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Families of relatively exact Lagrangians, free loop spaces and generalised homology 相对精确拉格朗日族、自由环空间和广义同源性

Selecta Mathematica Pub Date : 2024-02-03 DOI: 10.1007/s00029-023-00910-6

{"title":"Families of relatively exact Lagrangians, free loop spaces and generalised homology","authors":"","doi":"10.1007/s00029-023-00910-6","DOIUrl":"https://doi.org/10.1007/s00029-023-00910-6","url":null,"abstract":"<h3>Abstract</h3> We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy (psi ^1) of a symplectic manifold ((M, omega )) fixing a relatively exact Lagrangian L setwise must act trivially on (R_*(L)) , where (R_*) is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over ({mathbb {Z}}/2) and over ({mathbb {Z}}) under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, (psi ^1|_L) is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that (psi ^1|_L) acts trivially on (R_*({mathcal {L}}L)) , where ({mathcal {L}}L) is the free loop space of L. From this we deduce that when L is a surface or a (K(pi , 1)) , (psi ^1|_L) is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over ({mathbb {Z}}/2) .","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139678557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

q-Painlevé equations on cluster Poisson varieties via toric geometry 通过环几何研究群泊松变体上的 q-Painlevé 方程

Selecta Mathematica Pub Date : 2024-01-27 DOI: 10.1007/s00029-023-00906-2

引用次数: 0

Locally free representations of quivers over commutative Frobenius algebras 交换弗罗贝纽斯代数上四元组的局部自由表示

Selecta Mathematica Pub Date : 2024-01-27 DOI: 10.1007/s00029-023-00914-2

Tamás Hausel, Emmanuel Letellier, Fernando Rodriguez-Villegas

{"title":"Locally free representations of quivers over commutative Frobenius algebras","authors":"Tamás Hausel, Emmanuel Letellier, Fernando Rodriguez-Villegas","doi":"10.1007/s00029-023-00914-2","DOIUrl":"https://doi.org/10.1007/s00029-023-00914-2","url":null,"abstract":"In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra (textrm{R}) by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over (textrm{R}) equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over (textrm{R}[t]/(t^2)). Using these results together with results of Geiss, Leclerc and Schröer we give, when (textrm{k}) is algebraically closed, a classification of pairs ((Q,textrm{R})) such that the set of isomorphism classes of indecomposable locally free representations of Q over (textrm{R}) is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra (mathbb {F}_q[t]/(t^r)). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.\u0000","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the local-global principle for isogenies of abelian surfaces 论无常曲面同构的局部-全局原理

Selecta Mathematica Pub Date : 2024-01-26 DOI: 10.1007/s00029-023-00908-0

Davide Lombardo, Matteo Verzobio

引用次数: 0

Degenerating products of flag varieties and applications to the Breuil–Mézard conjecture 旗变体的退化积及布雷尔-梅扎尔猜想的应用

Selecta Mathematica Pub Date : 2024-01-19 DOI: 10.1007/s00029-023-00905-3

Robin Bartlett

引用次数: 0

Rational endomorphisms of Fano hypersurfaces 法诺超曲面的有理内定型

Selecta Mathematica Pub Date : 2024-01-18 DOI: 10.1007/s00029-023-00897-0

Nathan Chen, David Stapleton

引用次数: 0

On rank in algebraic closure 关于代数封闭中的秩

Selecta Mathematica Pub Date : 2024-01-13 DOI: 10.1007/s00029-023-00903-5

Amichai Lampert, Tamar Ziegler

{"title":"On rank in algebraic closure","authors":"Amichai Lampert, Tamar Ziegler","doi":"10.1007/s00029-023-00903-5","DOIUrl":"https://doi.org/10.1007/s00029-023-00903-5","url":null,"abstract":"Let ( {{textbf{k}}}) be a field and (Qin {{textbf{k}}}[x_1, ldots , x_s]) a form (homogeneous polynomial) of degree (d>1.) The ({{textbf{k}}})-Schmidt rank (text {rk}_{{textbf{k}}}(Q)) of Q is the minimal r such that (Q= sum _{i=1}^r R_iS_i) with (R_i, S_i in {{textbf{k}}}[x_1, ldots , x_s]) forms of degree (<d). When ( {{textbf{k}}}) is algebraically closed and ( text {char}({{textbf{k}}})) doesn’t divide d, this rank is closely related to ( text {codim}_{{mathbb {A}}^s} (nabla Q(x) = 0)) - also known as the Birch rank of Q. When ( {{textbf{k}}}) is a number field, a finite field or a function field, we give polynomial bounds for ( text {rk}_{{textbf{k}}}(Q) ) in terms of ( text {rk}_{{bar{{{textbf{k}}}}}} (Q) ) where ( {bar{{{textbf{k}}}}} ) is the algebraic closure of ( {{textbf{k}}}. ) Prior to this work no such bound (even ineffective) was known for (d>4). This result has immediate consequences for counting integer points (when ( {{textbf{k}}}) is a number field) or prime points (when ( {{textbf{k}}}= {mathbb {Q}})) of the variety ( (Q=0) ) assuming ( text {rk}_{{{textbf{k}}}} (Q) ) is large.\u0000","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139463906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Generic density of geodesic nets 测地网的一般密度

Selecta Mathematica Pub Date : 2024-01-12 DOI: 10.1007/s00029-023-00901-7

Yevgeny Liokumovich, Bruno Staffa

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Spectrum of p-adic linear differential equations I: the shape of the spectrum p-adic 线性微分方程谱 I：谱的形状

Selecta Mathematica Pub Date : 2024-01-11 DOI: 10.1007/s00029-023-00904-4

Tinhinane A. Azzouz

引用次数: 0

Strongly divisible lattices and crystalline cohomology in the imperfect residue field case 不完全残差域情况下的强可分晶格和结晶同调学

Selecta Mathematica Pub Date : 2023-12-27 DOI: 10.1007/s00029-023-00899-y

Yong Suk Moon

{"title":"Strongly divisible lattices and crystalline cohomology in the imperfect residue field case","authors":"Yong Suk Moon","doi":"10.1007/s00029-023-00899-y","DOIUrl":"https://doi.org/10.1007/s00029-023-00899-y","url":null,"abstract":"Let k be a perfect field of characteristic (p ge 3), and let K be a finite totally ramified extension of (K_0 = W(k)[p^{-1}]). Let (L_0) be a complete discrete valuation field over (K_0) whose residue field has a finite p-basis, and let (L = L_0otimes _{K_0} K). For (0 le r le p-2), we classify (textbf{Z}_p)-lattices of semistable representations of (textrm{Gal}(overline{L}/L)) with Hodge–Tate weights in [0, r] by strongly divisible lattices. This generalizes the result of Liu (Compos Math 144:61–88, 2008). Moreover, if (mathcal {X}) is a proper smooth formal scheme over (mathcal {O}_L), we give a cohomological description of the strongly divisible lattice associated to (H^i_{acute{text {e}}text {t}}(mathcal {X}_{overline{L}}, textbf{Z}_p)) for (i le p-2), under the assumption that the crystalline cohomology of the special fiber of (mathcal {X}) is torsion-free in degrees i and (i+1). This generalizes a result in Cais and Liu (Trans Am Math Soc 371:1199–1230, 2019).","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0