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Comparing Kahler cone and symplectic cone of one-point blowup of Enriques surface 比较恩里克曲面一点炸开的卡勒锥和交点锥
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-14 DOI: arxiv-2407.10217
Shengzhen Ning
{"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":null,"pages":null},"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}
引用次数: 0
On the blow-up formula of the Chow weights for polarized toric manifolds 论极化环流形的周权重吹胀公式
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-14 DOI: arxiv-2407.10082
King Leung Lee, Naoto Yotsutani
{"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":null,"pages":null},"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}
引用次数: 0
The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal 霍普夫链接常模的拉格朗日填充上的世界表矢量 D 模块和基本曲线
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-13 DOI: arxiv-2407.09836
Tobias Ekholm, Pietro Longhi, Lukas Nakamura
{"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":null,"pages":null},"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}
引用次数: 0
Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$ 刘维尔极化及其拉格朗日骨架在 4$ 维的刚性
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-12 DOI: arxiv-2407.09408
Emmanuel Opshtein, Felix Schlenk
{"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":null,"pages":null},"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}
引用次数: 0
String Geometry Theory and The String Vacuum 弦几何理论与弦真空
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-12 DOI: arxiv-2407.09049
Matsuo Sato
{"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":null,"pages":null},"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}
引用次数: 0
On Maslov-type index for general paths of symplectic matrices 论交映矩阵一般路径的马斯洛夫型指数
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-11 DOI: arxiv-2407.08433
Hai-Long Her, Qiyu Zhong
{"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":null,"pages":null},"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}
引用次数: 0
Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone 位移余切束、交映群像和法锥变形
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-11 DOI: arxiv-2407.08622
Damien Calaque, Pavel Safronov
{"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":null,"pages":null},"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}
引用次数: 0
Anchored symplectic embeddings 锚定交映嵌入
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-11 DOI: arxiv-2407.08512
Michael Hutchings, Agniva Roy, Morgan Weiler, Yuan Yao
{"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":null,"pages":null},"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}
引用次数: 0
Constraints on symplectic quasi-states 对称准态的约束条件
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-10 DOI: arxiv-2407.08014
Adi Dickstein, Frol Zapolsky
{"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":null,"pages":null},"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}
引用次数: 0
Square pegs between two graphs 两个图形之间的方钉
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-10 DOI: arxiv-2407.07798
Joshua Evan Greene, Andrew Lobb
{"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":null,"pages":null},"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}
引用次数: 0
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