Pub Date : 2020-02-06DOI: 10.4310/jsg.2021.v19.n5.a2
M. Cahen, Maxime G'erard, S. Gutt, Manar Hayyani
We look at methods to select triples $(M,omega,J)$ consisting of a symplectic manifold $(M,omega)$ endowed with a compatible positive almost complex structure $J$, in terms of the Nijenhuis tensor $N^J$ associated to $J$. We study in particular the image distribution $Image N^J$.
{"title":"Distributions associated to almost complex structures on symplectic manifolds","authors":"M. Cahen, Maxime G'erard, S. Gutt, Manar Hayyani","doi":"10.4310/jsg.2021.v19.n5.a2","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n5.a2","url":null,"abstract":"We look at methods to select triples $(M,omega,J)$ consisting of a symplectic manifold $(M,omega)$ endowed with a compatible positive almost complex structure $J$, in terms of the Nijenhuis tensor $N^J$ associated to $J$. We study in particular the image distribution $Image N^J$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87326406","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 : 2020-01-01DOI: 10.4310/jsg.2020.v18.n1.a7
Y. Savelyev, E. Shelukhin
{"title":"$mathrm{K}$-theoretic invariants of Hamiltonian fibrations","authors":"Y. Savelyev, E. Shelukhin","doi":"10.4310/jsg.2020.v18.n1.a7","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n1.a7","url":null,"abstract":"","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81271552","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 : 2020-01-01DOI: 10.4310/JSG.2020.V18.N3.A9
Guogang Liu
In this paper, we prove that there exist contractible positive loops of Legendrian embeddings based at any loose Legendrian submanifold. As an application, we define a partial order on Cont0(M, ξ), called strong orderability, and prove that overtwisted contact manifolds are not strongly orderable.
{"title":"Positive loops of loose Legendrian embeddings and applications","authors":"Guogang Liu","doi":"10.4310/JSG.2020.V18.N3.A9","DOIUrl":"https://doi.org/10.4310/JSG.2020.V18.N3.A9","url":null,"abstract":"In this paper, we prove that there exist contractible positive loops of Legendrian embeddings based at any loose Legendrian submanifold. As an application, we define a partial order on Cont0(M, ξ), called strong orderability, and prove that overtwisted contact manifolds are not strongly orderable.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84042817","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 : 2019-12-23DOI: 10.4310/jsg.2022.v20.n2.a1
B. An, Youngjin Bae, Tao Su
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_{infty}$-category, which lifts the set of augmentations of the associated Chekanov-Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infinity controlled by the (bordered) Legendrian graph. In other words, generalizing [21], we prove "augmentations are sheaves" in the singular case.
{"title":"Augmentations are sheaves for Legendrian graphs","authors":"B. An, Youngjin Bae, Tao Su","doi":"10.4310/jsg.2022.v20.n2.a1","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n2.a1","url":null,"abstract":"In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_{infty}$-category, which lifts the set of augmentations of the associated Chekanov-Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infinity controlled by the (bordered) Legendrian graph. In other words, generalizing [21], we prove \"augmentations are sheaves\" in the singular case.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81450369","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 : 2019-12-11DOI: 10.4310/jsg.2021.v19.n3.a1
D. Fiorenza, H. Lê
Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${rm Imm}_f(S,M)$ the space of all free immersions $varphi:S to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${rm Imm}_f(S,M)/{rm Diff}^+(S)$, where ${rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $varphi in Omega ^r(M, TM)$ and $dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally Kahler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.
设$S$为紧定向有限维流形，$M$为有限维黎曼流形，设${rm Imm}_f(S,M)$为所有自由浸入空间$varphi:S to M$，设$B^+_{i,f}(S,M)$为商空间${rm Imm}_f(S,M)/{rm Diff}^+(S)$，其中${rm Diff}^+(S)$表示$S$的保定向微分同态群。在本文中，我们证明了如果$M$允许一个平行的$r$ -折叠向量叉积$varphi in Omega ^r(M, TM)$与$dim S = r-1$，则$B^+_{i,f}(S,M)$是一个形式的Kahler流形。这推广了Brylinski, LeBrun和Verbitsky在$S$是$M$中的余维2子流形，$S = S^1$或$M$分别是无扭转$G_2$流形的情况下的结果。
{"title":"Formally integrable complex structures on higher dimensional knot spaces","authors":"D. Fiorenza, H. Lê","doi":"10.4310/jsg.2021.v19.n3.a1","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n3.a1","url":null,"abstract":"Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${rm Imm}_f(S,M)$ the space of all free immersions $varphi:S to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${rm Imm}_f(S,M)/{rm Diff}^+(S)$, where ${rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $varphi in Omega ^r(M, TM)$ and $dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally Kahler manifold. This generalizes Brylinski's, LeBrun's and Verbitsky's results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81125883","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 : 2019-12-05DOI: 10.4310/JSG.2021.v19.n4.a1
J. Asplund
Let S be a closed orientable spin manifold. Let K⊂S be a submanifold and denote its complement by MK. In this paper we prove that there exists an isomorphism between partially wrapped Floer cochain ...
{"title":"Fiber Floer cohomology and conormal stops","authors":"J. Asplund","doi":"10.4310/JSG.2021.v19.n4.a1","DOIUrl":"https://doi.org/10.4310/JSG.2021.v19.n4.a1","url":null,"abstract":"Let S be a closed orientable spin manifold. Let K⊂S be a submanifold and denote its complement by MK. In this paper we prove that there exists an isomorphism between partially wrapped Floer cochain ...","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75539656","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 : 2019-11-20DOI: 10.4310/jsg.2022.v20.n1.a2
B. An, Youngjin Bae, Tam'as K'alm'an
We define ruling invariants for even-valence Legendrian graphs in standard contact three-space. We prove that rulings exist if and only if the DGA of the graph, introduced by the first two authors, has an augmentation. We set up the usual ruling polynomials for various notions of gradedness and prove that if the graph is four-valent, then the ungraded ruling polynomial appears in Kauffman-Vogel's graph version of the Kauffman polynomial. Our ruling invariants are compatible with certain vertex-identifying operations as well as vertical cuts and gluings of front diagrams. We also show that Leverson's definition of a ruling of a Legendrian link in a connected sum of $S^1 times S^2$'s can be seen as a special case of ours.
{"title":"Ruling invariants for Legendrian graphs","authors":"B. An, Youngjin Bae, Tam'as K'alm'an","doi":"10.4310/jsg.2022.v20.n1.a2","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n1.a2","url":null,"abstract":"We define ruling invariants for even-valence Legendrian graphs in standard contact three-space. We prove that rulings exist if and only if the DGA of the graph, introduced by the first two authors, has an augmentation. We set up the usual ruling polynomials for various notions of gradedness and prove that if the graph is four-valent, then the ungraded ruling polynomial appears in Kauffman-Vogel's graph version of the Kauffman polynomial. Our ruling invariants are compatible with certain vertex-identifying operations as well as vertical cuts and gluings of front diagrams. We also show that Leverson's definition of a ruling of a Legendrian link in a connected sum of $S^1 times S^2$'s can be seen as a special case of ours.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74565886","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 : 2019-10-08DOI: 10.4310/jsg.2021.v19.n4.a3
Chris Gerig
We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg-Witten) invariants of various torus surgeries. We then recover a result of Morgan-Mrowka-Szabo on product formulas for the Seiberg-Witten invariants along 3-tori.
{"title":"Lagrangian torus invariants using $ECH = SWF$","authors":"Chris Gerig","doi":"10.4310/jsg.2021.v19.n4.a3","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n4.a3","url":null,"abstract":"We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg-Witten) invariants of various torus surgeries. We then recover a result of Morgan-Mrowka-Szabo on product formulas for the Seiberg-Witten invariants along 3-tori.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85728010","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 : 2019-09-15DOI: 10.4310/JSG.2020.V18.N6.A3
Kota Hattori
On a compact symplectic manifold $(X,omega)$ with a prequantum line bundle $(L,nabla,h)$, we consider the one-parameter family of $omega$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are several researches which show that the holomorphic sections of the line bundle localize at Bohr-Sommerfeld fibers. In this article we consider the one-parameter family of the Riemannian metrics on the frame bundle of $L$ determined by the complex structures and $nabla,h$, and we can see that their diameters diverge. If we fix a base point in some fibers of the Lagrangian fibration we can show that they measured Gromov-Hausdorff converge to some pointed metric measure spaces with the isometric $S^1$-actions, which may depend on the choice of the base point. We observe that the properties of the $S^1$-actions on the limit spaces actually depend on whether the base point is in the Bohr-Sommerfeld fibers or not.
{"title":"The geometric quantizations and the measured Gromov–Hausdorff convergences","authors":"Kota Hattori","doi":"10.4310/JSG.2020.V18.N6.A3","DOIUrl":"https://doi.org/10.4310/JSG.2020.V18.N6.A3","url":null,"abstract":"On a compact symplectic manifold $(X,omega)$ with a prequantum line bundle $(L,nabla,h)$, we consider the one-parameter family of $omega$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are several researches which show that the holomorphic sections of the line bundle localize at Bohr-Sommerfeld fibers. In this article we consider the one-parameter family of the Riemannian metrics on the frame bundle of $L$ determined by the complex structures and $nabla,h$, and we can see that their diameters diverge. If we fix a base point in some fibers of the Lagrangian fibration we can show that they measured Gromov-Hausdorff converge to some pointed metric measure spaces with the isometric $S^1$-actions, which may depend on the choice of the base point. We observe that the properties of the $S^1$-actions on the limit spaces actually depend on whether the base point is in the Bohr-Sommerfeld fibers or not.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72550940","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 : 2019-08-12DOI: 10.4310/jsg.2021.v19.n4.a4
Victor Mouquin
Given a standard complex semisimple Poisson Lie group $(G, pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson groupoid over $O^u$, extending a result from the aforementioned authors about double Bruhat cells in $(G, pi_{st})$. Our result on $G^{u,u}$ is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to twist a direct product of Poisson groupoids.
{"title":"Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells","authors":"Victor Mouquin","doi":"10.4310/jsg.2021.v19.n4.a4","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n4.a4","url":null,"abstract":"Given a standard complex semisimple Poisson Lie group $(G, pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson groupoid over $O^u$, extending a result from the aforementioned authors about double Bruhat cells in $(G, pi_{st})$. Our result on $G^{u,u}$ is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to twist a direct product of Poisson groupoids.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84296855","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}
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