a Docquier-Grauert holomorphic retraction plus the raising steps method I introduce earlier. This gives results in the case of domains with low regularity, $ {\mathcal{C}}^{3},$ for their boundary.
An odd-dimensional differentiable manifold is called \emph{holomorphically fillable} if it is diffeomorphic to the boundary of a compact strongly pseudoconvex complex manifold, \emph{Stein fillable} if this last manifold may be chosen to be Stein and \emph{Milnor fillable} if it is diffeomorphic to the abstract boundary of an isolated singularity of normal complex analytic space. We show that the homotopical dimension of a manifold-with-boundary of dimension at least 4 restricts the cohomology ring (with any coefficients) of its boundary. This gives restrictions on the cohomology rings of Stei…
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.
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