Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Th'el`ene, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the inverse Galois problem, and solves, at the same time, Grunwald's problem, for these groups. ----- Soit X une compactification lisse d'un espace homog`ene d'un groupe alg'ebrique lin'eaire G sur un corps de nombres k. Nous 'etablissons la conjecture de Colliot-Th'el`ene, Sansuc, Kato et Saito sur l'image du groupe de Chow des z'ero-cycles de X dans le produit des m^emes groupes sur tous les compl'et'es de k. Lorsque G est semi-simple et simplement connexe et que le stabilisateur g'eom'etrique est fini et hyper-r'esoluble, nous montrons que les points rationnels de X sont denses dans l'ensemble de Brauer-Manin. Pour les groupes finis hyper-r'esolubles, en particulier pour les groupes finis nilpotents, cela donne une nouvelle preuve du th'eor`eme de Shafarevich sur le probl`eme de Galois inverse et r'esout en m^eme temps, pour ces groupes, le probl`eme de Grunwald.
让X是一个光滑紧化的齐次线性代数空间G组数域k。我们建立的猜想Colliot-Th ' el '东北偏东,Sansuc,加藤和齐藤的形象Chow群zero-cycles X产品相同的组/ k的完成。当G是半单和单连通和几何稳定器是有限和supersolvable,我们表明,理性点X Brauer-Manin密集的集合。对于有限超可解群，特别是有限幂零群，给出了关于反伽罗瓦问题的Shafarevich定理的一个新的证明，同时解决了这些群的Grunwald问题。----- Soit X . une紧化lisse d'un空间同质性' ' ene ' 'ebrique lin ' 'eaire G . sur un corps de nombres k. Nous 'etablissons la conjecture de colliot - the 'el ' ene, sansue, Kato et Saito等人sur l'image du group de Chow ' z ' o-cycles de X ' s ' product ' m ' emes groups ' s les les compl 'et 'es de k. Lorsque G .半简单性'简单性' connexes et que le stabilisateur G 'eom 'etrique est fini et超可解性，现在，我想问一下X点的基本原理，我想问一下布劳尔-马宁的集合。把两个基团倒出超可溶物，更具体地说，把两个基团倒出零势物，把两个基团倒出零势物，把两个基团倒出，把两个基团倒出，把两个基团倒出，把两个基团倒出，把两个基团倒出，把两个基团倒出。
{"title":"Zéro-cycles sur les espaces homogènes et problème de Galois inverse","authors":"Yonatan Harpaz, Olivier Wittenberg","doi":"10.1090/jams/943","DOIUrl":"https://doi.org/10.1090/jams/943","url":null,"abstract":"Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Th'el`ene, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the inverse Galois problem, and solves, at the same time, Grunwald's problem, for these groups. \u0000----- \u0000Soit X une compactification lisse d'un espace homog`ene d'un groupe alg'ebrique lin'eaire G sur un corps de nombres k. Nous 'etablissons la conjecture de Colliot-Th'el`ene, Sansuc, Kato et Saito sur l'image du groupe de Chow des z'ero-cycles de X dans le produit des m^emes groupes sur tous les compl'et'es de k. Lorsque G est semi-simple et simplement connexe et que le stabilisateur g'eom'etrique est fini et hyper-r'esoluble, nous montrons que les points rationnels de X sont denses dans l'ensemble de Brauer-Manin. Pour les groupes finis hyper-r'esolubles, en particulier pour les groupes finis nilpotents, cela donne une nouvelle preuve du th'eor`eme de Shafarevich sur le probl`eme de Galois inverse et r'esout en m^eme temps, pour ces groupes, le probl`eme de Grunwald.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/943","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42336799","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}
<p>We give new examples of compact, negatively curved Einstein manifolds of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X_k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, <italic>Invent. Math.</italic> <bold>89</bold> (1987), no. 1, 1–12). The construction begins with a certain sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M_k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of hyperbolic four-manifolds, each containing a totally geodesic surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript k"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Sigma _k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is nullhomologous and whose normal injectivity radius tends to infinity with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For a fixed choice of natural number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we consider the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext
我们给出了4维负弯曲的紧致爱因斯坦流形的新例子。这些似乎是第一个不具有局部同质性的例子。我们的度量由四流形序列(xk) (X_k)承载，之前由Gromov和Thurston(双曲流形的捏紧常数，Invent)考虑。数学89(1987)，第1期。1、1 - 12)。构造从双曲四流形的一定序列(M k) (M_k)开始，每个序列包含一个完全测地曲面Σ k Sigma _k，该曲面是零同源的，其法向注入半径随k k趋于无穷大。对于一个固定选择的自然数l l，我们考虑l l折叠覆盖X k→M k X_k 到M_k沿Σ k Sigma _k分支。我们证明了对于任何ll的选择和所有足够大的k k(取决于ll)， X k X_k携带负截面曲率的爱因斯坦度规。证明的第一步是在X k X_k上找到一个近似的爱因斯坦度规，这是通过在分支轨迹附近的模型爱因斯坦度规和从M k M_k的双曲度规的回拉之间进行插值来完成的。证明的第二步是通过反函数定理的参数依赖版本，将其扰动为爱因斯坦方程的真正解。分析依赖于基于l2 L^2矫顽力估计的精细自举程序。
{"title":"Examples of compact Einstein four-manifolds with negative curvature","authors":"J. Fine, Bruno Premoselli","doi":"10.1090/jams/944","DOIUrl":"https://doi.org/10.1090/jams/944","url":null,"abstract":"<p>We give new examples of compact, negatively curved Einstein manifolds of dimension <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>. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X Subscript k Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X_k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, <italic>Invent. Math.</italic> <bold>89</bold> (1987), no. 1, 1–12). The construction begins with a certain sequence <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M Subscript k Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(M_k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of hyperbolic four-manifolds, each containing a totally geodesic surface <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma Subscript k\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Σ<!-- Σ --></mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">Sigma _k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which is nullhomologous and whose normal injectivity radius tends to infinity with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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>. For a fixed choice of natural number <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"l\">\u0000 <mml:semantics>\u0000 <mml:mi>l</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">l</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we consider the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/944","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48590677","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}
We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of “homological conjectures” in commutative algebra; namely, for any local homomorphism R → R ′ Rto R’ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay R R -algebra and some Cohen-Macaulay R ′ R’ -algebra. When R R contains a field, this is already known. When R R is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz’s refined treatment of weak functoriality of Cohen-Macaulay algebras in characteristic p p ; in fact, developing a “tilting argument” due to K. Shimomoto, we combine the perfectoid techniques of the author’s earlier work with Dietz’s result.
{"title":"Weak functoriality of Cohen-Macaulay algebras","authors":"Y. Andre","doi":"10.1090/jams/937","DOIUrl":"https://doi.org/10.1090/jams/937","url":null,"abstract":"We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of “homological conjectures” in commutative algebra; namely, for any local homomorphism \u0000\u0000 \u0000 \u0000 R\u0000 →\u0000 \u0000 R\u0000 ′\u0000 \u0000 \u0000 Rto R’\u0000 \u0000\u0000 of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay \u0000\u0000 \u0000 R\u0000 R\u0000 \u0000\u0000-algebra and some Cohen-Macaulay \u0000\u0000 \u0000 \u0000 R\u0000 ′\u0000 \u0000 R’\u0000 \u0000\u0000-algebra.\u0000\u0000When \u0000\u0000 \u0000 R\u0000 R\u0000 \u0000\u0000 contains a field, this is already known. When \u0000\u0000 \u0000 R\u0000 R\u0000 \u0000\u0000 is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz’s refined treatment of weak functoriality of Cohen-Macaulay algebras in characteristic \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000; in fact, developing a “tilting argument” due to K. Shimomoto, we combine the perfectoid techniques of the author’s earlier work with Dietz’s result.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/937","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42450319","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}
<p>We study the category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis script upper O right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(mathcal {O})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant perverse coherent sheaves on the affine Grassmannian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal r Subscript upper G"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">mathrm {Gr}_G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding.</p> <p>We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In particular, they can be used to deduce the existence of (categorified) cluster structures. We study the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals upper G upper L Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G = GL_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in detail and prove that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G Subscript m"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">mathbb {G}_m</
我们研究了仿射Grassmannian G r Gmathrm上G（O）G（mathcal｛O｝）-等变反常相干簇的范畴{Gr}_G。这个相干Satake范畴不是半单的，它的卷积积也不是对称的，与通常的可构造Satake类别相反。相反，我们使用Beilinson-Drinfeld-Grassmannian来构造重整化r-矩阵。这些是卷积乘积之间的正则非零映射，其满足比编织的公理弱的公理。我们还证明了相干Satake范畴是刚性的，这些结果加在一起强烈约束了它的卷积结构。特别地，它们可以用来推断（已分类的）簇结构的存在。我们详细研究了G=GLnG=GL_n的情形，并证明了Gmmathbb{G}_mGL_n的等变相干Satake范畴是显式量子簇代数的一个单oid范畴。更一般地说，我们在乘积与辅助手性范畴相容的任何单oid范畴中构造了重正化r-矩阵，并从这个角度解释了如何理解簇代数在4d N=2mathemical｛N｝=2场论中的出现。
{"title":"Cluster theory of the coherent Satake category","authors":"Sabin Cautis, H. Williams","doi":"10.1090/JAMS/918","DOIUrl":"https://doi.org/10.1090/JAMS/918","url":null,"abstract":"<p>We study the category of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G left-parenthesis script upper O right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">G(mathcal {O})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivariant perverse coherent sheaves on the affine Grassmannian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper G normal r Subscript upper G\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">G</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">r</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathrm {Gr}_G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\">\u0000 <mml:semantics>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">r</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding.</p>\u0000\u0000<p>We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In particular, they can be used to deduce the existence of (categorified) cluster structures. We study the case <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G equals upper G upper L Subscript n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">G = GL_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in detail and prove that the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper G Subscript m\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {G}_m</","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/918","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49643672","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}
We prove the test function conjecture of Kottwitz and the first-named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.
{"title":"The test function conjecture for parahoric local models","authors":"T. Haines, Timo Richarz","doi":"10.1090/jams/955","DOIUrl":"https://doi.org/10.1090/jams/955","url":null,"abstract":"We prove the test function conjecture of Kottwitz and the first-named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43784633","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}
Let kk be an uncountable field of characteristic different from two. We show that a very general hypersurface X⊂PkN+1Xsubset mathbb {P}^{N+1}_k of dimension N≥3Ngeq 3 and degree at least log2N+2log _2N +2 is not stably rational over the algebraic closure of kk.
设k k是一个不可数的具有不同于二的特征的域。我们证明了一个非常一般的超曲面X⊂P k N+1 Xsubet mathbb｛P｝^｛N+1｝_k的维数N≥3Ngeq3且度至少为log2 N+2log_2N+2在k k的代数闭包上是不稳定有理的。
{"title":"Stably irrational hypersurfaces of small slopes","authors":"Stefan Schreieder","doi":"10.1090/jams/928","DOIUrl":"https://doi.org/10.1090/jams/928","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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> be an uncountable field of characteristic different from two. We show that a very general hypersurface <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 Subscript k Superscript upper N plus 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>k</mml:mi>\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:msubsup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Xsubset mathbb {P}^{N+1}_k</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=\"upper N greater-than-or-equal-to 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\">Ngeq 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and degree at least <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"log Subscript 2 Baseline upper N plus 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">log _2N +2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is not stably rational over the algebraic closure of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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>.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/928","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49661592","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}
<p>In this paper, we apply our previous estimates in Chen and Cheng [<italic>On the constant scalar curvature Kähler metrics (I): a priori estimates</italic>, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-energy in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> geodesic distance <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d 1"> <mml:semantics> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">d_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-energy in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper E Superscript 1 Baseline comma d 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(mathcal {E}^1, d_1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> geodesic ray where the <inline-formula content-type="math/mathm
{"title":"On the constant scalar curvature Kähler metrics (II)—Existence results","authors":"Xiuxiong Chen, Jingrui Cheng","doi":"10.1090/jams/966","DOIUrl":"https://doi.org/10.1090/jams/966","url":null,"abstract":"<p>In this paper, we apply our previous estimates in Chen and Cheng [<italic>On the constant scalar curvature Kähler metrics (I): a priori estimates</italic>, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness 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>-energy in terms of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> geodesic distance <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">d_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the <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>-energy in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis script upper E Superscript 1 Baseline comma d 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">E</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(mathcal {E}^1, d_1)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> geodesic ray where the <inline-formula content-type=\"math/mathm","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2018-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47479415","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}
In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a C 0 C^0 bound for the Kähler potential.
{"title":"On the constant scalar curvature Kähler metrics (I)—A priori estimates","authors":"Xiuxiong Chen, Jingrui Cheng","doi":"10.1090/jams/967","DOIUrl":"https://doi.org/10.1090/jams/967","url":null,"abstract":"In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a \u0000\u0000 \u0000 \u0000 C\u0000 0\u0000 \u0000 C^0\u0000 \u0000\u0000 bound for the Kähler potential.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41965272","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}
We prove that a complete Kähler manifold with holomorphic curvature bounded between two negative constants admits a unique complete Kähler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kähler metric. Furthermore, all three metrics are shown to be uniformly equivalent to the Berg- man metric, if the complete Kähler manifold is simply-connected, with the sectional curvature bounded between two negative constants. In particular, we confirm two conjectures of R. E. Greene and H. Wu posted in 1979.
{"title":"Invariant metrics on negatively pinched complete Kähler manifolds","authors":"Damin Wu, S. Yau","doi":"10.1090/jams/933","DOIUrl":"https://doi.org/10.1090/jams/933","url":null,"abstract":"We prove that a complete Kähler manifold with holomorphic curvature bounded between two negative constants admits a unique complete Kähler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kähler metric. Furthermore, all three metrics are shown to be uniformly equivalent to the Berg- man metric, if the complete Kähler manifold is simply-connected, with the sectional curvature bounded between two negative constants. In particular, we confirm two conjectures of R. E. Greene and H. Wu posted in 1979.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/933","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46913598","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}
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an exceptional simple algebraic group over an algebraically closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals c h a r left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>char</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p={operatorname {char}}(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a good prime for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we classify the maximal Lie subalgebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German m"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathfrak {m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g equals upper L i e left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Lie</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathfrak {g}=operatorname {Lie}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Specifically, we show that either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German m equals upper L i e left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvarian
{"title":"Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic","authors":"A. Premet, David I. Stewart","doi":"10.1090/JAMS/926","DOIUrl":"https://doi.org/10.1090/JAMS/926","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be an exceptional simple algebraic group over an algebraically closed field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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> and suppose that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p equals c h a r left-parenthesis k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>char</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p={operatorname {char}}(k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a good prime for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this paper we classify the maximal Lie subalgebras <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German m\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">m</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathfrak {m}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of the Lie algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German g equals upper L i e left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">g</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>Lie</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathfrak {g}=operatorname {Lie}(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Specifically, we show that either <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German m equals upper L i e left-parenthesis upper M right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvarian","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/926","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42391043","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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