Pub Date : 2022-01-29DOI: 10.1515/crelle-2022-0049
Bo Zhu
Abstract In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.
{"title":"Geometry of positive scalar curvature on complete manifold","authors":"Bo Zhu","doi":"10.1515/crelle-2022-0049","DOIUrl":"https://doi.org/10.1515/crelle-2022-0049","url":null,"abstract":"Abstract In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72575938","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 : 2022-01-17DOI: 10.1515/crelle-2022-0088
W. Castryck, F. Vermeulen, Yongqiang Zhao
Abstract We give an explicit minimal graded free resolution, in terms of representations of the symmetric group S d {S_{d}} , of a Galois-theoretic configuration of d points in 𝐏 d - 2 {mathbf{P}^{d-2}} that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree d cover of 𝐏 1 {mathbf{P}^{1}} by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C → 𝐏 1 {Ctomathbf{P}^{1}} : one for each irreducible representation of S d {S_{d}} , i.e., one for each partition of d.
摘要本文给出了Bhargava在环参数化背景下研究的𝐏d-2 {mathbf{P}^{d-2}}中d点的伽罗瓦理论组态的显式最小梯度自由分辨率，其表示为对称群S d {S_{d}}。当我们的构造应用于𝐏1 {mathbf{P}^{1}}的简支度d覆盖的几何一般纤维时，我们给出了在其相对最小分辨率中出现的syzygy束的分裂类型的新解释。具体地说，我们的工作表明所有这些分裂类型都是由可解覆盖的滚动不变量组成的。这极大地推广了先前由于Casnati的观察，即4度覆盖的第一个syzygy束根据其三次解的滚动不变量分裂。我们的工作还表明，syzygy束的分裂类型，连同滚动不变量的多集，属于一个更大的多不变量集，可以附加到C→𝐏1 {Ctomathbf{P}^{1}}:对于S d {S_{d}}的每个不可约表示一个，即对于d的每个划分一个。
{"title":"Scrollar invariants, syzygies and representations of the symmetric group","authors":"W. Castryck, F. Vermeulen, Yongqiang Zhao","doi":"10.1515/crelle-2022-0088","DOIUrl":"https://doi.org/10.1515/crelle-2022-0088","url":null,"abstract":"Abstract We give an explicit minimal graded free resolution, in terms of representations of the symmetric group S d {S_{d}} , of a Galois-theoretic configuration of d points in 𝐏 d - 2 {mathbf{P}^{d-2}} that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree d cover of 𝐏 1 {mathbf{P}^{1}} by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C → 𝐏 1 {Ctomathbf{P}^{1}} : one for each irreducible representation of S d {S_{d}} , i.e., one for each partition of d.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82714407","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 : 2022-01-10DOI: 10.1515/crelle-2023-0012
Eusebio Gardella, S. Geffen, J. Kranz, P. Naryshkin
Abstract We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 {widetilde{A}_{2}} -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory.
{"title":"Classifiability of crossed products by nonamenable groups","authors":"Eusebio Gardella, S. Geffen, J. Kranz, P. Naryshkin","doi":"10.1515/crelle-2023-0012","DOIUrl":"https://doi.org/10.1515/crelle-2023-0012","url":null,"abstract":"Abstract We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 {widetilde{A}_{2}} -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81600903","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 : 2022-01-09DOI: 10.1515/crelle-2022-0069
B. Guo, Jian Song
Abstract We prove a local volume noncollapsing estimate for Kähler metrics induced from a family of complex Monge–Ampère equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various diameter and gradient estimate.
{"title":"Local noncollapsing for complex Monge–Ampère equations","authors":"B. Guo, Jian Song","doi":"10.1515/crelle-2022-0069","DOIUrl":"https://doi.org/10.1515/crelle-2022-0069","url":null,"abstract":"Abstract We prove a local volume noncollapsing estimate for Kähler metrics induced from a family of complex Monge–Ampère equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various diameter and gradient estimate.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77873705","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 : 2022-01-06DOI: 10.1515/crelle-2021-0079
Jun Ueki
Abstract We clarify the definition of the divisorial hull and recollect some basic facts. Then we correct Lemma 4.2 and Theorem 11.2 (1)–(2) in the original article.
{"title":"Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)","authors":"Jun Ueki","doi":"10.1515/crelle-2021-0079","DOIUrl":"https://doi.org/10.1515/crelle-2021-0079","url":null,"abstract":"Abstract We clarify the definition of the divisorial hull and recollect some basic facts. Then we correct Lemma 4.2 and Theorem 11.2 (1)–(2) in the original article.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72477435","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 : 2021-12-22DOI: 10.1515/crelle-2022-0066
O. Gorodetsky, Alexander P. Mangerel, B. Rodgers
Abstract We show that counts of squarefree integers up to X in short intervals of size H tend to a Gaussian distribution as long as H → ∞ {Htoinfty} and H = X o ( 1 ) {H=X^{o(1)}} . This answers a question posed by R. R. Hall in 1989. More generally, we prove a variant of Donsker’s theorem, showing that these counts scale to a fractional Brownian motion with Hurst parameter 1 / 4 {1/4} . In fact, we are able to prove these results hold in general for collections of B-free integers as long as the sieving set B satisfies a very mild regularity property, for Hurst parameter varying with the set B.
{"title":"Squarefrees are Gaussian in short intervals","authors":"O. Gorodetsky, Alexander P. Mangerel, B. Rodgers","doi":"10.1515/crelle-2022-0066","DOIUrl":"https://doi.org/10.1515/crelle-2022-0066","url":null,"abstract":"Abstract We show that counts of squarefree integers up to X in short intervals of size H tend to a Gaussian distribution as long as H → ∞ {Htoinfty} and H = X o ( 1 ) {H=X^{o(1)}} . This answers a question posed by R. R. Hall in 1989. More generally, we prove a variant of Donsker’s theorem, showing that these counts scale to a fractional Brownian motion with Hurst parameter 1 / 4 {1/4} . In fact, we are able to prove these results hold in general for collections of B-free integers as long as the sieving set B satisfies a very mild regularity property, for Hurst parameter varying with the set B.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79750099","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 : 2021-12-11DOI: 10.1515/crelle-2023-0041
Tom Bachmann, K. Wickelgren
Abstract We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We give example computations of quadratically enriched excess and residual intersections.
{"title":"On quadratically enriched excess and residual intersections","authors":"Tom Bachmann, K. Wickelgren","doi":"10.1515/crelle-2023-0041","DOIUrl":"https://doi.org/10.1515/crelle-2023-0041","url":null,"abstract":"Abstract We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We give example computations of quadratically enriched excess and residual intersections.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75716100","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 : 2021-11-24DOI: 10.1515/crelle-2021-0062
J. Kitagawa
{"title":"Extended Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160)","authors":"J. Kitagawa","doi":"10.1515/crelle-2021-0062","DOIUrl":"https://doi.org/10.1515/crelle-2021-0062","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77800624","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 : 2021-11-23DOI: 10.1515/crelle-2022-0019
D. Kazhdan, Alexander Yom Din
Abstract Let G be a unimodular locally compact group. We define a property of irreducible unitary G-representations V which we call c-temperedness, and which for the trivial V boils down to Følner’s condition (equivalent to the trivial V being tempered, i.e. to G being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered V’s, as well as for all tempered V’s in the cases of G:=SL2(ℝ){G:=mathrm{SL}_{2}({mathbb{R}})} and of G=PGL2(Ω){G=mathrm{PGL}_{2}(Omega)} for a non-Archimedean local field Ω of characteristic 0 and residual characteristic not 2. We also establish a weaker form of the conjecture, involving only K-finite vectors. In the non-Archimedean case, we give a formula expressing the character of a tempered V as an appropriately-weighted conjugation-average of a matrix coefficient of V, generalising a formula of Harish-Chandra from the case when V is square-integrable.
{"title":"On tempered representations","authors":"D. Kazhdan, Alexander Yom Din","doi":"10.1515/crelle-2022-0019","DOIUrl":"https://doi.org/10.1515/crelle-2022-0019","url":null,"abstract":"Abstract Let G be a unimodular locally compact group. We define a property of irreducible unitary G-representations V which we call c-temperedness, and which for the trivial V boils down to Følner’s condition (equivalent to the trivial V being tempered, i.e. to G being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered V’s, as well as for all tempered V’s in the cases of G:=SL2(ℝ){G:=mathrm{SL}_{2}({mathbb{R}})} and of G=PGL2(Ω){G=mathrm{PGL}_{2}(Omega)} for a non-Archimedean local field Ω of characteristic 0 and residual characteristic not 2. We also establish a weaker form of the conjecture, involving only K-finite vectors. In the non-Archimedean case, we give a formula expressing the character of a tempered V as an appropriately-weighted conjugation-average of a matrix coefficient of V, generalising a formula of Harish-Chandra from the case when V is square-integrable.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79456972","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 : 2021-11-18DOI: 10.1515/crelle-2022-0039
H. Cao, Tianbo Liu
Abstract In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm {mathrm{{Rm}}} and its covariant derivative ∇ Rm {nablamathrm{{Rm}}} can be bounded by the scalar curvature R by | Rm | ≤ C a R a {|mathrm{{Rm}}|leq C_{a}R^{a}} and | ∇ Rm | ≤ C a R a {|nablamathrm{{Rm}}|leq C_{a}R^{a}} (on M ∖ K {Msetminus K} ), for any 0 ≤ a < 1 {0leq a<1} and some constant C a > 0 {C_{a}>0} . Moreover, if the scalar curvature has at most polynomial decay at infinity, then | Rm | ≤ C R {|mathrm{{Rm}}|leq CR} (on M ∖ K {Msetminus K} ). As an application, it follows that if a 4-dimensional complete gradient expanding Ricci soliton ( M 4 , g , f ) {(M^{4},g,f)} has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, hence admits C 1 , α {C^{1,alpha}} asymptotic cones at infinity ( 0 < α < 1 ) {(0
{"title":"Curvature estimates for 4-dimensional complete gradient expanding Ricci solitons","authors":"H. Cao, Tianbo Liu","doi":"10.1515/crelle-2022-0039","DOIUrl":"https://doi.org/10.1515/crelle-2022-0039","url":null,"abstract":"Abstract In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm {mathrm{{Rm}}} and its covariant derivative ∇ Rm {nablamathrm{{Rm}}} can be bounded by the scalar curvature R by | Rm | ≤ C a R a {|mathrm{{Rm}}|leq C_{a}R^{a}} and | ∇ Rm | ≤ C a R a {|nablamathrm{{Rm}}|leq C_{a}R^{a}} (on M ∖ K {Msetminus K} ), for any 0 ≤ a < 1 {0leq a<1} and some constant C a > 0 {C_{a}>0} . Moreover, if the scalar curvature has at most polynomial decay at infinity, then | Rm | ≤ C R {|mathrm{{Rm}}|leq CR} (on M ∖ K {Msetminus K} ). As an application, it follows that if a 4-dimensional complete gradient expanding Ricci soliton ( M 4 , g , f ) {(M^{4},g,f)} has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, hence admits C 1 , α {C^{1,alpha}} asymptotic cones at infinity ( 0 < α < 1 ) {(0<alpha<1)} according to Chen and Deruelle (2015).[21].","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89908050","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：