A Kummer-type construction of self-dual 4-manifolds

Supported in part by NSF grant DMS-9204093     

The algebraic surfaces on which the classical Phragmén-Lindelöf theorem holds

Let V be an algebraic variety in . We say that V satisfies the strong Phragmén-Lindelöf property (SPL) or that the classical Phragmén-Lindelöf Theorem holds on V if the following is true: There exists a positive constant A such that each plurisubharmonic function u on V which is bounded above by |z|+o(|z|) on V and by 0 on the real points in V already is bounded by A| Im z|. For algebraic varieties V of pure dimension k we derive necessary conditions on V to satisfy (SPL) and we characterize the curves and surfaces in which satisfy (SPL). Several examples illustrate how these results can be applied.     

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     

The Poincaré-Bertrand Formula for the Bochner-Martinelli Integral

The Poincaré-Bertrand formula and the composition formula for the Bochner-Martinelli integral on piecewise smooth manifolds are obtained. As an application, the regularization problem for linear singular integral equation with Bochner-Martinelli kernel and variable coefficients is discussed.     

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.     

Equivariant connected sums of compact self-dual manifolds

Partially supported by NSF grant #DMS-9306950 and UC Riverside grant #5-510000-07427-5     

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

On the period function for a family of complex differential equations

We consider planar differential equations of the form being f(z) and g(z) holomorphic functions and prove that if g(z) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, that says that any continuum of periodic orbits has a constant period function.     

Interpolating varieties for entire functions of minimal type

We obtain both necessary and sufficient conditions for a discrete variety in Cⁿ to be an interpolating variety for entire functions of minimal type.     

The non-orientable genus of some metacyclic groups

We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which is arbitrarily large, where and are the orientable genus and the non-orientable genus of the Cayley graph.Work supported in part by the Research Council of Slovenia, Yugoslavia and NSF Contract DMS-8717441.Supported by NSF Contract DMS-8601760.     

Torus symmetry of compact self-dual manifolds

We classify compact anti-self-dual Hermitian surfaces and compact four-dimensional conformally flat manifolds for which the group of orientation preserving conformal transformations contains a two-dimensional torus. As a corollary, we derive a topological classification of compact self-dual manifolds for which the group of conformal transformations contains a two-dimensional torus.Partially supported by the National Science Foundation grant DMS-9306950.     

Problème du bord dans les variétés $q$-convexes et phénomène de Hartogs-Bochner

Résumé.   Soient E un fibré vectoriel holomorphe au dessus d'une variétéq-complète X et M une sous-variété réelle de dimension 2p-1 () de E. Nous montrons que le problème du bord pour M est résoluble si et seulement si M est maximalement complexe. Dans le cas où nous retrouvons le théorème de Harvey et Lawson [17]. Comme conséquence nous obtenons une généralisation du théorème de Hartogs-Bochner pour les applications CR-méromorphesà valeurs dans une variétéq-convexe et une généralisation au cas CR-méromorphe du théorème d'extension de Chazal [5].

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Let E be a holomorphic vector bundle over a -complete manifold X and M be a real submanifold of dimension 2p-1 () of E. We prove that M has a solution to the boundary problem in X if and only if M is maximally complex. In the case , this is a result of Harvey and Lawson [17]. As a consequence, we obtain a generalization of the Hartogs-Bochner theorem for CR-meromorphic maps taking their values in -convex manifolds and a generalization to the CR-meromorphic case of a theorem of Chazal [5].

Received: 27 April 2000 / Published online: 23 July 2001     

Combinatorics of rational singularities

A normal surface singularity is rational if and only if the dual intersection graph of a desingularization satisfies some combinatorial properties. In fact, the graphs defined in this way are trees. In this paper we give geometric features of these trees. In particular, we prove that the number of vertices of valency 3 in the dual intersection tree of the minimal desingularization of a rational singularity of multiplicity m 3 is at most m - 2.     

Peak curves in weakly pseudoconvex boundaries in C2

LetD be a pseudoconvex domain with real analytic boundary in C². A subsetE of ∂D is a local peak set for if for everyp ∈ ∂D, there exist a neighborhoodU ofp and a holomorphic functionf onU such thatf = 1 onE∩U and |f| < 1 on . We give conditions for the existence of real analytic LPι curves in ∂D through a point of finite type. On the other hand, we give examples showing that: (a) there exist a domainD and a real analytic curve γ in ∂D such that the complexification of γ intersectsD only along γ, but γ is not LPι, and (b) there exist a domain D and a pointp ∈ ∂D, which is LPι, of finite type, but such that ∂D contains no real analytic LP∂ curve throughp.