For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group Gmathsf {G} with compact, smooth orbit space, we show that the nilradical Nmathsf {N} of Gmathsf {G} acts polarly and that the Nmathsf {N}-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.
{"title":"Non-compact Einstein manifolds with symmetry","authors":"Christoph Böhm, Ramiro A. Lafuente","doi":"10.1090/jams/1022","DOIUrl":"https://doi.org/10.1090/jams/1022","url":null,"abstract":"For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper G\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathsf {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with compact, smooth orbit space, we show that the nilradical <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper N\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathsf {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper G\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathsf {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts polarly and that the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper N\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathsf {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135533619","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}
Olivier Debarre, Daniel Huybrechts, Emanuele Macrì, Claire Voisin
We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3[2]^{[2]} deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3[2]^{[2]} numerical type are of K3[2]^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3[2]^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.
{"title":"Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds","authors":"Olivier Debarre, Daniel Huybrechts, Emanuele Macrì, Claire Voisin","doi":"10.1090/jams/1016","DOIUrl":"https://doi.org/10.1090/jams/1016","url":null,"abstract":"We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript left-bracket 2 right-bracket\"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{[2]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript left-bracket 2 right-bracket\"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{[2]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> numerical type are of K3<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript left-bracket 2 right-bracket\"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{[2]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Superscript left-bracket 2 right-bracket\"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">^{[2]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135339733","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}
Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torus, i.e. <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row partial-differential Subscript t Baseline upper F left-parenthesis t comma x comma v right-parenthesis plus v Subscript i Baseline partial-differential Subscript x Sub Subscript i Subscript Baseline upper F left-parenthesis t comma x comma v right-parenthesis plus upper E Subscript i Baseline left-parenthesis t comma x right-parenthesis partial-differential Subscript v Sub Subscript i Subscript Baseline upper F left-parenthesis t comma x comma v right-parenthesis equals nu upper Q left-parenthesis upper F comma upper F right-parenthesis left-parenthesis t comma x comma v right-parenthesis comma 2nd Row upper E left-parenthesis t comma x right-parenthesis equals nabla normal upper Delta Superscript negative 1 Baseline left-parenthesis integral Underscript double-struck upper R cubed Endscripts upper F left-parenthesis t comma x comma v right-parenthesis normal d v minus integral minus Subscript double-struck upper T cubed Baseline integral Underscript double-struck upper R cubed Endscripts upper F left-parenthesis t comma x comma v right-parenthesis normal d v normal d x right-parenthesis comma EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> <m
考虑在33 -环面弱碰撞区具有库仑势的vlasov -泊松-朗道系统，即∂t F (t, x, v) +∂x i F (t, x, v) + E i (t, x)∂v i F (t, x, v) = ν Q (F, F) (t, x, v)， E (t, x) =∇Δ−1(∫R 3f (t, x, v) d v -∫R 3f (t, x, v) d v) begin{align*} partial _t F(t,x,v) + v_i partial _{x_i} F(t,x,v) + E_i(t,x) partial _{v_i} F(t,x,v) = nu Q(F,F)(t,x,v), E(t,x) = nabla Delta ^{-1} (int _{mathbb R^3} F(t,x,v), mathrm {d} v - {{int }llap {-}}_{mathbb T^3} int _{mathbb R^3} F(t,x,v), mathrm {d} v , mathrm {d} x), end{align*}， ν≪1 null 1。我们证明对于λ ＆gt;0 epsilon ＆gt;0足够小(但独立于ν nu)，初始数据为O(λ ν 1/3) O(epsilonnu ^1/3) -来自全局麦克斯韦方程组的sobolev空间摄动导致全局实时解收敛到全局麦克斯韦方程组为t→∞t {}toinfty。解具有均匀的ν nu朗道阻尼和增强的耗散。我们的主要结果类似于Bedrossian对具有相同阈值的Vlasov-Poisson-Fokker-Planck方程的早期结果。然而，与Fokker-Planck情况不同的是，由于朗道碰撞算子的复杂性，线性算子不能显式地反转。为此，我们开发了一个基于能量的框架，该框架将郭的加权能量方法与低强制能量方法和交换向量场方法相结合。该证明还依赖于线性化密度方程的逐点解析估计。
{"title":"The Vlasov–Poisson–Landau system in the weakly collisional regime","authors":"Sanchit Chaturvedi, Jonathan Luk, Toan T. Nguyen","doi":"10.1090/jams/1014","DOIUrl":"https://doi.org/10.1090/jams/1014","url":null,"abstract":"Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torus, i.e. <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartLayout 1st Row partial-differential Subscript t Baseline upper F left-parenthesis t comma x comma v right-parenthesis plus v Subscript i Baseline partial-differential Subscript x Sub Subscript i Subscript Baseline upper F left-parenthesis t comma x comma v right-parenthesis plus upper E Subscript i Baseline left-parenthesis t comma x right-parenthesis partial-differential Subscript v Sub Subscript i Subscript Baseline upper F left-parenthesis t comma x comma v right-parenthesis equals nu upper Q left-parenthesis upper F comma upper F right-parenthesis left-parenthesis t comma x comma v right-parenthesis comma 2nd Row upper E left-parenthesis t comma x right-parenthesis equals nabla normal upper Delta Superscript negative 1 Baseline left-parenthesis integral Underscript double-struck upper R cubed Endscripts upper F left-parenthesis t comma x comma v right-parenthesis normal d v minus integral minus Subscript double-struck upper T cubed Baseline integral Underscript double-struck upper R cubed Endscripts upper F left-parenthesis t comma x comma v right-parenthesis normal d v normal d x right-parenthesis comma EndLayout\"> <mml:semantics> <mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" side=\"left\" displaystyle=\"true\"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>F</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> <m","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136266828","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}
Abstract:In this article we prove a conjecture of Braverman-Kazhdan in [Geom. Funct. Anal. Special Volume (2000), pp. 237–278] on acyclicity of $rho$-Bessel sheaves on reductive groups. We do so by proving a vanishing conjecture proposed in our previous work [A vanishing conjecture: the GLn case, arXiv:1902.11190]. As a corollary, we obtain a geometric construction of the non-linear Fourier kernel for a finite reductive group as conjectured by Braverman and Kazhdan. The proof uses the theory of Mellin transforms, Drinfeld center of Harish-Chandra bimodules, and a construction of a class of character sheaves in mixed-characteristic.
{"title":"On a conjecture of Braverman-Kazhdan","authors":"Tsao-Hsien Chen","doi":"10.1090/jams/992","DOIUrl":"https://doi.org/10.1090/jams/992","url":null,"abstract":"Abstract:In this article we prove a conjecture of Braverman-Kazhdan in [Geom. Funct. Anal. Special Volume (2000), pp. 237–278] on acyclicity of $rho$-Bessel sheaves on reductive groups. We do so by proving a vanishing conjecture proposed in our previous work [A vanishing conjecture: the GLn case, arXiv:1902.11190]. As a corollary, we obtain a geometric construction of the non-linear Fourier kernel for a finite reductive group as conjectured by Braverman and Kazhdan. The proof uses the theory of Mellin transforms, Drinfeld center of Harish-Chandra bimodules, and a construction of a class of character sheaves in mixed-characteristic. <hr align=\"left\" noshade=\"noshade\" width=\"200\"/>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"47 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516414","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}
Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiring X mathcal {X} of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements of X mathcal {X} in terms of the asymptotic spectrum of X mathcal {X} , which is defined as the collection of semiring homomorphisms from X mathcal {X} to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information.Universal spectral points are elements in the asymptotic spectrum of the semiring of all tensors. Finding all universal spectral points suffices to find the asymptotic spectrum of any sub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, called quantum functionals. We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes.
{"title":"Universal points in the asymptotic spectrum of tensors","authors":"Matthias Christandl,Péter Vrana,Jeroen Zuiddam","doi":"10.1090/jams/996","DOIUrl":"https://doi.org/10.1090/jams/996","url":null,"abstract":"Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiring X mathcal {X} of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements of X mathcal {X} in terms of the asymptotic spectrum of X mathcal {X} , which is defined as the collection of semiring homomorphisms from X mathcal {X} to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information.Universal spectral points are elements in the asymptotic spectrum of the semiring of all tensors. Finding all universal spectral points suffices to find the asymptotic spectrum of any sub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, called quantum functionals. We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"21 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516415","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}
Abstract:Let $X$ be an affine spherical variety, possibly singular, and $mathsf L^+X$ its arc space. The intersection complex of $mathsf L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified $L$-functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and $L$-monoids. In this paper, we compute this intersection complex for the large class of those spherical $G$-varieties whose dual group is equal to $check G$, and the stalks of its nearby cycles on the horospherical degeneration of $X$. We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional $check G$-representation determined by the set of $B$-invariant valuations on $X$. We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of $mathsf L^+X$ as a ratio of local $L$-values for a large class of spherical varieties.
{"title":"Intersection complexes and unramified 𝐿-factors","authors":"Yiannis Sakellaridis, Jonathan Wang","doi":"10.1090/jams/990","DOIUrl":"https://doi.org/10.1090/jams/990","url":null,"abstract":"Abstract:Let $X$ be an affine spherical variety, possibly singular, and $mathsf L^+X$ its arc space. The intersection complex of $mathsf L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified $L$-functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and $L$-monoids. In this paper, we compute this intersection complex for the large class of those spherical $G$-varieties whose dual group is equal to $check G$, and the stalks of its nearby cycles on the horospherical degeneration of $X$. We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional $check G$-representation determined by the set of $B$-invariant valuations on $X$. We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of $mathsf L^+X$ as a ratio of local $L$-values for a large class of spherical varieties. <hr align=\"left\" noshade=\"noshade\" width=\"200\"/>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"22 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516421","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 axiomatise the theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal infinity comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories. We prove that the space of theories of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal infinity comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis double-struck upper Z slash 2 right-parenthesis Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B(mathbb {Z}/2)^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that Rezk’s complete Segal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Theta Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Θ<!-- Θ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Theta _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces, Simpson and Tamsamani’s Segal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories, the first author’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</m
我们公理化了（∞，n）（infty，n）-范畴的理论。我们证明了（∞，n）（infty，n）-范畴的理论空间是一个B（Z/2）nB（mathbb｛Z｝/2）^n。我们证明了Rezk的完全SegalΘnTheta _n空间，Simpson和Tamsamani的Segal n n-范畴，第一作者的n n-折叠完全Segal空间，Kan和第一作者的nn-相对范畴，和（∞，n−1）（infty，n-1）-范畴的任何模型中的完全分段空间对象都满足我们的公理。因此，这些理论都是等价的，在（Z/2）n（mathbb｛Z｝/2）^n的作用下是唯一的。
{"title":"On the unicity of the theory of higher categories","authors":"C. Barwick, Christopher J. Schommer-Pries","doi":"10.1090/JAMS/972","DOIUrl":"https://doi.org/10.1090/JAMS/972","url":null,"abstract":"<p>We axiomatise the theory of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis normal infinity comma n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories. We prove that the space of theories of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis normal infinity comma n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories is a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B left-parenthesis double-struck upper Z slash 2 right-parenthesis Superscript n\"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B(mathbb {Z}/2)^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that Rezk’s complete Segal <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript n\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Θ<!-- Θ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">Theta _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces, Simpson and Tamsamani’s Segal <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories, the first author’s <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</m","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":"1"},"PeriodicalIF":3.9,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49206433","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}
Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K to mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic rings. In addition, we show that $K$-theory with finite coefficients satisfies continuity for complete noetherian rings which are $F$-finite modulo $p$. Our main new ingredient is a basic finiteness property of $mathrm{TC}$ with finite coefficients.
{"title":"$K$-theory and topological cyclic homology of Henselian pairs","authors":"Dustin Clausen, Akhil Mathew, Matthew Morrow","doi":"10.1090/jams/961","DOIUrl":"https://doi.org/10.1090/jams/961","url":null,"abstract":"Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K to mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic rings. In addition, we show that $K$-theory with finite coefficients satisfies continuity for complete noetherian rings which are $F$-finite modulo $p$. Our main new ingredient is a basic finiteness property of $mathrm{TC}$ with finite coefficients.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"22 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516417","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 introduce a notion of complexity of a complex of ℓ ell -adic sheaves on a quasi-projective variety and prove that the six operations are “continuous”, in the sense that the complexity of the output sheaves is bounded solely in terms of the complexity of the input sheaves. A key feature of complexity is that it provides bounds for the sum of Betti numbers that, in many interesting cases, can be made uniform in the characteristic of the base field. As an illustration, we discuss a few simple applications to horizontal equidistribution results for exponential sums over finite fields.
{"title":"Quantitative sheaf theory","authors":"W. Sawin, A. Forey, J. Fres'an, E. Kowalski","doi":"10.1090/jams/1008","DOIUrl":"https://doi.org/10.1090/jams/1008","url":null,"abstract":"We introduce a notion of complexity of a complex of \u0000\u0000 \u0000 ℓ\u0000 ell\u0000 \u0000\u0000-adic sheaves on a quasi-projective variety and prove that the six operations are “continuous”, in the sense that the complexity of the output sheaves is bounded solely in terms of the complexity of the input sheaves. A key feature of complexity is that it provides bounds for the sum of Betti numbers that, in many interesting cases, can be made uniform in the characteristic of the base field. As an illustration, we discuss a few simple applications to horizontal equidistribution results for exponential sums over finite fields.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42065496","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}
{"title":"Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime","authors":"G. Fournodavlos, I. Rodnianski, Jared Speck","doi":"10.1090/jams/1015","DOIUrl":"https://doi.org/10.1090/jams/1015","url":null,"abstract":"<p>For <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis t comma x right-parenthesis element-of left-parenthesis 0 comma normal infinity right-parenthesis times double-struck upper T Superscript German upper D\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\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:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">D</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(t,x) in (0,infty )times mathbb {T}^{mathfrak {D}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t down-arrow 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">↓<!-- ↓ --></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">t downarrow 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q overTilde Subscript 1 Baseline comma midline-horizontal-ellipsis comma q overTilde Subscript German upper D Baseline element-of double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mo>⋯<!-- ⋯ --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 ","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44771248","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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