Infinite-dimensionality of the automorphism groups of homogeneous Stein manifolds

We show that the group of holomorphic automorphisms of a Stein manifold X with dim X ≥ 2 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.     

An application of q-concavity to increasing sequences of Stein open sets

We prove that in a normal Stein space a relatively compact open set is a domain of holomorphy provided that it is an increasing union of Stein open subspaces. Received: 13 December 2004     

Parametric H-principle for holomorphic immersions with approximation

We prove the parametric homotopy principle for holomorphic immersions of Stein manifolds into Euclidean 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 LfΣ_|=g, where L is a holomorphic vector field, semi-transversal to analytic variety Σ.     

Nondicritical $${\mathbb {C}^*}$$-actions on two-dimensional Stein manifolds

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f ⁿ g ^m where f and g are global holomorphic functions and . Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.     

On complex analytic compactifications of complex parallelizable manifolds

We show that for a quotient of a complex Lie group by a lattice there does not exist any open equivariant holomorphic embedding into a larger complex space. Received: 17 June 1999     

Geometry of domains with the uniform squeezing property

We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmüller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kähler-hyperbolicity, vanishing of cohomology groups and quasi-isometry of various invariant metrics. It also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of Serre about Stein properties of holomorphic fiber bundles.     

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

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 will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

Realizing connected Lie groups as automorphism groups of complex manifolds

We show that every connected real Lie group can be realized as the full automorphism group of a Stein hyperbolic complex manifold.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

Quantitative estimates for periodic points of reversible and symplectic holomorphic mappings

We show that reversible holomorphic mappings of have periodic points accumulating at an elliptic fixed point of general type. On the contrary, we also show the existence of holomorphic symplectic mappings that have no periodic points of certain periods in a sequence of deleted balls about an elliptic fixed point of general type. The radii of the balls are carefully chosen in terms of the periods, which allows us to show the existence of holomorphic mappi ngs of that are not reversible with respect to any involution with a holomorphic linear part, and that admit no invariant totally real and real surfaces. Received: 5 December 2000 / Published online: 4 April 2002     

An interpolation theorem for proper holomorphic embeddings

Given a Stein manifold x of dimension n > 1, a discrete sequence , and a discrete sequence where , there exists a proper holomorphic embedding satisfying f(a _j ) = b _j for every j = 1,2,... Forstnerič and Prezelj supported by grants P1-0291 and J1-6173, Republic of Slovenia. Kutzschebauch supported by Schweizerische National fonds grant 200021-107477/1. Ivarsson supported by The Wenner-Gren Foundations.     

The Levi problem for Riemann domains over Stein spaces with isolated singularities

We consider the following question: Let \({p:Y \rightarrow X}\) be an unbranched Riemann domain and assume that X is a Stein space and p is a Stein morphism. Does it follow that Y is Stein ? We show that the answer is affirmative if X has isolated singularities. This generalizes a result of Andreotti and Narasimhan.     

Legendrian tori and the semi-classical limit

Let X be CPn or a compact smooth quotient of the n-dimensional complex hyperbolic space, n>1. Let L be a hermitian holomorphic line bundle (with hermitian connection) on X chosen as follows: if X=CPn then L is the hyperplane bundle, and in the second case L is chosen so that L⊗(n+1)=KX⊗E, where KX is the canonical line bundle and E is a flat line bundle. The unit circle bundle P in L^∗ is a contact manifold. Let k^′ be a fixed positive integer. We construct certain Legendrian tori in P (the construction depends, in particular, on the choice of k^′) and sequences {uk}, k=k^′m, , of holomorphic sections of L⊗k associated to these tori. We study asymptotics of the norms ‖ukk_‖ as m→+∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n-dimensional complex hyperbolic space. We obtain an n>1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n=1).     

Characterization of the unit ball in C<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> among complex manifolds of dimension<Emphasis Type="BoldItalic">n</Emphasis>

We show that if the group of holomorphic automorphisms of a connected complex manifold M of dimension n is isomorphic as a topological group equipped with the compact-open topology to the automorphism group of the unit ball B ⁿ ⊂ ℂ ⁿ ,then M is biholomorphically equivalent to B ⁿ.     

Pseudoconvex domains spread over complex homogeneous manifolds

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.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

On the -equation¶over pseudoconvex Kähler manifolds

The Picard variety Pic⁰(? ⁿ ) of a complex n-dimensional torus? ⁿ is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ? ⁿ . The total space of a topologically trivial holomorphic (principal) line bundle on a compact K?hler manifold is weakly pseudoconvex. Thus we can regard Pic⁰(? ⁿ ) as a family of weakly pseudoconvex K?hler manifolds. We consider a problem whether the Kodaira's -Lemma holds on a total space of holomorphic line bundle belonging to Pic⁰(? ⁿ ). We get a criterion for this problem using a dynamical system of translations on Pic⁰(? ⁿ ). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex K?hler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete K?hler manifold on which the -Lemma does not hold. Received: 11 June 1999 / Revised version: 22 November 1999     

The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.