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Complete Calabi–Yau metrics from smoothing Calabi–Yau complete intersections 从平滑 Calabi-Yau 完全交点出发的完全 Calabi-Yau 度量
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-02-19 DOI: 10.1007/s10711-024-00886-3
Benjy J. Firester
{"title":"Complete Calabi–Yau metrics from smoothing Calabi–Yau complete intersections","authors":"Benjy J. Firester","doi":"10.1007/s10711-024-00886-3","DOIUrl":"https://doi.org/10.1007/s10711-024-00886-3","url":null,"abstract":"<p>We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection <span>(V_0)</span> that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by <span>({mathbb {C}}times V_0)</span>. This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139928208","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 branched coverings of singular (G, X)-manifolds 论奇异（G，X）-manifolds 的分支覆盖
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-02-17 DOI: 10.1007/s10711-023-00873-0
Léo Brunswic
{"title":"On branched coverings of singular (G, X)-manifolds","authors":"Léo Brunswic","doi":"10.1007/s10711-023-00873-0","DOIUrl":"https://doi.org/10.1007/s10711-023-00873-0","url":null,"abstract":"<p>Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (<i>G</i>, <i>X</i>)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (<i>G</i>, <i>X</i>)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (<i>G</i>, <i>X</i>)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (<i>G</i>, <i>X</i>)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"17 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752627","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
Coregularity of Fano varieties 法诺变种的内核性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-02-10 DOI: 10.1007/s10711-023-00882-z
{"title":"Coregularity of Fano varieties","authors":"","doi":"10.1007/s10711-023-00882-z","DOIUrl":"https://doi.org/10.1007/s10711-023-00882-z","url":null,"abstract":"<h3>Abstract</h3> <p>The absolute regularity of a Fano variety, denoted by <span> <span>(hat{textrm{reg}}(X))</span> </span>, is the largest dimension of the dual complex of a log Calabi–Yau structure on <em>X</em>. The absolute coregularity is defined to be <span> <span>$$begin{aligned} hat{textrm{coreg}}(X):= dim X - hat{textrm{reg}}(X)-1. end{aligned}$$</span> </span>The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of <em>X</em>. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"8 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752470","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
Topological and dynamical properties of Torelli groups of partitioned surfaces 分割曲面托雷利群的拓扑和动力学特性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-02-07 DOI: 10.1007/s10711-024-00889-0
Hyungryul Baik, Hyunshik Shin, Philippe Tranchida
{"title":"Topological and dynamical properties of Torelli groups of partitioned surfaces","authors":"Hyungryul Baik, Hyunshik Shin, Philippe Tranchida","doi":"10.1007/s10711-024-00889-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00889-0","url":null,"abstract":"<p>Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"18 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752564","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
Real structures on root stacks and parabolic connections 根栈和抛物线连接上的实结构
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-023-00880-1
Sujoy Chakraborty, Arjun Paul
{"title":"Real structures on root stacks and parabolic connections","authors":"Sujoy Chakraborty, Arjun Paul","doi":"10.1007/s10711-023-00880-1","DOIUrl":"https://doi.org/10.1007/s10711-023-00880-1","url":null,"abstract":"<p>Let <i>D</i> be a reduced effective strict normal crossing divisor on a smooth complex variety <i>X</i>, and let <span>(mathfrak {X}_D)</span> be the associated root stack over <span>(mathbb C)</span>. Suppose that <i>X</i> admits an anti-holomorphic involution (real structure) that keeps <i>D</i> invariant. We show that the root stack <span>(mathfrak {X}_D)</span> naturally admits a real structure compatible with <i>X</i>. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on <i>X</i>.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644594","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
Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces 包装、非正曲、格罗莫夫双曲度量空间的离散群
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-023-00874-z
Nicola Cavallucci, Andrea Sambusetti
{"title":"Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces","authors":"Nicola Cavallucci, Andrea Sambusetti","doi":"10.1007/s10711-023-00874-z","DOIUrl":"https://doi.org/10.1007/s10711-023-00874-z","url":null,"abstract":"<p>We prove a quantitative version of the classical Tits’ alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644583","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
Circumcenter extension maps for non-positively curved spaces 非正曲线空间的圆心扩展映射
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-30 DOI: 10.1007/s10711-023-00881-0
Merlin Incerti-Medici
{"title":"Circumcenter extension maps for non-positively curved spaces","authors":"Merlin Incerti-Medici","doi":"10.1007/s10711-023-00881-0","DOIUrl":"https://doi.org/10.1007/s10711-023-00881-0","url":null,"abstract":"<p>We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We describe regions on which this map is Hölder-continuous. Furthermore, we show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve previously known quasi-isometry constants, provided by Biswas, in the case of 2-dimensional <span>(mathrm {CAT(-1)})</span> manifolds. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"5 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644942","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
Homotopy equivalent boundaries of cube complexes 立方体复合物的同调等效边界
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-27 DOI: 10.1007/s10711-023-00877-w
Talia Fernós, David Futer, Mark Hagen
{"title":"Homotopy equivalent boundaries of cube complexes","authors":"Talia Fernós, David Futer, Mark Hagen","doi":"10.1007/s10711-023-00877-w","DOIUrl":"https://doi.org/10.1007/s10711-023-00877-w","url":null,"abstract":"<p>A finite-dimensional CAT(0) cube complex <i>X</i> is equipped with several well-studied boundaries. These include the <i>Tits boundary</i> <span>(partial _TX)</span> (which depends on the CAT(0) metric), the <i>Roller boundary</i> <span>({partial _R}X)</span> (which depends only on the combinatorial structure), and the <i>simplicial boundary</i> <span>(partial _triangle X)</span> (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of <span>({partial _R}X)</span> to define a simplicial Roller boundary <span>({mathfrak {R}}_triangle X)</span>. Then, we show that <span>(partial _TX)</span>, <span>(partial _triangle X)</span>, and <span>({mathfrak {R}}_triangle X)</span> are all homotopy equivalent, <span>(text {Aut}(X))</span>-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"28 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582817","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
From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds 从 $$L^p$$ 边界到黎曼流形的格罗莫夫-豪斯多夫收敛性
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-03 DOI: 10.1007/s10711-023-00875-y
Brian Allen
{"title":"From $$L^p$$ bounds to Gromov–Hausdorff convergence of Riemannian manifolds","authors":"Brian Allen","doi":"10.1007/s10711-023-00875-y","DOIUrl":"https://doi.org/10.1007/s10711-023-00875-y","url":null,"abstract":"<p>In this paper we provide a way of taking <span>(L^p)</span>, <span>(p &gt; frac{m}{2})</span> bounds on a <span>(m-)</span> dimensional Riemannian metric and transforming that into Hölder bounds for the corresponding distance function. One can think of this new estimate as a type of Morrey inequality for Riemannian manifolds where one thinks of a Riemannian metric as the gradient of the corresponding distance function so that the <span>(L^p)</span>, <span>(p &gt; frac{m}{2})</span> bound analogously implies Hölder control on the distance function. This new estimate is then used to state a compactness theorem, another theorem which guarantees convergence to a particular Riemmanian manifold, and a new scalar torus stability result. We expect these results to be useful for proving geometric stability results in the presence of scalar curvature bounds when Gromov–Hausdorff convergence can be achieved.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"22 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139082688","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
Cohomogeneity one solitons for the isometric flow of G 2 -structures. G 2 结构等距流的同质一孤子。
IF 0.5 4区 数学
Geometriae Dedicata Pub Date : 2024-01-01 Epub Date: 2024-09-30 DOI: 10.1007/s10711-024-00954-8
Thomas A Ivey, Spiro Karigiannis
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Cohomogeneity one solitons for the isometric flow of <ns0:math><ns0:msub><ns0:mtext>G</ns0:mtext> <ns0:mn>2</ns0:mn></ns0:msub> </ns0:math> -structures.","authors":"Thomas A Ivey, Spiro Karigiannis","doi":"10.1007/s10711-024-00954-8","DOIUrl":"https://doi.org/10.1007/s10711-024-00954-8","url":null,"abstract":"<p><p>We consider the existence of cohomogeneity one solitons for the isometric flow of <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -structures on the following classes of torsion-free <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -manifolds: the Euclidean <math> <msup><mrow><mi>R</mi></mrow> <mn>7</mn></msup> </math> with its standard <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon <math><msub><mtext>G</mtext> <mn>2</mn></msub> </math> -manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on <math> <msup><mrow><mi>R</mi></mrow> <mn>7</mn></msup> </math> is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"218 5","pages":"102"},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11442535/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142367550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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