Our purposes are to show the existence of a Calabi-Yau structure on the punctured cotangent bundle T ∗ 0 (P 2 O) of the Cayley projective plane P O and to construct a Bargmann type transformation between the L2-space on P 2 O and a space of holomorphic functions on T ∗ 0 (P 2 O), which corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on T ∗ 0 (P 2 O) was shown by identifying it with a quadrics in the complex space C{0} and the natural symplectic form of the cotangent bundle T ∗ 0 (P 2 O) is expressed as a Kähler form. Our method to construct the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map q : T ∗ 0 (P 2 O) −→ P O and the positive complex polarization defined by the Kähler structure. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turn out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.
{"title":"Calabi–Yau structure and Bargmann type transformation on the Cayley projective plane","authors":"Kurando Baba, Kenro Furutani","doi":"10.2969/jmsj/86638663","DOIUrl":"https://doi.org/10.2969/jmsj/86638663","url":null,"abstract":"Our purposes are to show the existence of a Calabi-Yau structure on the punctured cotangent bundle T ∗ 0 (P 2 O) of the Cayley projective plane P O and to construct a Bargmann type transformation between the L2-space on P 2 O and a space of holomorphic functions on T ∗ 0 (P 2 O), which corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on T ∗ 0 (P 2 O) was shown by identifying it with a quadrics in the complex space C{0} and the natural symplectic form of the cotangent bundle T ∗ 0 (P 2 O) is expressed as a Kähler form. Our method to construct the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map q : T ∗ 0 (P 2 O) −→ P O and the positive complex polarization defined by the Kähler structure. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turn out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46330968","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}
for polynomials P1, · · · , Pr, Q1, · · · , Qs on R . (We allow the cases r = 0 or s = 0.) A semialgebraic function φ : E → R is a function whose graph {(x, φ(x)) : x ∈ E} is a semialgebraic set. We define smoothness in terms of C and C loc. Here, C m ( R,R ) denotes the space of all R-valued functions on R whose derivatives up to order m are continuous and bounded on R. C loc ( R,R ) denotes the space of R-valued functions on R with continuous derivatives up to order m. If D = 1, we write C (R) and C loc (R ) in place of C ( R,R )
{"title":"$C^{m}$ semialgebraic sections over the plane","authors":"C. Fefferman, Garving K. Luli","doi":"10.2969/jmsj/86258625","DOIUrl":"https://doi.org/10.2969/jmsj/86258625","url":null,"abstract":"for polynomials P1, · · · , Pr, Q1, · · · , Qs on R . (We allow the cases r = 0 or s = 0.) A semialgebraic function φ : E → R is a function whose graph {(x, φ(x)) : x ∈ E} is a semialgebraic set. We define smoothness in terms of C and C loc. Here, C m ( R,R ) denotes the space of all R-valued functions on R whose derivatives up to order m are continuous and bounded on R. C loc ( R,R ) denotes the space of R-valued functions on R with continuous derivatives up to order m. If D = 1, we write C (R) and C loc (R ) in place of C ( R,R )","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43528747","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 nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${mathcal O}_{{mathbb P}^1}(1)^{oplus p} oplus {mathcal O}_{{mathbb P}^1}^{oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x in C,$ which is the germ of submanifolds ${mathcal C}^C_x subset {mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$. When $D subset TX$ is a contact distribution, a well-known necessary condition is that ${mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D subset TX$ and an unbendable rational curve $C subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$ at some point $xin C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.
{"title":"Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures","authors":"Jun-Muk Hwang","doi":"10.2969/JMSJ/85868586","DOIUrl":"https://doi.org/10.2969/JMSJ/85868586","url":null,"abstract":"A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${mathcal O}_{{mathbb P}^1}(1)^{oplus p} oplus {mathcal O}_{{mathbb P}^1}^{oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x in C,$ which is the germ of submanifolds ${mathcal C}^C_x subset {mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$. When $D subset TX$ is a contact distribution, a well-known necessary condition is that ${mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D subset TX$ and an unbendable rational curve $C subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$ at some point $xin C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43249400","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 construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.
{"title":"Stochastic integrals and Brownian motion on abstract nilpotent Lie groups","authors":"T. Melcher","doi":"10.2969/jmsj/84678467","DOIUrl":"https://doi.org/10.2969/jmsj/84678467","url":null,"abstract":"We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43483804","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}
Khodabakhsh Hessami Pilehrood, T. H. Pilehrood, R. Tauraso
We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.
我们主要回答了最近由H. Bachmann, Y. Takeyama和K. Tasaka提出的关于3-2-1指数上有限多重调和$q$-级数的两个开放性问题。本文还提供了关于循环和的两个猜想，它们推广了所给的结果。
{"title":"On 3-2-1 values of finite multiple harmonic $q$-series at roots of unity","authors":"Khodabakhsh Hessami Pilehrood, T. H. Pilehrood, R. Tauraso","doi":"10.2969/jmsj/86238623","DOIUrl":"https://doi.org/10.2969/jmsj/86238623","url":null,"abstract":"We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41437703","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}
Relations among fundamental invariants play an important role in algebraic geometry. It is known that an n-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image Y under the canonical map is of maximal dimension, satisfies K X ≥ 2(pg − n). We investigate the very interesting extremal situation K X = 2(pg−n), which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with in the works of Fujita [Fuj83] and Kobayashi [Kob92]. We carry out further studies of Horikawa varieties, proving new results on various geometric and topological issues concerning them. In particular, we prove that the geometric genus of those Horikawa varieties whose image under the canonical map is singular is bounded. We give an analogous result for polarized hyperelliptic subcanonical varieties, in particular, for polarized Calabi-Yau and Fano varieties. The pleasing numerology that emerges puts Horikawa’s result on surfaces in a broader perspective. We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected, even if Y is singular. We also prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2.
{"title":"On higher dimensional extremal varieties of general type","authors":"Purnaprajna Bangere, J. Chen, F. Gallego","doi":"10.2969/jmsj/88668866","DOIUrl":"https://doi.org/10.2969/jmsj/88668866","url":null,"abstract":"Relations among fundamental invariants play an important role in algebraic geometry. It is known that an n-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image Y under the canonical map is of maximal dimension, satisfies K X ≥ 2(pg − n). We investigate the very interesting extremal situation K X = 2(pg−n), which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with in the works of Fujita [Fuj83] and Kobayashi [Kob92]. We carry out further studies of Horikawa varieties, proving new results on various geometric and topological issues concerning them. In particular, we prove that the geometric genus of those Horikawa varieties whose image under the canonical map is singular is bounded. We give an analogous result for polarized hyperelliptic subcanonical varieties, in particular, for polarized Calabi-Yau and Fano varieties. The pleasing numerology that emerges puts Horikawa’s result on surfaces in a broader perspective. We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected, even if Y is singular. We also prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49188668","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 consider the multicanonical systems |mKS | of quasielliptic surfaces with Kodaira dimension 1 in characteristic 2. We show that for any m ≥ 6 |mKS | gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces.
{"title":"On the multicanonical systems of quasi-elliptic surfaces","authors":"Natsuo Saito, Bstract","doi":"10.2969/jmsj/85058505","DOIUrl":"https://doi.org/10.2969/jmsj/85058505","url":null,"abstract":"We consider the multicanonical systems |mKS | of quasielliptic surfaces with Kodaira dimension 1 in characteristic 2. We show that for any m ≥ 6 |mKS | gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69573956","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}
Dirichlet's theorem in Diophantine approximation is known to be closely related to geometry of the hyperbolic plane. In this paper we consider approximation in the setting of number fields and study relation between systems of linear forms and geometry of products of symmetric spaces.
{"title":"Diophantine approximation in number fields and geometry of products of symmetric spaces","authors":"T. Hattori","doi":"10.2969/JMSJ/81358135","DOIUrl":"https://doi.org/10.2969/JMSJ/81358135","url":null,"abstract":"Dirichlet's theorem in Diophantine approximation is known to be closely related to geometry of the hyperbolic plane. In this paper we consider approximation in the setting of number fields and study relation between systems of linear forms and geometry of products of symmetric spaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69574272","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":"On the Milnor fibration for $f(boldsymbol{z}) bar{g}(boldsymbol{z})$ II","authors":"M. Oka","doi":"10.2969/JMSJ/83328332","DOIUrl":"https://doi.org/10.2969/JMSJ/83328332","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69574346","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}
Relations between the hypergeometric function with a large parameter and Borel sums of WKB solutions of the hypergeometric differential equation with the large parameter are established. The confluent hypergeometric function is also investigated from the viewpoint of exact WKB analysis. As applications, asymptotic expansion formulas for those classical special functions with respect to parameters are obtained.
{"title":"The hypergeometric function, the confluent hypergeometric function and WKB solutions","authors":"T. Aoki, Toshinori Takahashi, M. Tanda","doi":"10.2969/jmsj/84528452","DOIUrl":"https://doi.org/10.2969/jmsj/84528452","url":null,"abstract":"Relations between the hypergeometric function with a large parameter and Borel sums of WKB solutions of the hypergeometric differential equation with the large parameter are established. The confluent hypergeometric function is also investigated from the viewpoint of exact WKB analysis. As applications, asymptotic expansion formulas for those classical special functions with respect to parameters are obtained.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69573916","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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