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":"1 1","pages":""},"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":" ","pages":""},"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":"1 1","pages":""},"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":"-1 1","pages":"1-48"},"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":"-1 1","pages":""},"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":"1 1","pages":""},"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}
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system [ mbox{(P)} qquad cases{ partial_t u=D_1Delta u+v^p, & $xin{bf R}^N,,,,t>0,$ partial_t v=D_2Delta v+u^q, & $xin{bf R}^N,,,,t>0,$ (u(cdot,0),v(cdot,0))=(mu,nu), & $xin{bf R}^N,$ } ] where $D_1$, $D_2>0$, $0 1$ and $(mu,nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
{"title":"Optimal singularities of initial functions for solvability of a semilinear parabolic system","authors":"Y. Fujishima, Kazuhiro Ishige","doi":"10.2969/jmsj/86058605","DOIUrl":"https://doi.org/10.2969/jmsj/86058605","url":null,"abstract":"Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system [ mbox{(P)} qquad cases{ partial_t u=D_1Delta u+v^p, & $xin{bf R}^N,,,,t>0,$ partial_t v=D_2Delta v+u^q, & $xin{bf R}^N,,,,t>0,$ (u(cdot,0),v(cdot,0))=(mu,nu), & $xin{bf R}^N,$ } ] where $D_1$, $D_2>0$, $0 1$ and $(mu,nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48612805","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}
Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps. Default functions were introduced in [7] and it is known that vanishing of the default function of a local martingale implies that it is a true martingale. Positivity of the default function then indicates the singularity of local martingale. Such local martingales are called strictly local martingales. Recently strictly local martingales are playing important roles in the theory of financial bubbles (cf. [19]), so the notion of default function becomes important in mathematical finance area. We consider them in a different context. In mathematical analysis of subharmonic functions it is classical and natural to consider the functions along Brownian motions. A stochastic process derived from a subharmonic function composed with Brownian motion is a local submartingale. Then if we know that the process is a true submartingale, which follows from vanishing of the default function of the submartingale, it effects simpleness and clearness in analysis of subharmonic functions. In this paper we intend to show that this probabilistic notion plays effective roles in some analysis such as L-Liouville type theorems of subharmonic functions and Liouville type theorems for functions satisfying some nonlinear differential inequalities. It covers and extends the precedent results about L-Liouville theorem for subharmonic functions 2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.
{"title":"Default functions and Liouville type theorems based on symmetric diffusions","authors":"A. Atsuji","doi":"10.2969/jmsj/82398239","DOIUrl":"https://doi.org/10.2969/jmsj/82398239","url":null,"abstract":"Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps. Default functions were introduced in [7] and it is known that vanishing of the default function of a local martingale implies that it is a true martingale. Positivity of the default function then indicates the singularity of local martingale. Such local martingales are called strictly local martingales. Recently strictly local martingales are playing important roles in the theory of financial bubbles (cf. [19]), so the notion of default function becomes important in mathematical finance area. We consider them in a different context. In mathematical analysis of subharmonic functions it is classical and natural to consider the functions along Brownian motions. A stochastic process derived from a subharmonic function composed with Brownian motion is a local submartingale. Then if we know that the process is a true submartingale, which follows from vanishing of the default function of the submartingale, it effects simpleness and clearness in analysis of subharmonic functions. In this paper we intend to show that this probabilistic notion plays effective roles in some analysis such as L-Liouville type theorems of subharmonic functions and Liouville type theorems for functions satisfying some nonlinear differential inequalities. It covers and extends the precedent results about L-Liouville theorem for subharmonic functions 2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43576089","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}
Beata Osińska-Ulrych, Grzegorz Skalski, S. Spodzieja
{"title":"Effective Łojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs","authors":"Beata Osińska-Ulrych, Grzegorz Skalski, S. Spodzieja","doi":"10.2969/jmsj/83378337","DOIUrl":"https://doi.org/10.2969/jmsj/83378337","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41454374","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 comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $Nin ]-infty,1]cup [n,+infty]$ with $varepsilon$-range introduced by Lu-Minguzzi-Ohta. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted $p$-Laplacian. Our results interpolate those for $Nin [n,+infty[$ and $varepsilon=1$, and for $Nin ]-infty,1]$ and $varepsilon=0$ by the second named author.
{"title":"Comparison geometry of manifolds with boundary under lower $N$-weighted Ricci curvature bounds with $varepsilon$-range","authors":"K. Kuwae, Y. Sakurai","doi":"10.2969/jmsj/87278727","DOIUrl":"https://doi.org/10.2969/jmsj/87278727","url":null,"abstract":"We study comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $Nin ]-infty,1]cup [n,+infty]$ with $varepsilon$-range introduced by Lu-Minguzzi-Ohta. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted $p$-Laplacian. Our results interpolate those for $Nin [n,+infty[$ and $varepsilon=1$, and for $Nin ]-infty,1]$ and $varepsilon=0$ by the second named author.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41582718","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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