We follow the study by Cascini-Panov on symplectic generic complex structures on Kahler surfaces with $p_g=0$, a question proposed by Tian-Jun Li, by demonstrating that the one-point blowup of an Enriques surface admits non-Kahler symplectic forms. This phenomenon relies on the abundance of elliptic fibrations on Enriques surfaces, characterized by various invariants from algebraic geometry. We also provide a quantitative comparison of these invariants to further give a detailed examination of the distinction between Kahler cone and symplectic cone.
{"title":"Comparing Kahler cone and symplectic cone of one-point blowup of Enriques surface","authors":"Shengzhen Ning","doi":"arxiv-2407.10217","DOIUrl":"https://doi.org/arxiv-2407.10217","url":null,"abstract":"We follow the study by Cascini-Panov on symplectic generic complex structures\u0000on Kahler surfaces with $p_g=0$, a question proposed by Tian-Jun Li, by\u0000demonstrating that the one-point blowup of an Enriques surface admits\u0000non-Kahler symplectic forms. This phenomenon relies on the abundance of\u0000elliptic fibrations on Enriques surfaces, characterized by various invariants\u0000from algebraic geometry. We also provide a quantitative comparison of these\u0000invariants to further give a detailed examination of the distinction between\u0000Kahler cone and symplectic cone.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721711","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}
Let $X$ be a smooth projective toric variety and let $widetilde{X}$ be the blow-up manifold of $X$ at finitely many distinct tours invariants points of $X$. In this paper, we give an explicit combinatorial formula of the Chow weight of $widetilde{X}$ in terms of the base toric manifold $X$ and the symplectic cuts of the Delzant polytope. We then apply this blow-up formula to the projective plane and see the difference of Chow stability between the toric blow-up manifolds and the manifolds of blow-ups at general points. Finally, we detect the blow-up formula of the Futaki-Ono invariant which is an obstruction for asymptotic Chow semistability of a polarized toric manifold.
{"title":"On the blow-up formula of the Chow weights for polarized toric manifolds","authors":"King Leung Lee, Naoto Yotsutani","doi":"arxiv-2407.10082","DOIUrl":"https://doi.org/arxiv-2407.10082","url":null,"abstract":"Let $X$ be a smooth projective toric variety and let $widetilde{X}$ be the\u0000blow-up manifold of $X$ at finitely many distinct tours invariants points of\u0000$X$. In this paper, we give an explicit combinatorial formula of the Chow\u0000weight of $widetilde{X}$ in terms of the base toric manifold $X$ and the\u0000symplectic cuts of the Delzant polytope. We then apply this blow-up formula to\u0000the projective plane and see the difference of Chow stability between the toric\u0000blow-up manifolds and the manifolds of blow-ups at general points. Finally, we\u0000detect the blow-up formula of the Futaki-Ono invariant which is an obstruction\u0000for asymptotic Chow semistability of a polarized toric manifold.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721700","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}
HOMFLYPT polynomials of knots in the 3-sphere in symmetric representations satisfy recursion relations. Their geometric origin is holomorphic curves at infinity on knot conormals that determine a $D$-module with characteristic variety the Legendrian knot conormal augmention variety and with the recursion relations as operator polynomial generators [arXiv:1304.5778, arXiv:1803.04011]. We consider skein lifts of recursions and $D$-modules corresponding to skein valued open curve counts [arXiv:1901.08027] that encode HOMFLYPT polynomials colored by arbitrary partitions. We define a worldsheet skein module which is the universal target for skein curve counts and a corresponding $D$-module. We then consider the concrete example of the Legendrian conormal of the Hopf link. We show that the worldsheet skein $D$-module for the Hopf link conormal is generated by three operator polynomials that annihilate the skein valued partition function for any choice of Lagrangian filling and recursively determine it uniquely. We find Lagrangian fillings for any point in the augmentation variety and show that their skein valued partition functions admit quiver-like expansions where all holomorphic curves are generated by a small number of basic holomorphic disks and annuli and their multiple covers.
{"title":"The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal","authors":"Tobias Ekholm, Pietro Longhi, Lukas Nakamura","doi":"arxiv-2407.09836","DOIUrl":"https://doi.org/arxiv-2407.09836","url":null,"abstract":"HOMFLYPT polynomials of knots in the 3-sphere in symmetric representations\u0000satisfy recursion relations. Their geometric origin is holomorphic curves at\u0000infinity on knot conormals that determine a $D$-module with characteristic\u0000variety the Legendrian knot conormal augmention variety and with the recursion\u0000relations as operator polynomial generators [arXiv:1304.5778,\u0000arXiv:1803.04011]. We consider skein lifts of recursions and $D$-modules\u0000corresponding to skein valued open curve counts [arXiv:1901.08027] that encode\u0000HOMFLYPT polynomials colored by arbitrary partitions. We define a worldsheet\u0000skein module which is the universal target for skein curve counts and a\u0000corresponding $D$-module. We then consider the concrete example of the Legendrian conormal of the Hopf\u0000link. We show that the worldsheet skein $D$-module for the Hopf link conormal\u0000is generated by three operator polynomials that annihilate the skein valued\u0000partition function for any choice of Lagrangian filling and recursively\u0000determine it uniquely. We find Lagrangian fillings for any point in the\u0000augmentation variety and show that their skein valued partition functions admit\u0000quiver-like expansions where all holomorphic curves are generated by a small\u0000number of basic holomorphic disks and annuli and their multiple covers.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721710","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}
The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian non-removable intersections at small scales, and a novel phenomenon of Legendrian barriers in contact geometry.
{"title":"Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$","authors":"Emmanuel Opshtein, Felix Schlenk","doi":"arxiv-2407.09408","DOIUrl":"https://doi.org/arxiv-2407.09408","url":null,"abstract":"The main theme of this paper is the introduction of a new type of\u0000polarizations, suited for some open symplectic manifolds, and their\u0000applications. These applications include symplectic embedding results that\u0000answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian\u0000non-removable intersections at small scales, and a novel phenomenon of\u0000Legendrian barriers in contact geometry.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721696","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}
String geometry theory is a candidate of the non-perturvative formulation of string theory. In this theory, strings constitute not only particles but also the space-time. In this review, we identify perturbative vacua, and derive the path-integrals of all order perturbative strings on the corresponding string backgrounds by considering the fluctuations around the vacua. On the other hand, the most dominant part of the path-integral of string geometry theory is the zeroth order part in the fluctuation of the action, which is obtained by substituting the perturbative vacua to the action. This part is identified with the effective potential of the string backgrounds and obtained explicitly. The global minimum of the potential is the string vacuum. The urgent problem is to find the global minimum. We introduce both analytical and numerical methods to solve it.
{"title":"String Geometry Theory and The String Vacuum","authors":"Matsuo Sato","doi":"arxiv-2407.09049","DOIUrl":"https://doi.org/arxiv-2407.09049","url":null,"abstract":"String geometry theory is a candidate of the non-perturvative formulation of\u0000string theory. In this theory, strings constitute not only particles but also\u0000the space-time. In this review, we identify perturbative vacua, and derive the\u0000path-integrals of all order perturbative strings on the corresponding string\u0000backgrounds by considering the fluctuations around the vacua. On the other\u0000hand, the most dominant part of the path-integral of string geometry theory is\u0000the zeroth order part in the fluctuation of the action, which is obtained by\u0000substituting the perturbative vacua to the action. This part is identified with\u0000the effective potential of the string backgrounds and obtained explicitly. The\u0000global minimum of the potential is the string vacuum. The urgent problem is to\u0000find the global minimum. We introduce both analytical and numerical methods to\u0000solve it.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721695","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}
In this article, we define an index of Maslov type for general symplectic paths which have two arbitrary end points. This Maslov-type index is a generalization of the Conley-Zehnder-Long index and the method of constructing the index is consistent no matter whether the starting point of the path is identity or not, which is different from the ones for Long's Maslov-type index and Liu's $L_0$-index. Some natural properties for the index still hold. We review other versions of Maslov indices and compare them with our definition. In particular, this Maslov-type index can be looked as a realization of Cappell-Lee-Miller's index for a pair of Lagrangian paths from the point of view of index for symplectic paths.
{"title":"On Maslov-type index for general paths of symplectic matrices","authors":"Hai-Long Her, Qiyu Zhong","doi":"arxiv-2407.08433","DOIUrl":"https://doi.org/arxiv-2407.08433","url":null,"abstract":"In this article, we define an index of Maslov type for general symplectic\u0000paths which have two arbitrary end points. This Maslov-type index is a\u0000generalization of the Conley-Zehnder-Long index and the method of constructing\u0000the index is consistent no matter whether the starting point of the path is\u0000identity or not, which is different from the ones for Long's Maslov-type index\u0000and Liu's $L_0$-index. Some natural properties for the index still hold. We\u0000review other versions of Maslov indices and compare them with our definition.\u0000In particular, this Maslov-type index can be looked as a realization of\u0000Cappell-Lee-Miller's index for a pair of Lagrangian paths from the point of\u0000view of index for symplectic paths.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141610104","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}
This article generalizes the theory of shifted symplectic structures to the relative context and non-geometric stacks. We describe basic constructions that naturally appear in this theory: shifted cotangent bundles and the AKSZ procedure. Along the way, we also develop the theory of shifted symplectic groupoids presenting shifted symplectic structures on quotients and define a deformation to the normal cone for shifted Lagrangian morphisms.
{"title":"Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone","authors":"Damien Calaque, Pavel Safronov","doi":"arxiv-2407.08622","DOIUrl":"https://doi.org/arxiv-2407.08622","url":null,"abstract":"This article generalizes the theory of shifted symplectic structures to the\u0000relative context and non-geometric stacks. We describe basic constructions that\u0000naturally appear in this theory: shifted cotangent bundles and the AKSZ\u0000procedure. Along the way, we also develop the theory of shifted symplectic\u0000groupoids presenting shifted symplectic structures on quotients and define a\u0000deformation to the normal cone for shifted Lagrangian morphisms.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141610106","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}
Michael Hutchings, Agniva Roy, Morgan Weiler, Yuan Yao
Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an ``anchored symplectic embedding'' to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain.
{"title":"Anchored symplectic embeddings","authors":"Michael Hutchings, Agniva Roy, Morgan Weiler, Yuan Yao","doi":"arxiv-2407.08512","DOIUrl":"https://doi.org/arxiv-2407.08512","url":null,"abstract":"Given two four-dimensional symplectic manifolds, together with knots in their\u0000boundaries, we define an ``anchored symplectic embedding'' to be a symplectic\u0000embedding, together with a two-dimensional symplectic cobordism between the\u0000knots (in the four-dimensional cobordism determined by the embedding). We use\u0000techniques from embedded contact homology to determine quantitative critera for\u0000when anchored symplectic embeddings exist, for many examples of toric domains.\u0000In particular we find examples where ordinarily symplectic embeddings exist,\u0000but they cannot be upgraded to anchored symplectic embeddings unless one\u0000enlarges the target domain.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141610102","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}
We prove that given a closed connected symplectic manifold equipped with a Borel probability measure, an arbitrarily large portion of the measure can be covered by a symplectically embedded polydisk, generalizing a result of Schlenk. We apply this to constraints on symplectic quasi-states. Quasi-states are a certain class of not necessarily linear functionals on the algebra of continuous functions of a compact space. When the space is a symplectic manifold, a more restrictive subclass of symplectic quasi-states was introduced by Entov--Polterovich. We use our embedding result to prove that a certain `soft' construction of quasi-states, which is due to Aarnes, cannot yield nonlinear symplectic quasi-states in dimension at least four.
{"title":"Constraints on symplectic quasi-states","authors":"Adi Dickstein, Frol Zapolsky","doi":"arxiv-2407.08014","DOIUrl":"https://doi.org/arxiv-2407.08014","url":null,"abstract":"We prove that given a closed connected symplectic manifold equipped with a\u0000Borel probability measure, an arbitrarily large portion of the measure can be\u0000covered by a symplectically embedded polydisk, generalizing a result of\u0000Schlenk. We apply this to constraints on symplectic quasi-states. Quasi-states\u0000are a certain class of not necessarily linear functionals on the algebra of\u0000continuous functions of a compact space. When the space is a symplectic\u0000manifold, a more restrictive subclass of symplectic quasi-states was introduced\u0000by Entov--Polterovich. We use our embedding result to prove that a certain\u0000`soft' construction of quasi-states, which is due to Aarnes, cannot yield\u0000nonlinear symplectic quasi-states in dimension at least four.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141610105","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}
We show that there always exists an inscribed square in a Jordan curve given as the union of two graphs of functions of Lipschitz constant less than $1 + sqrt{2}$. We are motivated by Tao's result that there exists such a square in the case of Lipschitz constant less than $1$. In the case of Lipschitz constant $1$, we show that the Jordan curve inscribes rectangles of every similarity class. Our approach involves analysing the change in the spectral invariants of the Jordan Floer homology under perturbations of the Jordan curve.
{"title":"Square pegs between two graphs","authors":"Joshua Evan Greene, Andrew Lobb","doi":"arxiv-2407.07798","DOIUrl":"https://doi.org/arxiv-2407.07798","url":null,"abstract":"We show that there always exists an inscribed square in a Jordan curve given\u0000as the union of two graphs of functions of Lipschitz constant less than $1 +\u0000sqrt{2}$. We are motivated by Tao's result that there exists such a square in\u0000the case of Lipschitz constant less than $1$. In the case of Lipschitz constant\u0000$1$, we show that the Jordan curve inscribes rectangles of every similarity\u0000class. Our approach involves analysing the change in the spectral invariants of\u0000the Jordan Floer homology under perturbations of the Jordan curve.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588418","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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