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Showing papers in "Tohoku Mathematical Journal in 2017"


Journal ArticleDOI
TL;DR: In this article, it was shown that a Chow group of algebraically closed fields is of abelian type if and only if it is spanned by the Chow groups of products of curves.
Abstract: A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. This paper contains three sections of independent interest. First, we show that a motive which becomes of abelian type after a base field extension of algebraically closed fields is of abelian type. Given a field extension $K/k$ and a motive $M$ over $k$, we also show that $M$ is finite-dimensional if and only if $M_K$ is finite-dimensional. As a corollary, we obtain Chow–Kunneth decompositions for varieties that become isomorphic to an abelian variety after some field extension. Second, let $\varOmega$ be a universal domain containing $k$. We show that Murre's conjectures for motives of abelian type over $k$ reduce to Murre's conjecture (D) for products of curves over $\varOmega$. In particular, we show that Murre's conjecture (D) for products of curves over $\varOmega$ implies Beauville's vanishing conjecture on abelian varieties over $k$. Finally, we give criteria on Chow groups for a motive to be of abelian type. For instance, we show that $M$ is of abelian type if and only if the total Chow group of algebraically trivial cycles $\mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ is spanned, via the action of correspondences, by the Chow groups of products of curves. We also show that a morphism of motives $f: N \rightarrow M$, with $N$ finite-dimensional, which induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{alg} \rightarrow \mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ also induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{hom} \rightarrow \mathrm{CH}_*(M_\varOmega)_\mathrm{hom}$ on homologically trivial cycles.

48 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated periodic magnetic curves in elliptic Sasakian space forms and obtained a quantization principle for periodic magnetic flowlines on Berger spheres, and gave a criterion for periodicity of magnetic curves on the unit sphere.
Abstract: It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe it. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criterion for periodicity of magnetic curves on the unit sphere ${\mathbb{S}}^3$.

27 citations


Journal ArticleDOI
TL;DR: The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms as mentioned in this paper.
Abstract: The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge and Tang (2008), independently. The present author established a generalized Wintgen inequality for Lagrangian submanifolds in complex space forms in 2014. In the present paper we obtain the DDVV inequality, also known as generalized Wintgen inequality, for Legendrian submanifolds in Sasakian space forms. Some geometric applications are derived. Also we state such an inequality for contact slant submanifolds in Sasakian space forms.

27 citations


Journal ArticleDOI
TL;DR: In this paper, the authors established the atomic decompositions for the weighted Hardy spaces with variable exponents, and revealed some intrinsic structures of atomic decomposition for Hardy type spaces also reveal some intrinsic properties of decomposition.
Abstract: We establish the atomic decompositions for the weighted Hardy spaces with variable exponents. These atomic decompositions also reveal some intrinsic structures of atomic decomposition for Hardy type spaces.

26 citations


Journal ArticleDOI
TL;DR: In this paper, the maximal operator on Musielak-orlicz-Morrey spaces was shown to be bounded by a bounded maximal operator, which is an improvement of [7, Theorem 4.1].
Abstract: We give the boundedness of the maximal operator on Musielak-Orlicz-Morrey spaces, which is an improvement of [7, Theorem 4.1]. We also discuss the sharpness of our conditions.

16 citations


Journal ArticleDOI
TL;DR: In this paper, the Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions and the spectral zeta function of regular trees is studied.
Abstract: We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.

15 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proved almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on a random initial data set below the energy space.
Abstract: In this note, we prove almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on $\mathbb{T}^d$, $d = 3, 4,$ and $5$, with random initial data below the energy space.

11 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived contiguity relations of Lauricella's hypergeometric function by using the twisted cohomology group and the intersection form, and constructed twisted cycles corresponding to a fundamental set of solutions to the system of differential equations satisfied by $F_D, which are expressed as Laurent series.
Abstract: We study contiguity relations of Lauricella's hypergeometric function $F_D$, by using the twisted cohomology group and the intersection form. We derive contiguity relations from those in the twisted cohomology group and give the coefficients in these relations by the intersection numbers. Furthermore, we construct twisted cycles corresponding to a fundamental set of solutions to the system of differential equations satisfied by $F_D$, which are expressed as Laurent series. We also give the contiguity relations of these solutions.

10 citations


Journal ArticleDOI
TL;DR: In this paper, the BMO-BLO boundedness of martingale maximal functions and Bennett type characterization of BLO martingales are shown, as well as a non-negative BMO Martingale that is not in BLO is constructed.
Abstract: Some new properties concerning BLO martingales are given. The BMO-BLO boundedness of martingale maximal functions and Bennett type characterization of BLO martingales are shown. Also, a non-negative BMO martingale that is not in BLO is constructed.

10 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the Tamagawa numbers of a crystalline representation over a tower of cyclotomic extensions under certain technical conditions on the representation, and showed that they may improve the asymptotic bounds given in the thesis of Arthur Laurent in certain cases.
Abstract: In this paper, we study the Tamagawa numbers of a crystalline representation over a tower of cyclotomic extensions under certain technical conditions on the representation. In particular, we show that we may improve the asymptotic bounds given in the thesis of Arthur Laurent in certain cases.

7 citations


Journal ArticleDOI
TL;DR: In this article, the existence of universal deformation of a given pseudo-SL_2$-representation of a finitely generated group over a perfect field whose characteristic is not 2 is proved.
Abstract: Based on the analogies between knot theory and number theory, we study a deformation theory for ${\rm SL}_2$-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-${\rm SL}_2$-representations, we prove the existence of the universal deformation of a given ${\rm SL}_2$-representation of a finitely generated group $\Pi$ over a perfect field $k$ whose characteristic is not 2. We then show its connection with the character scheme for ${\rm SL}_2$-representations of $\Pi$ when $k$ is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.

Journal ArticleDOI
TL;DR: In this article, the authors considered a semilinear Robin problem with an indefinite and unbounded potential and a reaction which exhibits asymmetric behavior as $x\rightarrow\pm\infty.
Abstract: We consider a semilinear Robin problem with an indefinite and unbounded potential and a reaction which exhibits asymmetric behavior as $x\rightarrow\pm\infty$. More precisely it is sublinear near $-\infty$ with possible resonance with respect to the principal eigenvalue of the negative Robin Laplacian and it is superlinear at $+\infty$. Resonance is also allowed at zero with respect to any nonprincipal eigenvalue. We prove two multiplicity results. In the first one, we obtain two nontrivial solutions and in the second, under stronger regularity conditions on the reaction, we produce three nontrivial solutions. Our work generalizes the recent one by Recova-Rumbos (Nonlin. Anal. 112 (2015), 181--198).

Journal ArticleDOI
TL;DR: In this article, the authors discuss holomorphic isometric embeddings of the projective line into quadrics using the generalisation of the theorem of do Carmo-Wallach in [14] to provide a description of their moduli spaces up to image and gauge equivalence.
Abstract: We discuss holomorphic isometric embeddings of the projective line into quadrics using the generalisation of the theorem of do Carmo-Wallach in [14] to provide a description of their moduli spaces up to image and gauge equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

Journal ArticleDOI
TL;DR: In this article, the authors characterize the potentials which correspond to conformally harmonic maps containing a light-like vector and precisely characterize those potentials that produce immersions into a Riemann surface that are not conformal to a minimal surface.
Abstract: The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\mathbb{R}^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\mathbb{R}^{1,n+3}$, contains a fixed lightlike vector or where it does not contain such a ``constant lightlike vector''. Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into $S^{n+2}$ which are not conformal to a minimal surface in $\mathbb{R}^{n+2}$. It turns out that our proof also works analogously for minimal immersions into the other space forms.

Journal ArticleDOI
Kōta Yoshioka1
TL;DR: In this article, the existence of stable sheaves on non-classical Enriques surfaces has been studied, and necessary and sufficient conditions for their existence have been given for the existence.
Abstract: We shall give a necessary and sufficient condition for the existence of stable sheaves on non-classical Enriques surfaces.

Journal ArticleDOI
TL;DR: The guiding thread of the present work is the following result, in the vain of Grothendieck's conjecture for differential equations: if the reduction modulo almost all prime $p$ of a given linear Mahler equation has a full set of algebraic solutions, then this equation has rational solutions as discussed by the authors.
Abstract: The guiding thread of the present work is the following result, in the vain of Grothendieck's conjecture for differential equations : if the reduction modulo almost all prime $p$ of a given linear Mahler equation with coefficients in $\mathbb{Q}(z)$ has a full set of algebraic solutions, then this equation has a full set of rational solutions. The proof of this result, given at the very end of the paper, relies on intermediate results of independent interest about Mahler equations in characteristic zero as well as in positive characteristic.

Journal ArticleDOI
TL;DR: In this paper, the structure of the Gauss map of a projective toric variety is described in terms of combinatorics in any characteristic, and two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic are given.
Abstract: We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

Journal ArticleDOI
TL;DR: In this article, the authors studied deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and proved that the deformed polynomial maps have only indefinite fold and cusp singularities in general.
Abstract: In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such a deformation. As a corollary, we show that a deformation of a complex Morse singularity with real linear terms has only indefinite folds and cusps in general and the number of cusps is 3.

Journal ArticleDOI
TL;DR: In this paper, a polynomial approximation of the Reidemeister torsion of a 3-manifold obtained by a $1/n$-Dehn surgery along a polygonal manifold was considered.
Abstract: Let $K$ be a $(2p,q)$-torus knot. Here $p$ and $q$ are coprime odd positive integers. Let $M_n$ be a 3-manifold obtained by a $1/n$-Dehn surgery along $K$. We consider a polynomial $\sigma_{(2p,q,n)}(t)$ whose zeros are the inverses of the Reidemeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations under some normalization. Johnson gave a formula for the case of the $(2,3)$-torus knot under another normalization. We generalize this formula for the case of $(2p,q)$-torus knots by using Tchebychev polynomials.

Journal ArticleDOI
TL;DR: In this paper, the theory of non-Archimedean uniformization is adapted to construct a smooth surface from a lattice in the PSL space that has nontrivial torsion.
Abstract: We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in ${\rm PSL}_3(\mathbb{Q}_2)$ that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake plane yet distinct from it and the other fake planes that arise from 2-adic uniformization by torsion-free groups. As part of the proof, and of independent interest, we compute the homotopy type of the Berkovich space of our plane.

Journal ArticleDOI
TL;DR: In this article, the authors discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than the usual curvature hypothesis in the generalized Toponogov theorems, but yet is strong enough to ensure that a geodesic triangle in the manifold has a corresponding triangle in a model with the same corresponding side lengths, but smaller corresponding angles.
Abstract: Toponogov's triangle comparison theorem and its generalizations are important tools for studying the topology of Riemannian manifolds. In these theorems, one assumes that the curvature of a given manifold is bounded from below by the curvature of a model surface. The models are either of constant curvature, or, in the generalizations, rotationally symmetric about some point. One concludes that geodesic triangles in the manifold correspond to geodesic triangles in the model surface which have the same corresponding side lengths, but smaller corresponding angles. In addition, a certain rigidity holds: Whenever there is equality in one of the corresponding angles, the geodesic triangle in the surface embeds totally geodesically and isometrically in the manifold. In this paper, we discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than the usual curvature hypothesis in the generalized Toponogov theorems, but yet is strong enough to ensure that a geodesic triangle in the manifold has a corresponding triangle in the model with the same corresponding side lengths, but smaller corresponding angles. In contrast, it is interesting that rigidity fails in this setting.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a projective surface of globally $F$-regular type defined over a field of characteristic zero is of Fano type, which is a special case of the Fano regularity.
Abstract: We prove that a projective surface of globally $F$-regular type defined over a field of characteristic zero is of Fano type.

Journal ArticleDOI
TL;DR: In this paper, the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in a Lagrangian submanifold was studied.
Abstract: Let $M$ be a symplectic manifold equipped with a Hamiltonian circle action and let $L$ be an invariant Lagrangian submanifold of $M$. We study the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in $L$ through degeneration. We obtain a conjectural relationship between the potential function of $L$ and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng [8, 9] relating certain correction terms appearing in the Seidel elements with the potential function.

Journal ArticleDOI
TL;DR: In this paper, a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels are given.
Abstract: In the study of the holomorphic automorphism groups, many researches have been carried out inside the category of bounded or hyperbolic domains. On the contrary to these cases, for unbounded non-hyperbolic cases, only a few results are known about the structure of the holomorphic automorphism groups. Main result of the present paper gives a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity theorem for finite volume Reinhardt domains is also given.

Journal ArticleDOI
TL;DR: The derivatives of the Bott class and those of the Godbillon-Vey class with respect to infinitesimal deformations of foliations are known to be represented by a formula in the projective Schwarzian derivatives of holonomies.
Abstract: The derivatives of the Bott class and those of the Godbillon-Vey class with respect to infinitesimal deformations of foliations, called infinitesimal derivatives, are known to be represented by a formula in the projective Schwarzian derivatives of holonomies [3], [1]. It is recently shown that these infinitesimal derivatives are represented by means of coefficients of transverse Thomas-Whitehead projective connections [2]. We will show that the formula can be also deduced from the latter representation.

Journal ArticleDOI
TL;DR: The 2-parameter family of homogeneous Lorentzian 3-manifolds with Minkowski motion groups is studied in this paper. But the authors focus on the 2-dimensional case.
Abstract: The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula which is the unification of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gaus map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.

Journal ArticleDOI
TL;DR: In this paper, the maximal ideal cycles and fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities, and it is shown that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.
Abstract: The maximal ideal cycles and the fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities. The former (resp. later) is determined by the analytic (resp. topological) structure of the singularities. We study such cycles for normal surface singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced homogeneous function of the minimal degree, we prove that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.

Journal ArticleDOI
TL;DR: In this paper, the expected number of components for a random link, and further, the most expected partition of the number of strings in a random braid for a given braid group was determined.
Abstract: We consider a random link, which is defined as the closure of a braid obtained from a random walk on the braid group. For such a random link, the expected value for the number of components was calculated by Jiming Ma. In this paper, we determine the most expected number of components for a random link, and further, consider the most expected partition of the number of strings for a random braid.

Journal ArticleDOI
TL;DR: In this article, the authors studied the monodromy representation of the generalized hypergeometric differential equation and that of Lauricella's $F_C$ system of hypergeometrical differential equations.
Abstract: We study the monodromy representation of the generalized hypergeometric differential equation and that of Lauricella's $F_C$ system of hypergeometric differential equations. We use fundamental systems of solutions expressed by the hypergeometric series. We express non-diagonal circuit matrices as reflections with respect to root vectors with all entries 1. We present a simple way to obtain circuit matrices.

Journal ArticleDOI
TL;DR: In this article, a smooth projective curve of genus G$ can be reconstructed from its polarized Jacobian as a certain locus in the Hilbert scheme, defined by geometric conditions in terms of the polarization of the Jacobian.
Abstract: We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $\Theta$. The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors.