Pub Date : 2022-12-16DOI: 10.1515/crelle-2022-0078
Laura Demarco, Niki Myrto Mavraki
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.
设f: 1→1 {f:mathbb{P} ^{1}tomathbb{P} ^{1}}是定义在函数域k= k≠(X) k= k (X)上的度{>1 >1}的映射，其中k是一个数域，X是k上的一个投影曲线。对于{满足动态稳定性条件的每个点a∈<}s:3> {1¹(k) a inmathbb{P} ^1(k)，我们证明了在点at {a＿t}处，}对于t∈X≠(π¯){t {}}{{}}{in X(overline{mathbb{Q}})在}有限集外，推导出曲线X上的韦尔高度;即，我们证明了在 {mathbb{Q}} -因子D= df,a {D=D＿f{,a，使得函数t∈h ^ f t¹(a t)-h D¹(t) t }}{mapstohat{h} ＿f＿t{(a＿t{)}}- h＿d{ (t)}对于任何与D相关的Weil高度的选择{都在}X²(π¯)X(}{overline{mathbb{Q}})上有界。我们还证明了一个局部版本，即局部正则高度t∈λ ^ f t，v≠(a t) t }{mapstohat{lambda} ＿f＿t{,{v}(a＿t)}与D的Weil函数不同，在{数域k的每个位置}v上，X≠(v) X(}{mathbb{C} ＿v{)上有一个连续函数}，这些结果对于多项式映射f和所有点a∈1≠(k) a }{inmathbb{P} ^{1}(k)是已知的，}没有稳定性假设，[21,14]，对于映射f，它是椭圆曲线E / k的自同态商和所有点a∈1∑(k){ a inmathbb{P} ^{1}(k)}。[32,29]。最后，我们用诱导映射f的几何特征来描述我们的稳定性条件:X X²1讲解X X²1{tilde{f}:X timesmathbb{P} ^{1}dashrightarrow X timesmathbb{P} ^{1}} / K;并且证明了(f,a) (f,a)对(f,a)的相对n录影带模型的存在性，当a是在k点γ处的Fatou点，其中局部正则高度λ ^ f， γ¹(a) {}{hat{lambda} ＿f{, gamma} (a)}可以计算为交点数。
{"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}
Pub Date : 2022-12-01DOI: 10.1515/crelle-2022-0074
Andrei Neguț
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.
{"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}
Pub Date : 2022-10-13DOI: 10.1515/crelle-2023-0027
Jaime Santos-Rodríguez, Sergio Zamora-Barrera
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.
{"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}
Pub Date : 2022-09-29DOI: 10.1515/crelle-2022-0048
A. E. Druzhinin
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.
Morel稳定连通性定理证明了负动力同伦群π¯is²(Y) {underline{pi} ^{s＿i}({Y)和π}¯i+j,j s²(Y) }{underline{pi} ^s＿i{+}j,{j(Y)}， i<}0 i<0{，对于任意}光滑方案(k){mathbf{SH} ^S²(k)和²(k) {{}}}{mathbf{SH} (k)上，最初在基方案S上的相对情况下推测出了相同的性质。鉴于Ayoub的反例，该猜想的修正版本说明了稳定动力同伦群π¯i S²(Y) }{underline{pi} ^s＿i(Y)(和π¯i + j)的消失。j {s}²(Y{) }}{underline{pi} ^s＿i+j,j(Y))对于i{<}-{d i<-d}，其中}d= dim (s) d{= }{dim s是}Krull维。在有限Krull维的noether域上，假设基格式的剩余域是无限的，证明了后一种猜想。这是J. Schmidt和F. Strunk对于Dedekind方案的结果，以及N. Deshmukh, A. Hogadi, G. Kulkarni和S. Yadavand对于任意维的noetherian域的结果。在本文中，我们证明了在不假设剩余域的情况下，对于任意有限Krull维的局部noetharian基格式的结果，特别是对于1s (0) {mathbf{SH} ^S^1({{}}mathbb{Z})}和²(0){mathbf{SH} (mathbb{Z})}。在附录中，我们修改了用于主要结果的参数，以获得有限域上Gabber的表示引理的独立证明。
{"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}
Pub Date : 2022-09-29DOI: 10.1515/crelle-2022-0058
Shin Hattori
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} .
{"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}
Pub Date : 2022-08-28DOI: 10.1515/crelle-2022-0086
Pak-Yeung Chan, B. Chow, Zilu Ma, Yongjia Zhang
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.
{"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}
Pub Date : 2022-08-28DOI: 10.1515/crelle-2023-0047
Marco Matassa, Robert Yuncken
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.
{"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}
Pub Date : 2022-08-24DOI: 10.1515/crelle-2023-0051
Søren Galatius, O. Randal-Williams
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} .
证明了拓扑Pontryagin类在B ^ Top ^ (d)的有理上同调中是代数独立的。 {bmathrm{Top}(d)} 对于所有d≥4 {dgeq 4} ．
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Pub Date : 2022-07-08DOI: 10.1515/crelle-2022-0055
C. Mantegazza, M. Novaga, Alessandra Pluda
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.
网络的曲率运动是对曲线缩短流的曲线有限并的推广。这种演变有几个特殊的特征，主要是由于曲线相交处存在连接点。在本文中，我们证明了当一条单曲线的长度消失并且两个三重结点合并时，进化网络的曲率保持有界。这种拓扑奇点不包含网络流，它可以被称为“0型”奇点，而不是众所周知的光滑曲线或超曲面的通常平均曲率流的“i型”和“ii型”奇点，其特征是曲率的膨胀率不同。因此，在假设所有“切线流”具有单位多重性的前提下，我们能够给出树状网络直到第一个奇异时间的进化的完整描述。如果这些解的寿命是有限的，那么网络的曲率仍然是有界的，我们可以应用[T]的结果。张建军，张建军，张建军，等。平面网络流的短时间存在性研究[J] .地球物理学报，2019，(1):39 - 39。Lira, R. Mazzeo, A. Pluda和M. Saez，网络流的短时存在，预印本2021]。
{"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}
Pub Date : 2022-06-27DOI: 10.1515/crelle-2023-0035
Junfu Yao
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.
{"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}
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