We prove that $(mathbb{RP}^{2n-1},xi_{std})$ is not exactly fillable for any $nne 2^k$ and there exist strongly fillable but not exactly fillable contact manifolds for all dimension $ge 5$.
{"title":"(ℝℙ2n−1,ξstd) is not exactly fillable for\u0000n≠2k","authors":"Zheng Zhou","doi":"10.2140/gt.2021.25.3013","DOIUrl":"https://doi.org/10.2140/gt.2021.25.3013","url":null,"abstract":"We prove that $(mathbb{RP}^{2n-1},xi_{std})$ is not exactly fillable for any $nne 2^k$ and there exist strongly fillable but not exactly fillable contact manifolds for all dimension $ge 5$.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"67 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81179748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.
{"title":"Transverse invariants and exotic surfaces in the\u00004–ball","authors":"Andr'as Juh'asz, Maggie Miller, Ian Zemke","doi":"10.2140/gt.2021.25.2963","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2963","url":null,"abstract":"Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"88 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2020-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79169404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathcal{C}_dsubset mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Vi`{e}te map $mathscr{V} : mathcal{C}_d rightarrow Sym_d(mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $mathscr{V}_*$ at the level of fundamental groups realises an isomorphism between $pi_1(mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $mathcal{C}$ of $mathcal{C}_d$ with a collection of coordinate hyperplanes in $mathbb{C}^{d+1}$, the image of the map $mathscr{V} _* : pi_1(mathcal{C}) rightarrow B_d$ is not known in general. In the present paper, we show that the map $mathscr{V} _*$ is surjective provided that the support of the corresponding polynomials spans $mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of $mathscr{V} _*$ is the expected wreath product of $mathbb{Z}/bmathbb{Z}$ with $B_{d/b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.
{"title":"Braid monodromy of univariate fewnomials","authors":"A. Esterov, Lionel Lang","doi":"10.2140/gt.2021.25.3053","DOIUrl":"https://doi.org/10.2140/gt.2021.25.3053","url":null,"abstract":"Let $mathcal{C}_dsubset mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Vi`{e}te map $mathscr{V} : mathcal{C}_d rightarrow Sym_d(mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $mathscr{V}_*$ at the level of fundamental groups realises an isomorphism between $pi_1(mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $mathcal{C}$ of $mathcal{C}_d$ with a collection of coordinate hyperplanes in $mathbb{C}^{d+1}$, the image of the map $mathscr{V} _* : pi_1(mathcal{C}) rightarrow B_d$ is not known in general. In the present paper, we show that the map $mathscr{V} _*$ is surjective provided that the support of the corresponding polynomials spans $mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of $mathscr{V} _*$ is the expected wreath product of $mathbb{Z}/bmathbb{Z}$ with $B_{d/b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"47 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72861792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kahler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations. Our main result relates this stability condition to Kahler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kahler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.
k -多稳定性一方面在推测上等价于某些正则Kahler度量在极化变体上的存在性，另一方面在推测上给出了模的正确概念。我们引入了k -聚稳定变种族的稳定性概念，通过相关的投影扩展了看作k -聚稳定变种族的束的边坡稳定性的经典概念。我们推测这是形成振动模量的正确条件。我们的主要结果将这种稳定性条件与Kahler几何联系起来:我们证明了最优辛连接的存在意味着纤维的半不稳定性。最优辛连接是指纤维向常数标量曲率Kahler度规的选择，满足一定的几何偏微分方程。我们推测这种连接的存在等价于纤维的多稳定性。我们通过描述一个嵌入在固定射影空间中的纤维的GIT问题，证明了这个猜想的有限维类比，并证明了GIT多稳定性等价于某个矩映射的零的存在性。
{"title":"Moduli theory, stability of fibrations and optimal symplectic connections","authors":"R. Dervan, Lars Martin Sektnan","doi":"10.2140/gt.2021.25.2643","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2643","url":null,"abstract":"K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kahler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations. \u0000Our main result relates this stability condition to Kahler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kahler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"60 7 Pt 2 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88620019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mixed curvature almost flat manifolds","authors":"V. Kapovitch","doi":"10.2140/gt.2021.25.2017","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2017","url":null,"abstract":"We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"54 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83316314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the minimum weight of a weighted blow-up of $mathbf A^d$ with $varepsilon$-log canonical singularities is bounded by a constant depending only on $varepsilon $ and $d$. This was conjectured by Birkar. Using the recent classification of $4$-dimensional empty simplices by Iglesias-Vali~no and Santos, we work out an explicit bound for blowups of $mathbf A^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.
我们证明了$mathbf a ^d$与$varepsilon$-log正则奇点的加权爆破的最小权值由一个仅依赖于$varepsilon$和$d$的常数所限定。这是比尔卡的推测。利用Iglesias-Vali~no和Santos最近对$ $4维空简式的分类，我们得到了$ $ mathbf A^4$具有端点奇点的膨胀的显式界:最小的权重总是最多$32$，除了有限多的情况外，在所有情况下最多$6$。
{"title":"Blowups with log canonical singularities","authors":"G. Sankaran, F. Santos","doi":"10.2140/gt.2021.25.2145","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2145","url":null,"abstract":"We show that the minimum weight of a weighted blow-up of $mathbf A^d$ with $varepsilon$-log canonical singularities is bounded by a constant depending only on $varepsilon $ and $d$. This was conjectured by Birkar. \u0000Using the recent classification of $4$-dimensional empty simplices by Iglesias-Vali~no and Santos, we work out an explicit bound for blowups of $mathbf A^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"76 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83854135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into CP is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into CP as a complete holomorphic Legendrian curve. Under the twistor projection π : CP → S onto the 4-sphere, immersed holomorphic Legendrian curves M → CP are in bijective correspondence with superminimal immersions M → S of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in S. In particular, superminimal immersions into S satisfy the Runge approximation theorem and the Calabi-Yau property.
{"title":"Holomorphic Legendrian curves in ℂℙ3 and\u0000superminimal surfaces in 𝕊4","authors":"A. Alarcón, F. Forstnerič, F. Lárusson","doi":"10.2140/gt.2021.25.3507","DOIUrl":"https://doi.org/10.2140/gt.2021.25.3507","url":null,"abstract":"We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into CP is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into CP as a complete holomorphic Legendrian curve. Under the twistor projection π : CP → S onto the 4-sphere, immersed holomorphic Legendrian curves M → CP are in bijective correspondence with superminimal immersions M → S of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in S. In particular, superminimal immersions into S satisfy the Runge approximation theorem and the Calabi-Yau property.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"59 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73014242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to the article An infinite-rank summand of topologically slice knots","authors":"Jennifer Hom","doi":"10.2140/gt.2019.23.2699","DOIUrl":"https://doi.org/10.2140/gt.2019.23.2699","url":null,"abstract":"","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"23 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85206056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we construct local and global solutions to the Kahler-Ricci flow from a non-collapsed Kahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kahler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kahler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth.
{"title":"Kähler manifolds with almost nonnegative\u0000curvature","authors":"Man-Chun Lee, Luen-Fai Tam","doi":"10.2140/gt.2021.25.1979","DOIUrl":"https://doi.org/10.2140/gt.2021.25.1979","url":null,"abstract":"In this paper, we construct local and global solutions to the Kahler-Ricci flow from a non-collapsed Kahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kahler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kahler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"82 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2019-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88466534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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