Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},dots,p_{r}$ there exists a continuous probability measure $mu $ on the unit circle $mathbb{T}$ such that [ inf_{k_{1}ge 0,dots,k_{r}ge 0}|widehat{mu }(p_{1}^{k_{1}}dots p_{r}^{k_{r}})|>0. ] This results applies in particular to the Furstenberg set $F={2^{k}3^{k'},;,kge 0, k'ge 0}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $times 2$-$times 3$ conjecture. We also estimate the modified Kazhdan constant of $F$ and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.
{"title":"Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences","authors":"C. Badea, S. Grivaux","doi":"10.4171/cmh/482","DOIUrl":"https://doi.org/10.4171/cmh/482","url":null,"abstract":"Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},dots,p_{r}$ there exists a continuous probability measure $mu $ on the unit circle $mathbb{T}$ such that [ inf_{k_{1}ge 0,dots,k_{r}ge 0}|widehat{mu }(p_{1}^{k_{1}}dots p_{r}^{k_{r}})|>0. ] This results applies in particular to the Furstenberg set $F={2^{k}3^{k'},;,kge 0, k'ge 0}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $times 2$-$times 3$ conjecture. We also estimate the modified Kazhdan constant of $F$ and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/482","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47812092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 field, $A$ a finitely generated associative $k$-algebra and $operatorname{Rep}_A[n]$ the functor $operatorname{Fields}_kto operatorname{Sets}$, which sends a field $K$ containing $k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n) := operatorname{ed}_k(operatorname{Rep}_A[n])$, as $ntoinfty$. In particular, we show that the rate of growth of $r_A(n)$ determines the representation type of $A$. That is, $r_A(n)$ is bounded from above if $A$ is of finite representation type, grows linearly if $A$ is of tame representation type and grows quadratically if A is of wild representation type. Moreover, $r_A(n)$ is a finer invariant of A, which allows us to distinguish among algebras of the same representation type.
{"title":"Essential dimension of representations of algebras","authors":"F. Scavia","doi":"10.4171/cmh/500","DOIUrl":"https://doi.org/10.4171/cmh/500","url":null,"abstract":"Let $k$ be a field, $A$ a finitely generated associative $k$-algebra and $operatorname{Rep}_A[n]$ the functor $operatorname{Fields}_kto operatorname{Sets}$, which sends a field $K$ containing $k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n) := operatorname{ed}_k(operatorname{Rep}_A[n])$, as $ntoinfty$. In particular, we show that the rate of growth of $r_A(n)$ determines the representation type of $A$. That is, $r_A(n)$ is bounded from above if $A$ is of finite representation type, grows linearly if $A$ is of tame representation type and grows quadratically if A is of wild representation type. Moreover, $r_A(n)$ is a finer invariant of A, which allows us to distinguish among algebras of the same representation type.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45563001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${mathbb R}^{n+p}$ is singularly genuinely rigid in ${mathbb R}^{n+q}$, for any $q < min{5,n} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.
{"title":"Singular genuine rigidity","authors":"L. Florit, Felippe Guimarães","doi":"10.4171/cmh/488","DOIUrl":"https://doi.org/10.4171/cmh/488","url":null,"abstract":"We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${mathbb R}^{n+p}$ is singularly genuinely rigid in ${mathbb R}^{n+q}$, for any $q < min{5,n} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/488","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46256434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify simple groups that act by birational transformations on compact complex K"ahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.
{"title":"Simple groups of birational transformations in dimension two","authors":"Christian Urech","doi":"10.4171/cmh/486","DOIUrl":"https://doi.org/10.4171/cmh/486","url":null,"abstract":"We classify simple groups that act by birational transformations on compact complex K\"ahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/486","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48094379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.
{"title":"Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero","authors":"Olivier Benoist, J. C. Ottem","doi":"10.4171/cmh/479","DOIUrl":"https://doi.org/10.4171/cmh/479","url":null,"abstract":"We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/479","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49334720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.
{"title":"Characterization of generic projective space bundles and algebraicity of foliations","authors":"Carolina Araujo, S. Druel","doi":"10.4171/cmh/475","DOIUrl":"https://doi.org/10.4171/cmh/475","url":null,"abstract":"In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/475","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45429026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane.
{"title":"Rigidity of Busemann convex Finsler metrics","authors":"S. Ivanov, A. Lytchak","doi":"10.4171/cmh/476","DOIUrl":"https://doi.org/10.4171/cmh/476","url":null,"abstract":"We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/476","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43105482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $pi$ be a cuspidal automorphic representation of $PGL_2(mathbb{A}_mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $phi$ be an $L^2$-normalized automorphic form in the space of $pi$. The sup-norm problem asks for bounds on $| phi |_infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|phi|^2 (g) , d g$ of $phi$. All previous work on these problems in the conductor-aspect has focused on the case that $phi$ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component $pi_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $phi$ for which the local components $phi_p in pi_p$ are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of $PGL_2(mathbb{R})$. For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if $pi_infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -epsilon} k^{1/4 - epsilon} ll_{epsilon} |phi|_infty ll_{epsilon} C^{1/8 + epsilon} k^{1/4+epsilon}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence for newforms of powerful level.
{"title":"Some analytic aspects of automorphic forms on GL(2) of minimal type","authors":"Yueke Hu, Paul D. Nelson, A. Saha","doi":"10.4171/cmh/473","DOIUrl":"https://doi.org/10.4171/cmh/473","url":null,"abstract":"Let $pi$ be a cuspidal automorphic representation of $PGL_2(mathbb{A}_mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $phi$ be an $L^2$-normalized automorphic form in the space of $pi$. The sup-norm problem asks for bounds on $| phi |_infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|phi|^2 (g) , d g$ of $phi$. All previous work on these problems in the conductor-aspect has focused on the case that $phi$ is a newform. \u0000In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component $pi_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $phi$ for which the local components $phi_p in pi_p$ are \"minimal\" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of $PGL_2(mathbb{R})$. \u0000For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if $pi_infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -epsilon} k^{1/4 - epsilon} ll_{epsilon} |phi|_infty ll_{epsilon} C^{1/8 + epsilon} k^{1/4+epsilon}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence for newforms of powerful level.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47689082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann hypothesis.
{"title":"Fourier optimization and prime gaps","authors":"E. Carneiro, M. Milinovich, K. Soundararajan","doi":"10.4171/CMH/467","DOIUrl":"https://doi.org/10.4171/CMH/467","url":null,"abstract":"We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann hypothesis.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/467","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48439160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao’s conjecture on the freeness of a line arrangement being determined by its combinatorics.
{"title":"On rational cuspidal plane curves and the local cohomology of Jacobian rings","authors":"A. Dimca","doi":"10.4171/cmh/471","DOIUrl":"https://doi.org/10.4171/cmh/471","url":null,"abstract":"This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao’s conjecture on the freeness of a line arrangement being determined by its combinatorics.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/471","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45348374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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