Pub Date : 2020-05-11DOI: 10.1215/00127094-2022-0078
Simon Eberle, H. Shahgholian, G. Weiss
Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.
假设维度上有一个下界，我们证明了一个关于无界重合集障碍问题全局解分类的长期猜想。
{"title":"On global solutions of the obstacle problem","authors":"Simon Eberle, H. Shahgholian, G. Weiss","doi":"10.1215/00127094-2022-0078","DOIUrl":"https://doi.org/10.1215/00127094-2022-0078","url":null,"abstract":"Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45303643","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 : 2020-05-06DOI: 10.1215/00127094-2021-0058
J. Lott
We give an analog of triangle comparison for Kaehler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov-Hausdorff limits. We discuss tangent cones and singular Kaehler spaces.
{"title":"Comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces","authors":"J. Lott","doi":"10.1215/00127094-2021-0058","DOIUrl":"https://doi.org/10.1215/00127094-2021-0058","url":null,"abstract":"We give an analog of triangle comparison for Kaehler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov-Hausdorff limits. We discuss tangent cones and singular Kaehler spaces.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47820429","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 : 2020-05-05DOI: 10.1215/00127094-2022-0038
V. Beresnevich, Erez Nesharim, Lei Yang
In this paper we prove that badly approximable points on any analytic non-degenerate curve in $mathbb{R}^n$ is an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani (2014) which represents a far-reaching generalisation of Davenport's problem from the 1960s. Amongst various consequences of our main result is a solution to Bugeaud's problem on real numbers badly approximable by algebraic numbers of arbitrary degree.
{"title":"Winning property of badly approximable points on curves","authors":"V. Beresnevich, Erez Nesharim, Lei Yang","doi":"10.1215/00127094-2022-0038","DOIUrl":"https://doi.org/10.1215/00127094-2022-0038","url":null,"abstract":"In this paper we prove that badly approximable points on any analytic non-degenerate curve in $mathbb{R}^n$ is an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani (2014) which represents a far-reaching generalisation of Davenport's problem from the 1960s. Amongst various consequences of our main result is a solution to Bugeaud's problem on real numbers badly approximable by algebraic numbers of arbitrary degree.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46161345","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 : 2020-05-05DOI: 10.1215/00127094-2023-0021
A. Berget, Hunter Spink, Dennis Tseng
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring $A^bullet(M)$ arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson whose proof was announced recently by Ardila, Denham and Huh.
对于任意矩阵 $M$，我们利用由格拉斯曼产生的组合周环 $A^bullet(M)$ 中某些同义类的混合交集数来计算图特多项式 $T_M(x,y)$。利用混合霍奇-黎曼关系，我们推导出了母题复数的 h 向量的对数凹性的加强，改进了道森的一个古老猜想，阿迪拉、德纳姆和胡最近公布了该猜想的证明。
{"title":"Log-concavity of matroid h-vectors and mixed Eulerian numbers","authors":"A. Berget, Hunter Spink, Dennis Tseng","doi":"10.1215/00127094-2023-0021","DOIUrl":"https://doi.org/10.1215/00127094-2023-0021","url":null,"abstract":"For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring $A^bullet(M)$ arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson whose proof was announced recently by Ardila, Denham and Huh.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141206465","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 : 2020-05-01DOI: 10.1215/00127094-2019-0080
J. Chen, Meng Chen, Chen Jiang
We establish the Noether inequality for projective $3$-folds. More precisely, we prove that the inequality $${rm vol}(X)geq tfrac{4}{3}p_g(X)-{tfrac{10}{3}}$$ holds for all projective $3$-folds $X$ of general type with either $p_g(X)leq 4$ or $p_g(X)geq 21$, where $p_g(X)$ is the geometric genus and ${rm vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.
{"title":"The Noether inequality for algebraic 3 -folds","authors":"J. Chen, Meng Chen, Chen Jiang","doi":"10.1215/00127094-2019-0080","DOIUrl":"https://doi.org/10.1215/00127094-2019-0080","url":null,"abstract":"We establish the Noether inequality for projective $3$-folds. More precisely, we prove that the inequality $${rm vol}(X)geq tfrac{4}{3}p_g(X)-{tfrac{10}{3}}$$ holds for all projective $3$-folds $X$ of general type with either $p_g(X)leq 4$ or $p_g(X)geq 21$, where $p_g(X)$ is the geometric genus and ${rm vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46035330","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 : 2020-04-15DOI: 10.1215/00127094-2019-0072
Hiroshi Iriyeh, Masataka Shibata
{"title":"Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case","authors":"Hiroshi Iriyeh, Masataka Shibata","doi":"10.1215/00127094-2019-0072","DOIUrl":"https://doi.org/10.1215/00127094-2019-0072","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48931057","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 : 2020-04-01DOI: 10.1215/00127094-2020-0016
{"title":"Errata for “Mapping class group and a global Torelli theorem for hyperkähler manifolds” by Misha Verbitsky","authors":"","doi":"10.1215/00127094-2020-0016","DOIUrl":"https://doi.org/10.1215/00127094-2020-0016","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47302919","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 : 2020-03-27DOI: 10.1215/00127094-2022-0037
Benjamin Antieau, A. Mathew, M. Morrow, T. Nikolaus
Using topological cyclic homology, we give a refinement of Beilinson's $p$-adic Goodwillie isomorphism between relative continuous $K$-theory and cyclic homology. As a result, we generalize results of Bloch-Esnault-Kerz and Beilinson on the $p$-adic deformations of $K$-theory classes. Furthermore, we prove structural results for the Bhatt-Morrow-Scholze filtration on $TC$ and identify the graded pieces with the syntomic cohomology of Fontaine-Messing.
{"title":"On the Beilinson fiber square","authors":"Benjamin Antieau, A. Mathew, M. Morrow, T. Nikolaus","doi":"10.1215/00127094-2022-0037","DOIUrl":"https://doi.org/10.1215/00127094-2022-0037","url":null,"abstract":"Using topological cyclic homology, we give a refinement of Beilinson's $p$-adic Goodwillie isomorphism between relative continuous $K$-theory and cyclic homology. As a result, we generalize results of Bloch-Esnault-Kerz and Beilinson on the $p$-adic deformations of $K$-theory classes. Furthermore, we prove structural results for the Bhatt-Morrow-Scholze filtration on $TC$ and identify the graded pieces with the syntomic cohomology of Fontaine-Messing.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47405170","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 : 2020-03-23DOI: 10.1215/00127094-2022-0030
D. Benson, P. Etingof, V. Ostrik
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(bf k)$ by the $n$-th Steinberg module, and ${rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(bf k)$-modules. We show that ${rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $mathbb{Z}[2cos(2pi/p^n)]$, and that ${rm Ver}_{p^n}$ embeds into ${rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $bf k$ admits a fiber functor to the union ${rm Ver}_{p^infty}$ of the nested sequence ${rm Ver}_{p}subset {rm Ver}_{p^2}subsetcdots$. This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${rm Ver}_p$.
{"title":"New incompressible symmetric tensor categories in positive characteristic","authors":"D. Benson, P. Etingof, V. Ostrik","doi":"10.1215/00127094-2022-0030","DOIUrl":"https://doi.org/10.1215/00127094-2022-0030","url":null,"abstract":"We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(bf k)$ by the $n$-th Steinberg module, and ${rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(bf k)$-modules. We show that ${rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $mathbb{Z}[2cos(2pi/p^n)]$, and that ${rm Ver}_{p^n}$ embeds into ${rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $bf k$ admits a fiber functor to the union ${rm Ver}_{p^infty}$ of the nested sequence ${rm Ver}_{p}subset {rm Ver}_{p^2}subsetcdots$. This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${rm Ver}_p$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46949399","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 : 2020-03-15DOI: 10.1215/00127094-2020-0001
R. Vakil, M. Wood
The definition ofM in Section 1.1 should be the quotient of K0(VarK) by relations of the form [X] − [Y ] whenever X → Y is a radicial surjective morphism of varieties over K, and all further statements in the paper should use this corrected definition. This quotient of the Grothendieck ring is often taken for applications to motivic integration (see [Mus11, Section 7.2] and [CNS18, Section 4.4]). When K has characteristic 0, these additional relations were already trivial in K0(VarK) (e.g. see [Mus11, Prop 7.25]). The motivic measure of point counting over a finite field still factors through this new definition ofM. This correction is necessary so that the proofs in the paper, in particular those of Theorem 1.13 and in Section 5, are correct. The arguments claim equality inM of [X] and [Y ] where we have a morphism X → Y that is bijective on points over any algebraically closed field. Such an argument is valid in the corrected definition ofM above ([Mus11, Remark A.22]), but is not known to be valid in K0(VarK). We thank Margaret Bilu and Sean Howe for pointing out this mistake and the necessary correction. See [BH19] for further discussion of this issue.
{"title":"Errata to “Discriminants in the Grothendieck ring”","authors":"R. Vakil, M. Wood","doi":"10.1215/00127094-2020-0001","DOIUrl":"https://doi.org/10.1215/00127094-2020-0001","url":null,"abstract":"The definition ofM in Section 1.1 should be the quotient of K0(VarK) by relations of the form [X] − [Y ] whenever X → Y is a radicial surjective morphism of varieties over K, and all further statements in the paper should use this corrected definition. This quotient of the Grothendieck ring is often taken for applications to motivic integration (see [Mus11, Section 7.2] and [CNS18, Section 4.4]). When K has characteristic 0, these additional relations were already trivial in K0(VarK) (e.g. see [Mus11, Prop 7.25]). The motivic measure of point counting over a finite field still factors through this new definition ofM. This correction is necessary so that the proofs in the paper, in particular those of Theorem 1.13 and in Section 5, are correct. The arguments claim equality inM of [X] and [Y ] where we have a morphism X → Y that is bijective on points over any algebraically closed field. Such an argument is valid in the corrected definition ofM above ([Mus11, Remark A.22]), but is not known to be valid in K0(VarK). We thank Margaret Bilu and Sean Howe for pointing out this mistake and the necessary correction. See [BH19] for further discussion of this issue.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49174324","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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