Pub Date : 2020-09-11DOI: 10.4310/jsg.2022.v20.n2.a4
B. Stratmann
Extracting isolated rays from a symplectic manifold result in a manifold symplectomorphic to the initial one. The same holds for higher dimensional parametrized rays under an additional condition. More precisely, let $(M,omega)$ be a symplectic manifold. Let $[0,infty)times Qsubsetmathbb{R}times Q$ be considered as parametrized rays $[0,infty)$ and let $varphi:[-1,infty)times Qto M$ be an injective, proper, continuous map immersive on $(-1,infty)times Q$. If for the standard vector field $frac{partial}{partial t}$ on $mathbb{R}$ and any further vector field $nu$ tangent to $(-1,infty)times Q$ the equation $varphi^*omega(frac{partial}{partial t},nu)=0$ holds then $M$ and $Msetminus varphi([0,infty)times Q)$ are symplectomorphic.
{"title":"Removing parametrized rays symplectically","authors":"B. Stratmann","doi":"10.4310/jsg.2022.v20.n2.a4","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n2.a4","url":null,"abstract":"Extracting isolated rays from a symplectic manifold result in a manifold symplectomorphic to the initial one. The same holds for higher dimensional parametrized rays under an additional condition. More precisely, let $(M,omega)$ be a symplectic manifold. Let $[0,infty)times Qsubsetmathbb{R}times Q$ be considered as parametrized rays $[0,infty)$ and let $varphi:[-1,infty)times Qto M$ be an injective, proper, continuous map immersive on $(-1,infty)times Q$. If for the standard vector field $frac{partial}{partial t}$ on $mathbb{R}$ and any further vector field $nu$ tangent to $(-1,infty)times Q$ the equation $varphi^*omega(frac{partial}{partial t},nu)=0$ holds then $M$ and $Msetminus varphi([0,infty)times Q)$ are symplectomorphic.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"105 6","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72371088","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 : 2020-09-07DOI: 10.4310/jsg.2023.v21.n1.a1
Hansol Hong
The Maurer-Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;Lambda)$ as an $A_infty$-algebra. We identify the Maurer-Cartan algebra with the $0$-th cohomology of the Koszul dual dga of $CF(L,L;Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer-Cartan algebra of $L$ and a certain analytic completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is emph{dual} to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.
{"title":"Maurer–Cartan deformation of Lagrangians","authors":"Hansol Hong","doi":"10.4310/jsg.2023.v21.n1.a1","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n1.a1","url":null,"abstract":"The Maurer-Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;Lambda)$ as an $A_infty$-algebra. We identify the Maurer-Cartan algebra with the $0$-th cohomology of the Koszul dual dga of $CF(L,L;Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer-Cartan algebra of $L$ and a certain analytic completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is emph{dual} to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83597825","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 : 2020-08-25DOI: 10.4310/JSG.2022.v20.n5.a4
Thomas Machon
Let $M$ be a smooth closed orientable manifold and $mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^Lambda$ and $d+ d^Lambda$ symplectic cohomology groups defined by Tseng and Yau.
{"title":"A Poisson bracket on the space of Poisson structures","authors":"Thomas Machon","doi":"10.4310/JSG.2022.v20.n5.a4","DOIUrl":"https://doi.org/10.4310/JSG.2022.v20.n5.a4","url":null,"abstract":"Let $M$ be a smooth closed orientable manifold and $mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^Lambda$ and $d+ d^Lambda$ symplectic cohomology groups defined by Tseng and Yau.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75357521","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 : 2020-08-14DOI: 10.4310/jsg.2022.v20.n1.a4
Peter Crooks, M. Roser
Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.
{"title":"The $log$ symplectic geometry of Poisson slices","authors":"Peter Crooks, M. Roser","doi":"10.4310/jsg.2022.v20.n1.a4","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n1.a4","url":null,"abstract":"Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"36 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76326941","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 : 2020-06-05DOI: 10.4310/jsg.2021.v19.n5.a1
Anar Akhmedov, cCaugri Karakurt, Sumeyra Sakalli
We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.
{"title":"Generalized chain surgeries and applications","authors":"Anar Akhmedov, cCaugri Karakurt, Sumeyra Sakalli","doi":"10.4310/jsg.2021.v19.n5.a1","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n5.a1","url":null,"abstract":"We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"80 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80950883","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 : 2020-05-12DOI: 10.4310/jsg.2022.v20.n3.a2
Penka V. Georgieva, Eleny-Nicoleta Ionel
Motivated by the real version of the Gopakumar-Vafa conjecture for 3-folds, the authors introduced in [GI] the notion of local real Gromov-Witten invariants. This article is devoted to the proof of a splitting formula for these invariants under target degenerations. It is used in [GI] to show that the invariants give rise to a 2-dimensional Klein TQFT and to prove the local version of the real Gopakumar-Vafa conjecture.
{"title":"Splitting formulas for the local real Gromov–Witten invariants","authors":"Penka V. Georgieva, Eleny-Nicoleta Ionel","doi":"10.4310/jsg.2022.v20.n3.a2","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n3.a2","url":null,"abstract":"Motivated by the real version of the Gopakumar-Vafa conjecture for 3-folds, the authors introduced in [GI] the notion of local real Gromov-Witten invariants. This article is devoted to the proof of a splitting formula for these invariants under target degenerations. It is used in [GI] to show that the invariants give rise to a 2-dimensional Klein TQFT and to prove the local version of the real Gopakumar-Vafa conjecture.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"87 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74858224","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 : 2020-04-19DOI: 10.4310/jsg.2023.v21.n1.a2
A. Tomassini, Xu Wang
Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $omega$. Then we have a natural double complex $overline{partial}+overline{partial}^Lambda$, where $overline{partial}^Lambda$ denotes the symplectic adjoint of the $overline{partial}$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $omega$. In cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $partialoverline{partial}$-Lemma, namely the $overline{partial}, overline{partial}^Lambda$-Lemma, which can be characterized in terms of Bott--Chern and Aeppli cohomologies associated to the above double complex. We obtain Nomizu type theorems for the Bott--Chern and Aeppli cohomologies and we show that the $overline{partial}, overline{partial}^Lambda$-Lemma is stable under small deformations of $omega$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $partialoverline{partial}$-Lemma then the $overline{partial}, overline{partial}^Lambda$-Lemma is stable.
{"title":"Cohomologies of complex manifolds with symplectic $(1,1)$-forms","authors":"A. Tomassini, Xu Wang","doi":"10.4310/jsg.2023.v21.n1.a2","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n1.a2","url":null,"abstract":"Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $omega$. Then we have a natural double complex $overline{partial}+overline{partial}^Lambda$, where $overline{partial}^Lambda$ denotes the symplectic adjoint of the $overline{partial}$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $omega$. In cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $partialoverline{partial}$-Lemma, namely the $overline{partial}, overline{partial}^Lambda$-Lemma, which can be characterized in terms of Bott--Chern and Aeppli cohomologies associated to the above double complex. We obtain Nomizu type theorems for the Bott--Chern and Aeppli cohomologies and we show that the $overline{partial}, overline{partial}^Lambda$-Lemma is stable under small deformations of $omega$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $partialoverline{partial}$-Lemma then the $overline{partial}, overline{partial}^Lambda$-Lemma is stable.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"140 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77781913","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 : 2020-04-16DOI: 10.4310/jsg.2022.v20.n4.a1
G. Cavalcanti, R. Klaasse, A. Witte
We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but also sheds some light into the smooth case in dimension four. We show that in four dimensions there is a natural connected sum operation for these structures as well as a smoothing operation which changes a self-crossing stable generalized complex structure into a smooth stable generalized complex structure on the same manifold. This allows us to construct large families of stable generalized complex manifolds.
{"title":"Self-crossing stable generalized complex structures","authors":"G. Cavalcanti, R. Klaasse, A. Witte","doi":"10.4310/jsg.2022.v20.n4.a1","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n4.a1","url":null,"abstract":"We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but also sheds some light into the smooth case in dimension four. We show that in four dimensions there is a natural connected sum operation for these structures as well as a smoothing operation which changes a self-crossing stable generalized complex structure into a smooth stable generalized complex structure on the same manifold. This allows us to construct large families of stable generalized complex manifolds.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"55 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83915754","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 : 2020-04-05DOI: 10.4310/JSG.2022.v20.n1.a1
Simon Allais
Inspired by the techniques of Givental and Theret, we provide a proof with generating functions of a recent result of Ginzburg-Gurel concerning the periodic points of Hamiltonian diffeomorphisms of $mathbb{C}text{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.
{"title":"On periodic points of Hamiltonian diffeomorphisms of $mathbb{C} mathrm{P}^d$ via generating functions","authors":"Simon Allais","doi":"10.4310/JSG.2022.v20.n1.a1","DOIUrl":"https://doi.org/10.4310/JSG.2022.v20.n1.a1","url":null,"abstract":"Inspired by the techniques of Givental and Theret, we provide a proof with generating functions of a recent result of Ginzburg-Gurel concerning the periodic points of Hamiltonian diffeomorphisms of $mathbb{C}text{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76995088","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 : 2020-02-06DOI: 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":"1 1","pages":""},"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}
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：