{"title":"Characteristic cycles of highest weight Harish-Chandra modules and the Weyl group action on the conormal variety","authors":"L. Barchini","doi":"10.2748/TMJ/1561082595","DOIUrl":"https://doi.org/10.2748/TMJ/1561082595","url":null,"abstract":"","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2748/TMJ/1561082595","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48758655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show explicitly that the compact flat Kähler manifold of complex dimension three with D8 holonomy studied by Dekimpe, Halenda and Szczepanski ([5] p. 367) possesses the structure of a nonsingular projective variety. This corrects a previous statement by H. Lange in [9] that the holonomy group of a hyperelliptic threefold is necessarily abelian. The study of flat Riemannian manifolds, begun by Bieberbach [2], has subsequently acquired a very extensive literature. See, for example, [3],[4],[13],[14]. In a paper published in the Tohoku Mathematical Journal [9], H. Lange investigated closed flat manifolds of real dimension six which, in addition, possess the structure of nonsingular complex projective varieties which have finite étale coverings by abelian varieties. In Lange’s terminology such varieties are called hyperelliptic three-folds. The significant claim of Lange’s paper is that the (finite) holonomy group of such a hyperelliptic three-fold is necessarily abelian. In particular, Lange claims that the dihedral group of order eight† does not occur as a holonomy group in this context. Lange’s claim is mistaken, however. In the present paper we show explicitly that the compact flat Kähler manifold of complex dimension three with D8 holonomy studied by Dekimpe, Halenda and Szczepanski ([5] p. 367) does indeed possess the structure of a nonsingular projective variety. In fact, the existence of this complex algebraic structure was previously shown, in a very general context, by the present author in the paper [7]. However, as Lange also makes a statement which explicitly claims to contradict the main result of [7] it seems appropriate, in setting the matter straight, to give a direct, and elementary, construction of the algebraic structure whose existence Lange denies. The present paper is organised as follows; in §1 we give a brief review of the theory of flat Riemannian manifolds as it pertains both to Kähler manifolds and projective varieties; in §2 we give a completely elementary criterion which guarantees that some flat Riemannian manifolds admit the structure of a nonsingular complex algebraic variety. Whilst this criterion does not immediately apply to the most general cases, it is quite sufficient to deal with all cases in which the holonomy group is D8. In §3 we construct an explicit complex algebraic structure for the Kähler manifold of Dekimpe, Halenda and Szczepanski. This can be checked by direct calculation and requires very little theory beyond an appeal to the criterion of §2. 2010 MSC Primary 53C29; Secondary 14F35, 14K02, 32J27.
我们明确地证明了Dekimpe, Halenda和Szczepanski ([5] p. 367)研究的具有D8完整度的复维三维紧平Kähler流形具有非奇异投影变的结构。这就纠正了H. Lange在1996年发表的关于超椭圆三倍体的完整群必然是阿贝尔的说法。平坦黎曼流形的研究，由比伯巴赫开始，随后获得了非常广泛的文献。例如，[3]，[4]，[13]，[14]。H. Lange在《Tohoku Mathematical Journal[9]》上发表的一篇论文中，研究了具有非奇异复射影变的实维六平面流形，该流形具有有限的阿贝尔变复复覆盖。在兰格的术语中，这种变体被称为超椭圆三折。Lange论文的一个重要论断是，这种超椭圆三倍的(有限)完整群必然是阿贝尔的。特别地，Lange声称八阶的二面体群在这种情况下不会作为一个完整群出现。然而，兰格的说法是错误的。在本文中，我们明确地证明了Dekimpe, Halenda和Szczepanski ([5] p. 367)研究的具有D8完整度的复维三维紧平Kähler流形确实具有非奇异投影变异体的结构。事实上，这个复杂的代数结构的存在性，在一个非常一般的背景下，由本作者先前在论文[7]中证明。然而，由于兰格也作了一个声明，明确地声称与[7]的主要结果相矛盾，因此，为了澄清问题，对兰格否认其存在的代数结构给出一个直接的、基本的构造，似乎是合适的。本论文组织如下:在§1中，我们简要回顾了平坦黎曼流形的理论，因为它既涉及Kähler流形也涉及射影变体;在§2中，我们给出了一个完全初等判据，以保证某些平坦黎曼流形承认一个非奇异复代数变的结构。虽然这个标准不能立即适用于最一般的情况，但它足以处理所有完整群为D8的情况。在§3中，我们构造了Dekimpe, Halenda和Szczepanski的Kähler流形的显式复代数结构。这一点可以用直接的计算来证明，而且只要借助于§2的判断就可以了。2010 MSC Primary 53C29;二级14F35、14K02、32J27。
{"title":"A flat projective variety with $D_8$-holonomy","authors":"F. Johnson","doi":"10.2748/TMJ/1561082601","DOIUrl":"https://doi.org/10.2748/TMJ/1561082601","url":null,"abstract":"We show explicitly that the compact flat Kähler manifold of complex dimension three with D8 holonomy studied by Dekimpe, Halenda and Szczepanski ([5] p. 367) possesses the structure of a nonsingular projective variety. This corrects a previous statement by H. Lange in [9] that the holonomy group of a hyperelliptic threefold is necessarily abelian. The study of flat Riemannian manifolds, begun by Bieberbach [2], has subsequently acquired a very extensive literature. See, for example, [3],[4],[13],[14]. In a paper published in the Tohoku Mathematical Journal [9], H. Lange investigated closed flat manifolds of real dimension six which, in addition, possess the structure of nonsingular complex projective varieties which have finite étale coverings by abelian varieties. In Lange’s terminology such varieties are called hyperelliptic three-folds. The significant claim of Lange’s paper is that the (finite) holonomy group of such a hyperelliptic three-fold is necessarily abelian. In particular, Lange claims that the dihedral group of order eight† does not occur as a holonomy group in this context. Lange’s claim is mistaken, however. In the present paper we show explicitly that the compact flat Kähler manifold of complex dimension three with D8 holonomy studied by Dekimpe, Halenda and Szczepanski ([5] p. 367) does indeed possess the structure of a nonsingular projective variety. In fact, the existence of this complex algebraic structure was previously shown, in a very general context, by the present author in the paper [7]. However, as Lange also makes a statement which explicitly claims to contradict the main result of [7] it seems appropriate, in setting the matter straight, to give a direct, and elementary, construction of the algebraic structure whose existence Lange denies. The present paper is organised as follows; in §1 we give a brief review of the theory of flat Riemannian manifolds as it pertains both to Kähler manifolds and projective varieties; in §2 we give a completely elementary criterion which guarantees that some flat Riemannian manifolds admit the structure of a nonsingular complex algebraic variety. Whilst this criterion does not immediately apply to the most general cases, it is quite sufficient to deal with all cases in which the holonomy group is D8. In §3 we construct an explicit complex algebraic structure for the Kähler manifold of Dekimpe, Halenda and Szczepanski. This can be checked by direct calculation and requires very little theory beyond an appeal to the criterion of §2. 2010 MSC Primary 53C29; Secondary 14F35, 14K02, 32J27.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46252553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
{"title":"The Leibniz rule for the Dirichlet and the Neumann Laplacian","authors":"T. Iwabuchi","doi":"10.2748/tmj.20211112","DOIUrl":"https://doi.org/10.2748/tmj.20211112","url":null,"abstract":"We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46922181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Emery Ricci tensor. Moreover, we improve a result from Soylu by using a weaker condition on a derivative $f'(t)$.
{"title":"Myers-type compactness theorem with the Bakry-Emery Ricci tensor","authors":"Seungsu Hwang, Sanghun Lee","doi":"10.2748/tmj.20200512","DOIUrl":"https://doi.org/10.2748/tmj.20200512","url":null,"abstract":"In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Emery Ricci tensor. Moreover, we improve a result from Soylu by using a weaker condition on a derivative $f'(t)$.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47372268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.
{"title":"Invariants of algebraic varieties over imperfect fields","authors":"Hiromu Tanaka","doi":"10.2748/tmj.20200611","DOIUrl":"https://doi.org/10.2748/tmj.20200611","url":null,"abstract":"We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43904808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We exhibit a particular free subarrangement of a certain restriction of the Weyl arrangement of type $E_7$ and use it to give an affirmative answer to a recent conjecture by T.~Abe on the nature of additionally free and stair-free arrangements.
{"title":"Some remarks on free arrangements","authors":"Torsten Hoge, G. Röhrle","doi":"10.2748/tmj.20200318","DOIUrl":"https://doi.org/10.2748/tmj.20200318","url":null,"abstract":"We exhibit a particular free subarrangement of a certain restriction of the Weyl arrangement of type $E_7$ and use it to give an affirmative answer to a recent conjecture by T.~Abe on the nature of additionally free and stair-free arrangements.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42827814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Obstacle problem for Musielak-Orlicz Dirichlet energy integral on metric measure spaces","authors":"F. Maeda, T. Ohno, T. Shimomura","doi":"10.2748/TMJ/1552100442","DOIUrl":"https://doi.org/10.2748/TMJ/1552100442","url":null,"abstract":"","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2748/TMJ/1552100442","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45936361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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