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Tautological rings of Hilbert modular varieties 希尔伯特模态变的同调环
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-08-02 DOI: 10.1007/s00209-024-03560-2
Simon Cooper
{"title":"Tautological rings of Hilbert modular varieties","authors":"Simon Cooper","doi":"10.1007/s00209-024-03560-2","DOIUrl":"https://doi.org/10.1007/s00209-024-03560-2","url":null,"abstract":"<p>In this note we compute the tautological ring of Hilbert modular varieties at an unramified prime. This is the first computation of the tautological ring of a non-compactified Shimura variety beyond the case of the Siegel modular variety <span>(mathcal {A}_{g})</span>. While the method generalises that of van der Geer for <span>(mathcal {A}_{g})</span>, there is an added difficulty in that the highest degree socle has <span>(d&gt;1)</span> generators rather than 1. To deal with this we prove that the <i>d</i> cycles obtained by taking closures of codimension one Ekedahl–Oort strata are linearly independent. In contrast, in the case of <span>(mathcal {A}_{g})</span> it suffices to prove that the class of the <i>p</i>-rank zero locus is non-zero. The limitations of this method for computing the tautological ring of other non-compactified Shimura varieties are demonstrated with an instructive example.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881678","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}
引用次数: 0
Algebraic characterizations of homeomorphisms between algebraic varieties 代数变体间同构的代数特征
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-31 DOI: 10.1007/s00209-024-03533-5
François Bernard, Goulwen Fichou, Jean-Philippe Monnier, Ronan Quarez
{"title":"Algebraic characterizations of homeomorphisms between algebraic varieties","authors":"François Bernard, Goulwen Fichou, Jean-Philippe Monnier, Ronan Quarez","doi":"10.1007/s00209-024-03533-5","DOIUrl":"https://doi.org/10.1007/s00209-024-03533-5","url":null,"abstract":"<p>We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886912","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}
引用次数: 0
Sequence of families of lattice polarized K3 surfaces, modular forms and degrees of complex reflection groups 晶格极化 K3 表面、模块形式和复反射群度的族序列
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-30 DOI: 10.1007/s00209-024-03562-0
Atsuhira Nagano
{"title":"Sequence of families of lattice polarized K3 surfaces, modular forms and degrees of complex reflection groups","authors":"Atsuhira Nagano","doi":"10.1007/s00209-024-03562-0","DOIUrl":"https://doi.org/10.1007/s00209-024-03562-0","url":null,"abstract":"<p>We introduce a sequence of families of lattice polarized <i>K</i>3 surfaces. This sequence is closely related to complex reflection groups of exceptional type. Namely, we obtain modular forms coming from the inverse correspondences of the period mappings attached to our sequence. We study a non-trivial relation between our modular forms and invariants of complex reflection groups. Especially, we consider a family concerned with the Shephard-Todd group No.34 based on arithmetic properties of lattices and algebro-geometric properties of the period mappings.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865134","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}
引用次数: 0
Gap Theorem on manifolds with small curvature concentration 小曲率集中流形上的差距定理
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-30 DOI: 10.1007/s00209-024-03570-0
Pak-Yeung Chan, Man-Chun Lee
{"title":"Gap Theorem on manifolds with small curvature concentration","authors":"Pak-Yeung Chan, Man-Chun Lee","doi":"10.1007/s00209-024-03570-0","DOIUrl":"https://doi.org/10.1007/s00209-024-03570-0","url":null,"abstract":"<p>In this work, we show that complete non-compact manifolds with non-negative Ricci curvature, Euclidean volume growth and sufficiently small curvature concentration are necessarily flat Euclidean space.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865197","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}
引用次数: 0
Synchronization rates and limit laws for random dynamical systems 随机动力系统的同步率和极限规律
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-30 DOI: 10.1007/s00209-024-03571-z
Katrin Gelfert, Graccyela Salcedo
{"title":"Synchronization rates and limit laws for random dynamical systems","authors":"Katrin Gelfert, Graccyela Salcedo","doi":"10.1007/s00209-024-03571-z","DOIUrl":"https://doi.org/10.1007/s00209-024-03571-z","url":null,"abstract":"<p>We study general random dynamical systems of continuous maps on some compact metric space. Assuming a local contraction condition and proximality, we establish probabilistic limit laws such as the (functional) central limit theorem, the strong law of large numbers, and the law of the iterated logarithm. Moreover, we study exponential synchronization and synchronization on average. In the particular case of iterated function systems on <span>({mathbb {S}}^1)</span>, we analyze synchronization rates and describe their large deviations. In the case of <span>(C^{1+beta })</span>-diffeomorphisms, these deviations on random orbits are obtained from the large deviations of the expected Lyapunov exponent.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865286","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}
引用次数: 0
Arboreal Galois groups for quadratic rational functions with colliding critical points 具有碰撞临界点的二次有理函数的阿贝尔伽罗瓦群
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-29 DOI: 10.1007/s00209-024-03566-w
Robert L. Benedetto, Anna Dietrich
{"title":"Arboreal Galois groups for quadratic rational functions with colliding critical points","authors":"Robert L. Benedetto, Anna Dietrich","doi":"10.1007/s00209-024-03566-w","DOIUrl":"https://doi.org/10.1007/s00209-024-03566-w","url":null,"abstract":"<p>Let <i>K</i> be a field, and let <span>(fin K(z))</span> be rational function. The preimages of a point <span>(x_0in mathbb {P}^1(K))</span> under iterates of <i>f</i> have a natural tree structure. As a result, the Galois group of the resulting field extension of <i>K</i> naturally embeds into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup <span>(M_{ell })</span> that this so-called arboreal Galois group <span>(G_{infty })</span> must lie in if <i>f</i> is quadratic and its two critical points collide at the <span>(ell )</span>-th iteration. After presenting a new description of <span>(M_{ell })</span> and a new proof of Pink’s theorem, we state and prove necessary and sufficient conditions for <span>(G_{infty })</span> to be the full group <span>(M_{ell })</span>.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865195","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}
引用次数: 0
Abelian envelopes of exact categories and highest weight categories 精确范畴的阿贝尔包络和最高权重范畴
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-29 DOI: 10.1007/s00209-024-03543-3
Agnieszka Bodzenta, Alexey Bondal
{"title":"Abelian envelopes of exact categories and highest weight categories","authors":"Agnieszka Bodzenta, Alexey Bondal","doi":"10.1007/s00209-024-03543-3","DOIUrl":"https://doi.org/10.1007/s00209-024-03543-3","url":null,"abstract":"<p>We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional sequences. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proven for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality on highest weight categories is interpreted as a duality between the right and left abelian envelopes of a thin exact category. A duality for thin exact categories compatible with Ringel duality is introduced by means of derived categories and Serre functor on them.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865196","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}
引用次数: 0
Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds CR 流形上托普利兹算子的函数微积分和量化与还原相通
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-27 DOI: 10.1007/s00209-024-03561-1
Andrea Galasso, Chin-Yu Hsiao
{"title":"Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds","authors":"Andrea Galasso, Chin-Yu Hsiao","doi":"10.1007/s00209-024-03561-1","DOIUrl":"https://doi.org/10.1007/s00209-024-03561-1","url":null,"abstract":"<p>Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szegő type. As an application, we establish semi-classical asymptotics for the dimension of the spectral spaces of Toeplitz operators. We then consider a CR manifold with a compact Lie group action <i>G</i> and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical asymptotics for the dimension of the spectral spaces of <i>G</i>-invariant Toeplitz operators.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785144","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}
引用次数: 0
Automorphism group functors of algebraic superschemes 代数超hemes的自形群函数
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-27 DOI: 10.1007/s00209-024-03572-y
A. N. Zubkov
{"title":"Automorphism group functors of algebraic superschemes","authors":"A. N. Zubkov","doi":"10.1007/s00209-024-03572-y","DOIUrl":"https://doi.org/10.1007/s00209-024-03572-y","url":null,"abstract":"<p>The famous theorem of Matsumura–Oort states that if <i>X</i> is a proper scheme, then the automorphism group functor <span>(mathfrak {Aut}(X))</span> of <i>X</i> is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if <span>({mathbb {X}})</span> is a proper superscheme, then the automorphism group functor <span>(mathfrak {Aut}({mathbb {X}}))</span> of <span>({mathbb {X}})</span> is a locally algebraic group superscheme. Moreover, we also show that if <span>(H^1(X, {mathchoice{text{ T }}{text{ T }}{text{ T }}{text{ T }}}_X)=0)</span>, where <i>X</i> is the geometric counterpart of <span>({mathbb {X}})</span> and <span>({mathchoice{text{ T }}{text{ T }}{text{ T }}{text{ T }}}_X)</span> is the tangent sheaf of <i>X</i>, then <span>(mathfrak {Aut}({mathbb {X}}))</span> is a smooth group superscheme.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785143","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}
引用次数: 0
The $$P=W$$ identity for isolated cluster varieties: full rank case 孤立群集品种的 $$P=W$$ 特性：全秩情况
IF 0.8 3区 数学
Mathematische Zeitschrift Pub Date : 2024-07-27 DOI: 10.1007/s00209-024-03568-8
Zili Zhang
{"title":"The $$P=W$$ identity for isolated cluster varieties: full rank case","authors":"Zili Zhang","doi":"10.1007/s00209-024-03568-8","DOIUrl":"https://doi.org/10.1007/s00209-024-03568-8","url":null,"abstract":"<p>We initiate a systematic construction of real analytic Lagrangian fibrations from integer matrices. We prove that when the matrix is of full column rank, the perverse filtration associated with the Lagrangian fibration matches the mixed Hodge-theoretic weight filtration of the isolated cluster variety associated with the matrix.</p>","PeriodicalId":18278,"journal":{"name":"Mathematische Zeitschrift","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785147","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}
引用次数: 0
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