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On positivity of the CM line bundle on K-moduli spaces 论k模空间上CM线束的正性
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-12-01 DOI: 10.4007/annals.2020.192.3.7
Chenyang Xu, Ziquan Zhuang
{"title":"On positivity of the CM line bundle on K-moduli spaces","authors":"Chenyang Xu, Ziquan Zhuang","doi":"10.4007/annals.2020.192.3.7","DOIUrl":"https://doi.org/10.4007/annals.2020.192.3.7","url":null,"abstract":"In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46527886","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}
引用次数: 49
Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios 平滑Jordan曲线中的内接矩形至少达到所有纵横比的三分之一
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-11-17 DOI: 10.4007/annals.2021.194.2.3
Cole Hugelmeyer
{"title":"Inscribed rectangles in a smooth Jordan curve attain at least one\u0000 third of all aspect ratios","authors":"Cole Hugelmeyer","doi":"10.4007/annals.2021.194.2.3","DOIUrl":"https://doi.org/10.4007/annals.2021.194.2.3","url":null,"abstract":"We prove that for every smooth Jordan curve $gamma$, if $X$ is the set of all $r in [0,1]$ so that there is an inscribed rectangle in $gamma$ of aspect ratio $tan(rcdot pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study disjoint Mobius strips bounding a $(2n,n)$-torus link in the solid torus times an interval. We prove that any such set of Mobius strips can be equipped with a natural total ordering. We then combine this total ordering with some additive combinatorics to prove that $1/3$ is a sharp lower bound on the probability that a Mobius strip bounding the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44833876","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}
引用次数: 11
Rough solutions of the 3-D compressible Euler equations 三维可压缩欧拉方程的粗糙解
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-11-12 DOI: 10.4007/annals.2022.195.2.3
Qian Wang
{"title":"Rough solutions of the 3-D compressible Euler equations","authors":"Qian Wang","doi":"10.4007/annals.2022.195.2.3","DOIUrl":"https://doi.org/10.4007/annals.2022.195.2.3","url":null,"abstract":"We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,varrho, fw) in H^stimes H^stimes H^{s'}$, $22$. Due to the works of Smith-Tataru and Wang for the irrotational isentropic case, the local well-posedness can be achieved if the data satisfy $v, varrho in H^{s}$, with $s>2$. In the incompressible case the solution is proven to be ill-posed for the datum $fwin H^frac{3}{2}$ by Bourgain-Li. The solution of the compressible Euler equations is not expected to be well-posed if the data merely satisfy $v, varrhoin H^{s}, s>2$ with a general rough vorticity. \u0000By decomposing the velocity into the term $(I-Delta_e)^{-1}curl fw$ and a wave function verifying an improved wave equation, with a series of cancellations for treating the latter, we achieve the $H^s$-energy bound and complete the linearization for the wave functions by using the $H^{s-f12}, , s>2$ norm for the vorticity. The propagation of energy for the vorticity typically requires $curl fwin C^{0, 0+}$ initially, stronger than our assumption by 1/2-derivative. We perform trilinear estimates to gain regularity by observing a div-curl structure when propagating the energy of the normalized double-curl of the vorticity, and also by spacetime integration by parts. To prove the Strichartz estimate for the linearized wave in the rough spacetime, we encounter a strong Ricci defect requiring the bound of $|curl fw|_{L_x^infty L_t^1}$ on null cones. This difficulty is solved by uncovering the cancellation structures due to the acoustic metric on the angular derivatives of Ricci and the second fundamental form.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44289686","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}
引用次数: 12
Unitriangular shape of decomposition matrices of unipotent blocks 幂块分解矩阵的单角形状
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-10-19 DOI: 10.4007/annals.2020.192.2.7
Olivier Brunat, O. Dudas, Jay Taylor
{"title":"Unitriangular shape of decomposition matrices of unipotent\u0000 blocks","authors":"Olivier Brunat, O. Dudas, Jay Taylor","doi":"10.4007/annals.2020.192.2.7","DOIUrl":"https://doi.org/10.4007/annals.2020.192.2.7","url":null,"abstract":"We show that the decomposition matrix of unipotent $ell$-blocks of a finite reductive group $mathbf{G}(mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $ell$ is very good for $mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44856185","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}
引用次数: 16
Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra Leech格顶点算子代数的维数公式和广义深孔
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-10-11 DOI: 10.4007/annals.2023.197.1.4
S. Møller, Nils R. Scheithauer
{"title":"Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra","authors":"S. Møller, Nils R. Scheithauer","doi":"10.4007/annals.2023.197.1.4","DOIUrl":"https://doi.org/10.4007/annals.2023.197.1.4","url":null,"abstract":"We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $operatorname{Aut}(V)$. \u0000We then give a construction of all 70 strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space as orbifolds of the Leech lattice vertex operator algebra $V_Lambda$ associated with generalised deep holes. This provides the first uniform construction of these vertex operator algebras and naturally generalises the construction of the 23 Niemeier lattices with non-vanishing root system from the deep holes of the Leech lattice $Lambda$ by Conway, Parker and Sloane.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43860069","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}
引用次数: 18
A proof of Carleson's 𝜀2-conjecture 证明了卡尔森的𝜀2-conjecture
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-09-18 DOI: 10.4007/annals.2021.194.1.2
B. Jaye, X. Tolsa, Michele Villa
{"title":"A proof of Carleson's 𝜀2-conjecture","authors":"B. Jaye, X. Tolsa, Michele Villa","doi":"10.4007/annals.2021.194.1.2","DOIUrl":"https://doi.org/10.4007/annals.2021.194.1.2","url":null,"abstract":"In this paper we provide a proof of the Carleson $varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $varepsilon^2$-square function.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43852231","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}
引用次数: 5
Weil representation and Arithmetic Fundamental Lemma Weil表示与算术基本引理
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-09-06 DOI: 10.4007/ANNALS.2021.193.3.5
Wei Zhang
{"title":"Weil representation and Arithmetic Fundamental Lemma","authors":"Wei Zhang","doi":"10.4007/ANNALS.2021.193.3.5","DOIUrl":"https://doi.org/10.4007/ANNALS.2021.193.3.5","url":null,"abstract":"By a global approach, we prove the arithmetic fundamental lemma conjecture for unitary groups in $n$ variables over $mathbb{Q}_p$ when $pgeq n$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41529540","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}
引用次数: 28
The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus 包含环面的仿射log Calabi-Yau变种的Frobenius结构定理
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-08-26 DOI: 10.4007/annals.2023.198.2.1
S. Keel, Tony Yue Yu
{"title":"The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus","authors":"S. Keel, Tony Yue Yu","doi":"10.4007/annals.2023.198.2.1","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.1","url":null,"abstract":"We show that the naive counts of rational curves in any affine log Calabi-Yau variety $U$, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a compatible multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of $U$. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when $U$ is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42898187","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}
引用次数: 22
Enumerating number fields 枚举数字字段
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-07-31 DOI: 10.4007/annals.2020.192.2.4
J. Couveignes
{"title":"Enumerating number fields","authors":"J. Couveignes","doi":"10.4007/annals.2020.192.2.4","DOIUrl":"https://doi.org/10.4007/annals.2020.192.2.4","url":null,"abstract":"We construct small models of number fields and deduce a better bound for the number of number fields of given degree and bounded discriminant.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44709404","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}
引用次数: 17
Equiangular lines with a fixed angle 具有固定角度的等角线
IF 4.9 1区 数学
Annals of Mathematics Pub Date : 2019-07-29 DOI: 10.4007/annals.2021.194.3.3
Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao
{"title":"Equiangular lines with a fixed angle","authors":"Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, Yufei Zhao","doi":"10.4007/annals.2021.194.3.3","DOIUrl":"https://doi.org/10.4007/annals.2021.194.3.3","url":null,"abstract":"Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. \u0000Fix $0 < alpha < 1$. Let $N_alpha(d)$ denote the maximum number of lines in $mathbb{R}^d$ with pairwise common angle $arccos alpha$. Let $k$ denote the minimum number (if it exists) of vertices of a graph whose adjacency matrix has spectral radius exactly $(1-alpha)/(2alpha)$. If $k < infty$, then $N_alpha(d) = lfloor k(d-1)/(k-1) rfloor$ for all sufficiently large $d$, and otherwise $N_alpha(d) = d + o(d)$. In particular, $N_{1/(2k-1)}(d) = lfloor k(d-1)/(k-1) rfloor$ for every integer $kgeq 2$ and all sufficiently large $d$. \u0000A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41497075","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}
引用次数: 43
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