Pub Date : 2018-10-05DOI: 10.1215/21562261-2022-0004
K. Namikawa
We explicitly give an inner product formula and a Bessel period formula for theta series on GSp(4), which was studied by Harris, Soudry and Taylor. As a consequence, we prove a non-vanishing of the theta series of small weights and we give a criterion for the non-vanishing of the theta series modulo a prime.
{"title":"Explicit inner product formulas and Bessel period formulas for HST lifts","authors":"K. Namikawa","doi":"10.1215/21562261-2022-0004","DOIUrl":"https://doi.org/10.1215/21562261-2022-0004","url":null,"abstract":"We explicitly give an inner product formula and a Bessel period formula for theta series on GSp(4), which was studied by Harris, Soudry and Taylor. As a consequence, we prove a non-vanishing of the theta series of small weights and we give a criterion for the non-vanishing of the theta series modulo a prime.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43421484","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}
Pub Date : 2018-10-02DOI: 10.1215/21562261-2021-0015
J. Lo
On a smooth projective threefold, we construct an essentially surjective functor $mathcal{F}$ from a category of two-term complexes to a category of quotients of coherent sheaves, and describe the fibers of this functor. Under a coprime assumption on rank and degree, the domain of $mathcal{F}$ coincides with the category of higher-rank PT stable objects, which appear on one side of Toda's higher-rank DT/PT correspondence formula. The codomain of $mathcal{F}$ is the category of objects that appear on one side of another correspondence formula by Gholampour-Kool, between the generating series of topological Euler characteristics of two types of quot schemes.
{"title":"A relation between higher-rank PT-stable objects and quotients of coherent sheaves","authors":"J. Lo","doi":"10.1215/21562261-2021-0015","DOIUrl":"https://doi.org/10.1215/21562261-2021-0015","url":null,"abstract":"On a smooth projective threefold, we construct an essentially surjective functor $mathcal{F}$ from a category of two-term complexes to a category of quotients of coherent sheaves, and describe the fibers of this functor. Under a coprime assumption on rank and degree, the domain of $mathcal{F}$ coincides with the category of higher-rank PT stable objects, which appear on one side of Toda's higher-rank DT/PT correspondence formula. The codomain of $mathcal{F}$ is the category of objects that appear on one side of another correspondence formula by Gholampour-Kool, between the generating series of topological Euler characteristics of two types of quot schemes.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45171541","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}
Pub Date : 2018-09-24DOI: 10.1215/21562261-2021-0005
Xujia Chen, A. Zinger
We first recall Solomon's relations for Welschinger's invariants counting real curves in real symplectic fourfolds, announced in cite{Jake2} and established in cite{RealWDVV}, and the WDVV-style relations for Welschinger's invariants counting real curves in real symplectic sixfolds with some symmetry established in cite{RealWDVV3}. We then explicitly demonstrate that in some important cases (projective spaces with standard conjugations, real blowups of the projective plane, and two- and three-fold products of the one-dimensional projective space with two involutions each), these relations provide complete recursions determining all Welschinger's invariants from basic input. We include extensive tables of Welschinger's invariants in low degrees obtained from these recursions with {it Mathematica}. These invariants provide lower bounds for counts of real rational curves, including with curve insertions in smooth algebraic threefolds.
{"title":"WDVV-type relations for Welschinger invariants: Applications","authors":"Xujia Chen, A. Zinger","doi":"10.1215/21562261-2021-0005","DOIUrl":"https://doi.org/10.1215/21562261-2021-0005","url":null,"abstract":"We first recall Solomon's relations for Welschinger's invariants counting real curves in real symplectic fourfolds, announced in cite{Jake2} and established in cite{RealWDVV}, and the WDVV-style relations for Welschinger's invariants counting real curves in real symplectic sixfolds with some symmetry established in cite{RealWDVV3}. We then explicitly demonstrate that in some important cases (projective spaces with standard conjugations, real blowups of the projective plane, and two- and three-fold products of the one-dimensional projective space with two involutions each), these relations provide complete recursions determining all Welschinger's invariants from basic input. We include extensive tables of Welschinger's invariants in low degrees obtained from these recursions with {it Mathematica}. These invariants provide lower bounds for counts of real rational curves, including with curve insertions in smooth algebraic threefolds.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44618010","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}
Pub Date : 2018-08-22DOI: 10.1215/21562261-2018-0010
A. Bahlekeh, Alireza Fallah, Shokrollah Salarian
Let $(R, m)$ be a $d$-dimensional commutative noetherian local ring. Let $M$ denote the morphism category of finitely generated $R$-modules and let $Sc$ be the submodule category of $M$. In this paper, we specify the Auslander transpose in submodule category $Sc$. It will turn out that the Auslander transpose in this category can be described explicitly within ${rm mod}R$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander-Reiten theory in $Sc$. Indeed, a characterization of horizontally linked morphisms in terms of module category is given. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander-Reiten translations in the subcategories $HH$ and $G$, consisting of all morphisms which are maximal Cohen-Macaulay $R$-modules and Gorenstein projective morphisms, respectively, may be computed within ${rm mod}R$ via $G$-covers. Corresponding result for subcategory of epimorphisms in $HH$ is also obtained.
{"title":"Specifying the Auslander transpose in submodule category and its applications","authors":"A. Bahlekeh, Alireza Fallah, Shokrollah Salarian","doi":"10.1215/21562261-2018-0010","DOIUrl":"https://doi.org/10.1215/21562261-2018-0010","url":null,"abstract":"Let $(R, m)$ be a $d$-dimensional commutative noetherian local ring. Let $M$ denote the morphism category of finitely generated $R$-modules and let $Sc$ be the submodule category of $M$. In this paper, we specify the Auslander transpose in submodule category $Sc$. It will turn out that the Auslander transpose in this category can be described explicitly within ${rm mod}R$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander-Reiten theory in $Sc$. Indeed, a characterization of horizontally linked morphisms in terms of module category is given. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander-Reiten translations in the subcategories $HH$ and $G$, consisting of all morphisms which are maximal Cohen-Macaulay $R$-modules and Gorenstein projective morphisms, respectively, may be computed within ${rm mod}R$ via $G$-covers. Corresponding result for subcategory of epimorphisms in $HH$ is also obtained.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/21562261-2018-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44976650","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}
Pub Date : 2018-08-22DOI: 10.1215/21562261-2022-0001
N. Bottman
Associated to a symplectic quotient $M/!/G$ is a Lagrangian correspondence $Lambda_G$ from $M/!/G$ to $M$. In this note, we construct in two examples quilts with seam condition on such a correspondence, in the case of $S^1$ acting on $mathbb{CP}^2$ with symplectic quotient $mathbb{CP}^2/!/ S^1 = mathbb{CP}^1$. First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain groups $CF(gamma,S_{text{Cl}}^1)$ and $CF(mathbb{RP}^2,T_{text{Cl}}^2)$, where $gamma$ is the connected double-cover of $mathbb{RP}^1$. Second, we answer a question due to Akveld-Cannas da Silva-Wehrheim by explicitly producing a figure eight bubble which obstructs an isomorphism between two Floer chain groups. The figure eight bubbles we construct in this paper are the first concrete examples of this phenomenon.
与辛商$M/!/G$相关的是一个拉格朗日对应$Lambda_G$从$M/!/G$到$M$。在这篇笔记中，我们构造了两个例子，在$S^1$作用于$mathbb{CP}^2$的辛商$mathbb{CP}^2/!/ S^1 = mathbb{CP}^1$的情况下，在这样一个对应上有接缝条件的被子。首先，我们研究了绗缝条，如果没有数字8冒泡，将识别花链基团$CF(gamma,S_{text{Cl}}^1)$和$CF(mathbb{RP}^2,T_{text{Cl}}^2)$，其中$gamma$是$mathbb{RP}^1$的连接双盖。其次，我们回答了Akveld-Cannas da Silva-Wehrheim提出的一个问题，明确地产生了一个阻碍两个flower链群之间同构的8字形气泡。我们在本文中构造的数字8气泡是这种现象的第一个具体例子。
{"title":"Explicit constructions of quilts with seam condition coming from symplectic reduction","authors":"N. Bottman","doi":"10.1215/21562261-2022-0001","DOIUrl":"https://doi.org/10.1215/21562261-2022-0001","url":null,"abstract":"Associated to a symplectic quotient $M/!/G$ is a Lagrangian correspondence $Lambda_G$ from $M/!/G$ to $M$. In this note, we construct in two examples quilts with seam condition on such a correspondence, in the case of $S^1$ acting on $mathbb{CP}^2$ with symplectic quotient $mathbb{CP}^2/!/ S^1 = mathbb{CP}^1$. First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain groups $CF(gamma,S_{text{Cl}}^1)$ and $CF(mathbb{RP}^2,T_{text{Cl}}^2)$, where $gamma$ is the connected double-cover of $mathbb{RP}^1$. Second, we answer a question due to Akveld-Cannas da Silva-Wehrheim by explicitly producing a figure eight bubble which obstructs an isomorphism between two Floer chain groups. The figure eight bubbles we construct in this paper are the first concrete examples of this phenomenon.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49023883","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}
Pub Date : 2018-08-06DOI: 10.1215/21562261-2020-0005
Kouhei Matsuura
In this paper, we investigate spectral properties of explosive symmetric Markov processes. Under a condition on its life time, we prove the $L^1$-semigroup of Markov processes become compact operators.
{"title":"Compactness of semigroups of explosive symmetric Markov processes","authors":"Kouhei Matsuura","doi":"10.1215/21562261-2020-0005","DOIUrl":"https://doi.org/10.1215/21562261-2020-0005","url":null,"abstract":"In this paper, we investigate spectral properties of explosive symmetric Markov processes. Under a condition on its life time, we prove the $L^1$-semigroup of Markov processes become compact operators.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44979680","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}
Pub Date : 2018-07-16DOI: 10.1215/21562261-2021-0021
Jens Kaad, Valerio Proietti
. We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Miščenko line bundle. In addition, we give a proof of Atiyah’s L 2 -index theorem in the general context of flat bundles of finitely generated projective Hilbert C ∗ -modules over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum-Connes assembly map implies the Kadison-Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric K -homology but rather on an explicit construction of Alexander-Spanier cohomology classes coming from a Chern character for tracial function algebras.
{"title":"Index theory on the Miščenko bundle","authors":"Jens Kaad, Valerio Proietti","doi":"10.1215/21562261-2021-0021","DOIUrl":"https://doi.org/10.1215/21562261-2021-0021","url":null,"abstract":". We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Miščenko line bundle. In addition, we give a proof of Atiyah’s L 2 -index theorem in the general context of flat bundles of finitely generated projective Hilbert C ∗ -modules over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum-Connes assembly map implies the Kadison-Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric K -homology but rather on an explicit construction of Alexander-Spanier cohomology classes coming from a Chern character for tracial function algebras.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47760437","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}
Pub Date : 2018-07-01DOI: 10.1215/21562261-2017-0028
Y. Yamamoto
{"title":"Divisorial contractions to cDV points with discrepancy greater than 1","authors":"Y. Yamamoto","doi":"10.1215/21562261-2017-0028","DOIUrl":"https://doi.org/10.1215/21562261-2017-0028","url":null,"abstract":"","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/21562261-2017-0028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43480072","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}
Pub Date : 2018-07-01DOI: 10.1215/21562261-2017-0029
G. Métivier, T. Nishitani
We consider the Cauchy problem in L2 for first order system. A necessary condition is that the system must be uniformly diagonalizable, or equivalently that it admits a bounded symmetrizer. A sufficient condition is that it admits a smooth (Lipschtitz) symmetrizer, which is true when the system is hyperbolic, diagonalizable with eigenvalues of constant multiplicities. Counterexamples show that uniform diagonalizability is not sufficient in general for systems with variable coefficients and indicate that the symplectic properties of the set Σ of the singular points of the characteristic variety are important. In this paper, give a new class of systems for which the Cauchy problem is well posed in L2. The main assumption is that Σ is a smooth involutive manifold and the system is transversally strictly hyperbolic.
{"title":"Note on strongly hyperbolic systems with involutive characteristics","authors":"G. Métivier, T. Nishitani","doi":"10.1215/21562261-2017-0029","DOIUrl":"https://doi.org/10.1215/21562261-2017-0029","url":null,"abstract":"We consider the Cauchy problem in L2 for first order system. A necessary condition is that the system must be uniformly diagonalizable, or equivalently that it admits a bounded symmetrizer. A sufficient condition is that it admits a smooth (Lipschtitz) symmetrizer, which is true when the system is hyperbolic, diagonalizable with eigenvalues of constant multiplicities. Counterexamples show that uniform diagonalizability is not sufficient in general for systems with variable coefficients and indicate that the symplectic properties of the set Σ of the singular points of the characteristic variety are important. In this paper, give a new class of systems for which the Cauchy problem is well posed in L2. The main assumption is that Σ is a smooth involutive manifold and the system is transversally strictly hyperbolic.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/21562261-2017-0029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43265411","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}
Pub Date : 2018-06-29DOI: 10.1215/21562261-2019-0065
Gr'egory Mendousse
We study $K$-types of degenerate principal series of ${rm Sp}(n,mathbb{C})$ by using two realisations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the non-compact picture via an appropriate partial Fourier transform. In the first case we find a family of $K$-finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of $K$-finite vectors whose expressions involve Bessel functions.
{"title":"Special functions associated with K -types of degenerate principal series of Sp ( n , C )","authors":"Gr'egory Mendousse","doi":"10.1215/21562261-2019-0065","DOIUrl":"https://doi.org/10.1215/21562261-2019-0065","url":null,"abstract":"We study $K$-types of degenerate principal series of ${rm Sp}(n,mathbb{C})$ by using two realisations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the non-compact picture via an appropriate partial Fourier transform. In the first case we find a family of $K$-finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of $K$-finite vectors whose expressions involve Bessel functions.","PeriodicalId":49149,"journal":{"name":"Kyoto Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44691711","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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