This is a continuation of the paper [K1] on the foundation of log geometry in the sense of Fontaine-Illusie. Here we discuss mainly log flat topologies, especially log flat descent theory. This paper was started around 1991, and was circulated as an incomplete preprint for a long time. Since then, some contents of this paper have been reproduced by several authors with proofs ([Ha], [KS], [Ni], [Na2], [Ol], ...). A. Moriwaki [M] also studied flat descents in the category of log schemes. In the parts which were incomplete in the circulated preprint, we sometimes referred to these papers instead of completing the original proofs. In particular, the author does not claim the results with ∗ (i.e., two theorems 7.1, 7.2 and one proposition 6.5) are his results. Since the paper is already referred to in many published works, in the other parts, we preferred to preserve the original, circulated form. In both parts, we tried to preserve the original numberings of definitions and propositions. The author wishes to express his special thanks to Chikara Nakayama who helped him a lot in the completion of this paper. He is also thankful to Luc Illusie, Takeshi Saito, Takeshi Kajiwara, and Takeshi Tsuji for helpful discussions. The author is partially supported by NSF Award 1601861.
{"title":"Logarithmic Structures of Fontaine-Illusie. II ---Logarithmic Flat Topology","authors":"Kazuya Kato","doi":"10.3836/tjm/1502179316","DOIUrl":"https://doi.org/10.3836/tjm/1502179316","url":null,"abstract":"This is a continuation of the paper [K1] on the foundation of log geometry in the sense of Fontaine-Illusie. Here we discuss mainly log flat topologies, especially log flat descent theory. This paper was started around 1991, and was circulated as an incomplete preprint for a long time. Since then, some contents of this paper have been reproduced by several authors with proofs ([Ha], [KS], [Ni], [Na2], [Ol], ...). A. Moriwaki [M] also studied flat descents in the category of log schemes. In the parts which were incomplete in the circulated preprint, we sometimes referred to these papers instead of completing the original proofs. In particular, the author does not claim the results with ∗ (i.e., two theorems 7.1, 7.2 and one proposition 6.5) are his results. Since the paper is already referred to in many published works, in the other parts, we preferred to preserve the original, circulated form. In both parts, we tried to preserve the original numberings of definitions and propositions. The author wishes to express his special thanks to Chikara Nakayama who helped him a lot in the completion of this paper. He is also thankful to Luc Illusie, Takeshi Saito, Takeshi Kajiwara, and Takeshi Tsuji for helpful discussions. The author is partially supported by NSF Award 1601861.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47849095","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}
L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension 2. In this note, instead of open cover consisting of displaceable sets in their work, we consider open cover constituted of topological discs and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.
{"title":"The Poisson Bracket Invariant for Open Covers Consisting of Topological Disks on Surfaces","authors":"Kun Shi, Guangcun Lu","doi":"10.3836/tjm/1502179384","DOIUrl":"https://doi.org/10.3836/tjm/1502179384","url":null,"abstract":"L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension 2. In this note, instead of open cover consisting of displaceable sets in their work, we consider open cover constituted of topological discs and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47728860","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}
Chunhua Li, Y. Nishii, Yuji Sagawa, Hideaki Sunagawa
This is a sequel to the paper "Large time asymptotics for a cubic nonlinear Schrodinger system in one space dimension" by the same authors. We continue to study the Cauchy problem for the two-component system of cubic nonlinear Schrodinger equations in one space dimension. We provide criteria for large time decay or non-decay in $L^2$ of the small amplitude solutions in terms of the Fourier transforms of the initial data.
{"title":"Large Time Asymptotics for a Cubic Nonlinear Schrödinger\u0000 System in One Space Dimension, II","authors":"Chunhua Li, Y. Nishii, Yuji Sagawa, Hideaki Sunagawa","doi":"10.1619/fesi.64.361","DOIUrl":"https://doi.org/10.1619/fesi.64.361","url":null,"abstract":"This is a sequel to the paper \"Large time asymptotics for a cubic nonlinear Schrodinger system in one space dimension\" by the same authors. We continue to study the Cauchy problem for the two-component system of cubic nonlinear Schrodinger equations in one space dimension. We provide criteria for large time decay or non-decay in $L^2$ of the small amplitude solutions in terms of the Fourier transforms of the initial data.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48828258","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 give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
{"title":"Rationally Elliptic Toric Varieties","authors":"I. Biswas, V. Muñoz, A. Murillo","doi":"10.3836/tjm/1502179327","DOIUrl":"https://doi.org/10.3836/tjm/1502179327","url":null,"abstract":"We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42415694","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 weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist so many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.
{"title":"On Weakly Reflective PF Submanifolds in Hilbert Spaces","authors":"M. Morimoto","doi":"10.3836/tjm/1502179323","DOIUrl":"https://doi.org/10.3836/tjm/1502179323","url":null,"abstract":"A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist so many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48328779","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 a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.
{"title":"Milnor Invariants, $2n$-moves and $V^{n}$-moves for Welded String Links","authors":"H. A. Miyazawa, K. Wada, A. Yasuhara","doi":"10.3836/tjm/1502179315","DOIUrl":"https://doi.org/10.3836/tjm/1502179315","url":null,"abstract":"In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44666210","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}
Let $K[X_d,Y_d]=K[x_1,ldots,x_d,y_1,ldots,y_d]$ be the polynomial algebra in $2d$ variables over a field $K$ of characteristic 0 and let $delta$ be the derivation of $K[X_d,Y_d]$ defined by $delta(y_i)=x_i$, $delta(x_i)=0$, $i=1,ldots,d$. In 1994 Nowicki conjectured that the algebra $K[X_d,Y_d]^{delta}$ of constants of $delta$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1leq i
{"title":"Another Proof of the Nowicki Conjecture","authors":"V. Drensky","doi":"10.3836/tjm/1502179320","DOIUrl":"https://doi.org/10.3836/tjm/1502179320","url":null,"abstract":"Let $K[X_d,Y_d]=K[x_1,ldots,x_d,y_1,ldots,y_d]$ be the polynomial algebra in $2d$ variables over a field $K$ of characteristic 0 and let $delta$ be the derivation of $K[X_d,Y_d]$ defined by $delta(y_i)=x_i$, $delta(x_i)=0$, $i=1,ldots,d$. In 1994 Nowicki conjectured that the algebra $K[X_d,Y_d]^{delta}$ of constants of $delta$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1leq i<jleq d$. The affirmative answer was given by several authors using different ideas. In the present paper we give another proof of the conjecture based on representation theory of the general linear group $GL_2(K)$.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45662106","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}
Ito-Miura-Okawa-Ueda have constructed a pair of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, but not stably birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.
Ito Miura Okawa Ueda构造了一对Calabi Yau三重$X$和$Y$，它们是L-等价的和派生等价的，但不是稳定的对偶。我们通过展示$X$和$Y$具有同构的Chow动机来完成这幅图。
{"title":"On the Motive of Ito-Miura-Okawa-Ueda Calabi-Yau Threefolds","authors":"R. Laterveer","doi":"10.3836/TJM/1502179303","DOIUrl":"https://doi.org/10.3836/TJM/1502179303","url":null,"abstract":"Ito-Miura-Okawa-Ueda have constructed a pair of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, but not stably birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43754906","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":"$L^1$ and $L^infty$-boundedness of Wave Operators for Three Dimensional Schrödinger Operators with Threshold Singularities","authors":"K. Yajima","doi":"10.3836/TJM/1502179271","DOIUrl":"https://doi.org/10.3836/TJM/1502179271","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41323242","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":"Precise Asymptotic Formulae for the First Hitting Times of Bessel Processes","authors":"Yuji Hamana, H. Matsumoto","doi":"10.3836/tjm/1502179246","DOIUrl":"https://doi.org/10.3836/tjm/1502179246","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49575749","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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