It is a theorem of Kim–Tamagawa that the Qℓ${mathbb {Q}}_ell$ ‐pro‐unipotent Kummer map associated to a smooth projective curve Y$Y$ over a finite extension of Qp${mathbb {Q}}_p$ is locally constant when ℓ≠p$ell ne p$ . This paper establishes two generalisations of this result. First, we extend the Kim–Tamagawa theorem to the case that Y$Y$ is a smooth variety of any dimension. Second, we formulate and prove the analogue of the Kim–Tamagawa theorem in the case ℓ=p$ell =p$ , again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale–de Rham comparison theorem for pro‐unipotent fundamental groupoids using methods of Scholze and Diao–Lan–Liu–Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.
{"title":"Local constancy of pro‐unipotent Kummer maps","authors":"L. A. Betts","doi":"10.1112/plms.12554","DOIUrl":"https://doi.org/10.1112/plms.12554","url":null,"abstract":"It is a theorem of Kim–Tamagawa that the Qℓ${mathbb {Q}}_ell$ ‐pro‐unipotent Kummer map associated to a smooth projective curve Y$Y$ over a finite extension of Qp${mathbb {Q}}_p$ is locally constant when ℓ≠p$ell ne p$ . This paper establishes two generalisations of this result. First, we extend the Kim–Tamagawa theorem to the case that Y$Y$ is a smooth variety of any dimension. Second, we formulate and prove the analogue of the Kim–Tamagawa theorem in the case ℓ=p$ell =p$ , again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale–de Rham comparison theorem for pro‐unipotent fundamental groupoids using methods of Scholze and Diao–Lan–Liu–Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49206673","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}
Let μ$mu$ be a positive compactly supported measure in the complex plane C$mathbb {C}$ , and for each p,1⩽p<∞$p,1leqslant p
设μ$mu$是复平面C$mathbb｛C｝$中的正紧支持测度，并且对于每个p，1⩽p<∞$p，1leqslant p
{"title":"Boundary values in spaces spanned by rational functions and the index of invariant subspaces","authors":"J. Brennan","doi":"10.1112/plms.12433","DOIUrl":"https://doi.org/10.1112/plms.12433","url":null,"abstract":"Let μ$mu$ be a positive compactly supported measure in the complex plane C$mathbb {C}$ , and for each p,1⩽p<∞$p,1leqslant p<infty$ , let Hp(μ)$H^p(mu )$ be the closed subspace of Lp(μ)$L^p(mu )$ spanned by the polynomials. In 1991, Thomson gave a complete description of its structure, expressing Hp(μ)$H^p(mu )$ as the direct sum of invariant subspaces, all but one of which is irreducible in the sense that it contains no non‐trivial characteristic function. Years later, Aleman, Richter and Sundberg gave a more detailed analysis of the invariant subspaces in any irreducible summand. Here we discuss the extent to which those earlier results can be extended to Rp(μ)$R^p(mu )$ , the closed subspace of Lp(μ)$L^p(mu )$ spanned by the rational functions having no poles on the support of μ$mu$ , by first establishing the existence of boundary values in these spaces. Our results all depend on the semiadditivity of analytic capacity, and ultimately on some form of the F. and M. Riesz theorem.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41821680","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12409","DOIUrl":"https://doi.org/10.1112/plms.12409","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48597064","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 no‐flux initial‐boundary value problem for the quasilinear Keller–Segel system * ut=∇·(D(u)∇u)−∇·(S(u)∇v),vt=Δv−v+u,begin{equation} hspace*{6pc}{leftlbrace defeqcellsep{&}begin{array}{l}u_t=nabla cdot (D(u)nabla u) - nabla cdot (S(u)nabla v), hspace*{-6pc}[3pt] v_t=Delta v-v+u, end{array} right.} end{equation}is considered in smoothly bounded domains Ω⊂Rn$Omega subset mathbb {R}^n$ , n⩾3$ngeqslant 3$ , where D∈C2([0,∞))$Din C^2([0,infty ))$ and S∈C2([0,∞))$Sin C^2([0,infty ))$ are such that D>0$D>0$ on [0,∞)$[0,infty )$ and that S(0)=00$s>0$ . A particular focus is on cases in which there exist κ>0,CSD>0$kappa >0, C_{SD}>0$ and f∈L1((1,∞))$fin L^1((1,infty ))$ such that ** −f(s)⩽D(s)S(s)−κs2/n⩽CSDsforalls⩾1.begin{equation} hspace*{6pc}- f(s) leqslant frac{D(s)}{S(s)} - frac{kappa }{s^{2/n}} leqslant frac{C_{SD}}{s} quad mbox{for all } sgeqslant 1.hspace*{-6pc} end{equation}It is first shown that then there exists m0>0$m_0>0$ such that whenever u0$u_0$ and v0$v_0$ are reasonably regular and nonnegative with ∫Ωu0
{"title":"A family of mass‐critical Keller–Segel systems","authors":"M. Winkler","doi":"10.1112/plms.12425","DOIUrl":"https://doi.org/10.1112/plms.12425","url":null,"abstract":"The no‐flux initial‐boundary value problem for the quasilinear Keller–Segel system * ut=∇·(D(u)∇u)−∇·(S(u)∇v),vt=Δv−v+u,begin{equation} hspace*{6pc}{leftlbrace defeqcellsep{&}begin{array}{l}u_t=nabla cdot (D(u)nabla u) - nabla cdot (S(u)nabla v), hspace*{-6pc}[3pt] v_t=Delta v-v+u, end{array} right.} end{equation}is considered in smoothly bounded domains Ω⊂Rn$Omega subset mathbb {R}^n$ , n⩾3$ngeqslant 3$ , where D∈C2([0,∞))$Din C^2([0,infty ))$ and S∈C2([0,∞))$Sin C^2([0,infty ))$ are such that D>0$D>0$ on [0,∞)$[0,infty )$ and that S(0)=00$s>0$ . A particular focus is on cases in which there exist κ>0,CSD>0$kappa >0, C_{SD}>0$ and f∈L1((1,∞))$fin L^1((1,infty ))$ such that ** −f(s)⩽D(s)S(s)−κs2/n⩽CSDsforalls⩾1.begin{equation} hspace*{6pc}- f(s) leqslant frac{D(s)}{S(s)} - frac{kappa }{s^{2/n}} leqslant frac{C_{SD}}{s} quad mbox{for all } sgeqslant 1.hspace*{-6pc} end{equation}It is first shown that then there exists m0>0$m_0>0$ such that whenever u0$u_0$ and v0$v_0$ are reasonably regular and nonnegative with ∫Ωu0","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42012552","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12408","DOIUrl":"https://doi.org/10.1112/plms.12408","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48272325","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 compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN 2022 (2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects Ed=Tr(Y1d1⋯YndnT1⋯Tn−1)$E_{mathbf {d}} = {rm Tr}(Y_1^{d_1} dots Y_n^{d_n} T_1 dots T_{n-1})$ as d=(d1,⋯,dn)∈Zn$mathbf {d}= (d_1,dots ,d_n) in mathbb {Z}^n$ , where Yi$Y_i$ denote the Wakimoto objects of Elias and Ti$T_i$ denote Rouquier complexes. We compute certain categorical commutators between the Ed$E_{mathbf {d}}$ 's and show that they match the categorical commutators between the sheaves Ed$mathcal {E}_{mathbf {d}}$ on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of K$K$ ‐theory, these commutators yield a certain integral form A∼$widetilde{mathcal {A}}$ of the elliptic Hall algebra, which we can thus map to the K$K$ ‐theory of the trace of the affine Hecke category.
{"title":"The trace of the affine Hecke category","authors":"E. Gorsky, Andrei Neguț","doi":"10.1112/plms.12523","DOIUrl":"https://doi.org/10.1112/plms.12523","url":null,"abstract":"We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN 2022 (2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects Ed=Tr(Y1d1⋯YndnT1⋯Tn−1)$E_{mathbf {d}} = {rm Tr}(Y_1^{d_1} dots Y_n^{d_n} T_1 dots T_{n-1})$ as d=(d1,⋯,dn)∈Zn$mathbf {d}= (d_1,dots ,d_n) in mathbb {Z}^n$ , where Yi$Y_i$ denote the Wakimoto objects of Elias and Ti$T_i$ denote Rouquier complexes. We compute certain categorical commutators between the Ed$E_{mathbf {d}}$ 's and show that they match the categorical commutators between the sheaves Ed$mathcal {E}_{mathbf {d}}$ on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of K$K$ ‐theory, these commutators yield a certain integral form A∼$widetilde{mathcal {A}}$ of the elliptic Hall algebra, which we can thus map to the K$K$ ‐theory of the trace of the affine Hecke category.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43250511","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 Gromov‐hyperbolic group G$G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of G$G$ . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.
{"title":"Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups","authors":"S. Cantrell, Cagri Sert","doi":"10.1112/plms.12550","DOIUrl":"https://doi.org/10.1112/plms.12550","url":null,"abstract":"Given a Gromov‐hyperbolic group G$G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of G$G$ . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45846873","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12407","DOIUrl":"https://doi.org/10.1112/plms.12407","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45626618","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}
G. Bellamy, C'edric Bonnaf'e, Baohua Fu, D. Juteau, Paul D. Levy, E. Sommers
We construct a new infinite family of four‐dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of C4$mathbb {C}^4$ by the dihedral group of order 2d$2d$ , (2) as singular points of Calogero–Moser spaces associated with dihedral groups of order 2d$2d$ at equal parameters, and (3) as singularities of a certain Slodowy slice in the d$d$ ‐fold cover of the nilpotent cone in sld${mathfrak {s}}{mathfrak {l}}_d$ .
{"title":"A new family of isolated symplectic singularities with trivial local fundamental group","authors":"G. Bellamy, C'edric Bonnaf'e, Baohua Fu, D. Juteau, Paul D. Levy, E. Sommers","doi":"10.1112/plms.12513","DOIUrl":"https://doi.org/10.1112/plms.12513","url":null,"abstract":"We construct a new infinite family of four‐dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of C4$mathbb {C}^4$ by the dihedral group of order 2d$2d$ , (2) as singular points of Calogero–Moser spaces associated with dihedral groups of order 2d$2d$ at equal parameters, and (3) as singularities of a certain Slodowy slice in the d$d$ ‐fold cover of the nilpotent cone in sld${mathfrak {s}}{mathfrak {l}}_d$ .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42084623","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12361","DOIUrl":"https://doi.org/10.1112/plms.12361","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42492344","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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