We prove duality theorems for the étale cohomology of split tori on smooth curves over a local field of positive characteristic. In particular, we show that the classical Brauer–Manin pairing between the Brauer and Picard groups of smooth projective curves over such a field extends to arbitrary smooth curves over the field. As another consequence, we obtain a description of the Brauer group of the function fields of curves over local fields in terms of the characters of the idele class groups.
{"title":"Duality theorems for curves over local fields","authors":"A. Krishna, Jitendra Rathore, Samiron Sadhukhan","doi":"10.1090/btran/187","DOIUrl":"https://doi.org/10.1090/btran/187","url":null,"abstract":"We prove duality theorems for the étale cohomology of split tori on smooth curves over a local field of positive characteristic. In particular, we show that the classical Brauer–Manin pairing between the Brauer and Picard groups of smooth projective curves over such a field extends to arbitrary smooth curves over the field. As another consequence, we obtain a description of the Brauer group of the function fields of curves over local fields in terms of the characters of the idele class groups.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"119 28","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141822267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if (X,d,μ)(X,d,mu ) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N1,p(X)N^{1,p}(X). Here the measure μmu is Borel and is finite and positive on all metric balls. In particular, we don’t assume properness of XX, doubling of μmu or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces XX an
我们证明，在局部完备且可分离的度量空间中，可以用局部 Lipschitz 函数计算容量。此外，我们还证明，如果 ( X , d , μ ) (X,d,mu ) 是局部完全且可分离的度量空间，那么连续函数在牛顿空间 N 1 , p ( X ) N^{1,p}(X) 中是密集的。这里的度量 μ mu 是玻尔的，并且在所有度量球上都是有限和正的。特别是，我们不假定 X X 的适当性、μ mu 的加倍或任何波恩卡雷不等式。这些都部分或全部解决了一些学者提出的问题，包括海诺宁（J. Heinonen）、比约恩（A. Björn ）和比约恩（J. Björn ）。与过去的许多工作不同，我们的结果适用于局部完全空间 X X，并免除了经常使用的正则性假设：加倍、适当性、Poincaré 不等式、Loewner 属性或准凸性。
{"title":"Density of continuous functions in Sobolev spaces with applications to capacity","authors":"S. Eriksson-Bique, Pietro Poggi-Corradini","doi":"10.1090/btran/188","DOIUrl":"https://doi.org/10.1090/btran/188","url":null,"abstract":"<p>We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma d comma mu right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>μ</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X,d,mu )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Superscript 1 comma p Baseline left-parenthesis upper X right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">N^{1,p}(X)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Here the measure <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\u0000 <mml:semantics>\u0000 <mml:mi>μ</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is Borel and is finite and positive on all metric balls. In particular, we don’t assume properness of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, doubling of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\u0000 <mml:semantics>\u0000 <mml:mi>μ</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to <italic>locally complete</italic> spaces <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> an","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"19 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141655136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Take a sequence of contactomorphisms of a contact three-manifold that C 0 C^0 -converges to a homeomorphism. If the images of a Legendrian knot limit to a smooth knot under this sequence, we show that it is contactomorphic to the original knot. We prove this by establishing that, on one hand, non–Legendrian knots admit a type of contact-squashing (similar to squeezing) onto transverse knots while, on the other hand, Legendrian knots do not admit such a squashing. The non-trivial input from contact topology that is needed is (a local version of) the Thurston–Bennequin inequality.
取一个接触三芒星的接触同构序列，C 0 C^0 -converges to a homeomorphism。如果在这个序列下，一个 Legendrian 结的图像极限到一个光滑结，我们就可以证明它与原来的结是接触同构的。我们证明这一点的方法是，一方面，非勒根结允许一种类型的接触挤压（类似于挤压）到横向结上，而另一方面，勒根结不允许这种挤压。这就需要从接触拓扑学中输入（局部版本的）瑟斯顿-贝内金不等式（Thurston-Bennequin inequality）。
{"title":"𝐶⁰-limits of Legendrian knots","authors":"Georgios Dimitroglou Rizell, Michael Sullivan","doi":"10.1090/btran/189","DOIUrl":"https://doi.org/10.1090/btran/189","url":null,"abstract":"Take a sequence of contactomorphisms of a contact three-manifold that \u0000\u0000 \u0000 \u0000 C\u0000 0\u0000 \u0000 C^0\u0000 \u0000\u0000-converges to a homeomorphism. If the images of a Legendrian knot limit to a smooth knot under this sequence, we show that it is contactomorphic to the original knot. We prove this by establishing that, on one hand, non–Legendrian knots admit a type of contact-squashing (similar to squeezing) onto transverse knots while, on the other hand, Legendrian knots do not admit such a squashing. The non-trivial input from contact topology that is needed is (a local version of) the Thurston–Bennequin inequality.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":" 46","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140683900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I analyze an unexpected connection between multiple orthogonal polynomials, d d -orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot’s combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.
我分析了多重正交多项式、d d 正交多项式、生成矩阵和支化续分之间意想不到的联系。这项工作可以看作是 Viennot 正交多项式组合理论的部分延伸，它适用于生成矩阵是下海森堡矩阵但不一定是三对角矩阵的情况。
{"title":"Multiple orthogonal polynomials, 𝑑-orthogonal polynomials, production matrices, and branched continued fractions","authors":"Alan Sokal","doi":"10.1090/btran/133","DOIUrl":"https://doi.org/10.1090/btran/133","url":null,"abstract":"I analyze an unexpected connection between multiple orthogonal polynomials, \u0000\u0000 \u0000 d\u0000 d\u0000 \u0000\u0000-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot’s combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"31 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140713373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the K K -theoretic k k -Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed k k -Schur Katalan functions is identified with the Schubert structure sheaves in the K K -homology of the affine Grassmannian. Our main result is a proof of this conjecture. We also study a K K -theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum K K -theory ring of the flag variety to a closed K K - k k -Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.
最近，布拉西亚克-莫尔斯-谢林格引入了称为卡塔兰函数的对称函数，并证明了林-席林-下野提出的 K K 理论 k k -Schur 函数构成了卡塔兰函数的一个亚族。他们猜想，被称为封闭 k k -Schur 卡塔兰函数的另一个卡塔兰函数亚族与仿射格拉斯曼的 K K -本构中的舒伯特结构剪子是一致的。我们还研究了池田（Ikeda）、岩尾（Iwao）和前野（Maeno）以非几何的方式，根据鲁伊塞纳斯（Ruijsenaars）相对论户田晶格的单能解构建的 K K 理论彼得森同构。我们证明，该映射将旗形变的量子 K K 理论环的舒伯特类发送到封闭的 K K - k k -Schur 卡塔兰函数，直到一个与平移元素相关的显式因子为止，而平移元素是相对于反显式角根的。事实上，我们证明了这个映射与一个映射重合，这个映射的存在性由 Lam、Li、Mihalcea、Shimozono 猜想，由 Kato 证明，最近由 Chow 和 Leung 证明。
{"title":"Closed 𝑘-Schur Katalan functions as 𝐾-homology Schubert representatives of the affine Grassmannian","authors":"Takeshi Ikeda, Shinsuke Iwao, Satoshi Naito","doi":"10.1090/btran/184","DOIUrl":"https://doi.org/10.1090/btran/184","url":null,"abstract":"Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-theoretic \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-Schur functions due to Lam–Schilling–Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-Schur Katalan functions is identified with the Schubert structure sheaves in the \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-homology of the affine Grassmannian. Our main result is a proof of this conjecture.\u0000\u0000We also study a \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-theory ring of the flag variety to a closed \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-\u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"133 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140251399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We say a null-homologous knot KK in a 33-manifold YY has Property G, if the Thurston norm and fiberedness of the complement of KK is preserved under the zero surgery on KK. In this paper, we will show that, if the smooth 44-genus of K×{0}Ktimes {0} (in a certain homology class) in (Y×[0,1]
如果 K K 的瑟斯顿规范和补集的纤维性在 K K 的零手术下得以保留，我们就说 3 3 -manifold Y Y 中的空同源结 K K 具有属性 G。本文将证明，如果 K × { 0 } 的光滑 4 4 - 属 在 ( Y × [ 0 , 1 ] ) # N C P 2 ¯ (Ytimes [0,1])#Noverline {mathbb CP^2} 中，Y Y 是有理同调。 当光滑的 4 4 - 属是 0 0 0 时，Y Y 可以看作是任何封闭的、定向的 3 3 -manifold。
{"title":"Property G and the 4-genus","authors":"Yi Ni","doi":"10.1090/btran/153","DOIUrl":"https://doi.org/10.1090/btran/153","url":null,"abstract":"<p>We say a null-homologous knot <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\u0000 <mml:semantics>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-manifold <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\">\u0000 <mml:semantics>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Y</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has Property G, if the Thurston norm and fiberedness of the complement of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is preserved under the zero surgery on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this paper, we will show that, if the smooth <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"4\">\u0000 <mml:semantics>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-genus of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K times StartSet 0 EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Ktimes {0}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (in a certain homology class) in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper Y times left-bracket 0 comma 1 right-bracket right-parenthesis number-sign upper N ModifyingAbove double-struck upper C upper P squared With bar\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 ","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"48 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139533209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a finite dimensional vector space VV of dimension nn, we consider the incidence correspondence (or partial flag variety) X⊂PV×PV∨Xsubset mathbb {P}V times mathbb {P}V^{vee }, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on XX in characteristic p>0p>0. If n=3n=3 then X
对于维数为 n n 的有限维向量空间 V V，我们考虑入射对应关系（或称偏旗形）X ⊂ P V × P V ∨ Xsubset mathbb {P}V times mathbb {P}V^{vee }。 ，参数对由一个点和包含该点的超平面组成。我们完全描述了特征 p > 0 p>0 时 X X 上线束同调群的消失与非消失行为。如果 n = 3 n=3，那么 X X 是 V V 的全旗变，特征描述包含在格里菲斯 70 年代的论文中。在特征 0 0 中，同调群是由 Borel-Weil-Bott 定理来描述所有 V V 的。我们的策略是用计算投影空间上余切剪切的（捻）分幂的同调来重构这个问题，然后利用弗罗贝纽斯诱导的自然截断以及对卡斯特努沃-芒福德正则性的仔细估计来研究这个问题。当 n = 3 n=3 时，我们恢复了刘林源近期工作中对特征的递归描述，而对于一般 n n，我们给出了线束受限集合同调的特征公式。我们的研究结果表明，截断舒尔函数是同调符的自然构件。
{"title":"Cohomology of line bundles on the incidence correspondence","authors":"Z. Gao, Claudiu Raicu","doi":"10.1090/btran/173","DOIUrl":"https://doi.org/10.1090/btran/173","url":null,"abstract":"<p>For a finite dimensional vector space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper V\">\u0000 <mml:semantics>\u0000 <mml:mi>V</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">V</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we consider the incidence correspondence (or partial flag variety) <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X subset-of double-struck upper P upper V times double-struck upper P upper V Superscript logical-or\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>V</mml:mi>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>V</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>∨<!-- ∨ --></mml:mo>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Xsubset mathbb {P}V times mathbb {P}V^{vee }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p>0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. If <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n=3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> then <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:m","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"21 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139382985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid {0,1,…,n}2{0,1,dots , n}^2 has L1L_1-distortion bounded below by a constant multiple of lognsqrt {log n}. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if {Gn}n=1∞{G_n}_{n=1}^infty is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number δ∈
{"title":"𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions","authors":"F. Baudier, C. Gartland, T. Schlumprecht","doi":"10.1090/btran/143","DOIUrl":"https://doi.org/10.1090/btran/143","url":null,"abstract":"<p>By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace 0 comma 1 comma ellipsis comma n right-brace squared\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mo>…<!-- … --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:msup>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{0,1,dots , n}^2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">L_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-distortion bounded below by a constant multiple of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartRoot log n EndRoot\">\u0000 <mml:semantics>\u0000 <mml:msqrt>\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msqrt>\u0000 <mml:annotation encoding=\"application/x-tex\">sqrt {log n}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace upper G Subscript n Baseline right-brace Subscript n equals 1 Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:msubsup>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msubsup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{G_n}_{n=1}^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta element-of left-bracket 2 comma normal infinity right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mo stretchy=\"fals","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122085201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Laurent Demonet, O. Iyama, Nathan Reading, I. Reiten, Hugh Thomas
The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set torsAmathsf {tors} A of torsion classes over a finite-dimensional algebra AA. We show that torsAmathsf {tors} A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of torsAmathsf {tors} A. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that
本文的目的是建立一个格理论框架来研究有限维代数a a上的偏序扭转类集合to o s a mathsf {tors} a。我们证明了A mathsf {tors} A是一个完备格，它具有很强的双代数性和完备半分布性。因此，它的Hasse颤振承载了其结构的重要部分，我们引入了它的Hasse颤振的砖标记，并利用它来研究它的格同余。特别地，我们给出了所谓的强迫序的一个表示理论解释，并证明了to o s a mathsf {tors} a是完全同余一致的。当I I是a a的双边理想时，tors (a /I) mathsf {tors} (a /I)是tors a mathsf {tors} a的格商，称为代数商，对应的格同余称为代数同余。本文的第二部分是对代数同余的研究。我们用砖标记的形式描述了由代数同余收缩的哈塞颤振的A / mathsf {tors} A的箭头。在第三部分中，我们详细地研究了预投影代数Π Pi的情况，其中t = 1 = Π mathsf {tors} Pi是弱阶Weyl群。特别地，当Q Q是Dynkin颤振时，我们给出了一个新的、更具代表性的理论证明，证明了Q Q与寒武纪晶格之间的同构。我们还证明了在A类型A中，tors Π mathsf {tors} Pi的代数商正是它的哈希正则格商。
{"title":"Lattice theory of torsion classes: Beyond 𝜏-tilting theory","authors":"Laurent Demonet, O. Iyama, Nathan Reading, I. Reiten, Hugh Thomas","doi":"10.1090/btran/100","DOIUrl":"https://doi.org/10.1090/btran/100","url":null,"abstract":"<p>The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif t sans-serif o sans-serif r sans-serif s upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"sans-serif\">t</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">o</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">r</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">s</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathsf {tors} A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of torsion classes over a finite-dimensional algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We show that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif t sans-serif o sans-serif r sans-serif s upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"sans-serif\">t</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">o</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">r</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">s</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathsf {tors} A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a complete lattice which enjoys very strong properties, as <italic>bialgebraicity</italic> and <italic>complete semidistributivity</italic>. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif t sans-serif o sans-serif r sans-serif s upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"sans-serif\">t</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">o</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">r</mml:mi>\u0000 <mml:mi mathvariant=\"sans-serif\">s</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathsf {tors} A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In particular, we give a representation-theoretical interpretation of the so-called <italic>forcing order</italic>, and we prove that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif t sans-serif o sans-serif r sans-serif s upper A","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126322906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m mathbb {C}^ntimes mathbb {C}^m . The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on M M which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the L p L^p and L p L^p -Sobolev mapping properties of the associated kernels, and (4) provide examples.
本文是继续研究n × cm mathbb {C} n乘以mathbb {C} m的一般二次子流形上的Kohn Laplacian及其逆的几何和解析性质的一部分。本文的目的是探讨在有向列维形式的特征值不消失的情况下的复格林算子。我们(1)研究了特征值条件在M M上的几何条件，(2)建立了复格林算子及其导数的最优点上界，(3)探索了相关核的L p L^p和L p L^p -Sobolev映射性质，(4)提供了示例。
{"title":"The fundamental solution to Box_{𝑏} on quadric manifolds with nonzero eigenvalues","authors":"A. Boggess, A. Raich","doi":"10.1090/btran/121","DOIUrl":"https://doi.org/10.1090/btran/121","url":null,"abstract":"This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of \u0000\u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 n\u0000 \u0000 ×\u0000 \u0000 \u0000 C\u0000 \u0000 m\u0000 \u0000 \u0000 mathbb {C}^ntimes mathbb {C}^m\u0000 \u0000\u0000. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on \u0000\u0000 \u0000 M\u0000 M\u0000 \u0000\u0000 which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the \u0000\u0000 \u0000 \u0000 L\u0000 p\u0000 \u0000 L^p\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 L\u0000 p\u0000 \u0000 L^p\u0000 \u0000\u0000-Sobolev mapping properties of the associated kernels, and (4) provide examples.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131185924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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