We study the monopole contribution to the refined Vafa-Witten invariant, recently defined by Maulik and Thomas [13]. We apply results of Gholampour and Thomas [7] to prove a universality result for the generating series of contributions of Higgs pairs with 1-dimensional weight spaces. For prime rank, these account for the entire monopole contribution, by a theorem of Thomas. We use toric computations to determine part of the generating series, and find agreement with the conjectures of G"ottsche and Kool [10] for rank 2 and 3.
{"title":"Monopole contributions to refined Vafa–Witten\u0000invariants","authors":"T. Laarakker","doi":"10.2140/gt.2020.24.2781","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2781","url":null,"abstract":"We study the monopole contribution to the refined Vafa-Witten invariant, recently defined by Maulik and Thomas [13]. We apply results of Gholampour and Thomas [7] to prove a universality result for the generating series of contributions of Higgs pairs with 1-dimensional weight spaces. For prime rank, these account for the entire monopole contribution, by a theorem of Thomas. We use toric computations to determine part of the generating series, and find agreement with the conjectures of G\"ottsche and Kool [10] for rank 2 and 3.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"16 4 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73420260","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 relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space.
{"title":"Complex algebraic compactifications\u0000of the moduli space of Hermitian Yang–Mills connections on a projective\u0000manifold","authors":"D. Greb, Benjamin Sibley, M. Toma, R. Wentworth","doi":"10.2140/gt.2021.25.1719","DOIUrl":"https://doi.org/10.2140/gt.2021.25.1719","url":null,"abstract":"In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"158 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77141372","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}
This is the first part of the project toward proving the BCOV's Feymann graph sum formula of all genera Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, we introduce the notion of N-Mixed-Spin-P fields, construct their moduli spaces, their virtual cycles, their virtual localization formulas, and a vanishing result associated with irregular graphs.
{"title":"The theory of N–mixed-spin-P fields","authors":"Huai-liang Chang, Shuai Guo, Jun Li, Wei-Ping Li","doi":"10.2140/gt.2021.25.775","DOIUrl":"https://doi.org/10.2140/gt.2021.25.775","url":null,"abstract":"This is the first part of the project toward proving the BCOV's Feymann graph sum formula of all genera Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, we introduce the notion of N-Mixed-Spin-P fields, construct their moduli spaces, their virtual cycles, their virtual localization formulas, and a vanishing result associated with irregular graphs.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"124 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85274764","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 a formula for the motive of the stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.
{"title":"A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve","authors":"Victoria Hoskins, Simon Pepin Lehalleur","doi":"10.2140/gt.2021.25.3555","DOIUrl":"https://doi.org/10.2140/gt.2021.25.3555","url":null,"abstract":"We prove a formula for the motive of the stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"8 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86329496","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}
M. Abouzaid, Sheel Ganatra, H. Iritani, Nick Sheridan
We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger-Yau-Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma Conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.
{"title":"The Gamma and Strominger–Yau–Zaslow\u0000conjectures : a tropical approach to periods","authors":"M. Abouzaid, Sheel Ganatra, H. Iritani, Nick Sheridan","doi":"10.2140/gt.2020.24.2547","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2547","url":null,"abstract":"We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger-Yau-Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma Conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"100 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76083175","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 will show that the projection $text{Homeo}^+(D^2_n)to B_n$ does not have a section; i.e. the braid group $B_n$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary point-wise and $n$ marked points in the interior as a set. We also give a new proof of a result of Markovic that the mapping class group of a closed surface cannot be geometrically realized as a group of homeomorphisms.
{"title":"On the nonrealizability of braid groups by homeomorphisms","authors":"Lei Chen","doi":"10.2140/gt.2019.23.3735","DOIUrl":"https://doi.org/10.2140/gt.2019.23.3735","url":null,"abstract":"In this paper, we will show that the projection $text{Homeo}^+(D^2_n)to B_n$ does not have a section; i.e. the braid group $B_n$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary point-wise and $n$ marked points in the interior as a set. We also give a new proof of a result of Markovic that the mapping class group of a closed surface cannot be geometrically realized as a group of homeomorphisms.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"32 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85388835","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 minimum number of critical points of a Weinstein Morse function on a Weinstein domain of dimension at least six is at most two more than the minimum number of critical points of a smooth Morse function on that domain; if the domain has non-zero middle-dimensional homology, these two numbers agree. As a corollary, we obtain a topological upper bound on the number of generators of the wrapped Fukaya category of the domain. We also show that there is an upper bound on the number of gradient trajectories between critical points in smoothly trivial Weinstein cobordisms.
{"title":"Simplifying Weinstein Morse functions","authors":"Oleg Lazarev","doi":"10.2140/gt.2020.24.2603","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2603","url":null,"abstract":"We prove that the minimum number of critical points of a Weinstein Morse function on a Weinstein domain of dimension at least six is at most two more than the minimum number of critical points of a smooth Morse function on that domain; if the domain has non-zero middle-dimensional homology, these two numbers agree. As a corollary, we obtain a topological upper bound on the number of generators of the wrapped Fukaya category of the domain. We also show that there is an upper bound on the number of gradient trajectories between critical points in smoothly trivial Weinstein cobordisms.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"10 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82015952","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}
The mapping class group $mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $mathrm{Aut}(pi_1 Sigma_g)$. For a surface of genus $g geq 2$, we show that any action of $mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $pi_1 Sigma_g$, or factors through a finite cyclic group. For $g geq 3$, all finite actions are trivial. This answers a question of Farb.
{"title":"Rigidity of mapping class group actions on\u0000S1","authors":"Kathryn Mann, M. Wolff","doi":"10.2140/GT.2020.24.1211","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1211","url":null,"abstract":"The mapping class group $mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $mathrm{Aut}(pi_1 Sigma_g)$. For a surface of genus $g geq 2$, we show that any action of $mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $pi_1 Sigma_g$, or factors through a finite cyclic group. For $g geq 3$, all finite actions are trivial. This answers a question of Farb.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"16 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84395942","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 establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.
我们建立了与给定叶形拟互补存在的h原理的一种形式;同样的方法也提供了经典的马瑟定理的证明。
{"title":"Quasicomplementary foliations and the\u0000Mather–Thurston theorem","authors":"G. Meigniez","doi":"10.2140/gt.2021.25.643","DOIUrl":"https://doi.org/10.2140/gt.2021.25.643","url":null,"abstract":"We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"19 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82698016","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 every positive definite closed 4-manifold with $b_2^+>1$ and without 1-handles has a vanishing stable cohomotopy Seiberg-Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented 4-manifold with $b_2^+notequiv 1$ and $b_2^-notequiv 1pmod{4}$ and without 1-handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the 1-handle condition, we prove these results under more general conditions which are much easier to verify.
{"title":"Geometrically simply connected 4–manifolds\u0000and stable cohomotopy Seiberg–Witten invariants","authors":"Kouichi Yasui","doi":"10.2140/gt.2019.23.2685","DOIUrl":"https://doi.org/10.2140/gt.2019.23.2685","url":null,"abstract":"We show that every positive definite closed 4-manifold with $b_2^+>1$ and without 1-handles has a vanishing stable cohomotopy Seiberg-Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented 4-manifold with $b_2^+notequiv 1$ and $b_2^-notequiv 1pmod{4}$ and without 1-handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the 1-handle condition, we prove these results under more general conditions which are much easier to verify.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"129 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79566447","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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