Pub Date : 2021-11-16DOI: 10.1515/crelle-2022-0056
Sarah Frei, B. Hassett, Anthony Várilly-Alvarado
Abstract Given a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.
{"title":"Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence","authors":"Sarah Frei, B. Hassett, Anthony Várilly-Alvarado","doi":"10.1515/crelle-2022-0056","DOIUrl":"https://doi.org/10.1515/crelle-2022-0056","url":null,"abstract":"Abstract Given a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75124129","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 : 2021-11-14DOI: 10.1515/crelle-2022-0061
M. Kashiwara, E. Park
Abstract In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 𝔤 ( ∞ ) {widehat{B}_{{mathfrak{g}}}(infty)} for an arbitrary quantum group U q ( 𝔤 ) {U_{q}({mathfrak{g}})} , which is the product of infinite copies of the crystal B ( ∞ ) {B(infty)} . For a complete duality datum 𝒟 {{mathcal{D}}} in the Hernandez–Leclerc category 𝒞 𝔤 0 {{mathscr{C}_{mathfrak{g}}^{0}}} of a quantum affine algebra U q ′ ( 𝔤 ) {U_{q}^{prime}({mathfrak{g}})} , we prove that the set ℬ 𝒟 ( 𝔤 ) {mathcal{B}_{{mathcal{D}}}({mathfrak{g}})} of the isomorphism classes of simple modules in 𝒞 𝔤 0 {{mathscr{C}_{mathfrak{g}}^{0}}} has an extended crystal structure isomorphic to B ^ 𝔤 fin ( ∞ ) {widehat{B}_{{{mathfrak{g}}_{mathrm{fin}}}}(infty)} . An explicit combinatorial description of the extended crystal ℬ 𝒟 ( 𝔤 ) {mathcal{B}_{{mathcal{D}}}({mathfrak{g}})} for affine type A n ( 1 ) {A_{n}^{(1)}} is given in terms of affine highest weights.
{"title":"Categorical crystals for quantum affine algebras","authors":"M. Kashiwara, E. Park","doi":"10.1515/crelle-2022-0061","DOIUrl":"https://doi.org/10.1515/crelle-2022-0061","url":null,"abstract":"Abstract In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 𝔤 ( ∞ ) {widehat{B}_{{mathfrak{g}}}(infty)} for an arbitrary quantum group U q ( 𝔤 ) {U_{q}({mathfrak{g}})} , which is the product of infinite copies of the crystal B ( ∞ ) {B(infty)} . For a complete duality datum 𝒟 {{mathcal{D}}} in the Hernandez–Leclerc category 𝒞 𝔤 0 {{mathscr{C}_{mathfrak{g}}^{0}}} of a quantum affine algebra U q ′ ( 𝔤 ) {U_{q}^{prime}({mathfrak{g}})} , we prove that the set ℬ 𝒟 ( 𝔤 ) {mathcal{B}_{{mathcal{D}}}({mathfrak{g}})} of the isomorphism classes of simple modules in 𝒞 𝔤 0 {{mathscr{C}_{mathfrak{g}}^{0}}} has an extended crystal structure isomorphic to B ^ 𝔤 fin ( ∞ ) {widehat{B}_{{{mathfrak{g}}_{mathrm{fin}}}}(infty)} . An explicit combinatorial description of the extended crystal ℬ 𝒟 ( 𝔤 ) {mathcal{B}_{{mathcal{D}}}({mathfrak{g}})} for affine type A n ( 1 ) {A_{n}^{(1)}} is given in terms of affine highest weights.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90435077","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 : 2021-11-02DOI: 10.1515/crelle-2022-0094
Kay Rülling, S. Saito
Abstract We prove a Zariski–Nagata purity theorem for the motivic ramification filtration of a reciprocity sheaf. An important tool in the proof is a generalization of the Kato-Saito reciprocity map from geometric global class field theory to all reciprocity sheaves. As a corollary we obtain cut-by-curves and cut-by-surfaces criteria for various ramification filtrations. In some cases this reproves known theorems, in some cases we obtain new results.
{"title":"Ramification theory of reciprocity sheaves, I: Zariski–Nagata purity","authors":"Kay Rülling, S. Saito","doi":"10.1515/crelle-2022-0094","DOIUrl":"https://doi.org/10.1515/crelle-2022-0094","url":null,"abstract":"Abstract We prove a Zariski–Nagata purity theorem for the motivic ramification filtration of a reciprocity sheaf. An important tool in the proof is a generalization of the Kato-Saito reciprocity map from geometric global class field theory to all reciprocity sheaves. As a corollary we obtain cut-by-curves and cut-by-surfaces criteria for various ramification filtrations. In some cases this reproves known theorems, in some cases we obtain new results.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81497995","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 : 2021-11-01DOI: 10.1515/crelle-2021-frontmatter780
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2021-frontmatter780","DOIUrl":"https://doi.org/10.1515/crelle-2021-frontmatter780","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75373268","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 : 2021-10-22DOI: 10.1515/crelle-2022-0085
P. Daskalopoulos, M. Sáez
Abstract In this paper we study the uniqueness of graphical mean curvature flow with locally Lipschitz initial data. We first prove that rotationally symmetric entire graphs are unique, without any further assumptions. Our methods also give an alternative simple proof of uniqueness in the one-dimensional case. In the general case, we establish the uniqueness of entire proper graphs that satisfy a uniform lower bound on the second fundamental form. The latter result extends to initial conditions that are proper graphs over subdomains of ℝ n {mathbb{R}^{n}} . A consequence of our result is the uniqueness of convex entire graphs, which allow us to prove that Hamilton’s Harnack estimate holds for mean curvature flow solutions that are convex entire graphs.
摘要本文研究了具有局部Lipschitz初始数据的图形平均曲率流的唯一性。我们首先证明了旋转对称全图是唯一的，没有任何进一步的假设。我们的方法在一维情况下也给出了另一种简单的唯一性证明。在一般情况下，我们建立了在第二种基本形式上满足一致下界的整个固有图的唯一性。后一种结果推广到初始条件，即在1 n {mathbb{R}^{n}}的子域上的固有图。我们的结果的一个结果是凸整图的唯一性，这使我们能够证明Hamilton的Harnack估计适用于凸整图的平均曲率流解。
{"title":"Uniqueness of entire graphs evolving by mean curvature flow","authors":"P. Daskalopoulos, M. Sáez","doi":"10.1515/crelle-2022-0085","DOIUrl":"https://doi.org/10.1515/crelle-2022-0085","url":null,"abstract":"Abstract In this paper we study the uniqueness of graphical mean curvature flow with locally Lipschitz initial data. We first prove that rotationally symmetric entire graphs are unique, without any further assumptions. Our methods also give an alternative simple proof of uniqueness in the one-dimensional case. In the general case, we establish the uniqueness of entire proper graphs that satisfy a uniform lower bound on the second fundamental form. The latter result extends to initial conditions that are proper graphs over subdomains of ℝ n {mathbb{R}^{n}} . A consequence of our result is the uniqueness of convex entire graphs, which allow us to prove that Hamilton’s Harnack estimate holds for mean curvature flow solutions that are convex entire graphs.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74520338","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 : 2021-10-16DOI: 10.1515/crelle-2023-0017
An-Min Li, Zhao Lian, Li Sheng
Abstract In this paper, we prove the Yau–Tian–Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles, thus for toric manifolds and homogeneous toric bundles, M admits an extremal metric in c 1 ( L ) {c_{1}(L)} if and only if ( M , L ) {(M,L)} is relatively K ^ {widehat{K}} -stable.
{"title":"Extremal metrics on toric manifolds and homogeneous toric bundles","authors":"An-Min Li, Zhao Lian, Li Sheng","doi":"10.1515/crelle-2023-0017","DOIUrl":"https://doi.org/10.1515/crelle-2023-0017","url":null,"abstract":"Abstract In this paper, we prove the Yau–Tian–Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles, thus for toric manifolds and homogeneous toric bundles, M admits an extremal metric in c 1 ( L ) {c_{1}(L)} if and only if ( M , L ) {(M,L)} is relatively K ^ {widehat{K}} -stable.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87890393","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 : 2021-10-05DOI: 10.1515/crelle-2022-0071
Camillo Brena, N. Gigli, Shouhei Honda, Xingyu Zhu
Abstract We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties tr ( Hess f ) = Δ f operatorname{tr}(operatorname{Hess}f)=Delta f on U ⊆ X Usubseteq{mathsf{X}} for every 𝑓 sufficiently regular, m = c H n mathfrak{m}=cmathscr{H}^{n} on U ⊆ X Usubseteq{mathsf{X}} for some c > 0 c>0 , where U ⊆ X Usubseteq{mathsf{X}} is open and 𝖷 is a – possibly collapsed – RCD space of essential dimension 𝑛.
{"title":"Weakly non-collapsed RCD spaces are strongly non-collapsed","authors":"Camillo Brena, N. Gigli, Shouhei Honda, Xingyu Zhu","doi":"10.1515/crelle-2022-0071","DOIUrl":"https://doi.org/10.1515/crelle-2022-0071","url":null,"abstract":"Abstract We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties tr ( Hess f ) = Δ f operatorname{tr}(operatorname{Hess}f)=Delta f on U ⊆ X Usubseteq{mathsf{X}} for every 𝑓 sufficiently regular, m = c H n mathfrak{m}=cmathscr{H}^{n} on U ⊆ X Usubseteq{mathsf{X}} for some c > 0 c>0 , where U ⊆ X Usubseteq{mathsf{X}} is open and 𝖷 is a – possibly collapsed – RCD space of essential dimension 𝑛.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88713341","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 : 2021-10-04DOI: 10.1515/crelle-2023-0026
C. Arezzo, A. Della Vedova, Yalong Shi
Abstract We give sufficient conditions for the existence of Kähler–Einstein and constant scalar curvature Kähler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kähler classes and the branching divisor. This result generalizes previous work on Kähler–Einstein metrics by Li and Sun [C. Li and S. Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 2014, 3, 927–973], and extends Chen–Cheng’s existence results for cscK metrics in [X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 2021, 4, 937–1009].
{"title":"Constant scalar curvature Kähler metrics on ramified Galois coverings","authors":"C. Arezzo, A. Della Vedova, Yalong Shi","doi":"10.1515/crelle-2023-0026","DOIUrl":"https://doi.org/10.1515/crelle-2023-0026","url":null,"abstract":"Abstract We give sufficient conditions for the existence of Kähler–Einstein and constant scalar curvature Kähler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kähler classes and the branching divisor. This result generalizes previous work on Kähler–Einstein metrics by Li and Sun [C. Li and S. Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 2014, 3, 927–973], and extends Chen–Cheng’s existence results for cscK metrics in [X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 2021, 4, 937–1009].","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87611421","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 : 2021-10-03DOI: 10.1515/crelle-2022-0024
Samir Canning, H. Larson
Abstract While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space ℋk,g{mathcal{H}_{k,g}} parametrizing smooth degree k, genus g covers of ℙ1{mathbb{P}^{1}}. Let k=3,4,5{k=3,4,5}. We prove that the rational Chow rings of ℋk,g{mathcal{H}_{k,g}} stabilize in a suitable sense as g tends to infinity. In the case k=3{k=3}, we completely determine the Chow rings for all g. We also prove that the rational Chow groups of the simply branched Hurwitz space ℋk,gs⊂ℋk,g{mathcal{H}^{s}_{k,g}subsetmathcal{H}_{k,g}} are zero in codimension up to roughly gk{frac{g}{k}}. In [S. Canning and H. Larson, The Chow rings of the moduli spaces of curves of genus 7,8{7,8} and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ7,ℳ8,{mathcal{M}_{7},mathcal{M}_{8},} and ℳ9{mathcal{M}_{9}} are tautological.
{"title":"Chow rings of low-degree Hurwitz spaces","authors":"Samir Canning, H. Larson","doi":"10.1515/crelle-2022-0024","DOIUrl":"https://doi.org/10.1515/crelle-2022-0024","url":null,"abstract":"Abstract While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space ℋk,g{mathcal{H}_{k,g}} parametrizing smooth degree k, genus g covers of ℙ1{mathbb{P}^{1}}. Let k=3,4,5{k=3,4,5}. We prove that the rational Chow rings of ℋk,g{mathcal{H}_{k,g}} stabilize in a suitable sense as g tends to infinity. In the case k=3{k=3}, we completely determine the Chow rings for all g. We also prove that the rational Chow groups of the simply branched Hurwitz space ℋk,gs⊂ℋk,g{mathcal{H}^{s}_{k,g}subsetmathcal{H}_{k,g}} are zero in codimension up to roughly gk{frac{g}{k}}. In [S. Canning and H. Larson, The Chow rings of the moduli spaces of curves of genus 7,8{7,8} and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ7,ℳ8,{mathcal{M}_{7},mathcal{M}_{8},} and ℳ9{mathcal{M}_{9}} are tautological.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78952336","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 : 2021-09-29DOI: 10.1515/crelle-2022-0096
Rostislav Akhmechet, Peter K. Johnson, Vyacheslav Krushkal
Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ( 2 ) {{rm SU}(2)} quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ {widehat{Z}} -invariant.
{"title":"Lattice cohomology and q-series invariants of 3-manifolds","authors":"Rostislav Akhmechet, Peter K. Johnson, Vyacheslav Krushkal","doi":"10.1515/crelle-2022-0096","DOIUrl":"https://doi.org/10.1515/crelle-2022-0096","url":null,"abstract":"Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ( 2 ) {{rm SU}(2)} quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ {widehat{Z}} -invariant.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83031566","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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