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The Nielsen realization problem for high degree del Pezzo surfaces 高阶德尔佩佐曲面的尼尔森实现问题
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-08 DOI: 10.1007/s10711-024-00912-4
Seraphina Eun Bi Lee
{"title":"The Nielsen realization problem for high degree del Pezzo surfaces","authors":"Seraphina Eun Bi Lee","doi":"10.1007/s10711-024-00912-4","DOIUrl":"https://doi.org/10.1007/s10711-024-00912-4","url":null,"abstract":"<p>Let <i>M</i> be a smooth 4-manifold underlying some del Pezzo surface of degree <span>(d ge 6)</span>. We consider the smooth Nielsen realization problem for <i>M</i>: which finite subgroups of <span>({{,textrm{Mod},}}(M) = pi _0({{,textrm{Homeo},}}^+(M)))</span> have lifts to <span>({{,textrm{Diff},}}^+(M) le {{,textrm{Homeo},}}^+(M))</span> under the quotient map <span>(pi : {{,textrm{Homeo},}}^+(M) rightarrow {{,textrm{Mod},}}(M))</span>? We give a complete classification of such finite subgroups of <span>({{,textrm{Mod},}}(M))</span> for <span>(d ge 7)</span> and a partial answer for <span>(d = 6)</span>. For the cases <span>(d ge 8)</span>, the quotient map <span>(pi )</span> admits a section with image contained in <span>({{,textrm{Diff},}}^+(M))</span>. For the case <span>(d = 7)</span>, we show that all finite order elements of <span>({{,textrm{Mod},}}(M))</span> have lifts to <span>({{,textrm{Diff},}}^+(M))</span>, but there are finite subgroups of <span>({{,textrm{Mod},}}(M))</span> that do not lift to <span>({{,textrm{Diff},}}^+(M))</span>. We prove that the condition of whether a finite subgroup <span>(G le {{,textrm{Mod},}}(M))</span> lifts to <span>({{,textrm{Diff},}}^+(M))</span> is equivalent to the existence of a certain equivariant connected sum realizing <i>G</i>. For the case <span>(d = 6)</span>, we show this equivalence for all maximal finite subgroups <span>(G le {{,textrm{Mod},}}(M))</span>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"149 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577830","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}
引用次数: 0
On the Teichmüller stack of compact quotients of $${text {SL}}_2({mathbb {C}})$$ 论 $${text {SL}}_2({mathbb {C}})$$ 的紧凑商的泰希米勒堆栈
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-04 DOI: 10.1007/s10711-024-00916-0
Théo Jamin
{"title":"On the Teichmüller stack of compact quotients of $${text {SL}}_2({mathbb {C}})$$","authors":"Théo Jamin","doi":"10.1007/s10711-024-00916-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00916-0","url":null,"abstract":"<p>This article aims to pursue and generalize, by using the global point of view offered by the stacks, the local study made by <span>Ghys</span> (J für die reine und angewandte Mathematik 468:113–138, 1995) concerning the deformations of complex structures of compact quotients of <span>({text {SL}}_2({mathbb {C}}))</span>. In his article, <span>Ghys</span> showed that the analytic germ of the representation variety <span>({mathcal {R}}(varGamma ):={text {Hom}}(varGamma ,{text {SL}}_2({mathbb {C}})))</span> of <span>(varGamma )</span> in <span>({text {SL}}_2({mathbb {C}}))</span>, pointed at the trivial morphism, determines the Kuranishi space of <span>({text {SL}}_2({mathbb {C}})/varGamma )</span>. In this note, we show that the tautological family above a Zariski analytic open subset <i>V</i> in <span>({mathcal {R}}(varGamma ))</span> remains complete. Moreover, the computation of the isotropy group of a complex structure in Teichmüller space, allows us to affirm that the quotient stack <span>([V/{text {SL}}_2({mathbb {C}})])</span> is an open substack of the Teichmüller stack of <span>({text {SL}}_2({mathbb {C}})/varGamma )</span>.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577949","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}
引用次数: 0
Boundary metric of Epstein-Penner convex hull and discrete conformality 爱泼斯坦-彭纳凸壳的边界度量和离散保角性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00901-7
Xin Nie
{"title":"Boundary metric of Epstein-Penner convex hull and discrete conformality","authors":"Xin Nie","doi":"10.1007/s10711-024-00901-7","DOIUrl":"https://doi.org/10.1007/s10711-024-00901-7","url":null,"abstract":"<p>The Epstein-Penner convex hull construction associates to every decorated punctured hyperbolic surface a convex set in the Minkowski space. It works in the de Sitter and anti-de Sitter spaces as well. In these three spaces, the quotient of the spacelike boundary part of the convex set has an induced Euclidean, spherical and hyperbolic metric, respectively, with conical singularities. We show that this gives a bijection from the decorated Teichmüller space to a moduli space of such metrics in the Euclidean and hyperbolic cases, as well as a bijection between specific subspaces of them in the spherical case. Moreover, varying the decoration of a fixed hyperbolic surface corresponds to a discrete conformal change of the metric. This gives a new 3-dimensional interpretation of discrete conformality which is in a sense inverse to the Bobenko-Pinkall-Springborn interpretation.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"54 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577732","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}
引用次数: 0
Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds 投影结构的整体性变化及其在钻孔双曲3-manifolds中的应用
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00908-0
Martin Bridgeman, Kenneth Bromberg
{"title":"Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds","authors":"Martin Bridgeman, Kenneth Bromberg","doi":"10.1007/s10711-024-00908-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00908-0","url":null,"abstract":"<p>We bound the derivative of complex length of a geodesic under variation of the projective structure on a closed surface in terms of the norm of the Schwarzian in a neighborhood of the geodesic. One application is to cone-manifold deformations of acylindrical hyperbolic 3-manifolds.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"90 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577738","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}
引用次数: 0
Halfspaces and hypersurfaces in the bidisk 双盘中的半空间和超曲面
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00917-z
Virginie Charette, Youngju Kim
{"title":"Halfspaces and hypersurfaces in the bidisk","authors":"Virginie Charette, Youngju Kim","doi":"10.1007/s10711-024-00917-z","DOIUrl":"https://doi.org/10.1007/s10711-024-00917-z","url":null,"abstract":"<p>We construct a halfspace in the bidisk, whose boundary acts like a bisector. As an application, we build a fundamental domain consisting of such halfspaces for the action of groups that project to Schottky groups in both factors.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"31 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577740","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}
引用次数: 0
Minimal surfaces and Colding-Minicozzi entropy in complex hyperbolic space 复双曲空间中的极小曲面和科尔丁-米尼科齐熵
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-03 DOI: 10.1007/s10711-024-00906-2
Jacob Bernstein, Arunima Bhattacharya
{"title":"Minimal surfaces and Colding-Minicozzi entropy in complex hyperbolic space","authors":"Jacob Bernstein, Arunima Bhattacharya","doi":"10.1007/s10711-024-00906-2","DOIUrl":"https://doi.org/10.1007/s10711-024-00906-2","url":null,"abstract":"<p>We study notions of asymptotic regularity for a class of minimal submanifolds of complex hyperbolic space that includes minimal Lagrangian submanifolds. As an application, we show a relationship between an appropriate formulation of Colding-Minicozzi entropy and a quantity we call the <i>CR</i>-volume that is computed from the asymptotic geometry of such submanifolds.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"43 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577950","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}
引用次数: 0
Continuous deformation of the Bowen-Series map associated to a cocompact triangle group 与协整三角形群相关的鲍温系列图的连续变形
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-04-02 DOI: 10.1007/s10711-024-00887-2
{"title":"Continuous deformation of the Bowen-Series map associated to a cocompact triangle group","authors":"","doi":"10.1007/s10711-024-00887-2","DOIUrl":"https://doi.org/10.1007/s10711-024-00887-2","url":null,"abstract":"<h3>Abstract</h3> <p>In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on <span> <span>({mathbb {S}}^1)</span> </span> tightly associated with this group. In general, their fundamental domain enjoys what has since been called both the ‘extension property’ and the ‘even corners property’. We determine the exact set of signatures for cocompact triangle groups for which this property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578025","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}
引用次数: 0
Prym representations of the handlebody group 手柄体群的普赖姆表征
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-03-29 DOI: 10.1007/s10711-024-00911-5
{"title":"Prym representations of the handlebody group","authors":"","doi":"10.1007/s10711-024-00911-5","DOIUrl":"https://doi.org/10.1007/s10711-024-00911-5","url":null,"abstract":"<h3>Abstract</h3> <p>Let <em>S</em> be an oriented, closed surface of genus <em>g</em>. The mapping class group of <em>S</em> is the group of orientation preserving homeomorphisms of <em>S</em> modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let <em>V</em> be a genus <em>g</em> handlebody with boundary <em>S</em>. The handlebody group is the subgroup of those mapping classes of <em>S</em> that extend over <em>V</em>. The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"47 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577943","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}
引用次数: 0
Mirror stabilizers for lattice complex hyperbolic triangle groups 晶格复双曲三角群的镜像稳定器
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-03-26 DOI: 10.1007/s10711-024-00910-6
Martin Deraux
{"title":"Mirror stabilizers for lattice complex hyperbolic triangle groups","authors":"Martin Deraux","doi":"10.1007/s10711-024-00910-6","DOIUrl":"https://doi.org/10.1007/s10711-024-00910-6","url":null,"abstract":"<p>For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"21 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315858","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}
引用次数: 0
K3 surfaces with two involutions and low Picard number 具有两个渐开线和低皮卡数的 K3 曲面
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-03-13 DOI: 10.1007/s10711-024-00900-8
Dino Festi, Wim Nijgh, Daniel Platt
{"title":"K3 surfaces with two involutions and low Picard number","authors":"Dino Festi, Wim Nijgh, Daniel Platt","doi":"10.1007/s10711-024-00900-8","DOIUrl":"https://doi.org/10.1007/s10711-024-00900-8","url":null,"abstract":"<p>Let <i>X</i> be a complex algebraic K3 surface of degree 2<i>d</i> and with Picard number <span>(rho )</span>. Assume that <i>X</i> admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, <span>(rho ge 1)</span> when <span>(d=1)</span> and <span>(rho ge 2)</span> when <span>(d ge 2)</span>. For <span>(d=1)</span>, the first example defined over <span>({mathbb {Q}})</span> with <span>(rho =1)</span> was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over <span>({mathbb {Q}})</span>, can be used to realise the minimum <span>(rho =2)</span> for all <span>(dge 2)</span>. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum <span>(rho =2)</span> for <span>(d=2,3,4)</span>. We also show that a nodal quartic surface can be used to realise the minimum <span>(rho =2)</span> for infinitely many different values of <i>d</i>. Finally, we strengthen a result of Morrison by showing that for any even lattice <i>N</i> of rank <span>(1le r le 10)</span> and signature <span>((1,r-1))</span> there exists a K3 surface <i>Y</i> defined over <span>({mathbb {R}})</span> such that <span>({{,textrm{Pic},}}Y_{mathbb {C}}={{,textrm{Pic},}}Y cong N)</span>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"26 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125414","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}
引用次数: 0
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