Pub Date : 2021-07-02DOI: 10.1515/crelle-2023-0025
Yang Li, Valentino Tosatti
Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.
{"title":"On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows","authors":"Yang Li, Valentino Tosatti","doi":"10.1515/crelle-2023-0025","DOIUrl":"https://doi.org/10.1515/crelle-2023-0025","url":null,"abstract":"Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75445820","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 : 2021-07-01DOI: 10.1515/crelle-2021-frontmatter776
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2021-frontmatter776","DOIUrl":"https://doi.org/10.1515/crelle-2021-frontmatter776","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74420187","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 : 2021-07-01DOI: 10.1515/crelle-2021-0033
Kai-Uwe Bux, Stefan Witzel, Matthew C. B. Zaremsky
{"title":"Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101)","authors":"Kai-Uwe Bux, Stefan Witzel, Matthew C. B. Zaremsky","doi":"10.1515/crelle-2021-0033","DOIUrl":"https://doi.org/10.1515/crelle-2021-0033","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89346495","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 : 2021-06-24DOI: 10.1515/crelle-2022-0018
Joachim Jelisiejew, Klemen Šivic
Abstract We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of 8×8{8times 8} matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki–Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of Quotd(𝒪𝔸n⊕r){operatorname{Quot}_{d}(mathcal{O}_{mathbb{A}^{n}}^{oplus r})} for d≤7{dleq 7} and all r, n.
{"title":"Components and singularities of Quot schemes and varieties of commuting matrices","authors":"Joachim Jelisiejew, Klemen Šivic","doi":"10.1515/crelle-2022-0018","DOIUrl":"https://doi.org/10.1515/crelle-2022-0018","url":null,"abstract":"Abstract We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of 8×8{8times 8} matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki–Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of Quotd(𝒪𝔸n⊕r){operatorname{Quot}_{d}(mathcal{O}_{mathbb{A}^{n}}^{oplus r})} for d≤7{dleq 7} and all r, n.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79923688","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 : 2021-06-21DOI: 10.1515/crelle-2022-0052
M. Andersson, David Witt Nyström, Elizabeth Wulcan
Abstract Given a domain Ω ⊂ ℂ n {Omegasubset{mathbb{C}}^{n}} we introduce a class of plurisubharmonic (psh) functions 𝒢 ( Ω ) {{mathcal{G}}(Omega)} and Monge–Ampère operators u ↦ [ d d c u ] p {umapsto[dd^{c}u]^{p}} , p ≤ n {pleq n} , on 𝒢 ( Ω ) {{mathcal{G}}(Omega)} that extend the Bedford–Taylor–Demailly Monge–Ampère operators. Here [ d d c u ] p {[dd^{c}u]^{p}} is a closed positive current of bidegree ( p , p ) {(p,p)} that dominates the non-pluripolar Monge–Ampère current 〈 d d c u 〉 p {langle dd^{c}urangle^{p}} . We prove that [ d d c u ] p {[dd^{c}u]^{p}} is the limit of Monge–Ampère currents of certain natural regularizations of u. On a compact Kähler manifold ( X , ω ) {(X,omega)} we introduce a notion of non-pluripolar energy and a corresponding finite energy class 𝒢 ( X , ω ) ⊂ PSH ( X , ω ) {{mathcal{G}}(X,omega)subsetoperatorname{PSH}(X,omega)} that is a global version of the class 𝒢 ( Ω ) {{mathcal{G}}(Omega)} . From the local construction we get global Monge–Ampère currents [ d d c φ + ω ] p {[dd^{c}varphi+omega]^{p}} for φ ∈ 𝒢 ( X , ω ) {varphiin{mathcal{G}}(X,omega)} that only depend on the current d d c φ + ω {dd^{c}varphi+omega} . The limits of Monge–Ampère currents of certain natural regularizations of φ can be expressed in terms of [ d d c φ + ω ] j {[dd^{c}varphi+omega]^{j}} for j ≤ p {jleq p} . We get a mass formula involving the currents [ d d c φ + ω ] p {[dd^{c}varphi+omega]^{p}} that describes the loss of mass of the non-pluripolar Monge–Ampère measure 〈 d d c φ + ω 〉 n {langle dd^{c}varphi+omegarangle^{n}} . The class 𝒢 ( X , ω ) {{mathcal{G}}(X,omega)} includes ω-psh functions with analytic singularities and the class ℰ ( X , ω ) {{mathcal{E}}(X,omega)} of ω-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.
给定一个域Ω∧∧n {Omegasubset{mathbb{C}}＾{n}} 我们引入了一类多次谐波(psh)函数𝒢¹(Ω) {{mathcal{G}}（Omega）} 和monge - ampontre算子u∈[d ^ d ^ c ^ u] p {你mapsto[dd^{c}u]＾{p}} ， p≤n {pleq n} ，网址:𝒢(Ω) {{mathcal{G}}（Omega）} 扩展了Bedford-Taylor-Demailly monge - ampitre算子。这里是[d²d²c²u] p {[dd^]{c}u]^{p}} 为双次(p, p)的闭合正电流 {(p,p)} 支配非多极蒙安培电流< d d cu > p {langle dd^{c}你rangle＾{p}} ． 我们证明了[d ^ d ^ c ^ u] p {[dd^]{c}u]^{p}} 在紧致Kähler流形(X， ω)上，u的某些自然正则化的monge - ampante电流的极限 {(x;omega）} 引入非多极能的概念和相应的有限能类𝒢(X， ω)∧PSH (X， ω) {{mathcal{G}}(x;omega）subsetoperatorname{PSH}(x;omega）} 这是一个全局版本的课程𝒢¹(Ω) {{mathcal{G}}（Omega）} ． 从局部构造中我们得到全局的蒙日-安培电流[d ^ d c ^ φ + ω] p {[dd^]{c}varphi+omega^{p}} 对于φ∈𝒢(X， ω) {varphiin{mathcal{G}}(x;omega）} 它只依赖于电流d²c²φ + ω {dd^{c}varphi+omega} ． φ的某些自然正则化的蒙日-安培电流的极限可以用[d ^ d ^ c ^ φ + ω] j来表示 {[dd^]{c}varphi+omega^{j}} 对于j≤p {jleq p} ． 我们得到一个质量公式包含电流[d²d²c²φ + ω] p {[dd^]{c}varphi+omega^{p}} 描述了非多极蒙日-安培量的质量损失< d ^ d c ^ φ + ω > n {langle dd^{c}varphi+omegarangle＾{n}} ． 该类𝒢(X， ω) {{mathcal{G}}(x;omega）} 包括具有解析奇异性的ω-psh函数和类e (X， ω) {{mathcal{E}}(x;omega）} 有限能量ω-psh函数和某些其他凸能类，尽管它本身不是凸。
{"title":"Non-pluripolar energy and the complex Monge–Ampère operator","authors":"M. Andersson, David Witt Nyström, Elizabeth Wulcan","doi":"10.1515/crelle-2022-0052","DOIUrl":"https://doi.org/10.1515/crelle-2022-0052","url":null,"abstract":"Abstract Given a domain Ω ⊂ ℂ n {Omegasubset{mathbb{C}}^{n}} we introduce a class of plurisubharmonic (psh) functions 𝒢 ( Ω ) {{mathcal{G}}(Omega)} and Monge–Ampère operators u ↦ [ d d c u ] p {umapsto[dd^{c}u]^{p}} , p ≤ n {pleq n} , on 𝒢 ( Ω ) {{mathcal{G}}(Omega)} that extend the Bedford–Taylor–Demailly Monge–Ampère operators. Here [ d d c u ] p {[dd^{c}u]^{p}} is a closed positive current of bidegree ( p , p ) {(p,p)} that dominates the non-pluripolar Monge–Ampère current 〈 d d c u 〉 p {langle dd^{c}urangle^{p}} . We prove that [ d d c u ] p {[dd^{c}u]^{p}} is the limit of Monge–Ampère currents of certain natural regularizations of u. On a compact Kähler manifold ( X , ω ) {(X,omega)} we introduce a notion of non-pluripolar energy and a corresponding finite energy class 𝒢 ( X , ω ) ⊂ PSH ( X , ω ) {{mathcal{G}}(X,omega)subsetoperatorname{PSH}(X,omega)} that is a global version of the class 𝒢 ( Ω ) {{mathcal{G}}(Omega)} . From the local construction we get global Monge–Ampère currents [ d d c φ + ω ] p {[dd^{c}varphi+omega]^{p}} for φ ∈ 𝒢 ( X , ω ) {varphiin{mathcal{G}}(X,omega)} that only depend on the current d d c φ + ω {dd^{c}varphi+omega} . The limits of Monge–Ampère currents of certain natural regularizations of φ can be expressed in terms of [ d d c φ + ω ] j {[dd^{c}varphi+omega]^{j}} for j ≤ p {jleq p} . We get a mass formula involving the currents [ d d c φ + ω ] p {[dd^{c}varphi+omega]^{p}} that describes the loss of mass of the non-pluripolar Monge–Ampère measure 〈 d d c φ + ω 〉 n {langle dd^{c}varphi+omegarangle^{n}} . The class 𝒢 ( X , ω ) {{mathcal{G}}(X,omega)} includes ω-psh functions with analytic singularities and the class ℰ ( X , ω ) {{mathcal{E}}(X,omega)} of ω-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83412427","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 : 2021-06-14DOI: 10.1515/crelle-2021-0074
Diego García-Lucas, L. Margolis, Á. del Río
Abstract We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.
摘要在任意特征为2的域上，给出了具有同构群代数的非同构有限2群，从而解决了模同构问题。
{"title":"Non-isomorphic 2-groups with isomorphic modular group algebras","authors":"Diego García-Lucas, L. Margolis, Á. del Río","doi":"10.1515/crelle-2021-0074","DOIUrl":"https://doi.org/10.1515/crelle-2021-0074","url":null,"abstract":"Abstract We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81515980","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 : 2021-06-11DOI: 10.1515/crelle-2023-0045
T. Bourni, Mathew T. Langford, S. Lynch
Abstract We provide several characterizations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane. As a consequence, we rule out collapsing singularity models in ( n - 1 ) {(n-1)} -convex mean curvature flow (even when the initial datum is only immersed). Explicit counterexamples show that ( n - 1 ) {(n-1)} -convexity is optimal. We are also able to rule out collapsing singularity models for suitably pinched solutions of higher codimension.
{"title":"Collapsing and noncollapsing in convex ancient mean curvature flow","authors":"T. Bourni, Mathew T. Langford, S. Lynch","doi":"10.1515/crelle-2023-0045","DOIUrl":"https://doi.org/10.1515/crelle-2023-0045","url":null,"abstract":"Abstract We provide several characterizations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane. As a consequence, we rule out collapsing singularity models in ( n - 1 ) {(n-1)} -convex mean curvature flow (even when the initial datum is only immersed). Explicit counterexamples show that ( n - 1 ) {(n-1)} -convexity is optimal. We are also able to rule out collapsing singularity models for suitably pinched solutions of higher codimension.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87570148","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 : 2021-06-10DOI: 10.1515/crelle-2021-0025
Thomas Bitoun, Andreas Bode
Abstract We investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D⏜Xoverparen{mathcal{D}}_{X}-module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.
{"title":"Extending meromorphic connections to coadmissible ̑𝒟-modules","authors":"Thomas Bitoun, Andreas Bode","doi":"10.1515/crelle-2021-0025","DOIUrl":"https://doi.org/10.1515/crelle-2021-0025","url":null,"abstract":"Abstract We investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D⏜Xoverparen{mathcal{D}}_{X}-module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78721646","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 : 2021-06-10DOI: 10.1515/crelle-2022-0087
T. Dinh, K. Oguiso, Xun Yu
Abstract We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long-standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.
{"title":"Smooth complex projective rational surfaces with infinitely many real forms","authors":"T. Dinh, K. Oguiso, Xun Yu","doi":"10.1515/crelle-2022-0087","DOIUrl":"https://doi.org/10.1515/crelle-2022-0087","url":null,"abstract":"Abstract We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long-standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90166596","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 : 2021-06-10DOI: 10.1515/crelle-2022-0095
V. Datar, X. Fu, Jian Song
Abstract We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry.
{"title":"Kähler–Einstein metrics near an isolated log-canonical singularity","authors":"V. Datar, X. Fu, Jian Song","doi":"10.1515/crelle-2022-0095","DOIUrl":"https://doi.org/10.1515/crelle-2022-0095","url":null,"abstract":"Abstract We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90955014","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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