Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ _ℂ ⁿ . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ _ℂ ³ for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.     

Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds

It is known [M4] that K_ℂ-orbits S and G_ℝ-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. It is possible to replace K_ℂ by some conjugate xK_ℂx⁻¹ so that the correspondence is preserved. We investigate the sets C(S) of such x for various orbits S and their relations with each other. We prove that for classical groups the intersection C = ∩_S C(S) equals D₀Z where D₀ = D₀/K_ℂ is the universal domain in G_ℂ/K_ℂ introduced in [AG] and Z is the center of G. As a corollary we prove that D₀ is Stein for classical groups. Moreover we conjecture that C(S)₀ = D₀ for generic S where C(S)₀ is the connected component of C(S) containing the identity.     

Residual kernels with singularities on coordinate planes

A finite collection of planes {E _v } in ℂ^d is called an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed differential form generating this group is called a residual kernel for the atomic family. We construct new residual kernels in the case when E _v are coordinate planes such that the complement ℂ^d/∪ E _v admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the well-known Bochner-Martinelli and Sorani differential forms. The kernels obtained are used to establish a new formula of integral representations for functions holomorphic in Reinhardt polyhedra.     

The homology and cohomology of the complements to some arrangements of codimension two complex planes

We consider the homology and cohomology of the complement to the arrangement Z = ∪_{1<|i−j|<d−1}{z _i = z _j = 0} of coordinate planes in ℂ ^d , and explicitly construct a basis for these groups as well as a basis for the homology groups of the one-point compactification of Z.     

Zur Theorie der Stetigen Teilbarkeitshalbgruppen

O. CallS:=(S,·,∩) a d-semigroup ifS satisfies the axioms (A1) (S,·) is a semigroup, (A2) (S,∩) is a semilattice (A3), (S,·,∩) is a semiring, (A4) a ≤b⇒bε aS ∩ Sa. Call tεS positive if ÅaεS: ta ≥a≤at. Let S⁺ denote the set {t‖t positive}. Every d-semigroup is closed under sup and (s,·,∪) is a semiring, (S, ∩, ∪) is a distributive lattice. Denote by D□X the implication s=Xa_i⇒x□s□y=X(x□a_i□y) where □ε{·,∩,∪} and Xε{∪,∩}. CallS continuous ifS satisfies all D□X. The theory of d-semigroups (divisibility-semigroups) was established in [3], [4], [5], and is continued here by some contributions to the theory of continuous d-semigroups the main results of which are the two propositions: (1) LetS be a d-semigroup with 1. ThenS satisfies D□X iffS ⁺ satisfies this axiom. (2) LetS be continuous. Then (S,·) is commutative. Obviously Proposition (2) is an improvement of Iwasawa's theorem concerning conditionally complete lattice ordered groups.

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Zur Theorie der Stetigen Teilbarkeitshalbgruppen

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Klaus Wagner zum 70. Geburtstag gewidmet     

Scalar curvature of a class of minimal submanifolds in rank 1 symmetric spaces

In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in ℂPⁿ, compact minimal quaternionic coisotropic submanifolds in ℚPⁿ and compact minimal hypersurfaces in P² (Cay).      

Algebraic Osculation and Application to Factorization of Sparse Polynomials

We prove a theorem on algebraic osculation and apply our result to the computer algebra problem of polynomial factorization. We consider X a smooth completion of ℂ² and D an effective divisor with support the boundary ∂X=X∖ℂ². Our main result gives explicit conditions that are necessary and sufficient for a given Cartier divisor on the subscheme (|D|,O_D)(|D|,\mathcal{O}_{D}) to extend to X. These osculation criteria are expressed with residues. When applied to the toric setting, our result gives rise to a new algorithm for factoring sparse bivariate polynomials which takes into account the geometry of the Newton polytope.     

Special points on products of two Shimura curves

Let S ₁ and S ₂ be two Shimura curves over ℚ attached to rational indefinite quaternion algebras B _{1 ℚ} and B _{1 ℚ} with maximal orders B ₁ and B ₂ respectively. We consider an irreducible closed algebraic curve C in the product (S ₁×S ₂)_ℂ such that C(ℂ) ∩ (S ₁×S ₂)(ℂ) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields, that C is of Hodge type. Received: 3 January 2000 / Revised version: 2 October 2000     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.     

On dimensions of the groups of biholomorphic automorphisms of real-analytic hypersurfaces

In this paper we study the dependence of the local geometry of real-analytic hypersuffaces in ℂ ⁿ on the dimension of the group of biholomorphic automorphisms of this surface. We also classify the hypersurfaces in terms of this group. We present some examples showing that the classes of the given construction are not empty. We find a new formulation of the Freeman theorem on the so-called straightening of a real-analytic CR-submanifold in ℂ ⁿ with degenerate Levi form of constant rank. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 349–358, March, 1997. Translated by E. G. Anisova     

Generalized Ham-Sandwich Cuts

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S ₁,…,S _d in R ^d and constants β _i ∈[0,1], there exists a unique hyperplane h with the property that Vol (h ⁺∩S _i )=β _i ⋅Vol (S _i ); h ⁺ is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut.” We give a discrete analogue for a set S of n points in R ^d which are partitioned into a family S=P ₁∪⋅⋅⋅∪P _d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n) ^d−3) algorithm which, given positive integers a _i ≤|P _i |, finds the unique hyperplane h incident with a point in each P _i and having |h ⁺∩P _i |=a _i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse.     

Viro theorem and topology of real and complex combinatorial hypersurfaces

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP ⁿ and are topologically “glued” out of algebraic hypersurfaces in (ℂ^*) ⁿ . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces. A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161. The work was completed during the stay of both authors at Max-Planck-Institu für Mathematik. The authors thank these funds and institutions for hospitality and financial support.     

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that X _p (ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S ^*(1) = {f ∈ S ^*: X ^* _f ⩽ 1} of the random conjugate space (S ^*,X ^*) of (S,X) is compact under the random weak star topology on (S ^*,X ^*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {A _n : n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪ _n=1^∞ A _n and for each element F in E ∩ A, there is an H in the σ-algebra generated by {A _n : n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: X _p ⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.     

A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3.     

Subvarieties of generic hypersurfaces in a nonsingular projective toric variety

We investigate the subvarieties contained in generic hypersurfaces of projective toric varieties and prove two main theorems. The first generalizes Clemens’ famous theorem on the genus of curves in hypersurfaces of projective spaces to curves in hypersurfaces of toric varieties and the second improves the bound in the special case of toric varieties in a theorem of Ein on the positivity of subvarieties contained in sufficiently ample generic hypersurfaces of projective varieties. Both depend on a hypothesis which deals with the surjectivity of multiplication maps of sections of line bundles on the toric variety. We also obtain an infinitesimal Torelli theorem for hypersurfaces of toric varieties.     

Homogeneous Hypersurfaces in ℂ<Superscript>3</Superscript>, Associated with a Model CR-Cubic

The model 4-dimensional CR-cubic in ℂ³ has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in ℂ³ with finite-dimensional infinitesimal automorphism algebra \mathfrakg\mathfrak{g} and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all \mathfrakg\mathfrak{g}-homogeneous hypersurfaces in ℂ³ and also classify the corresponding local transitive actions of the model algebra \mathfrakg\mathfrak{g} on hypersurfaces in ℂ³.     

On Cycles in 3-Connected Graphs

 Let G be a 3-connected graph of order n and S a subset of vertices. Denote by δ(S) the minimum degree (in G) of vertices of S. Then we prove that the circumference of G is at least min{|S|, 2δ(S)} if the degree sum of any four independent vertices of S is at least n+6. A cycle C is called S-maximum if there is no cycle C ^′ with |C ^′∩S|>|C∩S|. We also show that if ∑⁴ _i=1 d(a _i)≥n+3+|⋂⁴ _i=1 N(a _i)| for any four independent vertices a ₁, a ₂, a ₃, a ₄ in S, then G has an S-weak-dominating S-maximum cycle C, i.e. an S-maximum cycle such that every component in G−C contains at most one vertex in S. Received: March 9, 1998 Revised: January 7, 1999