Pub Date : 2024-05-10DOI: 10.4310/acta.2024.v232.n1.a2
Jeremy Kahn, François Labourie, Mozes Shahar
We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are Hölder. Using this notion, we show a quantitative version of our surface subgroup theorem, and in particular that one can obtain $K$-Sullivan limit maps, as close as one wants to smooth round circles. All these results use the coarse geometry of “path of triangles” in a certain flag manifold, and we prove an analogue to the Morse Lemma for quasi-geodesics in that context.
{"title":"Surface groups in uniform lattices of some semi-simple groups","authors":"Jeremy Kahn, François Labourie, Mozes Shahar","doi":"10.4310/acta.2024.v232.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2024.v232.n1.a2","url":null,"abstract":"We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are Hölder. Using this notion, we show a quantitative version of our surface subgroup theorem, and in particular that one can obtain $K$-Sullivan limit maps, as close as one wants to smooth round circles. All these results use the coarse geometry of “path of triangles” in a certain flag manifold, and we prove an analogue to the Morse Lemma for quasi-geodesics in that context.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-10DOI: 10.4310/acta.2024.v232.n1.a1
James Gabe, Gábor Szabó
Let $G$ be a second-countable, locally compact group. In this article we study amenable $G$-actions on Kirchberg algebras that admit an approximately central embedding of a canonical quasi-free action on the Cuntz algebra $mathcal{O}_{^infty}$. If $G$ is discrete, this coincides with the class of amenable and outer $G-$actions on Kirchberg algebras. We show that the resulting $G-C^ast$-dynamical systems are classified by equivariant Kasparov theory, up to cocycle conjugacy. This is the first classification theory of its kind applicable to actions of arbitrary locally compact groups. Among various applications, our main result solves a conjecture of Izumi for actions of discrete amenable torsion-free groups, and recovers the main results of recent work by Izumi–Matui for actions of poly-$mathbb{Z}$ groups.
{"title":"The dynamical Kirchberg–Phillips theorem","authors":"James Gabe, Gábor Szabó","doi":"10.4310/acta.2024.v232.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2024.v232.n1.a1","url":null,"abstract":"Let $G$ be a second-countable, locally compact group. In this article we study amenable $G$-actions on Kirchberg algebras that admit an approximately central embedding of a canonical quasi-free action on the Cuntz algebra $mathcal{O}_{^infty}$. If $G$ is discrete, this coincides with the class of amenable and outer $G-$actions on Kirchberg algebras. We show that the resulting $G-C^ast$-dynamical systems are classified by equivariant Kasparov theory, up to cocycle conjugacy. This is the first classification theory of its kind applicable to actions of arbitrary locally compact groups. Among various applications, our main result solves a conjecture of Izumi for actions of discrete amenable torsion-free groups, and recovers the main results of recent work by Izumi–Matui for actions of poly-$mathbb{Z}$ groups.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.4310/acta.2023.v231.n2.a3
Karl-Theodor Sturm
This is a correction to $href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (Acta Math.), as well as to the follow-up publications $href{https://doi.org/10.1016/j.jfa.2010.03.024}{[3]}$ and $href{https://doi.org/10.1016/j.jfa.2011.02.026}{[5]}$ (both in J. Funct. Anal.).
这是对 $href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (Acta Math.) 以及后续出版物 $href{https://doi.org/10.1016/j.jfa.2010.03.024}{[3]}$ 和 $href{https://doi.org/10.1016/j.jfa.2011.02.026}{[5]}$ (both in J. Funct. Anal.) 的更正。
{"title":"Correction to “On the geometry of metric measure spaces. I”","authors":"Karl-Theodor Sturm","doi":"10.4310/acta.2023.v231.n2.a3","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n2.a3","url":null,"abstract":"This is a correction to $href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (<i>Acta Math.</i>), as well as to the follow-up publications $href{https://doi.org/10.1016/j.jfa.2010.03.024}{[3]}$ and $href{https://doi.org/10.1016/j.jfa.2011.02.026}{[5]}$ (both in <i>J. Funct. Anal.</i>).","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.4310/acta.2023.v231.n2.a1
Robert Burklund, Jeremy Hahn, Andrew Senger
Building on work of Stolz, we prove for integers $0 leqslant d leqslant 3$ and $k gt 232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal–Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal–Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its $mathrm{H}mathbb{F}_p$-Adams filtrations for all primes $p$. We additionally prove new vanishing lines in the $mathrm{H}mathbb{F}_p$-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in $mathrm{BP}{langle n rangle}$-based Adams spectral sequences.
在斯托尔兹工作的基础上，我们证明了对于整数$0 leqslant d leqslant 3$和$k gt 232$，$(k-1)$连接的、几乎封闭的$(2k+d)$流形的边界也是可平行流形的边界。在有限多维之外，这解决了华尔（C.T.C. Wall）的长期问题，确定了所有斯坦因可填充同调球，并证明了加拉蒂乌斯和兰道尔-威廉姆斯的猜想。这对高连接流形的分类以及通过克雷克和克兰尼奇的工作计算其映射类群都有意义。我们的技术是用某个托达括号的消失来重构加拉蒂乌斯和兰道尔-威廉斯猜想，然后通过限定所有素数 $p$ 的 $mathrm{H}mathbb{F}_p$-Adams 滤波来分析这个托达括号。此外，我们还证明了球面和摩尔谱的 $mathrm{H}mathbb{F}_p$-Adams 谱序列中的新消失线，这些消失线很可能具有独立的意义。其中几条消失线依赖于罗伯特-伯克伦（Robert Burklund）的附录，该附录回答了马修关于基于 $mathrm{BP}{langle n rangle}$ 的亚当斯谱序列中的消失曲线的问题。
{"title":"On the boundaries of highly connected, almost closed manifolds","authors":"Robert Burklund, Jeremy Hahn, Andrew Senger","doi":"10.4310/acta.2023.v231.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n2.a1","url":null,"abstract":"Building on work of Stolz, we prove for integers $0 leqslant d leqslant 3$ and $k gt 232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal–Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal–Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its $mathrm{H}mathbb{F}_p$-Adams filtrations for all primes $p$. We additionally prove new vanishing lines in the $mathrm{H}mathbb{F}_p$-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in $mathrm{BP}{langle n rangle}$-based Adams spectral sequences.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.4310/acta.2023.v231.n2.a2
Michael Hartz
In the theory of complete Pick spaces, the column-row property has appeared in a variety of contexts. We show that it is satisfied by every complete Pick space in the following strong form: each sequence of multipliers that induces a contractive column multiplication operator also induces a contractive row multiplication operator. In combination with known results, this yields a number of consequences. Firstly, we obtain multiple applications to the theory of weak product spaces, including factorization, multipliers and invariant subspaces. Secondly, there is a short proof of the characterization of interpolating sequences in terms of separation and Carleson measure conditions, independent of the solution of the Kadison–Singer problem. Thirdly, we find that in the theory of de Branges–Rovnyak spaces on the ball, the column-extreme multipliers of Jury and Martin are precisely the extreme points of the unit ball of the multiplier algebra.
在完全皮克空间理论中，列-行性质出现在不同的语境中。我们证明，每一个完整 Pick 空间都满足以下强形式：每一个诱导收缩列乘法算子的乘法序列也诱导收缩行乘法算子。结合已知结果，这将产生一系列结果。首先，我们获得了弱乘空间理论的多种应用，包括因式分解、乘法器和不变子空间。其次，我们还简短地证明了插值序列在分离和卡列松度量条件下的特征，这与卡迪森-辛格问题的解无关。第三，我们发现在球上的 de Branges-Rovnyak 空间理论中，Jury 和 Martin 的列极值乘数正是乘数代数单位球的极值点。
{"title":"Every complete Pick space satisfies the column-row property","authors":"Michael Hartz","doi":"10.4310/acta.2023.v231.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n2.a2","url":null,"abstract":"In the theory of complete Pick spaces, the column-row property has appeared in a variety of contexts. We show that it is satisfied by every complete Pick space in the following strong form: each sequence of multipliers that induces a contractive column multiplication operator also induces a contractive row multiplication operator. In combination with known results, this yields a number of consequences. Firstly, we obtain multiple applications to the theory of weak product spaces, including factorization, multipliers and invariant subspaces. Secondly, there is a short proof of the characterization of interpolating sequences in terms of separation and Carleson measure conditions, independent of the solution of the Kadison–Singer problem. Thirdly, we find that in the theory of de Branges–Rovnyak spaces on the ball, the column-extreme multipliers of Jury and Martin are precisely the extreme points of the unit ball of the multiplier algebra.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.4310/acta.2023.v231.n1.a1
Victor Beresnevich, Lei Yang
In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary non-degenerate submanifolds of $mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $tau$-well approximable points lying on any non-degenerate submanifold for a range of Diophantine exponents $tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of ‘generic and special parts’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold. In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold. The latter uses a result of Bernik, Kleinbock and Margulis.
{"title":"Khintchine’s theorem and Diophantine approximation on manifolds","authors":"Victor Beresnevich, Lei Yang","doi":"10.4310/acta.2023.v231.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n1.a1","url":null,"abstract":"In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary non-degenerate submanifolds of $mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $tau$-well approximable points lying on any non-degenerate submanifold for a range of Diophantine exponents $tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of ‘generic and special parts’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold. In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold. The latter uses a result of Bernik, Kleinbock and Margulis.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/acta.2023.v231.n1.a2
Patrick Gérard, Thomas Kappeler, Peter Topalov
We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(mathbb{T},mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(mathbb{T},mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(mathbb{T},mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.
{"title":"Sharp well-posedness results of the Benjamin–Ono equation in $H^s (mathbb{T}, mathbb{R})$ and qualitative properties of its solutions","authors":"Patrick Gérard, Thomas Kappeler, Peter Topalov","doi":"10.4310/acta.2023.v231.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n1.a2","url":null,"abstract":"We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(mathbb{T},mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(mathbb{T},mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(mathbb{T},mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135844837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/acta.2023.v231.n1.a3
Yair Shenfeld, Ramon van Handel
The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov's original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of nonsmooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of Stanley on the extremal behavior of certain log-concave sequences that arise in the combinatorics of partially ordered sets.
{"title":"The extremals of the Alexandrov–Fenchel inequality for convex polytopes","authors":"Yair Shenfeld, Ramon van Handel","doi":"10.4310/acta.2023.v231.n1.a3","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n1.a3","url":null,"abstract":"The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov's original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of nonsmooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of Stanley on the extremal behavior of certain log-concave sequences that arise in the combinatorics of partially ordered sets.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136008443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-10DOI: 10.4310/acta.2021.v227.n2.a2
Gao Chen, Xiuxiong Chen
In this paper, we study gravitational instantons (i.e., complete hyperkähler $4$‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑$D_k$ case, it must have an $O(4)$‑multiplet.
{"title":"Gravitational instantons with faster than quadratic curvature decay. I","authors":"Gao Chen, Xiuxiong Chen","doi":"10.4310/acta.2021.v227.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n2.a2","url":null,"abstract":"In this paper, we study gravitational instantons (i.e., complete hyperkähler $4$‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑$D_k$ case, it must have an $O(4)$‑multiplet.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/acta.2022.v229.n1.a1
Yang Li
{"title":"Strominger–Yau–Zaslow conjecture for Calabi–Yau hypersurfaces in the Fermat family","authors":"Yang Li","doi":"10.4310/acta.2022.v229.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v229.n1.a1","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":3.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71153294","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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