(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.
{"title":"Rigid fibers of integrable systems on cotangent bundles","authors":"Morimichi Kawasaki, Ryuma Orita","doi":"10.2969/jmsj/84278427","DOIUrl":"https://doi.org/10.2969/jmsj/84278427","url":null,"abstract":"(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47935909","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}
A handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere $S^3$. A multiple conjugation quandle is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. In this paper, we introduce the notion of a presentation of a multiple conjugation quandle and define the fundamental multiple conjugation quandle of a handlebody-link. We also see that the fundamental multiple conjugation quandle is an invariant of handlebody-links.
{"title":"The fundamental multiple conjugation quandle of a handlebody-link","authors":"Atsushi Ishii","doi":"10.2969/jmsj/84308430","DOIUrl":"https://doi.org/10.2969/jmsj/84308430","url":null,"abstract":"A handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere $S^3$. A multiple conjugation quandle is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. In this paper, we introduce the notion of a presentation of a multiple conjugation quandle and define the fundamental multiple conjugation quandle of a handlebody-link. We also see that the fundamental multiple conjugation quandle is an invariant of handlebody-links.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47459818","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}
The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched m-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring (A,m). As an application, we present complete descriptions of the associated graded ring of stretched m-primary ideals with small reduction number.
{"title":"The reduction number of stretched ideals","authors":"K. Ozeki","doi":"10.2969/jmsj/86498649","DOIUrl":"https://doi.org/10.2969/jmsj/86498649","url":null,"abstract":"The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched m-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring (A,m). As an application, we present complete descriptions of the associated graded ring of stretched m-primary ideals with small reduction number.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48792082","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}
Hiraku Atobe, Masataka Chida, T. Ibukiyama, H. Katsurada, Takuya Yamauchi
Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.
{"title":"Harder's conjecture I","authors":"Hiraku Atobe, Masataka Chida, T. Ibukiyama, H. Katsurada, Takuya Yamauchi","doi":"10.2969/jmsj/87988798","DOIUrl":"https://doi.org/10.2969/jmsj/87988798","url":null,"abstract":"Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44438657","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}
Line congruences are 2-dimensional families of lines in 3space. The singularities that appear in generic line congruences are folds, cusps and swallowtails ([7]). In this paper we give a geometric description of these singularities. The main tool used is the existence of an equiaffine pair defining a generic line congruence. Mathematics Subject Classification (2010). 53A55, 57R45, 53A20.
{"title":"Singularities of generic line congruences","authors":"M. Craizer, Ronaldo Garcia","doi":"10.2969/jmsj/88348834","DOIUrl":"https://doi.org/10.2969/jmsj/88348834","url":null,"abstract":"Line congruences are 2-dimensional families of lines in 3space. The singularities that appear in generic line congruences are folds, cusps and swallowtails ([7]). In this paper we give a geometric description of these singularities. The main tool used is the existence of an equiaffine pair defining a generic line congruence. Mathematics Subject Classification (2010). 53A55, 57R45, 53A20.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44894972","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 Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such surfaces.
{"title":"Coble surfaces in characteristic two","authors":"T. Katsura, S. Kondō","doi":"10.2969/jmsj/87568756","DOIUrl":"https://doi.org/10.2969/jmsj/87568756","url":null,"abstract":"We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such surfaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46834662","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}
The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads f K ¥ ah and f re are constructed by using the quantum Hilbert spaces ℋ K ¥ ah and ℋ re , respectively. Moreover, the relationship between the two morphisms of operads f K ¥ ah and f re is studied and then the equality dim ℋ K ¥ ah = dim ℋ re is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama[6] for proving dim ℋ K ¥ ah = dim ℋ re in a special case.
{"title":"Operad structures in geometric quantization of the moduli space of spatial polygons","authors":"Yuya Takahashi","doi":"10.2969/jmsj/88548854","DOIUrl":"https://doi.org/10.2969/jmsj/88548854","url":null,"abstract":"The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads f K ¥ ah and f re are constructed by using the quantum Hilbert spaces ℋ K ¥ ah and ℋ re , respectively. Moreover, the relationship between the two morphisms of operads f K ¥ ah and f re is studied and then the equality dim ℋ K ¥ ah = dim ℋ re is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama[6] for proving dim ℋ K ¥ ah = dim ℋ re in a special case.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43946075","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}
. The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid–Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambro’s paper hold true. The proof is Kawamata’s X-method with the aid of the theory of basic slc-trivial fibrations. For the reader’s convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.
{"title":"On quasi-log schemes","authors":"O. Fujino","doi":"10.2969/jmsj/87348734","DOIUrl":"https://doi.org/10.2969/jmsj/87348734","url":null,"abstract":". The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid–Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambro’s paper hold true. The proof is Kawamata’s X-method with the aid of the theory of basic slc-trivial fibrations. For the reader’s convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43746865","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 prove that the energy density of uniformly continuous, quasiconformal mappings, omitting two points in CP, is equal to zero. We also prove the sharpness of this result, constructing a family of uniformly continuous, quasiconformal mappings, whose areas grow asymptotically quadratically. Finally, we prove that the energy density of pseudoholomorphic Brody curves, omitting three “complex lines” in general position in CP, is equal to zero. According to the Picard Theorem, a holomorphic function f defined on the complex plane C is constant as soon as f(C) omits at least three values in CP1. This result has different generalizations in at least two directions. S.Rickman [11] proved that for every n ≥ 2 and for every K > 1, a nonconstant entire Kquasiregular mapping in Rn omits at most m = m(n,K) values. M.Green [6] proved that a holomorphic map from C to the complex projective space CPn, omitting (2n+ 1) hyperplanes in general position, is constant. An almost complex version of that result was proved by J.Duval [5] for entire pseudoholomorphic curves in the complement of five J-lines, in general position in CP2 endowed with an almost complex structure J tamed by the Fubini Study metric ωFS . Let f be a mapping defined on C with values in CPn, f ∈ W 1,2 loc (C). We recall that if D ⊂⊂ C, then Area(f(D)) := ∫ D f ωFS is the area of f(D), counted with multiplicity. Then, the energy density E(f) defined by E(f) = lim sup R→∞ 1 πR2 Area(f(DR)) = lim sup R→∞ 1 πR2 ∫
{"title":"On the energy of quasiconformal mappings and pseudoholomorphic curves in complex projective spaces","authors":"H. Gaussier, M. Tsukamoto","doi":"10.2969/JMSJ/81238123","DOIUrl":"https://doi.org/10.2969/JMSJ/81238123","url":null,"abstract":"We prove that the energy density of uniformly continuous, quasiconformal mappings, omitting two points in CP, is equal to zero. We also prove the sharpness of this result, constructing a family of uniformly continuous, quasiconformal mappings, whose areas grow asymptotically quadratically. Finally, we prove that the energy density of pseudoholomorphic Brody curves, omitting three “complex lines” in general position in CP, is equal to zero. According to the Picard Theorem, a holomorphic function f defined on the complex plane C is constant as soon as f(C) omits at least three values in CP1. This result has different generalizations in at least two directions. S.Rickman [11] proved that for every n ≥ 2 and for every K > 1, a nonconstant entire Kquasiregular mapping in Rn omits at most m = m(n,K) values. M.Green [6] proved that a holomorphic map from C to the complex projective space CPn, omitting (2n+ 1) hyperplanes in general position, is constant. An almost complex version of that result was proved by J.Duval [5] for entire pseudoholomorphic curves in the complement of five J-lines, in general position in CP2 endowed with an almost complex structure J tamed by the Fubini Study metric ωFS . Let f be a mapping defined on C with values in CPn, f ∈ W 1,2 loc (C). We recall that if D ⊂⊂ C, then Area(f(D)) := ∫ D f ωFS is the area of f(D), counted with multiplicity. Then, the energy density E(f) defined by E(f) = lim sup R→∞ 1 πR2 Area(f(DR)) = lim sup R→∞ 1 πR2 ∫","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43869131","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 study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other, we extend Newelski’s Theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context. The Lascar (automorphism) group of a first-order complete theory and its quotient groups such as the Kim-Pillay group and the Shelah group have been central themes in contemporary model theory. The study on those groups enables us to develop Galois theoretic correspondence between the groups and their orbit-equivalence relations on a monster model such as Lascar types, Kim-Pillay types, and Shelah strong types. The notions of the Lascar group and its topology are introduced first by D. Lascar in [9] using ultraproducts. Later more favorable equivalent definition is suggested in [7] and [11], which is nowadays considered as a standard approach. However even a complete proof using the approach of the fundamental fact that the Lascar group is a topological group is not so well available. For example in [2], its proof is left to the readers, while the proof is not at all trivial. As far as we can see, only in [14], a detailed proof is written. Aforementioned results are for the Lascar group over ∅, or more generally over a real set A. In this paper we study the Lascar group over a hyperimaginary e and verify how results on the Lascar group over A can be extended to the case when the set is replaced by e. Indeed this attempt was made in [8] (and rewritten in [6, §5.1]). However those contain some errors, and moreover a proof of that the Lascar group over e is a topological group is also missing. In this paper we supply a proof of the fact in a detailed expository manner. Our proof is more direct and even simplifies that for the group over ∅ in [14]. We correct the mentioned errors in [6],[8], as well. In particular we correct the proof of that the orbit equivalence relation under a closed normal subgroup of the Lascar group over e is type-definable over e. Moreover we extend Newelski’s Theorem in [12] to the hyperimaginary context. Namely we show that if T is G-compact over e then there is 2020 Mathematics Subject Classification. Primary 03C60; Secondary 54H11.
{"title":"Automorphism groups over a hyperimaginary","authors":"Byunghan Kim, Hyoyoon Lee","doi":"10.2969/jmsj/87138713","DOIUrl":"https://doi.org/10.2969/jmsj/87138713","url":null,"abstract":"In this paper we study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other, we extend Newelski’s Theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context. The Lascar (automorphism) group of a first-order complete theory and its quotient groups such as the Kim-Pillay group and the Shelah group have been central themes in contemporary model theory. The study on those groups enables us to develop Galois theoretic correspondence between the groups and their orbit-equivalence relations on a monster model such as Lascar types, Kim-Pillay types, and Shelah strong types. The notions of the Lascar group and its topology are introduced first by D. Lascar in [9] using ultraproducts. Later more favorable equivalent definition is suggested in [7] and [11], which is nowadays considered as a standard approach. However even a complete proof using the approach of the fundamental fact that the Lascar group is a topological group is not so well available. For example in [2], its proof is left to the readers, while the proof is not at all trivial. As far as we can see, only in [14], a detailed proof is written. Aforementioned results are for the Lascar group over ∅, or more generally over a real set A. In this paper we study the Lascar group over a hyperimaginary e and verify how results on the Lascar group over A can be extended to the case when the set is replaced by e. Indeed this attempt was made in [8] (and rewritten in [6, §5.1]). However those contain some errors, and moreover a proof of that the Lascar group over e is a topological group is also missing. In this paper we supply a proof of the fact in a detailed expository manner. Our proof is more direct and even simplifies that for the group over ∅ in [14]. We correct the mentioned errors in [6],[8], as well. In particular we correct the proof of that the orbit equivalence relation under a closed normal subgroup of the Lascar group over e is type-definable over e. Moreover we extend Newelski’s Theorem in [12] to the hyperimaginary context. Namely we show that if T is G-compact over e then there is 2020 Mathematics Subject Classification. Primary 03C60; Secondary 54H11.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47377326","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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