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The Singular Support of Sheaves Is γ-Coisotropic 剪切的奇异支持是 γ-各向异性的
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-07-01 DOI: 10.1007/s00039-024-00682-x
Stéphane Guillermou, Claude Viterbo
{"title":"The Singular Support of Sheaves Is γ-Coisotropic","authors":"Stéphane Guillermou, Claude Viterbo","doi":"10.1007/s00039-024-00682-x","DOIUrl":"https://doi.org/10.1007/s00039-024-00682-x","url":null,"abstract":"<p>We prove that the singular support of an element in the derived category of sheaves is <i>γ</i>-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense of Kashiwara-Schapira, but being <i>γ</i>-coisotropic has the advantage to be invariant by symplectic homeomorphisms (while involutivity is only invariant by <i>C</i><sup>1</sup> diffeomorphisms) and we give an example of an involutive set that is not <i>γ</i>-coisotropic. Along the way we prove a number of results relating the singular support and the spectral norm <i>γ</i> and raise a number of new questions.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fusion and Positivity in Chiral Conformal Field Theory 手性共形场论中的融合与正向性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-06-27 DOI: 10.1007/s00039-024-00685-8
James E. Tener
{"title":"Fusion and Positivity in Chiral Conformal Field Theory","authors":"James E. Tener","doi":"10.1007/s00039-024-00685-8","DOIUrl":"https://doi.org/10.1007/s00039-024-00685-8","url":null,"abstract":"<p>In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann’s landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. ‘fusion’) of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of ‘bounded localized vertex operators,’ which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141461902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Growth of k-Dimensional Systoles in Congruence Coverings 全等覆盖中 k 维收缩的增长
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-06-05 DOI: 10.1007/s00039-024-00686-7
Mikhail Belolipetsky, Shmuel Weinberger
{"title":"Growth of k-Dimensional Systoles in Congruence Coverings","authors":"Mikhail Belolipetsky, Shmuel Weinberger","doi":"10.1007/s00039-024-00686-7","DOIUrl":"https://doi.org/10.1007/s00039-024-00686-7","url":null,"abstract":"<p>We study growth of absolute and homological <i>k</i>-dimensional systoles of arithmetic <i>n</i>-manifolds along congruence coverings. Our main interest is in the growth of systoles of manifolds whose real rank <i>r</i>≥2. We observe, in particular, that in some cases for <i>k</i>=<i>r</i> the growth function tends to oscillate between a power of a logarithm and a power function of the degree of the covering. This is a new phenomenon. We also prove the expected polylogarithmic and constant power bounds for small and large <i>k</i>, respectively.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rigidity Theorems for Higher Rank Lattice Actions 高阶晶格作用的刚性定理
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-29 DOI: 10.1007/s00039-024-00683-w
Homin Lee
{"title":"Rigidity Theorems for Higher Rank Lattice Actions","authors":"Homin Lee","doi":"10.1007/s00039-024-00683-w","DOIUrl":"https://doi.org/10.1007/s00039-024-00683-w","url":null,"abstract":"<p>Let Γ be a weakly irreducible lattice in a higher rank semisimple algebraic Lie group <i>G</i> without property (T) such as <span>(mathrm {SL}_{2}({mathbb{Z}}[sqrt{2}]))</span> in <span>(mathrm {SL}_{2}({mathbb{R}})times mathrm {SL}_{2}({mathbb{R}}))</span>.</p><p>In this paper, for such Γ, we prove a cocycle superrigidity, <i>Dynamical cocycle superrigidity</i>, for Γ-actions under <i>L</i><sup>2</sup> integrability and irreducibility assumptions. It gives a similar result to Zimmer’s cocycle superrigidity, so we can apply it instead of Zimmer’s cocycle superrigidity in many situations. For instance, in this paper, we obtain global rigidity of Anosov Γ-actions on nilmanifolds under the irreducibility assumption on a fully supported invariant measure.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows 抛物线 U(1)-Higgs 方程与二维平均曲率流
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-29 DOI: 10.1007/s00039-024-00684-9
Davide Parise, Alessandro Pigati, Daniel Stern
{"title":"The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows","authors":"Davide Parise, Alessandro Pigati, Daniel Stern","doi":"10.1007/s00039-024-00684-9","DOIUrl":"https://doi.org/10.1007/s00039-024-00684-9","url":null,"abstract":"<p>We develop the asymptotic analysis as <i>ε</i>→0 for the natural gradient flow of the self-dual <i>U</i>(1)-Higgs energies </p><span>$$ E_{varepsilon }(u,nabla )=int _{M}left (|nabla u|^{2}+ varepsilon ^{2}|F_{nabla }|^{2}+ frac{(1-|u|^{2})^{2}}{4varepsilon ^{2}}right ) $$</span><p> on Hermitian line bundles over closed manifolds (<i>M</i><sup><i>n</i></sup>,<i>g</i>) of dimension <i>n</i>≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (<i>n</i>−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (<i>n</i>−2)-cycle Γ<sub>0</sub> in <i>M</i>, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual <i>U</i>(1)-Yang–Mills–Higgs energies to the (<i>n</i>−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ<sub>0</sub> with additional structure, similar to those produced via Ilmanen’s elliptic regularization.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Equilibrium States of Endomorphisms of $mathbb{P}^{k}$ : Spectral Stability and Limit Theorems $mathbb{P}^{k}$的均衡状态：谱稳定性与极限定理
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-13 DOI: 10.1007/s00039-024-00678-7
Fabrizio Bianchi, Tien-Cuong Dinh
{"title":"Equilibrium States of Endomorphisms of $mathbb{P}^{k}$ : Spectral Stability and Limit Theorems","authors":"Fabrizio Bianchi, Tien-Cuong Dinh","doi":"10.1007/s00039-024-00678-7","DOIUrl":"https://doi.org/10.1007/s00039-024-00678-7","url":null,"abstract":"<p>We establish the existence of a spectral gap for the transfer operator induced on <span>(mathbb{P}^{k} = mathbb{P}^{k} (mathbb{C}))</span> by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140914941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lagrangians, SO(3)-Instantons and Mixed Equation 拉格朗日、SO(3)-等式和混合方程
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-02 DOI: 10.1007/s00039-024-00677-8
Aliakbar Daemi, Kenji Fukaya, Maksim Lipyanskiy
{"title":"Lagrangians, SO(3)-Instantons and Mixed Equation","authors":"Aliakbar Daemi, Kenji Fukaya, Maksim Lipyanskiy","doi":"10.1007/s00039-024-00677-8","DOIUrl":"https://doi.org/10.1007/s00039-024-00677-8","url":null,"abstract":"<p>The <i>mixed equation</i>, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140819341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Commensurations of Aut(FN) and Its Torelli Subgroup Aut(FN) 及其 Torelli 子群的共形
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-04-22 DOI: 10.1007/s00039-024-00681-y
Martin R. Bridson, Richard D. Wade
{"title":"Commensurations of Aut(FN) and Its Torelli Subgroup","authors":"Martin R. Bridson, Richard D. Wade","doi":"10.1007/s00039-024-00681-y","DOIUrl":"https://doi.org/10.1007/s00039-024-00681-y","url":null,"abstract":"<p>For <i>N</i>≥3, the abstract commensurators of both Aut(<i>F</i><sub><i>N</i></sub>) and its Torelli subgroup IA<sub><i>N</i></sub> are isomorphic to Aut(<i>F</i><sub><i>N</i></sub>) itself.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140632140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sections and Unirulings of Families over $mathbb{P}^{1}$ $mathbb{P}^{1}$上族的分段和单圈
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-04-18 DOI: 10.1007/s00039-024-00679-6
Alex Pieloch
{"title":"Sections and Unirulings of Families over $mathbb{P}^{1}$","authors":"Alex Pieloch","doi":"10.1007/s00039-024-00679-6","DOIUrl":"https://doi.org/10.1007/s00039-024-00679-6","url":null,"abstract":"<p>We consider morphisms <span>(pi : X to mathbb{P}^{1})</span> of smooth projective varieties over <span>(mathbb{C})</span>. We show that if <i>π</i> has at most one singular fibre, then <i>X</i> is uniruled and <i>π</i> admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if <i>π</i> has at most two singular fibres, and the first Chern class of <i>X</i> is supported in a single fibre of <i>π</i>.</p><p>To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140608063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms 恒定周期数据的支配分裂与阿诺索夫自动形的全局刚性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-04-15 DOI: 10.1007/s00039-024-00680-z
Jonathan DeWitt, Andrey Gogolev
{"title":"Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms","authors":"Jonathan DeWitt, Andrey Gogolev","doi":"10.1007/s00039-024-00680-z","DOIUrl":"https://doi.org/10.1007/s00039-024-00680-z","url":null,"abstract":"<p>We show that a <span>(operatorname{GL}(d,mathbb{R}))</span> cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of <span>(mathbb{T}^{d})</span>. Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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