Pub Date : 2024-08-19DOI: 10.4310/jsg.2024.v22.n1.a3
Nicki Magill
This paper continues the study of the ellipsoid embedding function of symplectic Hirzebruch surfaces parametrized by $b in (0, 1)$, the size of the symplectic blowup. Cristofaro–Gardiner, et al. $href{https://doi.org/10.48550/arXiv.2004.13062}{textrm{arXiv:2004.13062}}$ found that if the embedding function for a Hirzebruch surface has an infinite staircase, then the function is equal to the volume curve at the accumulation point of the staircase. Here, we use almost toric fibrations to construct full-fillings at the accumulation points for an infinite family of recursively defined irrational $b$-values implying these $b$ are potential staircase values. The $b$-values are defined via a family of obstructive classes defined in Magill–McDuff–Weiler (arXiv:2203.06453). There is a correspondence between the recursive, interwoven structure of the obstructive classes and the sequence of possible mutations in the almost toric fibrations. This result is used in Magill–McDuff–Weiler $href{ https://ui.adsabs.harvard.edu/link_gateway/2022arXiv220306453M/doi:10.48550/arXiv.2203.06453}{textrm{(arXiv:2203.06453)}}$ to show that these classes are exceptional and that these $b$-values do have infinite staircases.
{"title":"Unobstructed embeddings in Hirzebruch surfaces","authors":"Nicki Magill","doi":"10.4310/jsg.2024.v22.n1.a3","DOIUrl":"https://doi.org/10.4310/jsg.2024.v22.n1.a3","url":null,"abstract":"This paper continues the study of the ellipsoid embedding function of symplectic Hirzebruch surfaces parametrized by $b in (0, 1)$, the size of the symplectic blowup. Cristofaro–Gardiner, <i>et al.</i> $href{https://doi.org/10.48550/arXiv.2004.13062}{textrm{arXiv:2004.13062}}$ found that if the embedding function for a Hirzebruch surface has an infinite staircase, then the function is equal to the volume curve at the accumulation point of the staircase. Here, we use almost toric fibrations to construct full-fillings at the accumulation points for an infinite family of recursively defined irrational $b$-values implying these $b$ are potential staircase values. The $b$-values are defined via a family of obstructive classes defined in Magill–McDuff–Weiler (arXiv:2203.06453). There is a correspondence between the recursive, interwoven structure of the obstructive classes and the sequence of possible mutations in the almost toric fibrations. This result is used in Magill–McDuff–Weiler $href{ https://ui.adsabs.harvard.edu/link_gateway/2022arXiv220306453M/doi:10.48550/arXiv.2203.06453}{textrm{(arXiv:2203.06453)}}$ to show that these classes are exceptional and that these $b$-values do have infinite staircases.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216898","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}
Pub Date : 2024-08-19DOI: 10.4310/jsg.2024.v22.n1.a1
Dylan Cant
We construct a contactomorphism of $(S^{2n-1}, alpha_{mathrm{std}})$ which does not have any translated points, providing a negative answer to a conjecture posed in $href{https://doi.org/10.1007/s10711-012-9741-1}{textrm{[San13]}}$.
{"title":"Contactomorphisms of the sphere without translated points","authors":"Dylan Cant","doi":"10.4310/jsg.2024.v22.n1.a1","DOIUrl":"https://doi.org/10.4310/jsg.2024.v22.n1.a1","url":null,"abstract":"We construct a contactomorphism of $(S^{2n-1}, alpha_{mathrm{std}})$ which does not have any translated points, providing a negative answer to a conjecture posed in $href{https://doi.org/10.1007/s10711-012-9741-1}{textrm{[San13]}}$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216897","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}
Pub Date : 2024-08-19DOI: 10.4310/jsg.2024.v22.n1.a2
Jennifer Dalton, John B. Etnyre, Lisa Traynor
We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal Thurston–Bennequin invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links. We also give a classification of Legendrian and transversal cable links of knot types that are uniformly thick and Legendrian simple. Here we see some similarities with the classification of Legendrian torus links but also some differences. In particular, we show that there are Legendrian representatives of cable links of any uniformly thick knot type for which no symmetries of the components can be realized by a Legendrian isotopy, others where only cyclic permutations of the components can be realized, and yet others where all smooth symmetries are realizable.
{"title":"Legendrian torus and cable links","authors":"Jennifer Dalton, John B. Etnyre, Lisa Traynor","doi":"10.4310/jsg.2024.v22.n1.a2","DOIUrl":"https://doi.org/10.4310/jsg.2024.v22.n1.a2","url":null,"abstract":"We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal Thurston–Bennequin invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links. We also give a classification of Legendrian and transversal cable links of knot types that are uniformly thick and Legendrian simple. Here we see some similarities with the classification of Legendrian torus links but also some differences. In particular, we show that there are Legendrian representatives of cable links of any uniformly thick knot type for which no symmetries of the components can be realized by a Legendrian isotopy, others where only cyclic permutations of the components can be realized, and yet others where all smooth symmetries are realizable.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216896","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}
Pub Date : 2024-08-19DOI: 10.4310/jsg.2024.v22.n1.a4
María Amelia Salazar, Daniele Sepe, Camilo Angulo
In this paper we prove Gray stability for compact contact groupoids and we use it to prove stability results for deformations of the induced Jacobi bundles.
在本文中，我们证明了紧凑接触群集的格雷稳定性，并用它证明了诱导雅可比束变形的稳定性结果。
{"title":"Multiplicative gray stability","authors":"María Amelia Salazar, Daniele Sepe, Camilo Angulo","doi":"10.4310/jsg.2024.v22.n1.a4","DOIUrl":"https://doi.org/10.4310/jsg.2024.v22.n1.a4","url":null,"abstract":"In this paper we prove Gray stability for compact contact groupoids and we use it to prove stability results for deformations of the induced Jacobi bundles.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216899","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}
Pub Date : 2024-06-06DOI: 10.4310/jsg.2023.v21.n6.a4
John Etnyre, Marc Kegel, Sinem Onaran
It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration. It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $3$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $S^1 times S^2$, the Poincaré homology sphere and the Brieskorn sphere $Sigma(2,3,7)$.We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $3$-sphere. We further obtain results for the $3$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.
{"title":"Contact surgery numbers","authors":"John Etnyre, Marc Kegel, Sinem Onaran","doi":"10.4310/jsg.2023.v21.n6.a4","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n6.a4","url":null,"abstract":"It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration. It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $3$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $S^1 times S^2$, the Poincaré homology sphere and the Brieskorn sphere $Sigma(2,3,7)$.We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $3$-sphere. We further obtain results for the $3$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552905","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}
Pub Date : 2024-06-06DOI: 10.4310/jsg.2023.v21.n6.a2
Kota Hattori, Mayuko Yamashita
This paper is a sequel to $href{https://dx.doi.org/10.4310/JSG.2020.v18.n6.a3}{[11]}$. We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $overline{partial}$-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost Kähler quantization for compatible almost complex structures, and show the analogous convergence results.
{"title":"Spectral convergence in geometric quantization — the case of non-singular Langrangian fibrations","authors":"Kota Hattori, Mayuko Yamashita","doi":"10.4310/jsg.2023.v21.n6.a2","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n6.a2","url":null,"abstract":"This paper is a sequel to $href{https://dx.doi.org/10.4310/JSG.2020.v18.n6.a3}{[11]}$. We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $overline{partial}$-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost Kähler quantization for compatible almost complex structures, and show the analogous convergence results.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550013","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}
Pub Date : 2024-06-06DOI: 10.4310/jsg.2023.v21.n6.a3
Tommaso Pacini
We show that, in toric Kähler geometry, the sign of the Ricci curvature corresponds exactly to convexity properties of the volume functional.We also discuss analogous relationships in the more general context of quasi-homogeneous manifolds, and existence results for minimal Lagrangian submanifolds.
我们还讨论了准均质流形更一般情况下的类似关系，以及最小拉格朗日子流形的存在性结果。
{"title":"Ricci curvature, the convexity of volume and minimal Lagrangian submanifolds","authors":"Tommaso Pacini","doi":"10.4310/jsg.2023.v21.n6.a3","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n6.a3","url":null,"abstract":"We show that, in toric Kähler geometry, the sign of the Ricci curvature corresponds exactly to convexity properties of the volume functional.We also discuss analogous relationships in the more general context of quasi-homogeneous manifolds, and existence results for minimal Lagrangian submanifolds.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552904","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}
Pub Date : 2024-06-06DOI: 10.4310/jsg.2023.v21.n6.a1
Jo Nelson, Morgan Weiler
The 2011 PhD thesis of Farris [Fa] demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a $mathbb{Z}^2$-graded group to the exterior algebra of the homology of its base. We extend this result by computing the $ mathbb{Z}$-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse–Bott direct limit argument and the classification of certain $J$-holomorphic buildings. The former requires the isomorphism between filtered Seiberg–Witten Floer cohomology and filtered ECH as established by Hutchings–Taubes [HT13]. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner–Hutchings–Zhang [CGHZ] to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings–Nelson [HN16].
{"title":"Embedded contact homology of prequantization bundles","authors":"Jo Nelson, Morgan Weiler","doi":"10.4310/jsg.2023.v21.n6.a1","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n6.a1","url":null,"abstract":"The 2011 PhD thesis of Farris [Fa] demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a $mathbb{Z}^2$-graded group to the exterior algebra of the homology of its base. We extend this result by computing the $ mathbb{Z}$-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse–Bott direct limit argument and the classification of certain $J$-holomorphic buildings. The former requires the isomorphism between filtered Seiberg–Witten Floer cohomology and filtered ECH as established by Hutchings–Taubes [HT13]. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner–Hutchings–Zhang [CGHZ] to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings–Nelson [HN16].","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552903","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}
Pub Date : 2024-06-03DOI: 10.4310/jsg.2023.v21.n5.a1
Laurent,Charles
In the setting of geometric quantization, we associate to any prequantum bundle automorphism a unitary map of the corresponding quantum space. These maps are controlled in the semiclassical limit by two invariants of symplectic topology: the Calabi–Weinstein morphism and a quasimorphism on the universal cover of the Hamiltonian diffeomorphism group introduced by Entov, Py, Shelukhin.
{"title":"On a quasimorphism of Hamiltonian diffeomorphisms and quantization","authors":"Laurent,Charles","doi":"10.4310/jsg.2023.v21.n5.a1","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n5.a1","url":null,"abstract":"In the setting of geometric quantization, we associate to any prequantum bundle automorphism a unitary map of the corresponding quantum space. These maps are controlled in the semiclassical limit by two invariants of symplectic topology: the Calabi–Weinstein morphism and a quasimorphism on the universal cover of the Hamiltonian diffeomorphism group introduced by Entov, Py, Shelukhin.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252122","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}
Pub Date : 2024-06-03DOI: 10.4310/jsg.2023.v21.n5.a2
Wilmer,Smilde
By considering suitable Poisson groupoids, we develop an approach to obtain Lie group structures on (subgroups of) the Poisson diffeomorphism groups of various classes of Poisson manifolds. As applications, we show that the Poisson diffeomorphism groups of (normal-crossing) log-symplectic, elliptic symplectic, scattering-symplectic and cosymplectic manifolds are regular infinite-dimensional Lie groups.
{"title":"Lie groups of Poisson diffeomorphisms","authors":"Wilmer,Smilde","doi":"10.4310/jsg.2023.v21.n5.a2","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n5.a2","url":null,"abstract":"By considering suitable Poisson groupoids, we develop an approach to obtain Lie group structures on (subgroups of) the Poisson diffeomorphism groups of various classes of Poisson manifolds. As applications, we show that the Poisson diffeomorphism groups of (normal-crossing) log-symplectic, elliptic symplectic, scattering-symplectic and cosymplectic manifolds are regular infinite-dimensional Lie groups.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251962","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：