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The quantum Witten–Kontsevich series andone-part double Hurwitz numbers 量子Witten-Kontsevich级数和一元双Hurwitz数
Geometry & Topology Pub Date : 2020-04-16 DOI: 10.2140/gt.2022.26.1669
X. Blot
{"title":"The quantum Witten–Kontsevich series and\u0000one-part double Hurwitz numbers","authors":"X. Blot","doi":"10.2140/gt.2022.26.1669","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1669","url":null,"abstract":"We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Guere and Rossi in cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $epsilon$ and a quantum parameter $hbar$. When $hbar=0$, this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves. We establish a link between the $epsilon=0$ part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number non-equivalent holomorphic maps from a Riemann surface of genus $g$ to $mathbb{P}^{1}$ with a prescribe ramification profile over $0$, a complete ramification over $infty$ and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil proved in cite{goulden2005towards} that these numbers have the property to be polynomial in the orders of ramification over $0$. We prove that the coefficients of these polynomials are the coefficients of the quantum Witten-Kontsevich series. We also present some partial results about the full quantum Witten-Kontsevich power series.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123088809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Anosov groups: local mixing, counting and equidistribution 阿诺索夫群:局部混合、计数和均分
Geometry & Topology Pub Date : 2020-03-31 DOI: 10.2140/gt.2023.27.513
Samuel Edwards, Minju M. Lee, H. Oh
{"title":"Anosov groups: local mixing, counting and equidistribution","authors":"Samuel Edwards, Minju M. Lee, H. Oh","doi":"10.2140/gt.2023.27.513","DOIUrl":"https://doi.org/10.2140/gt.2023.27.513","url":null,"abstract":"For a Zariski dense Anosov subgroup $Gamma$ of a semisimple real Lie group $G$, we describe the asymptotic behavior of matrix coefficients $Phi(g)=langle g f_1, f_2rangle$ in $L^2(Gammabackslash G)$ for local functions $f_1, f_2in C_c(Gammabackslash G)$. These asymptotics involve higher rank analogues of Burger-Roblin measures. As an application, for any symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $Gamma$-orbits in affine symmetric spaces $Hbackslash G$. The link between mixing and counting is provided by an equidistribution result for the translates $Gammabackslash Gamma H a$ as $ato infty$ in $Hbackslash G$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128875934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 45
Isotopy of the Dehn twist on K3 # K3 after asingle stabilization 一次稳定后K3 # K3上Dehn捻的同位素
Geometry & Topology Pub Date : 2020-03-09 DOI: 10.2140/gt.2023.27.1987
Jianfeng Lin
{"title":"Isotopy of the Dehn twist on K3 # K3 after a\u0000single stabilization","authors":"Jianfeng Lin","doi":"10.2140/gt.2023.27.1987","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1987","url":null,"abstract":"Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of $K3#K3$ is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.) In this paper, we use the Pin(2)-equivariant Bauer-Furuta invariant to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on $S^{2}times S^{2}$). This gives the first example of exotic phenomena on simply connected smooth 4-manifolds that do not disappear after a single stabilization.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"152 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132904550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
Counting hyperbolic multigeodesics with respectto the lengths of individual components and asymptotics of Weil–Peterssonvolumes 关于单个分量长度和Weil-Peterssonvolumes渐近性的双曲多重测地线计数
Geometry & Topology Pub Date : 2020-02-25 DOI: 10.2140/gt.2022.26.1291
Francisco Arana-Herrera
{"title":"Counting hyperbolic multigeodesics with respect\u0000to the lengths of individual components and asymptotics of Weil–Petersson\u0000volumes","authors":"Francisco Arana-Herrera","doi":"10.2140/gt.2022.26.1291","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1291","url":null,"abstract":"Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L to infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a conjecture of Wolpert. In the simple case we consider more precise countings that also keep track of the class of the multi-geodesics in the space of projective measured geodesic laminations. We provide a unified geometric and topological description of the leading terms of the asymptotics of all the countings considered. Our proofs combine techniques and results from several papers of Mirzakhani as well as ideas introduced by Margulis in his thesis.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132312238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
A homological model for Uq𝔰𝔩2 Verma modulesand their braid representations Uq𝔩2 Verma模的同调模型及其辫状表示
Geometry & Topology Pub Date : 2020-02-20 DOI: 10.2140/gt.2022.26.1225
Jules Martel
{"title":"A homological model for Uq𝔰𝔩2 Verma modules\u0000and their braid representations","authors":"Jules Martel","doi":"10.2140/gt.2022.26.1225","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1225","url":null,"abstract":"We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. We define homological operators and we show that they actually provide a representation for an integral version for $U_q mathfrak{sl}(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules, and we show it to preserve the integral ring of coefficients, the action of $U_q mathfrak{sl}(2)$, the braid group representations and their grading. This recovers an integral version for Kohno's theorem relating absolute Lawrence representations with quantum braid representation on highest weight vectors. It is an extension of the latter theorem as we get rid of generic conditions on parameters, and as we recover the entire product of Verma-modules as a braid group and a $U_q mathfrak{sl}(2)$-module.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114557806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Compactifications of moduli of elliptic K3 surfaces: Stable pair and toroidal 椭圆型K3曲面模的紧化:稳定对和环面
Geometry & Topology Pub Date : 2020-02-17 DOI: 10.2140/gt.2022.26.3525
V. Alexeev, Adrian Brunyate, P. Engel
{"title":"Compactifications of moduli of elliptic K3 surfaces: Stable pair and toroidal","authors":"V. Alexeev, Adrian Brunyate, P. Engel","doi":"10.2140/gt.2022.26.3525","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3525","url":null,"abstract":"We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129174064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 20
Birational geometry of the intermediate Jacobian fibration of a cubic fourfold 三次四重的中间雅可比颤振的双几何
Geometry & Topology Pub Date : 2020-02-04 DOI: 10.2140/gt.2023.27.1479
Giulia Saccà, with an appendix by Claire Voisin
{"title":"Birational geometry of the intermediate Jacobian fibration of a cubic fourfold","authors":"Giulia Saccà, with an appendix by Claire Voisin","doi":"10.2140/gt.2023.27.1479","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1479","url":null,"abstract":"We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold $X$ admits a hyper-K\"ahler compactification $J(X)$ with a regular Lagrangian fibration $J to mathbb P^5$. This builds upon arXiv:1602.05534, where the result is proved for general $X$, as well as on the degeneration techniques on arXiv:1704.02731 and techniques from the minimal model program. We then study some aspects of the birational geometry of $J(X)$: for very general $X$ we compute the movable and nef cones of $J(X)$, showing that $J(X)$ is not birational to the twisted version of the intermediate Jacobian fibration arXiv:1611.06679, nor to an OG$10$-type moduli space of objects in the Kuznetsov component of $X$; for any smooth $X$ we show, using normal functions, that the Mordell-Weil group $MW(pi)$ of the abelian fibration $pi: J to mathbb P^5$ is isomorphic to the integral degree $4$ primitive algebraic cohomology of $X$, i.e., $MW(pi) = H^{2,2}(X, mathbb Z)_0$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126315563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 18
From flops to diffeomorphism groups 从簇到微分同构群
Geometry & Topology Pub Date : 2020-02-04 DOI: 10.2140/gt.2022.26.875
G. Smirnov
{"title":"From flops to diffeomorphism groups","authors":"G. Smirnov","doi":"10.2140/gt.2022.26.875","DOIUrl":"https://doi.org/10.2140/gt.2022.26.875","url":null,"abstract":"We exhibit many examples of closed complex surfaces whose diffeomorphism groups are not simply-connected and contain loops that are not homotopic to loops of symplectomorphisms.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115660984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Abelian quotients of the Y –filtration on thehomology cylinders via the LMO functor 通过LMO函子在同调圆柱体上的Y过滤的阿贝尔商
Geometry & Topology Pub Date : 2020-01-27 DOI: 10.2140/gt.2022.26.221
Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki
{"title":"Abelian quotients of the Y –filtration on the\u0000homology cylinders via the LMO functor","authors":"Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki","doi":"10.2140/gt.2022.26.221","DOIUrl":"https://doi.org/10.2140/gt.2022.26.221","url":null,"abstract":"We construct a series of homomorphisms on the $Y$-filtration on the homology cylinders via the mod $mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3mathcal{C}_{g,1}/Y_4$ of the $Y$-filtration.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125632921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
On the existence of minimal hypersurfaces with arbitrarily large area and Morse index 关于任意大面积和莫尔斯指数的极小超曲面的存在性
Geometry & Topology Pub Date : 2020-01-21 DOI: 10.2140/gt.2022.26.2713
Yangyang Li
{"title":"On the existence of minimal hypersurfaces with arbitrarily large area and Morse index","authors":"Yangyang Li","doi":"10.2140/gt.2022.26.2713","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2713","url":null,"abstract":"We show that a bumpy closed Riemannian manifold $(M^{n+1}, g)$ $(3 leq n+1 leq 7)$ admits a sequence of connected closed embedded two-sided minimal hypersurfaces whose areas and Morse indices both tend to infinity. This improves a previous result by O. Chodosh and C. Mantoulidis on connected minimal hypersurfaces with arbitrarily large area.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124153648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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