Separate holomorphic extension of CR functions

We consider a real analytic foliation of by complex analytic manifolds of dimension m issued transversally from a CR generic submanifold of codimension m. We prove that a continuous CR function f on M which has separate holomorphic extension along each leaf, is holomorphic. When the leaves are cartesian straight planes, separate holomorphic extension along suitable selections of these planes suffices and f turns out to be holomorphic in a neighbourhood of their union. If M is a hypersurface we can also specify the side of the extension, regardless the leaves are straight or not.     

On the Cowen-Douglas Class for Banach Space Operators

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class if and only if T is unitarily equivalent with the operator M _z on a Hilbert space -valued analytic functions, where M _z denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators.      

Pull-back of currents by holomorphic maps

We define the pull-back operator, associated to a meromorphic transform, on several types of currents. We also give a simple proof to a version of a classical theorem on the extension of currents.     

A generalized flat extension theorem for moment matrices

In this note we prove a generalization of the flat extension theorem of Curto and Fialkow (Memoirs of the American Mathematical Society, vol. 119. American Mathematical Society, Providence, 1996) for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.      

On the disk theorem

We construct an example of a 2-dimensional Stein normal space X with one singular point x ₀ such that X\{x ₀} is simply connected and it satisfies the disk condition. This answers a question raised by Forn?ss and Narasimhan. We also prove that any increasing union of Stein open sets contained in a Stein space of dimension 2 satisfies the disk condition. Starting from the above example we exhibit, without using deformation theory, a new type of 2-dimensional holes which cannot be filled.     

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

Covering spheres of Banach spaces by balls

If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as translates of the same ball. Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel.     

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.     

An interpolation theorem for holomorphic automorphisms ofC n

We construct automorphisms of C ⁿ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set.     

Banach algebra techniques for spectral multiplicity

In this paper, we use Banach algebra techniques to prove addition formulas for spectral multiplicities of direct sums of operator families. This work has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) with project 107T649.     

A structure theorem of Dirac-harmonic maps between spheres

For an arbitrary Dirac-harmonic map (φ,ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N. On the basis, we could clarify all of non-trivial Dirac-harmonic maps from S ² to S ².     

Parabolic Raynaud bundles

Let X be an irreducible smooth projective curve defined over the field of complex numbers, a finite set of closed points and N ≥ 2 a fixed integer. For any pair , there exists a parabolic vector bundle on X, with parabolic structure over S and all parabolic weights in , that has the following property: Take any parabolic vector bundle of rank r on X whose parabolic points are contained in S, all the parabolic weights are in and the parabolic degree is d. Then is parabolically semistable if and only if there is no nonzero parabolic homomorphism from to .     

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.     

A degree bound for globally generated vector bundles

Let E be a globally generated vector bundle of rank e ≥ 2 over a reduced irreducible projective variety X of dimension n defined over an algebraically closed field of characteristic zero. Let L := det(E). If deg(E) := deg(L) = L ⁿ  > 0 and E is not isomorphic to , we obtain a sharp bound

Fixed point indices and periodic points of holomorphic mappings

Let Δ ⁿ be the ball |x| <  1 in the complex vector space , let be a holomorphic mapping and let M be a positive integer. Assume that the origin is an isolated fixed point of both f and the Mth iteration f ^M of f. Then for each factor m of M, the origin is again an isolated fixed point of f ^m and the fixed point index of f ^m at the origin is well defined, and so is the (local) Dold’s index [Invent. Math. 74(3), 419–435 (1983)] at the origin:

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

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.     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.