{"title":"Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets","authors":"Yann Guggisberg, Fabian Ziltener","doi":"arxiv-2408.17444","DOIUrl":"https://doi.org/arxiv-2408.17444","url":null,"abstract":"We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff\u0000measure vanishes. We show that every countably $m$-rectifiable subset of\u0000$mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a\u0000Hamiltonian diffeomorphism that is arbitrarily $C^infty$-close to the\u0000identity. As a consequence, every countably $n$-rectifiable and $n$-negligible\u0000subset of $mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both\u0000results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding\u0000method can be modified to show that the Gromov width of $B^{2n}_1setminus A$\u0000equals $pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the\u0000open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Categorical quantization on Kähler manifolds","authors":"YuTung Yau","doi":"arxiv-2408.17201","DOIUrl":"https://doi.org/arxiv-2408.17201","url":null,"abstract":"Generalizing deformation quantizations with separation of variables of a\u0000K\"ahler manifold $M$, we adopt Fedosov's gluing argument to construct a\u0000category $mathsf{DQ}$, enriched over sheaves of $mathbb{C}[[hbar]]$-modules\u0000on $M$, as a quantization of the category of Hermitian holomorphic vector\u0000bundles over $M$ with morphisms being smooth sections of hom-bundles. We then define quantizable morphisms among objects in $mathsf{DQ}$,\u0000generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon\u0000evaluation of quantizable morphisms at $hbar = tfrac{sqrt{-1}}{k}$, we\u0000obtain an enriched category $mathsf{DQ}_{operatorname{qu}, k}$. We show that,\u0000when $M$ is prequantizable, $mathsf{DQ}_{operatorname{qu}, k}$ is equivalent\u0000to the category $mathsf{GQ}$ of holomorphic vector bundles over $M$ with\u0000morphisms being holomorphic differential operators, via a functor obtained from\u0000Bargmann-Fock actions.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Semiclassical Hodge theory for log Poisson manifolds","authors":"Aidan Lindberg, Brent Pym","doi":"arxiv-2408.16685","DOIUrl":"https://doi.org/arxiv-2408.16685","url":null,"abstract":"We construct a mixed Hodge structure on the topological K-theory of smooth\u0000Poisson varieties, depending weakly on a choice of compactification. We\u0000establish a package of tools for calculations with these structures, such as\u0000functoriality statements, projective bundle formulae, Gysin sequences and\u0000Torelli properties. We show that for varieties with trivial A-hat class, the\u0000corresponding period maps for families can be written as exponential maps for\u0000bundles of tori, which we call the \"quantum parameters\". As justification for\u0000the terminology, we show that in many interesting examples, the quantum\u0000parameters of a Poisson variety coincide with the parameters appearing in its\u0000known deformation quantizations. In particular, we give a detailed\u0000implementation of an argument of Kontsevich, to prove that his canonical\u0000quantization formula, when applied to Poisson tori, yields noncommutative tori\u0000with parameter \"$q = e^hbar$\".","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lagrangian Approximation of Totally Real Concordances","authors":"Georgios Dimitroglou Rizell","doi":"arxiv-2408.16614","DOIUrl":"https://doi.org/arxiv-2408.16614","url":null,"abstract":"We show that two-dimensional totally real concordances can be approximated by\u0000Lagrangian concordances after sufficiently many positive and negative\u0000stabilisations of the Legendrian boundaries. The applications of this result\u0000are the construction of knotted Lagrangian concordances, and knotted Lagrangian\u0000tori in symplectisations of overtwisted contact manifolds.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Relative Equilibria for Scaling Symmetries and Central Configurations","authors":"Giovanni Rastelli, Manuele Santoprete","doi":"arxiv-2408.15191","DOIUrl":"https://doi.org/arxiv-2408.15191","url":null,"abstract":"In this paper, we explore scaling symmetries within the framework of\u0000symplectic geometry. We focus on the action $Phi$ of the multiplicative group\u0000$G = mathbb{R}^+$ on exact symplectic manifolds $(M, omega,theta)$, with\u0000$omega = -dtheta$, where $ theta $ is a given primitive one-form. Extending\u0000established results in symplectic geometry and Hamiltonian dynamics, we\u0000introduce conformally symplectic maps, conformally Hamiltonian systems,\u0000conformally symplectic group actions, and the notion of conformal invariance.\u0000This framework allows us to generalize the momentum map to the conformal\u0000momentum map, which is crucial for understanding scaling symmetries.\u0000Additionally, we provide a generalized Hamiltonian Noether's theorem for these\u0000symmetries. We introduce the (conformal) augmented Hamiltonian $H_{xi}$ and prove that\u0000the relative equilibria of scaling symmetries are solutions to equations\u0000involving $ H _{ xi } $ and the primitive one-form $theta$. We derive their\u0000main properties, emphasizing the differences from relative equilibria in\u0000traditional symplectic actions. For cotangent bundles, we define a scaled cotangent lifted action and derive\u0000explicit formulas for the conformal momentum map. We also provide a general\u0000definition of central configurations for Hamiltonian systems on cotangent\u0000bundles that admit scaling symmetries. Applying these results to simple\u0000mechanical systems, we introduce the augmented potential $U_{xi}$ and show\u0000that the relative equilibria of scaling symmetries are solutions to an equation\u0000involving $ U _{ xi } $ and the Lagrangian one-form $theta_L$. Finally, we apply our general theory to the Newtonian $n$-body problem,\u0000recovering the classical equations for central configurations.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lagrangian surplusection phenomena","authors":"Georgios Dimitroglou Rizell, Jonathan David Evans","doi":"arxiv-2408.14883","DOIUrl":"https://doi.org/arxiv-2408.14883","url":null,"abstract":"In this paper, we introduce a broad class of phenomena which appear when you\u0000intersect a given Lagrangian submanifold $K$ with a family of Lagrangian\u0000submanifolds $L_t$ (all Hamiltonian isotopic to one another). We establish that\u0000this phenomenon occurs in a particular situation, which lets us give a lower\u0000bound for the volume of any Lagrangian torus in $mathbb{CP}^2$ which is\u0000Hamiltonian isotopic to the Chekanov torus. The rest of the paper is a\u0000discussion of why we should expect these phenomena to be very common, motivated\u0000by Oh's conjecture on the volume-minimising property of the Clifford torus and\u0000the concurrent normals conjecture in convex geometry. We pose many open\u0000questions.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Superheavy Skeleta for non-Normal Crossings Divisors","authors":"Elliot Gathercole","doi":"arxiv-2408.13187","DOIUrl":"https://doi.org/arxiv-2408.13187","url":null,"abstract":"Given an anticanonical divisor in a projective variety, one naturally obtains\u0000a monotone K\"ahler manifold. In this paper, for divisors in a certain class\u0000(larger than normal crossings), we construct smoothing families of contact\u0000hypersurfaces with controlled Reeb dynamics. We use these to adapt arguments of\u0000Borman, Sheridan and Varolgunes to obtain analogous results about symplectic\u0000cohomology with supports in the divisor complement. In particular, we will show\u0000that several examples of Lagrangian skeleta of such divisor complements are\u0000superheavy, in cases where applying Lagrangian Floer theory may be intractable.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scheme-theoretic coisotropic reduction","authors":"Peter Crooks, Maxence Mayrand","doi":"arxiv-2408.11932","DOIUrl":"https://doi.org/arxiv-2408.11932","url":null,"abstract":"We develop an affine scheme-theoretic version of Hamiltonian reduction by\u0000symplectic groupoids. It works over $Bbbk=mathbb{R}$ or $Bbbk=mathbb{C}$,\u0000and is formulated for an affine symplectic groupoid\u0000$mathcal{G}rightrightarrows X$, an affine Hamiltonian $mathcal{G}$-scheme\u0000$mu:Mlongrightarrow X$, a coisotropic subvariety $Ssubseteq X$, and a\u0000stabilizer subgroupoid $mathcal{H}rightrightarrows S$. Our first main result\u0000is that the Poisson bracket on $Bbbk[M]$ induces a Poisson bracket on the\u0000subquotient $Bbbk[mu^{-1}(S)]^{mathcal{H}}$. The Poisson scheme\u0000$mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ is then declared to be a\u0000Hamiltonian reduction of $M$. Other main results include sufficient conditions\u0000for $mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ to inherit a residual\u0000Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to\u0000an earlier paper, where we simultaneously generalize several Hamiltonian\u0000reduction processes. In this way, the present work yields scheme-theoretic\u0000analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction,\u0000'{S}niatycki-Weinstein reduction, and symplectic reduction along general\u0000coisotropic submanifolds. The initial impetus for this work was its utility in\u0000formulating and proving generalizations of the Moore-Tachikawa conjecture.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"From curve shortening to flat link stability and Birkhoff sections of geodesic flows","authors":"Marcelo R. R. Alves, Marco Mazzucchelli","doi":"arxiv-2408.11938","DOIUrl":"https://doi.org/arxiv-2408.11938","url":null,"abstract":"We employ the curve shortening flow to establish three new theorems on the\u0000dynamics of geodesic flows of closed Riemannian surfaces. The first one is the\u0000stability, under $C^0$-small perturbations of the Riemannian metric, of certain\u0000flat links of closed geodesics. The second one is a forced existence theorem\u0000for orientable closed Riemannian surfaces of positive genus, asserting that the\u0000existence of a contractible simple closed geodesic $gamma$ forces the\u0000existence of infinitely many closed geodesics intersecting $gamma$ in every\u0000primitive free homotopy class of loops. The third theorem asserts the existence\u0000of Birkhoff sections for the geodesic flow of any closed orientable Riemannian\u0000surface of positive genus.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unknotting Lagrangian $mathrm{S}^1timesmathrm{S}^{n-1}$ in $mathbb{R}^{2n}$","authors":"Stefan Nemirovski","doi":"arxiv-2408.10916","DOIUrl":"https://doi.org/arxiv-2408.10916","url":null,"abstract":"Lagrangian embeddings\u0000$mathrm{S}^1timesmathrm{S}^{n-1}hookrightarrowmathbb{R}^{2n}$ are\u0000classified up to smooth isotopy for all $nge 3$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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