Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
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Showing 1-12 articles out of 12 articles from the selected issue
  • Joe Kamimoto
    2021 Volume 73 Issue 1 Pages 1-39
    Published: 2021
    Released: January 22, 2021
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    The purpose of this paper is to investigate order of contact on real hypersurfaces in ℂ𝑛 by using Newton polyhedra which are important notion in the study of singularity theory. To be more precise, an equivalence condition for the equality of regular type and singular type is given by using the Newton polyhedron of a defining function for the respective hypersurface. Furthermore, a sufficient condition for this condition, which is more useful, is also given. This sufficient condition is satisfied by many earlier known cases (convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4, etc.). Under the above conditions, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

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  • Marco Bonatto, David Stanovský
    2021 Volume 73 Issue 1 Pages 41-75
    Published: 2021
    Released: January 22, 2021
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    We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties, such as abelianness and centrality, are reflected by the corresponding relative displacement groups, and the global properties, solvability and nilpotence, are reflected by the properties of the whole displacement group. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order 2𝑘, no latin quandles of order ≡ 2 (mod 4)), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by solvable quandles; in particular, by finite latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

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  • Michael Reiter, Duong Ngoc Son
    2021 Volume 73 Issue 1 Pages 77-98
    Published: 2021
    Released: January 22, 2021
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    We derive an explicit formula for the well-known Chern–Moser–Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to “pluriharmonic perturbations” of the sphere or to a Fefferman approximate solution to the complex Monge–Ampère equation. As an application, we show that the CR invariant one-form 𝑋𝛼 constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in ℂ3, unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the ℐ'-pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.

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  • Toshio Mikami
    2021 Volume 73 Issue 1 Pages 99-123
    Published: 2021
    Released: January 22, 2021
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    We study the continuity and the measurability of the solution to Schrödinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This is a continuation of our previous result where the space of all Borel probability measures was endowed with the strong topology. As an application, we construct a convex function of which the moment measure is a given probability measure, by the zero noise limit of a class of stochastic optimal transportation problems.

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  • Hiro-aki Narita
    2021 Volume 73 Issue 1 Pages 125-159
    Published: 2021
    Released: January 22, 2021
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    This paper is a continuation of the first paper. The aim of this second paper is to discuss the non-vanishing of the theta lifts to the indefinite symplectic group 𝐺𝑆𝑝(1,1), which have been shown to be involved in the Jacquet–Langlands–Shimizu correspondence with some theta lifts to the ℚ-split symplectic group 𝐺𝑆𝑝(2) of degree two. We study an explicit formula for the square norms of the Bessel periods of the theta lifts to 𝐺𝑆𝑝(1,1) in terms of central 𝐿-values. This study involves two aspects in proving the non-vanishing of the theta lifts. One aspect is to apply the results by Hsieh and Chida–Hsieh on “non-vanishing modulo 𝑝” of central 𝐿-values for some Rankin 𝐿-functions. The other is to relate such non-vanishing with studies on some special values of hypergeometric functions. We also take up the theta lifts to the compact inner form 𝐺𝑆𝑝*(2). We provide examples of the non-vanishing theta lifts to 𝐺𝑆𝑝*(2), which are essentially due to Ibukiyama and Ihara.

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  • Zifei Shen, Bin Shang, Chenyin Qian
    2021 Volume 73 Issue 1 Pages 161-183
    Published: 2021
    Released: January 22, 2021
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    The 𝑝(𝑥)-Laplacian Kirchhoff type equation involving the nonlocal term 𝑏 ∫𝑁 (1/𝑝(𝑥)) |∇𝑢|𝑝(𝑥)𝑑𝑥 is investigated. Based on the variational methods, deformation lemma and other technique of analysis, it is proved that the problem possesses one least energy sign-changing solution 𝑢𝑏 which has precisely two nodal domains. Moreover, the convergence property of 𝑢𝑏 as the parameter 𝑏 ↘ 0 is also obtained.

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  • Tomoyuki Arakawa, Hiromichi Yamada, Hiroshi Yamauchi
    2021 Volume 73 Issue 1 Pages 185-209
    Published: 2021
    Released: January 22, 2021
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    We introduce a simple, self-dual, rational, and 𝐶2-cofinite vertex operator algebra of CFT-type associated with a ℤ𝑘-code for 𝑘 ≥ 2. Our argument is based on the ℤ𝑘-symmetry among the simple current modules for the parafermion vertex operator algebra 𝐾(𝔰𝔩2, 𝑘). We show that it is naturally realized as the commutant of a certain subalgebra in a lattice vertex operator algebra. Furthermore, we construct all the irreducible modules inside a module for the lattice vertex operator algebra.

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  • Michał Farnik, Zbigniew Jelonek, Maria Aparecida Soares Ruas
    2021 Volume 73 Issue 1 Pages 211-220
    Published: 2021
    Released: January 22, 2021
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    Denote by 𝐻(𝑑1, 𝑑2, 𝑑3) the set of all homogeneous polynomial mappings 𝐹 = (𝑓1, 𝑓2, 𝑓3) : ℂ3 → ℂ3, such that deg 𝑓𝑖 = 𝑑𝑖. We show that if gcd(𝑑𝑖, 𝑑𝑗) ≤ 2 for 1 ≤ 𝑖 < 𝑗 ≤ 3 and gcd(𝑑1, 𝑑2, 𝑑3) = 1, then there is a non-empty Zariski open subset 𝑈 ⊂ 𝐻(𝑑1, 𝑑2, 𝑑3) such that for every mapping 𝐹 ∈ 𝑈 the map germ (𝐹, 0) is 𝒜-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping (𝑓1, 𝑓2, 𝑓3): ℂ3 → ℂ3, where deg 𝑓𝑖 = 𝑑𝑖.

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  • Matthew Westaway
    2021 Volume 73 Issue 1 Pages 221-261
    Published: 2021
    Released: January 22, 2021
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    In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.

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  • Yuhei Suzuki
    2021 Volume 73 Issue 1 Pages 263-275
    Published: 2021
    Released: January 22, 2021
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    We extend a theorem of Haagerup and Kraus in the C*-algebra context: for a locally compact group with the approximation property (AP), the reduced C*-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group C*-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.

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  • Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja
    2021 Volume 73 Issue 1 Pages 277-299
    Published: 2021
    Released: January 22, 2021
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    Let 𝑋 ⊂ ℝ𝑛 be a compact semialgebraic set and let 𝑓 : 𝑋 → ℝ be a nonzero Nash function. We give a Solernó and D'Acunto–Kurdyka type estimation of the exponent ϱ ∈ [0,1) in the Łojasiewicz gradient inequality |∇𝑓(𝑥)| ≥ 𝐶|𝑓(𝑥)|ϱ for 𝑥 ∈ 𝑋, |𝑓(𝑥)| < 𝜀 for some constants 𝐶,𝜀 > 0, in terms of the degree of a polynomial 𝑃 such that 𝑃(𝑥, 𝑓(𝑥)) = 0, 𝑥 ∈ 𝑋. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities.

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  • Shigeyuki Kondō
    2021 Volume 73 Issue 1 Pages 301-328
    Published: 2021
    Released: January 22, 2021
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    We classify Enriques surfaces covered by the supersingular 𝐾3 surface with the Artin invariant 1 in characteristic 2. There are exactly three types of such Enriques surfaces.

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