Tohoku Mathematical Journal, Second Series
Online ISSN : 2186-585X
Print ISSN : 0040-8735
ISSN-L : 0040-8735
Current issue
Displaying 1-8 of 8 articles from this issue
  • Satoru Fukasawa, Kei Miura, Takeshi Takahashi
    2019 Volume 71 Issue 4 Pages 487-494
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    We investigate the automorphism group of a plane curve, introducing the notion of a quasi-Galois point. We show that the automorphism group of several curves, for example, Klein quartic, Wiman sextic and Fermat curves, is generated by the groups associated with quasi-Galois points.

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  • Naoto Yotsutani, Bin Zhou
    2019 Volume 71 Issue 4 Pages 495-524
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    In this paper we study the relative Chow and $K$-stability of toric manifolds. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [34], which fits into the relative GIT stability detected by Székelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [13, 36]. In the end, we determine the relative $K$-stability of all toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable.

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  • Satoshi Nakamura
    2019 Volume 71 Issue 4 Pages 525-532
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.

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  • Kazuyoshi Watanabe
    2019 Volume 71 Issue 4 Pages 533-547
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    In this paper, we introduce a new definition of the Ricci curvature on cell-complexes and prove the Gauss-Bonnnet type theorem for graphs and 2-complexes that decompose closed surfaces. The differential forms on a cell complex are defined as linear maps on the chain complex, and the Laplacian operates this differential forms. Our Ricci curvature is defined by the combinatorial Bochner-Weitzenböck formula. We prove some propositionerties of combinatorial vector fields on a cell complex.

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  • Daniel Litt
    2019 Volume 71 Issue 4 Pages 549-557
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we improve the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.

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  • Akio Kodama
    2019 Volume 71 Issue 4 Pages 559-580
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    Let $M$ be a connected Stein manifold of dimension $N$ and let $D$ be a Fock-Bargmann-Hartogs domain in $\mathbb{C}^N$. Let Aut$(M)$ and Aut$(D)$ denote the groups of all biholomorphic automorphisms of $M$ and $D$, respectively, equipped with the compact-open topology. Note that Aut$(M)$ cannot have the structure of a Lie group, in general; while it is known that Aut$(D)$ has the structure of a connected Lie group. In this paper, we show that if the identity component of Aut$(M)$ is isomorphic to Aut$(D)$ as topological groups, then $M$ is biholomorphically equivalent to $D$. As a consequence of this, we obtain a fundamental result on the topological group structure of Aut$(D)$.

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  • Yuxin Dong, Hezi Lin, Shihshu Walter Wei
    2019 Volume 71 Issue 4 Pages 581-607
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar curvature, under curvature pinching conditions. Secondly, we prove vanishing results for $L^2$ and some non-$L^2$ harmonic $p$-forms on LCF manifolds, by assuming that the underlying manifolds satisfy pointwise or integral curvature conditions. Moreover, by a theorem of Li-Tam for harmonic functions, we show that the underlying manifold must have only one end. Finally, we obtain Liouville theorems for $p$-harmonic functions on LCF manifolds under pointwise Ricci curvature conditions.

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  • Boo Rim Choe, Hyungwoon Koo, Jongho Yang
    2019 Volume 71 Issue 4 Pages 609-637
    Published: December 30, 2019
    Released on J-STAGE: February 07, 2025
    JOURNAL FREE ACCESS

    Choe et al. have recently characterized compact double differences formed by four composition operators acting on the standard weighted Bergman spaces over the disk of the complex plane. In this paper, we extend such a result to the ball setting. Our characterization is obtained under a suitable restriction on inducing maps, which is automatically satisfied in the case of the disk. We exhibit concrete examples, for the first time even for single composition operators, which shows that such a restriction is essential in the case of the ball.

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