On the automorphisms of Siegel domains

We prove that every holomorphic automorphism of a strongly pseudoconvex Siegel domain with no pole on the edge is an affine automorphism. Some related results are also given. This work was supported in part by a grant from the Ministry of Science and Technology of the Republic of Slovenia.     

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.     

A note on the instability of embeddings of Cauchy-Riemann manifolds

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding. Both authors were partially supported by NSF grants.     

Obstructions to embeddability into hyperquadrics and explicit examples

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstnerič. The author was supported in part by the RCBS grant of Trinity College Dublin and by the Science Foundation Ireland.     

A note on plurisubharmonic defining functions in \mathbbCn{\mathbb{C}^{n}}

Let , n ≥ 3, be a smoothly bounded domain. Suppose that Ω admits a smooth defining function which is plurisubharmonic on the boundary of Ω. Then a Diederich–Forn?ss exponent can be chosen arbitrarily close to 1, and the closure of Ω admits a Stein neighborhood basis. Research of J. E. Forn?ss was partially supported by an NSF grant. Research of A.-K. Herbig was supported by FWF grant P19147.     

Integral Operators Induced by the Fock Kernel

We study the L ^p boundedness and find the norm of a class of integral operators induced by the reproducing kernel of Fock spaces over . Dostanić is partialy supported by MNZZS Grant N°ON144010, Zhu is partially supported by the US National Science Foundation.     

Non-existence of the Artin function for Henselian pairs

Research supported by NSF, NSERC and the Connaught Fund     

Invariant metrics and the boundary behavior of holomorphic functions on domains in\mathbb{C}^n

Invariant metrics are used to provide a unified approach to the study of holomorphic functions in Hardy classes on domains in one and several complex variables. Both approach regions and boundary measures are constructed from the metric. Examples are provided to show how diverse theories can be unified with this approach. The Hartogs extension phenomenon and Fatou’s theorem are seen to be two aspects of the same circle of ideas. Author supported in part by a grant from the National Science Foundation     

The equation of complex Monge-Ampère type and stability of solutions

We prove that in a family of plurisubharmonic functions with Monge-Ampère measures bounded from above by such a measure of one function weak convergence is equivalent to convergence in capacity. We also show a very general statement on the existence of solutions of the complex Monge-Ampère type equation. First author partially supported by the Swedish Research Council contract 621-2002-5308. Second author partially supported by Polish ministerial Grant No. 1 PO3A 037 27.     

The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements

The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an ℓ-dimensional Euclidean space was introduced and studied by K. Saito [5] [6]. It is closely related to the flat structure or the Frobenius manifold structure. We will show that the Hodge filtration coincides with the filtration by the order of contacts to the reflecting hyperplanes. Moreover, a standard basis for the Hodge filtration is explicitly given.Partially supported by the Grant-in-aid for scientific research (No. 14340018 and 13874005), the Ministry of Education, Sports, Science and Technology, Japan     

A local version of Levenberg's theorem on determining measures and Leja's polynomial condition

A local version of Levenberg's theorem characterizing Leja's polynomial condition in terms of determining measures is given. This yields a very short proof of the invariance of Leja's property under non-degenerate holomorphic maps.Research supported, in part, by KBN grant 2 1077 91 01.     

Regularity of the Bergman Projection on Forms and Plurisubharmonicity Conditions

Let be a smoothly bounded domain. Suppose Ω has a defining function, such that the sum of any q eigenvalues of its complex Hessian is non-negative. We show that this implies global regularity of the Bergman projection, B _j-1, and the -Neumann operator, N _j , acting on (0,j)-forms, for .Research of the first author was partially supported by a Rackham Fellowship.Research of the second author was partially supported by an NSF grant.     

Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into such that the values on any given discrete sequence can be arbitrarily prescribed. Kutzschebauch supported by Schweizerische Nationalfonds grant 200021-107477/1.     

Reduction theorem of the codimension of minimal generic submanifolds in a complex projective space

We study compact minimal generic submanifolds of a complex projective space with flat normal connection and prove a reduction theorem of codimension under the condition on the Ricci tensor.The present studies were supported by the Basic Science Research Institute Program, Korea Ministry of Education, 1993-114.     

On a problem of Bremermann concerning Runge domains

In this paper we give an example of a bounded Stein domain in , with smooth boundary, which is not Runge and whose intersection with every complex line is simply connected.This work was supported by Marie Curie International Reintegration Grant no. 013023 and by the Romanian Ministry of Education and Research grant 2-CEx06-11-10/25.07.06.     

Fibre cycles of holomorphic maps

This and the second part of the paper are to a great extent parts of my thesis [Si] which also appeared as preprint Mathematica Gottingensis 5/92. This work was partly supported by the S7B170 Geometrie und Analysis in Göttingen     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Cartan decomposition of the moment map

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070.     

A Robin formula for the Fekete-Leja transfinite diameter

We generalize the classical Robin formula to higher dimensions. Work supported in part by NSF grant DMS-0300784.     

Upper semicontinuity of automorphism groups

The second author was partially supported by NSF#DMS-9101826