We point out a gap in Murre's proof of the existence of a universal regular homomorphism for codimension $2$ cycles on a smooth projective variety, and offer two arguments to fill this gap.
{"title":"On the universal regular homomorphism in codimension 2","authors":"B. Kahn","doi":"10.5802/aif.3408","DOIUrl":"https://doi.org/10.5802/aif.3408","url":null,"abstract":"We point out a gap in Murre's proof of the existence of a universal regular homomorphism for codimension $2$ cycles on a smooth projective variety, and offer two arguments to fill this gap.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45082482","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 this short note, we study the bordism problem for group actions on the torus and give examples of groups acting on the torus by diffeomorphisms isotopic to the identity that cannot be extended to an action on a bounding 3-manifold. This solves a question raised in the previous work of the authors.
{"title":"On the bordism group for group actions on the torus","authors":"Kathryn Mann, Sam Nariman","doi":"10.5802/aif.3480","DOIUrl":"https://doi.org/10.5802/aif.3480","url":null,"abstract":"In this short note, we study the bordism problem for group actions on the torus and give examples of groups acting on the torus by diffeomorphisms isotopic to the identity that cannot be extended to an action on a bounding 3-manifold. This solves a question raised in the previous work of the authors.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44541790","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 characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety $X_x(b)$ in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of b has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements x by explicitly constructing such galleries.
{"title":"Affine Deligne–Lusztig varieties and folded galleries governed by chimneys","authors":"E. Milicevic, Petra Schwer, Anne Thomas","doi":"10.5802/aif.3578","DOIUrl":"https://doi.org/10.5802/aif.3578","url":null,"abstract":"We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety $X_x(b)$ in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of b has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements x by explicitly constructing such galleries.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46925328","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}
— An extension of a fundamental conjecture by R. Bott suggests that all simply connected closed almost non-negatively curved manifolds M are rationally elliptic, i.e., all but finitely many homotopy groups of such M are finite. We confirm this conjecture when in addition M supports an isometric action with orbits of codimension at most two. Our proof uses the geometry of the orbit space to control the topology of the homotopy fiber of the inclusion map of an orbit in M , and is applicable to more general contexts. Résumé. — D’après une extension d’une conjecture fondamentale de R. Bott, toute variété compacte (sans bord) simplement connexe M à courbure positive est rationellement elliptique, i.e., seul un nombre fini de groupes d’homotopie de M sont infinis. On montre cette conjecture dans le cas où M admet une action par isométries dont l’orbite principale a codimension au plus est de deux. Notre preuve utilise la géométrie de l’espace quotient pour contrôler la topologie de la fibre homotopique de l’inclusion d’une orbite dans M , et s’applique à des contextes plus généraux.
{"title":"Almost non-negative curvature and rational ellipticity in cohomogeneity two","authors":"K. Grove, Burkhard Wilking, Joseph E. Yeager","doi":"10.5802/aif.3340","DOIUrl":"https://doi.org/10.5802/aif.3340","url":null,"abstract":"— An extension of a fundamental conjecture by R. Bott suggests that all simply connected closed almost non-negatively curved manifolds M are rationally elliptic, i.e., all but finitely many homotopy groups of such M are finite. We confirm this conjecture when in addition M supports an isometric action with orbits of codimension at most two. Our proof uses the geometry of the orbit space to control the topology of the homotopy fiber of the inclusion map of an orbit in M , and is applicable to more general contexts. Résumé. — D’après une extension d’une conjecture fondamentale de R. Bott, toute variété compacte (sans bord) simplement connexe M à courbure positive est rationellement elliptique, i.e., seul un nombre fini de groupes d’homotopie de M sont infinis. On montre cette conjecture dans le cas où M admet une action par isométries dont l’orbite principale a codimension au plus est de deux. Notre preuve utilise la géométrie de l’espace quotient pour contrôler la topologie de la fibre homotopique de l’inclusion d’une orbite dans M , et s’applique à des contextes plus généraux.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47580188","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 $k$ be a number field. We give an explicit bound, depending only on $[k:mathbf{Q}]$, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer-Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on $[k:mathbf{Q}]$, on the number of $mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$.
{"title":"Explicit uniform bounds for Brauer groups of singular K3 surfaces","authors":"F. Balestrieri, Alexis Johnson, Rachel Newton","doi":"10.5802/aif.3526","DOIUrl":"https://doi.org/10.5802/aif.3526","url":null,"abstract":"Let $k$ be a number field. We give an explicit bound, depending only on $[k:mathbf{Q}]$, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer-Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on $[k:mathbf{Q}]$, on the number of $mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41636911","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 prove that the harmonic measures on the spheres of a pinched Hadamard manifold admit uniform upper and lower bounds. Résumé. — Nous prouvons que les mesures harmoniques sur les sphères des variétés Hadamard pincées admettent des bornes supérieures et infériueures uniformes.
—《我们骄傲的谐波情绪,the spheres of a pinched Hadamard流形saints制服upper and下城出生的。摘要。-我们证明了捏捏的阿达玛流形球体上的调和测量具有均匀的上下边界。
{"title":"Harmonic measures on negatively curved manifolds","authors":"Y. Benoist, D. Hulin","doi":"10.5802/aif.3342","DOIUrl":"https://doi.org/10.5802/aif.3342","url":null,"abstract":"— We prove that the harmonic measures on the spheres of a pinched Hadamard manifold admit uniform upper and lower bounds. Résumé. — Nous prouvons que les mesures harmoniques sur les sphères des variétés Hadamard pincées admettent des bornes supérieures et infériueures uniformes.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47167452","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 present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen–Margulis measure and whose Poincaré series converges at the critical exponent δΓ. We obtain an explicit asymptotic for their orbital growth function. Namely, for any α ∈ ]1, 2[ and any smooth slowly varying function L : R → (0,+∞), we construct N dimensional Hadamard manifolds (X, g) of negative and pinched curvature, whose group of oriented isometries possesses convergent geometrically finite subgroups Γ such that, as R→ +∞, NΓ(R) := ]{γ ∈ Γ | d(o, γ · o) 6 R} ∼ CΓ(o) L(R) Rα eΓ, for some CΓ(o) > 0 depending on the base point o. Résumé. — Nous construisons des variétés géométriquement finies à courbure strictement négative pincée, dont le flot géodésique possède une mesure de BowenMargulis non ergodique infinie, et dont la série de Poincaré converge à l’exposant δΓ, et nous obtenons une estimation précise du comportement asymptotique de la fonction orbitale de ce groupe. Plus précisément, pour tout α ∈ ]1, 2[ et toute fonction à variations lentes L : R → (0,+∞), nous construisons des variétés de Hadamard (X, g) de dimension N > 2 dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis Γ tels que, lorsque R→ +∞, NΓ(R) := ]{γ ∈ Γ | d(o, γ · o) 6 R} ∼ CΓ(o) L(R) Rα eΓ, où CΓ(o) est une constante strictement positive qui dépend du point o.
{"title":"Orbital counting for some convergent groups","authors":"M. Peigné, S. Tapie, Pierre Vidotto","doi":"10.5802/AIF.3335","DOIUrl":"https://doi.org/10.5802/AIF.3335","url":null,"abstract":"— We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen–Margulis measure and whose Poincaré series converges at the critical exponent δΓ. We obtain an explicit asymptotic for their orbital growth function. Namely, for any α ∈ ]1, 2[ and any smooth slowly varying function L : R → (0,+∞), we construct N dimensional Hadamard manifolds (X, g) of negative and pinched curvature, whose group of oriented isometries possesses convergent geometrically finite subgroups Γ such that, as R→ +∞, NΓ(R) := ]{γ ∈ Γ | d(o, γ · o) 6 R} ∼ CΓ(o) L(R) Rα eΓ, for some CΓ(o) > 0 depending on the base point o. Résumé. — Nous construisons des variétés géométriquement finies à courbure strictement négative pincée, dont le flot géodésique possède une mesure de BowenMargulis non ergodique infinie, et dont la série de Poincaré converge à l’exposant δΓ, et nous obtenons une estimation précise du comportement asymptotique de la fonction orbitale de ce groupe. Plus précisément, pour tout α ∈ ]1, 2[ et toute fonction à variations lentes L : R → (0,+∞), nous construisons des variétés de Hadamard (X, g) de dimension N > 2 dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis Γ tels que, lorsque R→ +∞, NΓ(R) := ]{γ ∈ Γ | d(o, γ · o) 6 R} ∼ CΓ(o) L(R) Rα eΓ, où CΓ(o) est une constante strictement positive qui dépend du point o.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43110974","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 provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group $Gamma := mathbf{SL}_2(A)$, where $A := mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with coefficients in $mathbb{F}_q$, endowed with the length function coming from a natural action of $Gamma$ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.
{"title":"Radial rapid decay does not imply rapid decay","authors":"A. Boyer, Antoine Pinochet Lobos, C. Pittet","doi":"10.5802/aif.3552","DOIUrl":"https://doi.org/10.5802/aif.3552","url":null,"abstract":"We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group $Gamma := mathbf{SL}_2(A)$, where $A := mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with coefficients in $mathbb{F}_q$, endowed with the length function coming from a natural action of $Gamma$ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71210510","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 families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.
{"title":"Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces","authors":"J. Lyczak","doi":"10.5802/aif.3529","DOIUrl":"https://doi.org/10.5802/aif.3529","url":null,"abstract":"We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43934346","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 sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over $S^1$.
{"title":"Non-unimodular transversely homogeneous foliations","authors":"E. Mac'ias-Virg'os, P. L. Mart'in-M'endez","doi":"10.5802/aif.3412","DOIUrl":"https://doi.org/10.5802/aif.3412","url":null,"abstract":"We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over $S^1$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42726172","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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