Pub Date : 2023-10-10DOI: 10.1112/s0010437x23007492
Yoel Groman, Umut Varolgunes
A complete embedding is a symplectic embedding $iota :Yto M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $iota (K)$ inside $M$ . Under the assumption that the torsion exponents of $K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.
{"title":"Locality of relative symplectic cohomology for complete embeddings","authors":"Yoel Groman, Umut Varolgunes","doi":"10.1112/s0010437x23007492","DOIUrl":"https://doi.org/10.1112/s0010437x23007492","url":null,"abstract":"A complete embedding is a symplectic embedding $iota :Yto M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $iota (K)$ inside $M$ . Under the assumption that the torsion exponents of $K$ are bounded, we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254980","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 : 2023-10-09DOI: 10.1112/s0010437x22008144
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"COM volume 159 issue 11 Cover and Front matter","authors":"","doi":"10.1112/s0010437x22008144","DOIUrl":"https://doi.org/10.1112/s0010437x22008144","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141188","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 : 2023-10-09DOI: 10.1112/s0010437x22008156
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"COM volume 159 issue 11 Cover and Back matter","authors":"","doi":"10.1112/s0010437x22008156","DOIUrl":"https://doi.org/10.1112/s0010437x22008156","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135142193","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 : 2023-10-06DOI: 10.1112/s0010437x23007443
N. I. Shepherd-Barron
Suppose that $f:Xto C$ is a general Jacobian elliptic surface over ${mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$ . Assume that $10 h>12(q-1)$ , that $h>0$ and let $overline {mathcal {E}ell ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{ain Z}$ , where $Zsubset C$ is the ramification locus of the classifying morphism $phi :Cto overline {mathcal {E}ell ell }$ that corresponds to $Xto C$ . (2) For each $ain Z$ the map $overline {nabla }_{v_a}:H^{2,0}(X)to H^{1,1}_{rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${mathbb {C}}[eta _a]$ and its kernel is $H^0(X,Omega ^2_X(-E_a))$ , where $E_a=f^{-1}(a)$ . (3) The classes $[eta _a]$ form an orthogonal basis of $H^{1,1}_{rm prim}(X)$ and $[eta _a]$ is represented by a meromorphic $2$ -form $eta _a$ in $H^0(X,Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $hge q+3$ . (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$ . (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)
{"title":"Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces","authors":"N. I. Shepherd-Barron","doi":"10.1112/s0010437x23007443","DOIUrl":"https://doi.org/10.1112/s0010437x23007443","url":null,"abstract":"Suppose that $f:Xto C$ is a general Jacobian elliptic surface over ${mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$ . Assume that $10 h>12(q-1)$ , that $h>0$ and let $overline {mathcal {E}ell ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{ain Z}$ , where $Zsubset C$ is the ramification locus of the classifying morphism $phi :Cto overline {mathcal {E}ell ell }$ that corresponds to $Xto C$ . (2) For each $ain Z$ the map $overline {nabla }_{v_a}:H^{2,0}(X)to H^{1,1}_{rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${mathbb {C}}[eta _a]$ and its kernel is $H^0(X,Omega ^2_X(-E_a))$ , where $E_a=f^{-1}(a)$ . (3) The classes $[eta _a]$ form an orthogonal basis of $H^{1,1}_{rm prim}(X)$ and $[eta _a]$ is represented by a meromorphic $2$ -form $eta _a$ in $H^0(X,Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $hge q+3$ . (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$ . (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351181","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 : 2023-09-18DOI: 10.1112/s0010437x23007388
Elden Elmanto, Rune Haugseng
Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ( $infty$ -)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(infty,2)$ -categories of bispans, characterized by a universal property: they corepresent functors out of $infty$ -categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $infty$ -categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$ -sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$ -theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$ -theory spectra.
{"title":"On distributivity in higher algebra I: the universal property of bispans","authors":"Elden Elmanto, Rune Haugseng","doi":"10.1112/s0010437x23007388","DOIUrl":"https://doi.org/10.1112/s0010437x23007388","url":null,"abstract":"Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ( $infty$ -)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(infty,2)$ -categories of bispans, characterized by a universal property: they corepresent functors out of $infty$ -categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $infty$ -categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$ -sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$ -theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$ -theory spectra.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135109327","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 : 2023-09-18DOI: 10.1112/s0010437x23007479
Richard Hind, Jun Zhang
The Hamiltonian shape invariant of a domain $X subset mathbb R^4$, as a subset of $mathbb R^2$, describes the product Lagrangian tori which may be embedded in $X$. We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when $X$ is a basic $4$-dimensional toric domain such as a ball $B^4(R)$, an ellipsoid $E(a,b)$ with $frac{b}{a} in {mathbb N}_{geq 2}$, or a polydisk $P(c,d)$. As applications, via the path lifting, we can detect knotted embeddings of product Lagrangian tori in many toric $X$. We also obtain novel obstructions to symplectic embeddings between domains that are more general than toric concave or toric convex.
作为$mathbb R^2$的子集，域$X subset mathbb R^4$的哈密顿形状不变量描述了可嵌入到$X$中的积拉格朗日环面。当$X$是一个基本的$4$维环面域，如球$B^4(R)$、带$frac{b}{a} in {mathbb N}_{geq 2}$的椭球$E(a,b)$或多盘$P(c,d)$时，我们提供了确定形状不变量中的路径是否可以提升的充分必要条件，即实现为嵌入拉格朗日环面的光滑族。作为应用，通过路径提升，我们可以在许多环面中检测乘积拉格朗日环面的打结嵌入$X$。我们还获得了比环面凹或环面凸更一般的域间辛嵌入的新障碍。
{"title":"Hamiltonian knottedness and lifting paths from the shape invariant","authors":"Richard Hind, Jun Zhang","doi":"10.1112/s0010437x23007479","DOIUrl":"https://doi.org/10.1112/s0010437x23007479","url":null,"abstract":"The Hamiltonian shape invariant of a domain $X subset mathbb R^4$, as a subset of $mathbb R^2$, describes the product Lagrangian tori which may be embedded in $X$. We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when $X$ is a basic $4$-dimensional toric domain such as a ball $B^4(R)$, an ellipsoid $E(a,b)$ with $frac{b}{a} in {mathbb N}_{geq 2}$, or a polydisk $P(c,d)$. As applications, via the path lifting, we can detect knotted embeddings of product Lagrangian tori in many toric $X$. We also obtain novel obstructions to symplectic embeddings between domains that are more general than toric concave or toric convex.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135109516","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 : 2023-09-18DOI: 10.1112/s0010437x23007431
Ana Caraiani, Matteo Tamiozzo
We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$ -adic local Langlands correspondence for $mathrm {GL}_2(mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.
{"title":"On the étale cohomology of Hilbert modular varieties with torsion coefficients","authors":"Ana Caraiani, Matteo Tamiozzo","doi":"10.1112/s0010437x23007431","DOIUrl":"https://doi.org/10.1112/s0010437x23007431","url":null,"abstract":"We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$ -adic local Langlands correspondence for $mathrm {GL}_2(mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135109517","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 : 2023-09-18DOI: 10.1112/s0010437x23007455
Leonid Polterovich, Egor Shelukhin
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.
{"title":"Lagrangian configurations and Hamiltonian maps","authors":"Leonid Polterovich, Egor Shelukhin","doi":"10.1112/s0010437x23007455","DOIUrl":"https://doi.org/10.1112/s0010437x23007455","url":null,"abstract":"We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135109322","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 : 2023-09-18DOI: 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":null,"pages":null},"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：