In this paper, we develop new techniques for understanding surfaces in $mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques.
{"title":"Bridge trisections in ℂℙ2 and the Thom\u0000conjecture","authors":"Peter Lambert-Cole","doi":"10.2140/GT.2020.24.1571","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1571","url":null,"abstract":"In this paper, we develop new techniques for understanding surfaces in $mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"25 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75295323","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 compute the Euler characteristics of the recently discovered series of Gothic Teichmuller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichmuller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Euler characteristic of the ambient Hilbert modular surfaces. This results in interesting 'varying' phenomena for Lyapunov exponents.
{"title":"Euler characteristics of Gothic Teichmüller\u0000curves","authors":"M. Moller, David Torres-Teigell","doi":"10.2140/GT.2020.24.1149","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1149","url":null,"abstract":"We compute the Euler characteristics of the recently discovered series of Gothic Teichmuller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichmuller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Euler characteristic of the ambient Hilbert modular surfaces. This results in interesting 'varying' phenomena for Lyapunov exponents.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"45 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78932379","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 study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.
{"title":"Analytic tangent cones of admissible Hermitian\u0000Yang–Mills connections","authors":"Xuemiao Chen, Song Sun","doi":"10.2140/GT.2021.25.2061","DOIUrl":"https://doi.org/10.2140/GT.2021.25.2061","url":null,"abstract":"In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"14 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78683356","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}
Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the $q$-refined 2-dimensional Kontsevich-Soibelman scattering diagrams compute, after the change of variables $q=e^{i hbar}$, generating series of certain higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces. This result provides a mathematically rigorous realization of the physical derivation of the refined wall-crossing formula from topological string theory proposed by Cecotti-Vafa, and in particular can be seen as a non-trivial mathematical check of the connection suggested by Witten between higher genus open A-model and Chern-Simons theory. We also prove some new BPS integrality results and propose some other BPS integrality conjectures.
{"title":"The quantum tropical vertex","authors":"Pierrick Bousseau","doi":"10.2140/gt.2020.24.1297","DOIUrl":"https://doi.org/10.2140/gt.2020.24.1297","url":null,"abstract":"Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the $q$-refined 2-dimensional Kontsevich-Soibelman scattering diagrams compute, after the change of variables $q=e^{i hbar}$, generating series of certain higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces. \u0000This result provides a mathematically rigorous realization of the physical derivation of the refined wall-crossing formula from topological string theory proposed by Cecotti-Vafa, and in particular can be seen as a non-trivial mathematical check of the connection suggested by Witten between higher genus open A-model and Chern-Simons theory. \u0000We also prove some new BPS integrality results and propose some other BPS integrality conjectures.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"11 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86567497","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 introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.
{"title":"Edge stabilization in the homology of graph braid groups","authors":"B. An, Gabriel C. Drummond-Cole, Ben Knudsen","doi":"10.2140/gt.2020.24.421","DOIUrl":"https://doi.org/10.2140/gt.2020.24.421","url":null,"abstract":"We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"21 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87256375","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 establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit $2$-disk in $mathbf{R}^2$ into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit $2$-disk proved by Colding and Minicozzi.
{"title":"Min-max minimal disks with free boundary in Riemannian manifolds","authors":"Longzhi Lin, Ao Sun, Xin Zhou","doi":"10.2140/gt.2020.24.471","DOIUrl":"https://doi.org/10.2140/gt.2020.24.471","url":null,"abstract":"In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $mathbf{R}^n$ to the Riemannian setting. \u0000More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit $2$-disk in $mathbf{R}^2$ into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit $2$-disk proved by Colding and Minicozzi.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"31 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86384242","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 a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of restriction maps, and prove some basic properties. Our main contribution is to identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 coisotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors.
{"title":"Mayer–Vietoris property for relative symplectic\u0000cohomology","authors":"Umut Varolgunes","doi":"10.2140/GT.2021.25.547","DOIUrl":"https://doi.org/10.2140/GT.2021.25.547","url":null,"abstract":"In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of restriction maps, and prove some basic properties. Our main contribution is to identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 coisotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"24 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79123177","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}
Given a smooth target curve $X$, we explore the relationship between Gromov-Witten invariants of $X$ relative to a smooth divisor and orbifold Gromov-Witten invariants of the $r$-th root stack along the divisor. We proved that relative invariants are equal to the $r^0$-coefficient of the corresponding orbifold Gromov-Witten invariants of $r$-th root stack for $r$ sufficiently large. Our result provides a precise relation between relative and orbifold invariants of target curves generalizing the result of Abramovich-Cadman-Wise to higher genus invariants of curves. Moreover, when $r$ is sufficiently large, we proved that relative stationary invariants of $X$ are equal to the orbifold stationary invariants in all genera. Our results lead to some interesting applications: a new proof of genus zero equality between relative and orbifold invariants of $X$ via localization; a new proof of the formula of Johnson-Pandharipande-Tseng for double Hurwitz numbers; a version of GW/H correspondence for stationary orbifold invariants.
{"title":"Higher genus relative and orbifold Gromov–Witten\u0000invariants","authors":"Hsian-Hua Tseng, F. You","doi":"10.2140/gt.2020.24.2749","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2749","url":null,"abstract":"Given a smooth target curve $X$, we explore the relationship between Gromov-Witten invariants of $X$ relative to a smooth divisor and orbifold Gromov-Witten invariants of the $r$-th root stack along the divisor. We proved that relative invariants are equal to the $r^0$-coefficient of the corresponding orbifold Gromov-Witten invariants of $r$-th root stack for $r$ sufficiently large. Our result provides a precise relation between relative and orbifold invariants of target curves generalizing the result of Abramovich-Cadman-Wise to higher genus invariants of curves. Moreover, when $r$ is sufficiently large, we proved that relative stationary invariants of $X$ are equal to the orbifold stationary invariants in all genera. \u0000Our results lead to some interesting applications: a new proof of genus zero equality between relative and orbifold invariants of $X$ via localization; a new proof of the formula of Johnson-Pandharipande-Tseng for double Hurwitz numbers; a version of GW/H correspondence for stationary orbifold invariants.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"56 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84857794","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 study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected $4$-manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $mathbb{C}P^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $mathbb{C}P^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.
{"title":"Isotopies of surfaces in 4–manifolds via banded\u0000unlink diagrams","authors":"M. Hughes, Seungwon Kim, Maggie Miller","doi":"10.2140/GT.2020.24.1519","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1519","url":null,"abstract":"In this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected $4$-manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $mathbb{C}P^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $mathbb{C}P^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"10 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88028397","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 prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $mathrm{H}underline{mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $mathrm{H}mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.
{"title":"Eilenberg–Mac Lane spectra as equivariant\u0000Thom spectra","authors":"Jeremy Hahn, D. Wilson","doi":"10.2140/gt.2020.24.2709","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2709","url":null,"abstract":"We prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $mathrm{H}underline{mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $mathrm{H}mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"106 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81097310","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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