Pub Date : 2020-12-04DOI: 10.4007/annals.2022.195.2.5
Tom Bachmann, Hana Jia Kong, Guozhen Wang, Zhouli Xu
We define the Chow $t$-structure on the $infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{cheartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, cheartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.
{"title":"The Chow $t$-structure on the $infty$-category of motivic spectra","authors":"Tom Bachmann, Hana Jia Kong, Guozhen Wang, Zhouli Xu","doi":"10.4007/annals.2022.195.2.5","DOIUrl":"https://doi.org/10.4007/annals.2022.195.2.5","url":null,"abstract":"We define the Chow $t$-structure on the $infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{cheartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, cheartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47932345","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 : 2020-12-01DOI: 10.4007/annals.2022.196.3.6
Jeremy Hahn, D. Wilson
We equip $mathrm{BP} langle n rangle$ with an $mathbb{E}_3$-$mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this $mathbb{E}_3$-ring is of chromatic height exactly $n+1$.
{"title":"Redshift and multiplication for truncated Brown--Peterson spectra","authors":"Jeremy Hahn, D. Wilson","doi":"10.4007/annals.2022.196.3.6","DOIUrl":"https://doi.org/10.4007/annals.2022.196.3.6","url":null,"abstract":"We equip $mathrm{BP} langle n rangle$ with an $mathbb{E}_3$-$mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this $mathbb{E}_3$-ring is of chromatic height exactly $n+1$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42022295","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 : 2020-10-01DOI: 10.4007/annals.2023.197.1.5
S. Dasgupta, M. Kakde
Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $Theta^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $text{Cl}^T(H) otimes mathbf{Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer-Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross-Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.
{"title":"On the Brumer--Stark conjecture","authors":"S. Dasgupta, M. Kakde","doi":"10.4007/annals.2023.197.1.5","DOIUrl":"https://doi.org/10.4007/annals.2023.197.1.5","url":null,"abstract":"Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $Theta^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $text{Cl}^T(H) otimes mathbf{Z}[1/2]$ in terms of Stickelberger elements. \u0000We also show that this stronger result implies Rubin's higher rank version of the Brumer-Stark conjecture, again away from 2. \u0000Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross-Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45241115","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 : 2020-08-08DOI: 10.4007/annals.2022.196.1.2
M. Bialy, A. Mironov
In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $gamma$ admits a $C^1$-smooth first integral which is not singular on $mathcal A$, then the curve $gamma$ is an ellipse. } The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in [8], [10]; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in [6], [7] for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.
{"title":"The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables","authors":"M. Bialy, A. Mironov","doi":"10.4007/annals.2022.196.1.2","DOIUrl":"https://doi.org/10.4007/annals.2022.196.1.2","url":null,"abstract":"In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $gamma$ admits a $C^1$-smooth first integral which is not singular on $mathcal A$, then the curve $gamma$ is an ellipse. } \u0000The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in [8], [10]; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in [6], [7] for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44514204","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 : 2020-08-03DOI: 10.4007/annals.2022.195.3.4
Ben Krause, Mariusz Mirek, T. Tao
We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages [ A_N(f,g)(x) := frac{1}{N} sum_{n =1}^N f(T^nx) g(T^{P(n)}x)] as $N to infty$, where $T colon X to X$ is a measure-preserving transformation of a $sigma$-finite measure space $(X,mu)$, $P(mathrm{n}) in mathbb Z[mathrm{n}]$ is a polynomial of degree $d geq 2$, and $f in L^{p_1}(X), g in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $frac{1}{p_1} + frac{1}{p_2} leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to ``break duality'' by handling some ranges of exponents $p_1,p_2$ with $frac{1}{p_1}+frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly. This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(mathrm{n})=mathrm{n}^2$), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $mathbb A_{mathbb Z}$ also plays a role.
对于非常规(在Furstenberg意义上)双线性多项式遍历平均，我们几乎处处建立了范数收敛性和点向收敛性 [ A_N(f,g)(x) := frac{1}{N} sum_{n =1}^N f(T^nx) g(T^{P(n)}x)] as $N to infty$，其中 $T colon X to X$ 是a的保测度变换吗 $sigma$-有限测度空间 $(X,mu)$， $P(mathrm{n}) in mathbb Z[mathrm{n}]$ 是次数的多项式吗 $d geq 2$，和 $f in L^{p_1}(X), g in L^{p_2}(X)$ 对一些人来说 $p_1,p_2 > 1$ 有 $frac{1}{p_1} + frac{1}{p_2} leq 1$． 我们还建立了 $r$-这些平均值(在空白尺度下)在最佳范围内的变分不等式 $r > 2$． 我们还可以通过处理指数的某些范围来“打破对偶性” $p_1,p_2$ 有 $frac{1}{p_1}+frac{1}{p_2} > 1$，代价是不断增长 $r$ 稍微。这就给出了Frantzikinakis为Furstenberg- Weiss平均值所做的开放性问题调查中的第11个问题的肯定答案 $P(mathrm{n})=mathrm{n}^2$)，这是Bergelson在1996年对遍历拉姆齐理论(Ergodic Ramsey Theory)的调查中考虑的问题9的双线性变体。我们的方法结合了调和分析技术和最近的加性组合学中的Peluse和Prendiville逆定理。在大尺度下，阿德利克整数的调和分析 $mathbb A_{mathbb Z}$ 也发挥了作用。
{"title":"Pointwise ergodic theorems for non-conventional bilinear polynomial averages","authors":"Ben Krause, Mariusz Mirek, T. Tao","doi":"10.4007/annals.2022.195.3.4","DOIUrl":"https://doi.org/10.4007/annals.2022.195.3.4","url":null,"abstract":"We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages [ A_N(f,g)(x) := frac{1}{N} sum_{n =1}^N f(T^nx) g(T^{P(n)}x)] as $N to infty$, where $T colon X to X$ is a measure-preserving transformation of a $sigma$-finite measure space $(X,mu)$, $P(mathrm{n}) in mathbb Z[mathrm{n}]$ is a polynomial of degree $d geq 2$, and $f in L^{p_1}(X), g in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $frac{1}{p_1} + frac{1}{p_2} leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to ``break duality'' by handling some ranges of exponents $p_1,p_2$ with $frac{1}{p_1}+frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly. \u0000This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(mathrm{n})=mathrm{n}^2$), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $mathbb A_{mathbb Z}$ also plays a role.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48469027","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 : 2020-06-11DOI: 10.4007/annals.2021.194.3.6
Chao Li, Yifeng Liu
In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $pi$, if the central derivative $L'(1/2,pi)$ is nonvanishing, then the $pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and $L$-functions, which generalizes the Birch and Swinnerton-Dyer conjecture. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,pi)$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by one of us, which generalizes the Gross--Zagier formula to higher dimensional motives.
{"title":"Chow groups and $L$-derivatives of automorphic motives for unitary groups","authors":"Chao Li, Yifeng Liu","doi":"10.4007/annals.2021.194.3.6","DOIUrl":"https://doi.org/10.4007/annals.2021.194.3.6","url":null,"abstract":"In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $pi$, if the central derivative $L'(1/2,pi)$ is nonvanishing, then the $pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and $L$-functions, which generalizes the Birch and Swinnerton-Dyer conjecture. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,pi)$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by one of us, which generalizes the Gross--Zagier formula to higher dimensional motives.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49324603","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 : 2020-06-08DOI: 10.4007/annals.2023.197.1.2
M. Tsujii, Zhiyuan Zhang
We show that a topologically mixing $C^infty$ Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential.
{"title":"Smooth mixing Anosov flows in dimension three are exponentially mixing","authors":"M. Tsujii, Zhiyuan Zhang","doi":"10.4007/annals.2023.197.1.2","DOIUrl":"https://doi.org/10.4007/annals.2023.197.1.2","url":null,"abstract":"We show that a topologically mixing $C^infty$ Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48441808","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 : 2020-06-03DOI: 10.4007/annals.2023.197.3.3
T. Browning, Pierre Le Boudec, W. Sawin
It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thelene that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.
{"title":"The Hasse principle for random Fano hypersurfaces","authors":"T. Browning, Pierre Le Boudec, W. Sawin","doi":"10.4007/annals.2023.197.3.3","DOIUrl":"https://doi.org/10.4007/annals.2023.197.3.3","url":null,"abstract":"It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thelene that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41393535","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 : 2020-05-26DOI: 10.4007/annals.2022.195.3.3
P. Belkale
We give a construction which produces irreducible complex rigid local systems on $Bbb{P}_{Bbb{C}}^1-{p_1,dots,p_s}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $operatorname{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give (all) possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems (including those with finite global monodromy) on $Bbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S={p_1,dots,p_s}$ and require that the local monodromies have orders which divide $n$, for a fixed $n$. Answering a question of N. Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number. We also show that all unitary irreducible rigid local systems on $Bbb{P}^1_{Bbb{C}} -S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalising previous works of the author and J. Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $operatorname{SU}(n)$.
{"title":"Rigid local systems and the multiplicative eigenvalue problem","authors":"P. Belkale","doi":"10.4007/annals.2022.195.3.3","DOIUrl":"https://doi.org/10.4007/annals.2022.195.3.3","url":null,"abstract":"We give a construction which produces irreducible complex rigid local systems on $Bbb{P}_{Bbb{C}}^1-{p_1,dots,p_s}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $operatorname{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give (all) possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems (including those with finite global monodromy) on $Bbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S={p_1,dots,p_s}$ and require that the local monodromies have orders which divide $n$, for a fixed $n$. Answering a question of N. Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number. \u0000We also show that all unitary irreducible rigid local systems on $Bbb{P}^1_{Bbb{C}} -S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalising previous works of the author and J. Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $operatorname{SU}(n)$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42682066","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 : 2020-05-19DOI: 10.4007/annals.2021.194.2.4
J. Greene, A. Lobb
For every smooth Jordan curve $gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $gamma$. The proof relies on Shevchishin's theorem that the Klein bottle does not admit a smooth Lagrangian embedding in $mathbb{C}^2$.
{"title":"The rectangular peg problem","authors":"J. Greene, A. Lobb","doi":"10.4007/annals.2021.194.2.4","DOIUrl":"https://doi.org/10.4007/annals.2021.194.2.4","url":null,"abstract":"For every smooth Jordan curve $gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $gamma$. The proof relies on Shevchishin's theorem that the Klein bottle does not admit a smooth Lagrangian embedding in $mathbb{C}^2$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46354561","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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