Showing papers in "Tohoku Mathematical Journal in 2016"

Matrix valued orthogonal polynomials for Gelfand pairs of rank one

Abstract: In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair $(G,K)$ of rank one, as a generalization of earlier work by Koornwinder [30] and subsequently by Koelink, van Pruijssen and Roman [28], [29] for the pair (SU(2)$\times$SU(2), SU(2)), and by Grunbaum, Pacharoni and Tirao [13] for the pair (SU(3), U(2)). Our method is based on representation theory using an explicit determination of the relevant branching rules. Our matrix valued orthogonal polynomials have the Sturm-Liouville property of being eigenfunctions of a second order matrix valued linear differential operator coming from the Casimir operator, and in fact are eigenfunctions of a commutative algebra of matrix valued linear differential operators coming from the $K$-invariant elements in the universal enveloping algebra of the Lie algebra of $G$.

Isometric deformations of cuspidal edges

Abstract: Along cuspidal edge singularities on a given surface in Euclidean 3-space $\boldsymbol{R}^3$, which can be parametrized by a regular space curve $\hat\gamma(t)$, a unit normal vector field $ u$ is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of $\hat\gamma(t)$ is not orthogonal to $ u$. This genericity is equivalent to the condition that its limiting normal curvature $\kappa_ u$ takes a non-zero value. In this paper, we show that a given generic (real analytic) cuspidal edge $f$ can be isometrically deformed preserving $\kappa_ u$ into a cuspidal edge whose singular set lies in a plane. Such a limiting cuspidal edge is uniquely determined from the initial germ of the cuspidal edge.

Wedge operations and torus symmetries

Abstract: A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some classes of manifolds having well-behaved torus actions, say toric objects, can be classified in terms of combinatorial data containing simplicial complexes. In this paper, we investigate the relationship between the topological toric manifolds over a simplicial complex $K$ and those over the complex obtained by simplicial wedge operations from $K$. Our result provides a systematic way to classify toric objects associated with the class of simplicial complexes obtained from a given $K$ by wedge operations. As applications, we completely classify smooth toric varieties with a few generators and show their projectivity. We also study smooth real toric varieties.

Bowman-Bradley type theorem for finite multiple zeta values

Abstract: The multiple zeta values are multivariate generalizations of the values of the Riemann zeta function at positive integers. The Bowman-Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between $3,1,\dots,3,1$ add up to a rational multiple of a power of $\pi$. We show that an analogous theorem holds in a very strong sense for finite multiple zeta values, which have been investigated by Hoffman and Zhao among others and recently recast by Zagier.

Homotopy theory of mixed Hodge complexes

Abstract: We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced by Guillen-Navarro-Pascual-Roig leading to a good calculation of the homotopy category in terms of (co)fibrant objects. Using Deligne's decalage, we show that the homotopy categories associated with the two notions of mixed Hodge complex introduced by Deligne and Beilinson respectively, are equivalent. The results provide a conceptual framework from which Beilinson's and Carlson's results on mixed Hodge complexes and extensions of mixed Hodge structures follow easily.

Large deviation principles for generalized Feynman-Kac functionals and its applications

Abstract: Large deviation principles of occupation distribution for generalized Feyn-man-Kac functionals are presented in the framework of symmetric Markov processes having doubly Feller or strong Feller property. As a consequence, we obtain the $L^p$-independence of spectral radius of our generalized Feynman-Kac functionals. We also prove Fukushima's decomposition in the strict sense for functions locally in the domain of Dirichlet form having energy measure of Dynkin class without assuming no inside killing.

On the geometry of the cross-cap in the Minkoswki 3-space and binary differential equations

Abstract: We initiate in this paper the study of the geometry of the cross-cap in Minkowski 3-space $\mathbb{R}^3_1$. We distinguish between three types of cross caps according to their tangential line being spacelike, timelike or lightlike. For each of these types, the principal plane which is generated by the tangential line and the limiting tangent direction to the curve of self-intersection of the cross-cap plays a key role. We obtain special parametrisations for the three types of cross-caps and consider their affine properties. The pseudo-metric on the cross-cap changes signature along a curve and the singularities of this curve depend on the type of the cross-cap. We also study the binary differential equations of the lightlike curves and of the principal curves in the parameters space and obtain their topological models as well as the configurations of their solution curves.

The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces

Abstract: We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex K-theory ring of a divisive weighted projective space (which is singular for nontrivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we conrm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant K-theory and complex cobordism rings of more general singular toric varieties, namely those whose integral cohomology is concentrated in even dimensions, are isomorphic to rings of appropriate piecewise formal power series. Finally, we conrm the corresponding descriptions for any smooth, compact, projective toric variety, and rewrite them in a face ring context. In many cases our results agree with those of Vezzosi and Vistoli for algebraic K-theory, Anderson and Payne for operational K-theory, Krishna and Uma for algebraic cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we summarize the details of these coincidences.

Construction of isolated left orderings via partially central cyclic amalgamation

Abstract: We give a new method to construct isolated left orderings of groups whose positive cones are finitely generated. Our construction uses an amalgamated free product of two groups having an isolated ordering. We construct a lot of new examples of isolated orderings, and give an example of isolated left orderings with various properties which previously known isolated orderings do not have.

Calabi–Yau 3-folds of Borcea–Voisin type and elliptic fibrations

Abstract: We consider Calabi–Yau 3-folds of Borcea–Voisin type, i.e. Calabi–Yau 3-folds obtained as crepant resolutions of a quotient $(S\times E)/(\alpha_S\times \alpha_E)$, where $S$ is a K3 surface, $E$ is an elliptic curve, $\alpha_S\in \operatorname{Aut}(S)$ and $\alpha_E\in \operatorname{Aut}(E)$ act on the period of $S$ and $E$ respectively with order $n=2,3,4,6$. The case $n=2$ is very classical, the case $n=3$ was recently studied by Rohde, the other cases are less known. First, we construct explicitly a crepant resolution, $X$, of $(S\times E)/(\alpha_S\times \alpha_E)$ and we compute its Hodge numbers; some pairs of Hodge numbers we found are new. Then, we discuss the presence of maximal automorphisms and of a point with maximal unipotent monodromy for the family of $X$. Finally, we describe the map $\mathcal{E}_n: X \rightarrow S/\alpha_S$ whose generic fiber is isomorphic to $E$.

Wintgen ideal submanifolds of codimension two, complex curves, and Möbius geometry

Abstract: Wintgen ideal submanifolds in space forms are those ones attaining the equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Mobius geometry, we show that in the codimension two case, the mean curvature spheres of the Wintgen ideal submanifold correspond to a 1-isotropic holomorphic curve in a complex quadric. Conversely, any 1-isotropic complex curve in this complex quadric describes a 2-parameter family of spheres whose envelope is always a Wintgen ideal submanifold of codimension two at the regular points. Via a complex stereographic projection, we show that our characterization is equivalent to Dajczer and Tojeiro's previous description of these submanifolds in terms of minimal surfaces in the Euclidean space.

The Lê-Greuel formula for functions on analytic spaces

Abstract: In this article we give an extension of the Le-Greuel formula to the general setting of function germs $(f,g)$ defined on a complex analytic variety $X$ with arbitrary singular set, where $f = (f_1,\ldots,f_k): (X,\underline{0}) \to (\mathbb{C}^k,\underline{0})$ is generically a submersion with respect to some Whitney stratification on $X$. We assume further that the dimension of the zero set $V(f)$ is larger than 0, that $f$ has the Thom $a_f$-property with respect to this stratification, and $g: (X,\underline{0}) \to (\mathbb{C},0)$ has an isolated critical point in the stratified sense, both on $X$ and on $V(f)$.

Compact sums of Toeplitz products and Toeplitz algebra on the Dirichlet space

Abstract: In this paper we consider Toeplitz operators on the Dirichlet space of the ball. We first characterize the compactness of operators which are finite sums of products of two Toeplitz operators. We also characterize Fredholm Toeplitz operators and describe the essential norm of Toeplitz operators. By using these results, we establish a short exact sequence associated with the $C^*$-algebra generated by all Toeplitz operators.

Holomorphic submersions onto Kähler or balanced manifolds

Abstract: We study many properties concerning weak Kahlerianity on compact complex manifolds which admits a holomorphic submersion onto a Kahler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the Kahler case), Michelsohn (the balanced case), Popovici (the sG case) and others.

Umbilical surfaces of products of space forms

Abstract: We give a complete classification of umbilical surfaces of arbitrary codimension of a product Q n1 k1 × Q n2 k2 of space forms whose curvatures satisfy k1 + k2 6 0. MSC 2010: 53C40, 53C42

On the holomorphic automorphism group of a generalized Hartogs triangle

Abstract: In this paper, we completely determine the structure of the holomorphic automorphism group of a generalized Hartogs triangle and obtain natural generalizations of some results due to Landucci and Chen-Xu. These give affirmative answers to some open problems posed by Jarnicki and Pflug.

Crossed actions of matched pairs of groups on tensor categories

Abstract: We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category, where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category is called a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G, \Gamma)$-crossed action. We show that every $(G,\Gamma)$-crossed tensor category $\mathcal{C}$ gives rise to a tensor category $\mathcal{C}^{(G, \Gamma)}$ that fits into an exact sequence of tensor categories $\operatorname{Rep} G \longrightarrow \mathcal{C}^{(G, \Gamma)} \longrightarrow \mathcal{C}$. We also define the notion of a $(G, \Gamma)$-braiding in a $(G, \Gamma)$-crossed tensor category, which is connected with certain set-theoretical solutions of the QYBE. This extends the notion of $G$-crossed braided tensor category due to Turaev. We show that if $\mathcal{C}$ is a $(G, \Gamma)$-crossed tensor category equipped with a $(G, \Gamma)$-braiding, then the tensor category $\mathcal{C}^{(G, \Gamma)}$ is a braided tensor category in a canonical way.

Fukushima type decomposition for semi-Dirichlet forms

Abstract: We present a Fukushima type decomposition in the setting of general quasi-regular semi-Dirichlet forms. The decomposition is then employed to give a transformation formula for martingale additive functionals. Applications of the results to some concrete examples of semi-Dirichlet forms are given at the end of the paper. We discuss also the uniqueness question about the Doob-Meyer decomposition on optional sets of interval type.

Richness of smith equivalent modules for finite gap oliver groups

Abstract: Let $G$ be a finite group not of prime power order. Two real $G$-modules $U$ and $V$ are $\mathcal{P}(G)$-connectively Smith equivalent if there exists a homotopy sphere with smooth $G$-action such that the fixed point set by $P$ is connected for all Sylow subgroups $P$ of $G$, it has just two fixed points, and $U$ and $V$ are isomorphic to the tangential representations as real $G$-modules respectively. We study the $\mathcal{P}(G)$-connective Smith set for a finite Oliver group $G$ of the real representation ring consisting of all differences of $\mathcal{P}(G)$-connectively Smith equivalent $G$-modules, and determine this set for certain nonsolvable groups $G$.

Weighted Hamiltonian stationary Lagrangian submanifolds and generalized Lagrangian mean curvature flows in toric almost Calabi-Yau manifolds

Abstract: In this paper, we generalize examples of Lagrangian mean curvature flows constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. To be more precise, we construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized Lagrangian mean curvature flows starting from these examples. We allow these flows to have some singularities and topological changes.

Codimension one connectedness of the graph of associated varieties

Abstract: Let $ \pi $ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ \mathcal{AV}(\pi) $. If $ \mathcal{AV}(\pi) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ \pi $, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connected for even nilpotent orbits. Finally, for indefinite unitary group $ U(p, q) $, we prove that for each connected component of the orbit graph $ \Gamma_K(\mathcal{O}^G_\lambda) $ thus defined, there is an irreducible Harish-Chandra module $ \pi $ whose associated graph is exactly equal to the connected component.

Holomorphic submersions onto K\"ahler or balanced manifolds

Abstract: We study many properties concerning weak K\"ahlerianity on compact complex manifolds which admits a holomorphic submersion onto a K\"ahler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the K\"ahler case), Michelson (the balanced case), Popovici (the sG case) and others.

Homogeneous Ricci soliton hypersurfaces in the complex hyperbolic spaces

Abstract: A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.

A note on the Kakeya maximal operator and radial weights on the plane

Abstract: We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume that a radial weight $w$ satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths $a$ and $aN$ with $N$ fixed. We are interested in its eccentricity $N$ with $a$ fixed. We give an example of a non-constant weight showing that $\sqrt{\log N}$ cannot be removed.

Construction of sign-changing solutions for a subcritical problem on the four dimensional half sphere

Abstract: This paper is devoted to studying the nonlinear problem with subcritical exponent $(S_\varepsilon) : -\Delta_g u+2u = K|u|^{2-\varepsilon}u$, in $ S^4_+ $, ${\partial u}/{\partial u} =0$, on $\partial S^4_+,$ where $g$ is the standard metric of $S^4_+$ and $K$ is a $C^3$ positive Morse function on $\overline{S_+^4}$. We construct some sign-changing solutions which blow up at two different critical points of $K$ in interior. Furthermore, we construct sign-changing solutions of $(S_\varepsilon)$ having two bubbles and blowing up at the same critical point of $K$.