Pub Date : 2023-09-18DOI: 10.1112/s0010437x23007480
Igor Uljarević
We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology , that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to $+infty$ on the open subset but remain close to $0$ and positive on the rest of the boundary.
{"title":"Selective symplectic homology with applications to contact non-squeezing","authors":"Igor Uljarević","doi":"10.1112/s0010437x23007480","DOIUrl":"https://doi.org/10.1112/s0010437x23007480","url":null,"abstract":"We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology , that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to $+infty$ on the open subset but remain close to $0$ and positive on the rest of the boundary.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135202829","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}
Pub Date : 2023-08-17DOI: 10.1112/S0010437X23007339
S. Nie
For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.
{"title":"Connected components of affine Deligne–Lusztig varieties for unramified groups","authors":"S. Nie","doi":"10.1112/S0010437X23007339","DOIUrl":"https://doi.org/10.1112/S0010437X23007339","url":null,"abstract":"For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"159 1","pages":"2051 - 2088"},"PeriodicalIF":1.8,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42989753","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}
Pub Date : 2023-08-03DOI: 10.1112/S0010437X23007406
Andrew Senger
Given a height at most two Landweber exact $mathbb {E}_infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $mathbb {E}_infty$-complex orientation $mathrm {MU} to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $mathbb {E}_infty$-complex orientation $mathrm {MU} to mathrm {tmf}_1 (n)$ for all $n, {geq}, 2$.
{"title":"Obstruction theory and the level n elliptic genus","authors":"Andrew Senger","doi":"10.1112/S0010437X23007406","DOIUrl":"https://doi.org/10.1112/S0010437X23007406","url":null,"abstract":"Given a height at most two Landweber exact $mathbb {E}_infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $mathbb {E}_infty$-complex orientation $mathrm {MU} to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $mathbb {E}_infty$-complex orientation $mathrm {MU} to mathrm {tmf}_1 (n)$ for all $n, {geq}, 2$.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"159 1","pages":"2000 - 2021"},"PeriodicalIF":1.8,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47099002","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}
Pub Date : 2023-07-31DOI: 10.1112/S0010437X23007327
W. Gan, B. Gross, Dipendra Prasad
In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case: that of a pair of skew-Hermitian spaces $W = V$.
{"title":"Twisted GGP problems and conjectures","authors":"W. Gan, B. Gross, Dipendra Prasad","doi":"10.1112/S0010437X23007327","DOIUrl":"https://doi.org/10.1112/S0010437X23007327","url":null,"abstract":"In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case: that of a pair of skew-Hermitian spaces $W = V$.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"159 1","pages":"1916 - 1973"},"PeriodicalIF":1.8,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41318824","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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