A bstract . We determine the mod 2 cohomology over the Steenrod algebra A 2 of the classifying spaces of the free loop groups LG for compact groups G = Spin (7), Spin (8), Spin (9), and F 4 . Then, we show that they are isomorphic as algebras over A 2 to the mod 2 cohomology of the corresponding Chevalley groups of type G ( q ), where q is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over BDI (4) and show that it is isomorphic to that of BSol ( q ) as algebras over A 2 . This note is a revised version of [8].
{"title":"Mod 2 cohomology of 2-compact groups of low rank","authors":"S. Kaji","doi":"10.1215/kjm/1250281055","DOIUrl":"https://doi.org/10.1215/kjm/1250281055","url":null,"abstract":"A bstract . We determine the mod 2 cohomology over the Steenrod algebra A 2 of the classifying spaces of the free loop groups LG for compact groups G = Spin (7), Spin (8), Spin (9), and F 4 . Then, we show that they are isomorphic as algebras over A 2 to the mod 2 cohomology of the corresponding Chevalley groups of type G ( q ), where q is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over BDI (4) and show that it is isomorphic to that of BSol ( q ) as algebras over A 2 . This note is a revised version of [8].","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"47 1","pages":"441-450"},"PeriodicalIF":0.0,"publicationDate":"2007-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66088659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.
{"title":"A Babylonian tower theorem for principal bundles over projective spaces","authors":"I. Biswas, I. Coandă, G. Trautmann","doi":"10.1215/KJM/1248983030","DOIUrl":"https://doi.org/10.1215/KJM/1248983030","url":null,"abstract":"We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"69-82"},"PeriodicalIF":0.0,"publicationDate":"2007-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66087207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for any such lifting, the endormorphism algebra of the transcendental cycles, as a Hodge module, is a CM field. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type.
{"title":"K3 surfaces of finite height over finite fields","authors":"J.-D. Yu, N. Yui","doi":"10.1215/KJM/1250271381","DOIUrl":"https://doi.org/10.1215/KJM/1250271381","url":null,"abstract":"Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for any such lifting, the endormorphism algebra of the transcendental cycles, as a Hodge module, is a CM field. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"48 1","pages":"499-519"},"PeriodicalIF":0.0,"publicationDate":"2007-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66088073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.
{"title":"A sequence of blowing-ups connecting moduli of sheaves and the Donaldson Polynomial under change of polarization","authors":"Kimiko Yamada","doi":"10.1215/KJM/1250281738","DOIUrl":"https://doi.org/10.1215/KJM/1250281738","url":null,"abstract":"Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"43 1","pages":"829-878"},"PeriodicalIF":0.0,"publicationDate":"2007-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66089348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or in- trinsically, free from local coordinates) many of the most important and most com- monly used special Finsler manifolds: locally Minkowskian, Berwald, Landesberg, general Landesberg, P-reducible, C-reducible, semi-C-reducible, quasi-C-reducible, P ∗ -Finsler, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, S v -recurrent of the second order, C2-like, S3-like, S4-like, P2-like, R3-like, P-symmetric, h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-curvature. The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced. Although our investigation is entirely global, we provide; for comparison rea- sons, an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered. 1
本文的目的是提供特殊的芬斯勒流形理论的一个全局表示。我们从全局(或者从本质上讲，不依赖于局部坐标)引入和研究了许多最重要和最常用的特殊Finsler流形:局部minkowski, Berwald, Landesberg，一般Landesberg, P-可约，C-可约，半C-可约，拟C-可约，P * -Finsler, C h-递归，C v -递归，C 0 -递归，S v -递归，S v -二阶递归，c2类，s3类，s4类，p2类，r3类，P对称，h各向同性，标量曲率，常数曲率，P-标量曲率，S -ps曲率。给出了这些特殊的Finsler流形的全局定义。发现了不同类型的特殊芬斯勒流形之间的各种关系。许多文献中已知的局部结果得到了全局证明，并获得了一些新的结果。作为一个副产品，我们推导出了关于扭转张量场和曲率张量场的有趣的恒等式和性质。虽然我们的调查完全是全球性的，但我们提供;为了便于比较，在附录中给出了我们的全局方法的局部对应物和所考虑的特殊Finsler空间的局部定义。1
{"title":"A global approach to the theory of special finsler manifolds","authors":"N. L. Youssef, S. H. Abed, A. Soleiman","doi":"10.1215/KJM/1250271321","DOIUrl":"https://doi.org/10.1215/KJM/1250271321","url":null,"abstract":"The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or in- trinsically, free from local coordinates) many of the most important and most com- monly used special Finsler manifolds: locally Minkowskian, Berwald, Landesberg, general Landesberg, P-reducible, C-reducible, semi-C-reducible, quasi-C-reducible, P ∗ -Finsler, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, S v -recurrent of the second order, C2-like, S3-like, S4-like, P2-like, R3-like, P-symmetric, h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-curvature. The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced. Although our investigation is entirely global, we provide; for comparison rea- sons, an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered. 1","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"5 1","pages":"857-893"},"PeriodicalIF":0.0,"publicationDate":"2007-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66087897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce an algorithm which transforms a finitely presented group G into another one G Ψ . By using this, we can get many finitely presented groups whose group homology with coefficients in the group von Neumann algebra vanish, that is, many counterexamples to an algebraic version of the zero-in-the-spectrum conjecture. Moreover we prove that the Baum-Connes conjecture does not imply the algebraic version of the zero-in-the-spectrum conjecture for finitely presented groups. Also we will show that for any p ≥ 3 the p -th group homology of G Ψ coming from free groups has infinite rank.
{"title":"The group homology and an algebraic version of the zero-in-the-spectrum conjecture","authors":"Shin-ichi Oguni","doi":"10.1215/KJM/1250281050","DOIUrl":"https://doi.org/10.1215/KJM/1250281050","url":null,"abstract":"We introduce an algorithm which transforms a finitely presented group G into another one G Ψ . By using this, we can get many finitely presented groups whose group homology with coefficients in the group von Neumann algebra vanish, that is, many counterexamples to an algebraic version of the zero-in-the-spectrum conjecture. Moreover we prove that the Baum-Connes conjecture does not imply the algebraic version of the zero-in-the-spectrum conjecture for finitely presented groups. Also we will show that for any p ≥ 3 the p -th group homology of G Ψ coming from free groups has infinite rank.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"47 1","pages":"359-369"},"PeriodicalIF":0.0,"publicationDate":"2007-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66089067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fujita’s approximation theorem in positive charactristics","authors":"S. Takagi","doi":"10.1215/KJM/1250281075","DOIUrl":"https://doi.org/10.1215/KJM/1250281075","url":null,"abstract":"","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"47 1","pages":"179-202"},"PeriodicalIF":0.0,"publicationDate":"2007-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1250281075","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66088889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For each positive integer k, we describe a map f from the complex plane to a suitable non-complete complex locally convex space such that f is k times continuously complex differentiable but not k+1 times, and hence not complex analytic. We also describe a complex analytic map from l^1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l^1. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduct, we find that free locally convex spaces over subsets of the complex plane with non-empty interior are not Mackey complete.
{"title":"Instructive examples of smooth complex differentiable and complex analytic mappings into locally convex spaces","authors":"Helge Glöckner","doi":"10.1215/KJM/1250281028","DOIUrl":"https://doi.org/10.1215/KJM/1250281028","url":null,"abstract":"For each positive integer k, we describe a map f from the complex plane to a suitable non-complete complex locally convex space such that f is k times continuously complex differentiable but not k+1 times, and hence not complex analytic. We also describe a complex analytic map from l^1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l^1. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduct, we find that free locally convex spaces over subsets of the complex plane with non-empty interior are not Mackey complete.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"47 1","pages":"631-642"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66088528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the exact controllability of a Timoshenko beam system, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by two partial differential equations and two ordinary differential equations. Using the HUM method, we show that the system is exactly controllable in the usual energy space.
{"title":"Exact controllability of a Timoshenko beam with dynamical boundary","authors":"Chun-guo Zhang, Hongrui Hu","doi":"10.1215/KJM/1250281029","DOIUrl":"https://doi.org/10.1215/KJM/1250281029","url":null,"abstract":"We consider the exact controllability of a Timoshenko beam system, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by two partial differential equations and two ordinary differential equations. Using the HUM method, we show that the system is exactly controllable in the usual energy space.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"47 1","pages":"643-655"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1250281029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66088539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an effective divisor A with support B in a compact K¨ahler manifold M of dimension ≥ 3, the following are antinomic. a) M B has a C ∞ plurisubharmonic exhaustion function whose Levi form has pointwise at least 3 positive eigenvalues outside a compact subset of M B . b) [ A ] | B , the normal bundle of A , is topologically trivial.
对于维数≥3的紧化K¨ahler流形M中具有支撑点B的有效因子A，下列是反律的。a) M B具有一个C∞多次调和耗尽函数，其Levi形式在M B的紧子集外点方向上至少有3个正特征值。b) [A] | A的正规束b是拓扑平凡的。
{"title":"A remark on pseudoconvex domains with analytic complements in compact Kähler manifolds","authors":"T. Ohsawa","doi":"10.1215/KJM/1250281070","DOIUrl":"https://doi.org/10.1215/KJM/1250281070","url":null,"abstract":"For an effective divisor A with support B in a compact K¨ahler manifold M of dimension ≥ 3, the following are antinomic. a) M B has a C ∞ plurisubharmonic exhaustion function whose Levi form has pointwise at least 3 positive eigenvalues outside a compact subset of M B . b) [ A ] | B , the normal bundle of A , is topologically trivial.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"47 1","pages":"115-119"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1250281070","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66088799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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