An argument by Hassell proving the existence of a Bunimovich stadium for which there are semiclassical measures giving positive mass to the submanifold of bouncing ball trajectories uses a notion of non-gliding points. However, this notion is defined only for domains with $C^2$-boundaries. The purpose of this note is to clarify the argument.
{"title":"A note on the semiclassical measure at singular points of the boundary of the Bunimovich stadium","authors":"D. Mangoubi, Adi Weller Weiser","doi":"10.5802/aif.3601","DOIUrl":"https://doi.org/10.5802/aif.3601","url":null,"abstract":"An argument by Hassell proving the existence of a Bunimovich stadium for which there are semiclassical measures giving positive mass to the submanifold of bouncing ball trajectories uses a notion of non-gliding points. However, this notion is defined only for domains with $C^2$-boundaries. The purpose of this note is to clarify the argument.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43002703","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 generalize the well known Künneth formula for Chow groups to an arbitrary oriented Borel–Moore homology theory satisfying localization and descent (e.g. algebraic bordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a new, homological, description for the homology groups of smooth toric varieties, which allows us to compute the algebraic bordism groups of some singular toric varieties. Résumé. — Nous généralisons la formule de Künneth bien connue pour les groupes de Chow au cas d’une théorie homologique orientée de Borel–Moore arbitraire qui vérifient des propriétés de localisation et de descente (par exemple le bordisme algébrique) pour les produits avec une variété torique. En corollaire, nous obtenons un théorème de coefficients universels pour les anneaux de cohomologie opérationnelle. Nous donnons également une nouvelle description, de nature homologique, des groupes d’homologie des variétés toriques lisses, qui nous permet de calculer les groupes de bordisme algébrique de quelques variétés toriques singulières.
{"title":"Oriented Borel–Moore homologies of toric varieties","authors":"Toni M. Annala","doi":"10.5802/aif.3452","DOIUrl":"https://doi.org/10.5802/aif.3452","url":null,"abstract":"— We generalize the well known Künneth formula for Chow groups to an arbitrary oriented Borel–Moore homology theory satisfying localization and descent (e.g. algebraic bordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a new, homological, description for the homology groups of smooth toric varieties, which allows us to compute the algebraic bordism groups of some singular toric varieties. Résumé. — Nous généralisons la formule de Künneth bien connue pour les groupes de Chow au cas d’une théorie homologique orientée de Borel–Moore arbitraire qui vérifient des propriétés de localisation et de descente (par exemple le bordisme algébrique) pour les produits avec une variété torique. En corollaire, nous obtenons un théorème de coefficients universels pour les anneaux de cohomologie opérationnelle. Nous donnons également une nouvelle description, de nature homologique, des groupes d’homologie des variétés toriques lisses, qui nous permet de calculer les groupes de bordisme algébrique de quelques variétés toriques singulières.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44048369","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 set F(3) of foliations of degree three on the complex projective plane can be identified with a Zariski’s open set of a projective space of dimension 23 on which acts Aut(PC). The subset FP(3) of F(3) consisting of foliations of F(3) with a flat Legendre transform (dual web) is a Zariski closed subset of F(3). We classify up to automorphism of PC the elements of FP(3). More precisely, we show that up to an automorphism there are 16 foliations of degree three with a flat Legendre transform. From this classification we deduce that FP(3) has exactly 12 irreducible components. We also deduce that up to an automorphism there are 4 convex foliations of degree three on P2. Résumé. — L’ensemble F(3) des feuilletages de degré trois du plan projectif complexe s’identifie à un ouvert de Zariski dans un espace projectif de dimension 23 sur lequel agit le groupe Aut(PC). Le sous-ensemble FP(3) de F(3) formé des feuilletages de F(3) ayant une transformée de Legendre (tissu dual) plate est un fermé de Zariski de F(3). Nous classifions à automorphisme de PC près les éléments de F(3); plus précisément, nous montrons qu’à automorphisme près il y a 16 feuilletages de degré 3 ayant une transformée de Legendre plate. De cette classification nous obtenons la décomposition de F(3) en ses composantes irréductibles. Nous en déduisons aussi la classification à automorphisme près des feuilletages convexes de degré 3 de PC.
{"title":"Classification of foliations of degree three on ℙ ℂ 2 with a flat Legendre transform","authors":"Samir Bedrouni, D. Marín","doi":"10.5802/aif.3431","DOIUrl":"https://doi.org/10.5802/aif.3431","url":null,"abstract":"— The set F(3) of foliations of degree three on the complex projective plane can be identified with a Zariski’s open set of a projective space of dimension 23 on which acts Aut(PC). The subset FP(3) of F(3) consisting of foliations of F(3) with a flat Legendre transform (dual web) is a Zariski closed subset of F(3). We classify up to automorphism of PC the elements of FP(3). More precisely, we show that up to an automorphism there are 16 foliations of degree three with a flat Legendre transform. From this classification we deduce that FP(3) has exactly 12 irreducible components. We also deduce that up to an automorphism there are 4 convex foliations of degree three on P2. Résumé. — L’ensemble F(3) des feuilletages de degré trois du plan projectif complexe s’identifie à un ouvert de Zariski dans un espace projectif de dimension 23 sur lequel agit le groupe Aut(PC). Le sous-ensemble FP(3) de F(3) formé des feuilletages de F(3) ayant une transformée de Legendre (tissu dual) plate est un fermé de Zariski de F(3). Nous classifions à automorphisme de PC près les éléments de F(3); plus précisément, nous montrons qu’à automorphisme près il y a 16 feuilletages de degré 3 ayant une transformée de Legendre plate. De cette classification nous obtenons la décomposition de F(3) en ses composantes irréductibles. Nous en déduisons aussi la classification à automorphisme près des feuilletages convexes de degré 3 de PC.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49071654","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 method to resolve a quotient surface singularity which arises as the quotient of a product action of a finite group on two curves. In the characteristic zero case, the singularity is resolved by means of a continued fraction, which is known as the Hirzebruch–Jung desingularization. We develop the method in the positive characteristic case where the square of the characteristic does not divide the order of the group. Résumé. — Nous donnons une méthode pour résoudre une singularité quotient de surface qui se présente comme le quotient d’une action produit d’un groupe fini sur deux courbes. En caractéristique nulle, la singularité est résolue au moyen d’une fraction continue (désingularisation de Hirzebruch–Jung). Nous développons la méthode dans le cas de la caractéristique strictement positive où le carré de la caractéristique ne divise pas l’ordre du groupe.
{"title":"Quotient singularities of products of two curves","authors":"Kentaro Mitsui","doi":"10.5802/aif.3434","DOIUrl":"https://doi.org/10.5802/aif.3434","url":null,"abstract":"— We give a method to resolve a quotient surface singularity which arises as the quotient of a product action of a finite group on two curves. In the characteristic zero case, the singularity is resolved by means of a continued fraction, which is known as the Hirzebruch–Jung desingularization. We develop the method in the positive characteristic case where the square of the characteristic does not divide the order of the group. Résumé. — Nous donnons une méthode pour résoudre une singularité quotient de surface qui se présente comme le quotient d’une action produit d’un groupe fini sur deux courbes. En caractéristique nulle, la singularité est résolue au moyen d’une fraction continue (désingularisation de Hirzebruch–Jung). Nous développons la méthode dans le cas de la caractéristique strictement positive où le carré de la caractéristique ne divise pas l’ordre du groupe.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45665419","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":"Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators","authors":"J. J. Morales, Sonia L. Rueda, M. Zurro","doi":"10.5802/aif.3425","DOIUrl":"https://doi.org/10.5802/aif.3425","url":null,"abstract":"","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42097788","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 show that any fibration of a 'special' compact K{"a}hler manifold X onto an Abelian variety has no multiple fibre in codimension one. This statement strengthens and extends previous results of Kawamata and Viehweg when $kappa$(X) = 0. This also corrects the proof given in [2], 5.3 which was incomplete.
{"title":"Albanese map of special manifolds: a correction","authors":"Frederic Campana","doi":"10.5802/aif.3563","DOIUrl":"https://doi.org/10.5802/aif.3563","url":null,"abstract":"We show that any fibration of a 'special' compact K{\"a}hler manifold X onto an Abelian variety has no multiple fibre in codimension one. This statement strengthens and extends previous results of Kawamata and Viehweg when $kappa$(X) = 0. This also corrects the proof given in [2], 5.3 which was incomplete.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44804110","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 established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal K"ahler metrics on a large class of toric fibrations, introduced by Apostolov--Calderbank--Gauduchon--Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature K"ahler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal K"ahler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle $mathbb{P}(mathcal{L}_0oplusmathcal{L}_1 oplus mathcal{L}_2)$, where $mathcal{L}_i$ are holomorphic line bundles over an elliptic curve, admits an extremal metric in every K"ahler class.
{"title":"A Yau–Tian–Donaldson correspondence on a class of toric fibrations","authors":"S. Jubert","doi":"10.5802/aif.3580","DOIUrl":"https://doi.org/10.5802/aif.3580","url":null,"abstract":"We established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal K\"ahler metrics on a large class of toric fibrations, introduced by Apostolov--Calderbank--Gauduchon--Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature K\"ahler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal K\"ahler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle $mathbb{P}(mathcal{L}_0oplusmathcal{L}_1 oplus mathcal{L}_2)$, where $mathcal{L}_i$ are holomorphic line bundles over an elliptic curve, admits an extremal metric in every K\"ahler class.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41487148","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 $theta = [0; a_1, a_2, dots]$ be the continued fraction expansion of an irrational real number $theta in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $theta$ is the limit of a sequence of finite words $(M_k)_{k ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $theta$) being a suitable concatenation of $a_k$ copies of $M_{k-1}$ and one copy of $M_{k-2}$. Our first result extends this to any Sturmian word. Let $b ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $xi$ whose $b$-ary expansion is a Sturmian word ${bf s}$ over the alphabet ${0, b-1}$. This extends a classical result of B"ohmer who considered only the case where ${bf s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $xi$ in terms of the slope and the intercept of ${bf s}$.
{"title":"Combinatorial structure of Sturmian words and continued fraction expansion of Sturmian numbers","authors":"Y. Bugeaud, M. Laurent","doi":"10.5802/aif.3561","DOIUrl":"https://doi.org/10.5802/aif.3561","url":null,"abstract":"Let $theta = [0; a_1, a_2, dots]$ be the continued fraction expansion of an irrational real number $theta in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $theta$ is the limit of a sequence of finite words $(M_k)_{k ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $theta$) being a suitable concatenation of $a_k$ copies of $M_{k-1}$ and one copy of $M_{k-2}$. Our first result extends this to any Sturmian word. Let $b ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $xi$ whose $b$-ary expansion is a Sturmian word ${bf s}$ over the alphabet ${0, b-1}$. This extends a classical result of B\"ohmer who considered only the case where ${bf s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $xi$ in terms of the slope and the intercept of ${bf s}$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47190667","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 compute the second (and the first) cohomology groups of $^*$-algebras associated to the universal quantum unitary groups of not neccesarily Kac type, extending our earlier results for the free unitary group $U_d^+$. The extended setup forces us to use infinite-dimensional representations to construct the cocycles.
{"title":"Second cohomology groups of the Hopf * -algebras associated to universal unitary quantum groups","authors":"Biswarup Das, U. Franz, A. Kula, Adam G. Skalski","doi":"10.5802/aif.3527","DOIUrl":"https://doi.org/10.5802/aif.3527","url":null,"abstract":"We compute the second (and the first) cohomology groups of $^*$-algebras associated to the universal quantum unitary groups of not neccesarily Kac type, extending our earlier results for the free unitary group $U_d^+$. The extended setup forces us to use infinite-dimensional representations to construct the cocycles.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47859693","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 apply the concept of braiding sequences to the Conway and skein polynomial, and some geometric invariants of positive links. Using degree and coefficient growth properties of the Conway polynomial, estimates of braid index and Legendrian invariants are given. We enumerate alternating (and some other classes of) links of given genus asymptotically up to constants by braid index. Résumé. — Nous appliquons le concept de séquences de tressage aux polynômes de skein et de Conway, mais aussi à quelques invariants géométriques des entrelacs positifs. On donne des estimations pour l’indice des tresses et pour des invariants legendriens, en utilisant le degré et des propriétés de croissance des coefficients du polynôme de Conway. Nous énumérons asymptotiquement à une constante près les entrelacs alternants (et quelques autres) de genre donné par leur indice de tresses.
= =地理= =根据美国人口普查，这个县的面积为，其中土地面积为，其中土地面积为。利用学位和多项式系数growth properties of the Conway,估计of braid索引Legendrian不变量这些人。= =地理= =根据美国人口普查，这个县的面积为，其中土地面积为。摘要。-我们将编织序列的概念应用于skein和Conway多项式，以及一些正交错的几何不变量。利用康威多项式系数的度和生长特性，给出了辫子指数和legendrian不变量的估计。我们渐近地列出了一个常数附近的交替交错(和其他一些)的类型，由他们的辫子指数给出。
{"title":"Application of braiding sequences IV: link polynomials and geometric invariants","authors":"A. Stoimenow","doi":"10.5802/AIF.3371","DOIUrl":"https://doi.org/10.5802/AIF.3371","url":null,"abstract":"— We apply the concept of braiding sequences to the Conway and skein polynomial, and some geometric invariants of positive links. Using degree and coefficient growth properties of the Conway polynomial, estimates of braid index and Legendrian invariants are given. We enumerate alternating (and some other classes of) links of given genus asymptotically up to constants by braid index. Résumé. — Nous appliquons le concept de séquences de tressage aux polynômes de skein et de Conway, mais aussi à quelques invariants géométriques des entrelacs positifs. On donne des estimations pour l’indice des tresses et pour des invariants legendriens, en utilisant le degré et des propriétés de croissance des coefficients du polynôme de Conway. Nous énumérons asymptotiquement à une constante près les entrelacs alternants (et quelques autres) de genre donné par leur indice de tresses.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48189984","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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