Journal fur die Reine und Angewandte Mathematik最新文献

{"title":"Variation of canonical height forbreak Fatou points on ℙ1","authors":"Laura Demarco, Niki Myrto Mavraki","doi":"10.1515/crelle-2022-0078","DOIUrl":"https://doi.org/10.1515/crelle-2022-0078","url":null,"abstract":"Abstract Let f : ℙ 1 → ℙ 1 {f:mathbb{P}^{1}tomathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ⁢ ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K. For each point a ∈ ℙ 1 ⁢ ( k ) {ainmathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ⁢ ( ℚ ¯ ) {tin X(overline{mathbb{Q}})} outside a finite set, induces a Weil height on the curve X; i.e., we prove the existence of a ℚ {mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ⁢ ( a t ) - h D ⁢ ( t ) {tmapstohat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ⁢ ( ℚ ¯ ) {X(overline{mathbb{Q}})} for any choice of Weil height associated to D. We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ⁢ ( a t ) {tmapstohat{lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ⁢ ( ℂ v ) {X(mathbb{C}_{v})} , at each place v of the number field K. These results were known for polynomial maps f and all points a ∈ ℙ 1 ⁢ ( k ) {ainmathbb{P}^{1}(k)} without the stability hypothesis, [21, 14], and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ⁢ ( k ) {ainmathbb{P}^{1}(k)} . [32, 29]. Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {tilde{f}:Xtimesmathbb{P}^{1}dashrightarrow Xtimesmathbb{P}^{1}} over K; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k, where the local canonical height λ ^ f , γ ⁢ ( a ) {hat{lambda}_{f,gamma}(a)} can be computed as an intersection number.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80926524","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":"Shuffle algebras for quivers and wheel conditions","authors":"Andrei Neguț","doi":"10.1515/crelle-2022-0074","DOIUrl":"https://doi.org/10.1515/crelle-2022-0074","url":null,"abstract":"Abstract We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo–Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized 𝐾-theoretic Hall algebra associated to the quiver by Grojnowski, Schiffmann–Vasserot and Yang–Zhao. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in joint work with Sala and Schiffmann to give a generators-and-relations description of the Hall algebra of any curve over a finite field (which is a shuffle algebra due to Kapranov–Schiffmann–Vasserot). When the quiver has no edge loops or multiple edges, we show that the shuffle algebra, localized 𝐾-theoretic Hall algebra, and the positive half of the corresponding quantum loop group are all isomorphic; we also obtain the non-degeneracy of the Hopf pairing on the latter quantum loop group.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86947223","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":"On fundamental groups of RCD spaces","authors":"Jaime Santos-Rodríguez, Sergio Zamora-Barrera","doi":"10.1515/crelle-2023-0027","DOIUrl":"https://doi.org/10.1515/crelle-2023-0027","url":null,"abstract":"Abstract We obtain results about fundamental groups of RCD ∗ ⁢ ( K , N ) {mathrm{RCD}^{ast}(K,N)} spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed K ∈ ℝ {Kinmathbb{R}} , N ∈ [ 1 , ∞ ) {Nin[1,infty)} , D > 0 {D>0} , we show the following: • There is C > 0 {C>0} such that for each RCD ∗ ⁢ ( K , N ) {mathrm{RCD}^{ast}(K,N)} space X of diameter ≤ D {leq D} , its fundamental group π 1 ⁢ ( X ) {pi_{1}(X)} is generated by at most C elements. • There is D ~ > 0 {tilde{D}>0} such that for each RCD ∗ ⁢ ( K , N ) {mathrm{RCD}^{ast}(K,N)} space X of diameter ≤ D {leq D} with compact universal cover X ~ {tilde{X}} , one has diam ⁡ ( X ~ ) ≤ D ~ {operatorname{diam}(tilde{X})leqtilde{D}} . • If a sequence of RCD ∗ ⁢ ( 0 , N ) {mathrm{RCD}^{ast}(0,N)} spaces X i {X_{i}} of diameter ≤ D {leq D} and rectifiable dimension n is such that their universal covers X ~ i {tilde{X}_{i}} converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n, then there is C > 0 {C>0} such that for each i, the fundamental group π 1 ⁢ ( X i ) {pi_{1}(X_{i})} contains an abelian subgroup of index ≤ C {leq C} . • If a sequence of RCD ∗ ⁢ ( K , N ) {mathrm{RCD}^{ast}(K,N)} spaces X i {X_{i}} of diameter ≤ D {leq D} and rectifiable dimension n is such that their universal covers X ~ i {tilde{X}_{i}} are compact and converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n, then there is C > 0 {C>0} such that for each i, the fundamental group π 1 ⁢ ( X i ) {pi_{1}(X_{i})} contains an abelian subgroup of index ≤ C {leq C} . • If a sequence of RCD ∗ ⁢ ( K , N ) {mathrm{RCD}^{ast}(K,N)} spaces X i {X_{i}} with first Betti number ≥ r {geq r} and rectifiable dimension n converges in the Gromov–Hausdorff sense to a compact space X of rectifiable dimension m, then the first Betti number of X is at least r + m - n {r+m-n} . The main tools are the splitting theorem by Gigli, the splitting blow-up property by Mondino and Naber, the semi-locally-simple-connectedness of RCD ∗ ⁢ ( K , N ) {mathrm{RCD}^{ast}(K,N)} spaces by Wang, the isometry group structure by Guijarro and the first author, and the structure of approximate subgroups by Breuillard, Green and Tao.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74661540","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":"Stable 𝔸1-connectivity over a base","authors":"A. E. Druzhinin","doi":"10.1515/crelle-2022-0048","DOIUrl":"https://doi.org/10.1515/crelle-2022-0048","url":null,"abstract":"Abstract Morel’s stable connectivity theorem states the vanishing of the sheaves of the negative motivic homotopy groups π ¯ i s ⁢ ( Y ) {underline{pi}^{s}_{i}(Y)} and π ¯ i + j , j s ⁢ ( Y ) {underline{pi}^{s}_{i+j,j}(Y)} , i < 0 {i<0} , in the stable motivic homotopy categories 𝐒𝐇 S 1 ⁢ ( k ) {mathbf{SH}^{S^{1}}(k)} and 𝐒𝐇 ⁢ ( k ) {mathbf{SH}(k)} for an arbitrary smooth scheme Y over a field k. Originally the same property was conjectured in the relative case over a base scheme S. In view of Ayoub’s conterexamples the modified version of the conjecture states the vanishing of stable motivic homotopy groups π ¯ i s ⁢ ( Y ) {underline{pi}^{s}_{i}(Y)} (and π ¯ i + j , j s ⁢ ( Y ) {underline{pi}^{s}_{i+j,j}(Y)} ) for i < - d {i<-d} , where d = dim ⁡ S {d=dim S} is the Krull dimension. The latter version of the conjecture is proven over noetherian domains of finite Krull dimension under the assumption that residue fields of the base scheme are infinite. This is the result by J. Schmidt and F. Strunk for Dedekind schemes case, and the result by N. Deshmukh, A. Hogadi, G. Kulkarni and S. Yadavand for the case of noetherian domains of an arbitrary dimension. In the article, we prove the result for any locally noetharian base scheme of finite Krull dimension without the assumption on the residue fields, in particular for 𝐒𝐇 S 1 ⁢ ( ℤ ) {mathbf{SH}^{S^{1}}(mathbb{Z})} and 𝐒𝐇 ⁢ ( ℤ ) {mathbf{SH}(mathbb{Z})} . In the appendix, we modify the arguments used for the main result to obtain the independent proof of Gabber’s Presentation Lemma over finite fields.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72631464","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":"Triviality of the Hecke action on ordinary Drinfeld cuspforms of level Γ1(tn )","authors":"Shin Hattori","doi":"10.1515/crelle-2022-0058","DOIUrl":"https://doi.org/10.1515/crelle-2022-0058","url":null,"abstract":"Abstract Let k ≥ 2 {kgeq 2} and n ≥ 1 {ngeq 1} be any integers. In this paper, we prove that all Hecke operators act trivially on the space of ordinary Drinfeld cuspforms of level Γ 1 ⁢ ( t n ) {hskip-0.569055ptGamma_{1}(t^{n})hskip-0.284528pt} and weight k {hskip-0.284528ptkhskip-0.569055pt} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75343968","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":"Lower bounds for the scalar curvatures of Ricci flow singularity models","authors":"Pak-Yeung Chan, B. Chow, Zilu Ma, Yongjia Zhang","doi":"10.1515/crelle-2022-0086","DOIUrl":"https://doi.org/10.1515/crelle-2022-0086","url":null,"abstract":"Abstract In a series of papers, Bamler [5, 4, 6] further developed the high-dimensional theory of Hamilton’s Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger–Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of 4-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77765313","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":"Crystal limits of compact semisimple quantum groups as higher-rank graph algebras","authors":"Marco Matassa, Robert Yuncken","doi":"10.1515/crelle-2023-0047","DOIUrl":"https://doi.org/10.1515/crelle-2023-0047","url":null,"abstract":"Abstract Let O q ⁢ [ K ] mathcal{O}_{q}[K] be the quantized coordinate ring over the field C ⁢ ( q ) mathbb{C}(q) of rational functions corresponding to a compact semisimple Lie group 𝐾, equipped with its ∗-structure. Let A 0 ⊂ C ⁢ ( q ) {mathbf{A}_{0}}subsetmathbb{C}(q) denote the subring of regular functions at q = 0 q=0 . We introduce an A 0 mathbf{A}_{0} -subalgebra O q A 0 ⁢ [ K ] ⊂ O q ⁢ [ K ] mathcal{O}_{q}^{{mathbf{A}_{0}}}[K]subsetmathcal{O}_{q}[K] which is stable with respect to the ∗-structure and which has the following properties with respect to the crystal limit q → 0 qto 0 . The specialization of O q ⁢ [ K ] mathcal{O}_{q}[K] at each q ∈ ( 0 , ∞ ) ∖ { 1 } qin(0,infty)setminus{1} admits a faithful ∗-representation π q pi_{q} on a fixed Hilbert space, a result due to Soibelman. We show that, for every element a ∈ O q A 0 ⁢ [ K ] ainmathcal{O}_{q}^{{mathbf{A}_{0}}}[K] , the family of operators π q ⁢ ( a ) pi_{q}(a) admits a norm limit as q → 0 qto 0 . These limits define a ∗-representation π 0 pi_{0} of O q A 0 ⁢ [ K ] mathcal{O}_{q}^{{mathbf{A}_{0}}}[K] . We show that the resulting ∗-algebra O ⁢ [ K 0 ] = π 0 ⁢ ( O q A 0 ⁢ [ K ] ) mathcal{O}[K_{0}]=pi_{0}(mathcal{O}_{q}^{{mathbf{A}_{0}}}[K]) is a Kumjian–Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of C * C^{*} -algebras ( C ⁢ ( K q ) ) q ∈ [ 0 , ∞ ] (C(K_{q}))_{qin[0,infty]} , where the fibres at q = 0 q=0 and ∞ are explicitly defined higher-rank graph algebras.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80219111","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":"Algebraic independence of topological Pontryagin classes","authors":"Søren Galatius, O. Randal-Williams","doi":"10.1515/crelle-2023-0051","DOIUrl":"https://doi.org/10.1515/crelle-2023-0051","url":null,"abstract":"Abstract We show that the topological Pontryagin classes are algebraically independent in the rationalised cohomology of B ⁢ Top ⁢ ( d ) {Bmathrm{Top}(d)} for all d ≥ 4 {dgeq 4} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88596612","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":"Type-0 singularities in the network flow – Evolution of trees","authors":"C. Mantegazza, M. Novaga, Alessandra Pluda","doi":"10.1515/crelle-2022-0055","DOIUrl":"https://doi.org/10.1515/crelle-2022-0055","url":null,"abstract":"Abstract The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a “Type-0” singularity, in contrast to the well known “Type-I” and “Type-II” ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the “tangents flows” have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results from [T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, J. Differential Geom. 111 2019, 1, 39–89] and [J. Lira, R. Mazzeo, A. Pluda and M. Saez, Short–time existence for the network flow, preprint 2021] to “restart” the flow after the singularity.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85522923","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":"Relative entropy of hypersurfaces in hyperbolic space","authors":"Junfu Yao","doi":"10.1515/crelle-2023-0035","DOIUrl":"https://doi.org/10.1515/crelle-2023-0035","url":null,"abstract":"Abstract We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham–Witten. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80733803","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}