Removable singularities of CR functions on singular boundaries

The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature of singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. Received: 8 December 2000; in final form: 24 June 2001/Published online: 1 February 2002     

Equicontinuity of families of plurisubharmonic functions with bounds on their Monge-Ampère masses

It is proved that families of solutions of the complex Monge-Ampère equation with fixed boundary values and certain bounds on the right hand side are equicontinuous. Received: 24 April 2001; in final form: 10 August 2001 / Published online: 28 February 2002     

The overdetermined Cauchy problem in some classes of ultradifferentiable functions

In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of ultradifferentiable functions of class (M _p ). We show that evolution is equivalent to the validity of a Phragmén-Lindel?f principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties, and make applications in different situations. We find necessary and sufficient conditions for well posedness, and relate the hyperbolicity of a given system to that of its principal part. Received: January 19, 1999?Published online: May 10, 2001     

Oka's principle for holomorphic submersions with sprays

We classify, up to a local isometry, all non-K?hler almost K?hler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost K?hler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. Received: 1 October 2001 / Published online: 17 June 2002     

A little microlocal Morse theory

If a complex analytic function, f, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of f has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along f have isolated support. We also investigate an index-theoretic proof of this fact. Received: 15 September 2000 / Published online: 18 June 2001     

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     

Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains

We prove that any proper holomorphic mapping from to is necessarily a totally geodesic isometric embedding with respect to their Bergman metrics and therefore is the standard linear embedding up to their automorphisms. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics. Received: 31 October 2000; in final form: 2 July 2001/ Published online: 4 April 2002     

Relative morsification theory

In this paper we develop a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known morsification results for non-isolated singularities and generalise them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalisations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalised jet-space we show how to interpret the newly defined invariants as certain intersection multiplicities; finally, we characterise which invariants can be interpreted as intersection multiplicities in the above mentioned generalised jet space.     

Boundary Behaviour of -Harmonic Functions and Non-Isotropic Hausdorff Measure

 In this paper we consider weighted boundary behaviour of -harmonic functions on the unit ball of . In particular, we show that a measure on the unit sphere of admits a component along the non-isotropic Hausdorff measure on that sphere and that this component gives rise to certain weighted boundary behaviour of the -harmonic extension of the original measure. (Received 24 July 2000; in revised form 10 August 2001)     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

An analog of the Hartogs theorem in a ball of

In this paper we consider continuous functions given on the boundary of a ball B in , , and having a one‐dimensional property of holomorphic extension along the families of complex lines, passing through a finite number of points of B. We study the problem of existence of holomorphic extension of such functions in the ball B.     

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     

On functions with one‐dimensional holomorphic extension property in circular domains

In this paper we consider continuous functions given on the boundary of a circular bounded domain D in , , and having the one‐dimensional holomorphic extension property along family of complex lines, passing through a finite number of points of D. We study the problem of existence of holomorphic extension of such functions into D.     

Bouquet and join theorems for disentanglements

The disentanglement of certain augmentations is shown to be the topological join of a disentanglement and a Milnor fibre. The kth disentanglement of a finite map is defined and for corank 1 maps from ℂ ⁿ to ℂ ^{n +1} it is shown that they are homotopically equivalent to a wedge of spheres. Applications to the Mond conjecture are given. Oblatum 24-VII-2000 & 5-VII-2001?Published online: 12 October 2001     

On the string-theoretic Euler number of a class of absolutely isolated singularities

An explicit computation of the so-called string-theoretic E-function of a normal complex variety X with at most log-terminal singularities can be achieved by constructing one snc-desingularization of X, accompanied with the intersection graph of the exceptional prime divisors, and with the precise knowledge of their structure. In the present paper, it is shown that this is feasible for the case in which X is the underlying space of a class of absolutely isolated singularities (including both usual ? _n -singularities and Fermat singularities of arbitrary dimension). As byproduct of the exact evaluation of lim, for this class of singularities, one gets counterexamples to a conjecture of Batyrev concerning the boundedness of the string-theoretic index. Finally, the string-theoretic Euler number is also computed for global complete intersections in ℙ_ℂ ^N with prescribed singularities of the above type. Received: 2 January 2001 / Revised version: 22 May 2001     

On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003     

On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces. Received: 8 February 2001; in final form: 17 September 2001 / Published online: 1 February 2002     

Dimensions of very ample pluricanonical subsystems for compact quotients of bounded symmetric domains

A compact complex manifold X obtained by taking quotient of a bounded symmetric domain has an ample canonical line bundle. We prove that the dimension of very ample pluricanonical subsystem is strictly bigger than 2n, where n is the dimension of X. Received: 23 June 2000 / Revised version: 30 March 2001