{"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}
{"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}
{"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}
{"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}
{"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}
{"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}