Pub Date : 2019-09-06DOI: 10.17323/1609-4514-2021-21-2-401-412
Dominique Mattei
In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.
{"title":"Categorical vs Topological Entropy of Autoequivalences of Surfaces","authors":"Dominique Mattei","doi":"10.17323/1609-4514-2021-21-2-401-412","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-401-412","url":null,"abstract":"In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44056340","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}
Pub Date : 2019-07-08DOI: 10.17323/1609-4514-2022-22-1-69-81
Alexander Dunaykin, V. Zhukov
To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. For a given 4-regular graph, we can build a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying 4-term relations and determining thus a knot invariant. We extend our function to ribbon graphs and further to binary delta-matroids and show that 4-term relations are satisfied for it.
{"title":"Transition Polynomial as a Weight System for Binary Delta-Matroids","authors":"Alexander Dunaykin, V. Zhukov","doi":"10.17323/1609-4514-2022-22-1-69-81","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-1-69-81","url":null,"abstract":"To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. For a given 4-regular graph, we can build a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying 4-term relations and determining thus a knot invariant. We extend our function to ribbon graphs and further to binary delta-matroids and show that 4-term relations are satisfied for it.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42900074","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}
Pub Date : 2019-05-21DOI: 10.17323/1609-4514-2021-21-4-659-694
T. Assiotis, J. Najnudel
The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov chain with given transition probabilities. This formulation of the problem makes sense for general $beta$-ensembles when one takes as the transition probabilities the Dixon-Anderson conditional probability distribution. In this paper we determine the boundary of this Markov chain for any $beta in (0,infty]$, also giving in this way a new proof of the classical $beta=2$ case. Finally, as a by-product of our results we obtain alternative proofs of the almost sure convergence of the rescaled Hua-Pickrell and Laguerre $beta$-ensembles to the general $beta$ Hua-Pickrell and $beta$ Bessel point processes respectively.
Pickrell、Olshanski和Vershik对无穷厄米矩阵上的酉不变概率测度进行了分类。这种分类等价于确定具有给定转移概率的非齐次马尔可夫链的边界。当将狄克逊-安德森条件概率分布作为转移概率时，这种问题的表述对一般$beta$ -系综是有意义的。本文确定了任意$beta in (0,infty]$情况下马尔可夫链的边界，并由此给出了经典$beta=2$情况的一个新的证明。最后，作为我们的结果的一个副产品，我们分别获得了重新标定的Hua-Pickrell和Laguerre $beta$ -系综对一般的$beta$ Hua-Pickrell和$beta$ Bessel点过程几乎肯定收敛的替代证明。
{"title":"The Boundary of the Orbital Beta Process","authors":"T. Assiotis, J. Najnudel","doi":"10.17323/1609-4514-2021-21-4-659-694","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-4-659-694","url":null,"abstract":"The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov chain with given transition probabilities. This formulation of the problem makes sense for general $beta$-ensembles when one takes as the transition probabilities the Dixon-Anderson conditional probability distribution. In this paper we determine the boundary of this Markov chain for any $beta in (0,infty]$, also giving in this way a new proof of the classical $beta=2$ case. Finally, as a by-product of our results we obtain alternative proofs of the almost sure convergence of the rescaled Hua-Pickrell and Laguerre $beta$-ensembles to the general $beta$ Hua-Pickrell and $beta$ Bessel point processes respectively.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44121323","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}
Pub Date : 2019-05-13DOI: 10.17323/1609-4514-2019-19-2-189-216
Kristian Bjerklöv
We investigate the dynamics of certain homeomorphisms F: T-2 -> T-2 of the form F(x, y) = (x + omega , h(x)+ f (y)), where omega is an element of RQ, f: T -> T is a circle diffeomorphism wit ...
{"title":"Quasi-Periodic Kicking of Circle Diffeomorphisms Having Unique Fixed Points","authors":"Kristian Bjerklöv","doi":"10.17323/1609-4514-2019-19-2-189-216","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-2-189-216","url":null,"abstract":"We investigate the dynamics of certain homeomorphisms F: T-2 -> T-2 of the form F(x, y) = (x + omega , h(x)+ f (y)), where omega is an element of RQ, f: T -> T is a circle diffeomorphism wit ...","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42587700","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}
Pub Date : 2019-05-06DOI: 10.17323/1609-4514-2021-21-4-695-736
V. Cort'es, L. David
We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators for regular Courant algebroids. As an application we provide a criterion for the integrability of generalized almost Hermitian structures and generalized almost hyper-Hermitian structures defined on a regular Courant algebroid E, in terms of canonically defined differential operators on spinor bundles associated to E.
{"title":"Generalized Connections, Spinors, and Integrability of Generalized Structures on Courant Algebroids","authors":"V. Cort'es, L. David","doi":"10.17323/1609-4514-2021-21-4-695-736","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-4-695-736","url":null,"abstract":"We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators for regular Courant algebroids. As an application we provide a criterion for the integrability of generalized almost Hermitian structures and generalized almost hyper-Hermitian structures defined on a regular Courant algebroid E, in terms of canonically defined differential operators on spinor bundles associated to E.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41985351","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}
Pub Date : 2019-03-09DOI: 10.17323/1609-4514-2022-22-1-83-102
A. Gaifullin
The action of the mapping class group $mathrm{Mod}_g$ of an oriented surface $Sigma_g$ on the lower central series of $pi_1(Sigma_g)$ defines the descending filtration in $mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $mathcal{I}_g$ and the Johnson kernel $mathcal{K}_g$. By a fundamental result of Johnson (1985), $mathcal{K}_g$ is the subgroup of $mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(mathcal{K}_g,mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(mathcal{K}_g,mathbb{Q})$ is not finitely generated as a module over the group ring $mathbb{Q}[mathcal{I}_g]$.
{"title":"On the Top Homology Group of the Johnson Kernel","authors":"A. Gaifullin","doi":"10.17323/1609-4514-2022-22-1-83-102","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-1-83-102","url":null,"abstract":"The action of the mapping class group $mathrm{Mod}_g$ of an oriented surface $Sigma_g$ on the lower central series of $pi_1(Sigma_g)$ defines the descending filtration in $mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $mathcal{I}_g$ and the Johnson kernel $mathcal{K}_g$. By a fundamental result of Johnson (1985), $mathcal{K}_g$ is the subgroup of $mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(mathcal{K}_g,mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(mathcal{K}_g,mathbb{Q})$ is not finitely generated as a module over the group ring $mathbb{Q}[mathcal{I}_g]$.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46345034","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}
Pub Date : 2019-02-18DOI: 10.17323/1609-4514-2022-22-2-239-263
R. Bandiera, M. Manetti, Francesco Meazzini
We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.
{"title":"Deformations of Polystable Sheaves on Surfaces: Quadraticity Implies Formality","authors":"R. Bandiera, M. Manetti, Francesco Meazzini","doi":"10.17323/1609-4514-2022-22-2-239-263","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-2-239-263","url":null,"abstract":"We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44897934","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}
Pub Date : 2019-01-24DOI: 10.17323/1609-4514-2020-20-4-711-740
H. Duminil-Copin, V. Tassion
The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including speed of mixing, tails of decay of the connectivity probabilities, scaling relations, etc. In this article, we develop a renormalization scheme for crossing probabilities in the two-dimensional random-cluster model. The outcome of the process is a precise description of an alternative between four behaviors: - Subcritical: Crossing probabilities, even with favorable boundary conditions, converge exponentially fast to 0. - Supercritical: Crossing probabilities, even with unfavorable boundary conditions, converge exponentially fast to 1. - Critical discontinuous: Crossing probabilities converge to 0 exponentially fast with unfavorable boundary conditions and to 1 with favorable boundary conditions. - Critical continuous: Crossing probabilities remain bounded away from 0 and 1 uniformly in the boundary conditions. The approach does not rely on self-duality, enabling it to apply in a much larger generality, including the random-cluster model on arbitrary graphs with sufficient symmetry, but also other models like certain random height models.
{"title":"Renormalization of Crossing Probabilities in the Planar Random-Cluster Model","authors":"H. Duminil-Copin, V. Tassion","doi":"10.17323/1609-4514-2020-20-4-711-740","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-20-4-711-740","url":null,"abstract":"The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including speed of mixing, tails of decay of the connectivity probabilities, scaling relations, etc. In this article, we develop a renormalization scheme for crossing probabilities in the two-dimensional random-cluster model. The outcome of the process is a precise description of an alternative between four behaviors: \u0000- Subcritical: Crossing probabilities, even with favorable boundary conditions, converge exponentially fast to 0. \u0000- Supercritical: Crossing probabilities, even with unfavorable boundary conditions, converge exponentially fast to 1. \u0000- Critical discontinuous: Crossing probabilities converge to 0 exponentially fast with unfavorable boundary conditions and to 1 with favorable boundary conditions. \u0000- Critical continuous: Crossing probabilities remain bounded away from 0 and 1 uniformly in the boundary conditions. \u0000The approach does not rely on self-duality, enabling it to apply in a much larger generality, including the random-cluster model on arbitrary graphs with sufficient symmetry, but also other models like certain random height models.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44753109","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}
Pub Date : 2019-01-21DOI: 10.17323/1609-4514-2022-22-2-225-237
Robert Auffarth, G. Arteche
We give an explicit characterization of all principally polarized abelian varieties $(A,Theta)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Theta$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobian varieties of curves.
{"title":"Smooth Quotients of Principally Polarized Abelian Varieties","authors":"Robert Auffarth, G. Arteche","doi":"10.17323/1609-4514-2022-22-2-225-237","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-2-225-237","url":null,"abstract":"We give an explicit characterization of all principally polarized abelian varieties $(A,Theta)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Theta$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobian varieties of curves.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42109438","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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