Notes on trimodal singularities

Singularities of functions of modality 1 or 2 were classified by Arnold [3]. Types of those of modality 3 were listed in [14]. Here we show that these are all Newton non-degenerate in the extended sense introduced in [17], and use this fact to obtain explicit regular bases and precise normal forms for right and for contact equivalence. This requires care since the only existing method for non-semiquasihomogeneous germs (using Arnold's “Condition A”) does not apply to all our cases. Received: 8 February 1999 / Revised version: 28 May 1999     

Non-analytic Bergman and Szegö kernels for weakly pseudoconvex tube domains in $\mathbb C^2$

For any weakly pseudoconvex tube domain in with real analytic boundary, there exist points on the boundary off the diagonal where the Bergman kernel and the Szeg? kernel fail to be real analytic. Received April 6, 1999; in final form August 28, 1999 / Published online December 8, 2000     

On partial analyticity of CR mappings

We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions. Received November 26, 1998; in final form March 23, 1999 / Published online October 11, 2000     

Globalizations, fiber bundles, and envelopes of holomorphy

We describe a relationship between globalizations of local holomorphic actions on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for particular Stein Riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms. We then derive some consequences on the existence of Stein globalizations. Received August 11, 1998; in final form March 8, 1999     

On the meromorphic convexity of normality domains in a Stein manifold

Let X be a Stein manifold. Then we prove that for any family ℱ⊂?(X) the normality domain Dℱ) is a meromorphically ?(X)-convex open set of X. Received: 4 November 1999     

Large discrete sets in Stein manifolds

We construct infinite discrete subsets of Stein manifolds with remarkable properties. These generalize results of Rosay and Rudin on discrete subsets of . Received August 21, 1998; in final form October 11, 1999 / Accepted Published online February 5, 2001     

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     

An Andreotti–Vesentini separation theorem on real hypersurfaces

Using Serre duality in CR manifolds and integral operators for the solution of the tangential Cauchy–Riemann equation with compact support, we prove a separation theorem of Andreotti–Vesentini type for the -cohomology in q-concave real hypersurfaces. Received: February 17, 1999?Published online: May 10, 2001     

The algebraic torus theorem

A structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup. Oblatum 12-II-1999 & 18-XI-1999?Published online: 21 February 2000     

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     

Quadrature rules for rational functions

Summary. It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turán, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. Received June 21, 1999 / Revised version received September 14, 1999 / Published online June 21, 2000     

Boundedness of Bergman projections on tube domains over light cones

Let be the future light cone in , and be the associated tube domain. We prove that the weighted Bergman projection is bounded on for , where Q denotes the Lorentz quadratic form. This theorem extends previous results by Bekollé and Bonami [BB]. Our proof relies on the analysis of the projection on mixed norm spaces, which allows us to exploit the oscillation of the Bergman kernel using the Laplace-Fourier transform. Received October 8, 1999 / Published online February 5, 2001     

On Pitt’s Theorem for Operators between Scalar and Vector-Valued Quasi-Banach Sequence Spaces

 We find natural conditions under which all continuous linear operators between two scalar or vector-valued quasi-Banach sequence spaces are compact. In the case of scalar-valued Banach sequence spaces we show that all such operators essentially factorize through diagonal operators between suitable -spaces. (Received 21 June 1999; in revised form 27 September 1999)     

Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000     

Dwork cohomology and algebraic ${\Cal D}$-modules

Using local cohomology and algebraic -Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, P. Monsky, A. Adolphson and S. Sperber. Received June 10, 1999 / Published online July 20, 2000     

A Sobolev mapping property of the Bergman kernel

We prove that if D is a pseudoconvex domain with Lipschitz boundary having an exhaustion function such that is plurisubharmonic, then the Bergman projection maps the Sobolev space boundedly to itself for any . Received March 10, 1999 / Published online May 8, 2000     

Mixed Hodge complexes on algebraic varieties

Using the theory of mixed Hodge Modules, we introduce the notion of mixed Hodge complex on an algebraic variety, and establish the relation between the filtered complex of Du Bois and the corresponding complex of mixed Hodge Modules. Some application to the Du Bois singularity is given. Received: 20 February 1999     

The weak lower semicontinuity of the Kobayashi distance and its applications

In this paper we prove the lower semicontinuity with respect to the weak topology of the Kobayashi distance in a bounded, convex and open subset of a reflexive Banach space. We apply this result to the Denjoy-Wolff theorem for condensing mappings in the unit open ball in a strictly convex reflexive Banach space with the Kadec-Klee property. Received June 30, 1998 / in final form December 20, 1999 / Published online July 20, 2000     

Milnor number at infinity, topology and Newton boundary of a polynomial function

Abstract. In this paper we show that the Euler characteristic of the generic fibre of a complex polynomial function can be easily computed using the Newton number of f. We apply this result to study polynomials with a finite number of critical points. Received May 25, 1998; in final form January 28, 1999     

Real Principal Value Integrals

 We study principal value integrals of multi-valued differential forms on compact spaces, as introduced by Langlands. Using resolutions of singularities we extend Langlands definition to the case in which the differential form may have no normal crossings. We show by an example that for non-compact spaces the principal value integral associated to a compactification may depend on the compactification. Principal value integrals appear as residues of poles of distributions and as coefficients of asymptotic expansions of oscillating integrals and fibre integrals. (Received 11 November 1999)