Pub Date : 2020-11-18DOI: 10.4310/cjm.2022.v10.n4.a4
Paul Apisa, A. Wright
We classify a natural collection of GL(2,R)-invariant subvarieties which includes loci of double covers as well as the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces. This is motivated in part by a forthcoming application to another classification result, the classification of "high rank" invariant subvarieties. We also give new examples, which negatively resolve two questions of Mirzakhani and Wright, clarify the complex geometry of Teichmuller space, and illustrate new behavior relevant to the finite blocking problem.
{"title":"Generalizations of the Eierlegende–Wollmilchsau","authors":"Paul Apisa, A. Wright","doi":"10.4310/cjm.2022.v10.n4.a4","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n4.a4","url":null,"abstract":"We classify a natural collection of GL(2,R)-invariant subvarieties which includes loci of double covers as well as the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces. This is motivated in part by a forthcoming application to another classification result, the classification of \"high rank\" invariant subvarieties. We also give new examples, which negatively resolve two questions of Mirzakhani and Wright, clarify the complex geometry of Teichmuller space, and illustrate new behavior relevant to the finite blocking problem.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47048803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-07DOI: 10.4310/cjm.2021.v9.n3.a3
Teng Fei, D. Phong, Sebastien Picard, Xiangwen Zhang
A geometric flow on $6$-dimensional symplectic manifolds is introduced which is motivated by supersymmetric compactifications of the Type IIA string. The underlying structure turns out to be SU(3) holonomy, but with respect to the projected Levi-Civita connection of an almost-Hermitian structure. The short-time existence is established, and new identities for the Nijenhuis tensor are found which are crucial for Shi-type estimates. The integrable case can be completely solved, giving an alternative proof of Yau's theorem on Ricci-flat K"ahler metrics. In the non-integrable case, models are worked out which suggest that the flow should lead to optimal almost-complex structures compatible with the given symplectic form.
{"title":"Geometric flows for the Type IIA string","authors":"Teng Fei, D. Phong, Sebastien Picard, Xiangwen Zhang","doi":"10.4310/cjm.2021.v9.n3.a3","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n3.a3","url":null,"abstract":"A geometric flow on $6$-dimensional symplectic manifolds is introduced which is motivated by supersymmetric compactifications of the Type IIA string. The underlying structure turns out to be SU(3) holonomy, but with respect to the projected Levi-Civita connection of an almost-Hermitian structure. The short-time existence is established, and new identities for the Nijenhuis tensor are found which are crucial for Shi-type estimates. The integrable case can be completely solved, giving an alternative proof of Yau's theorem on Ricci-flat K\"ahler metrics. In the non-integrable case, models are worked out which suggest that the flow should lead to optimal almost-complex structures compatible with the given symplectic form.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44877101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-02DOI: 10.4310/cjm.2022.v10.n4.a5
N. Yang
. We study the Milnor-Witt motives which are a finite direct sum of Z ( q )[ p ] and Z /η ( q )[ p ]. We show that for MW-motives of this type, we can determine an MW-motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroups of Witt cohomology, if the MW-motive splits as above. As an application, we give the splitting formula of Milnor-Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real complete flags has only 2-torsions.
{"title":"Split Milnor–Witt motives and its applications to fiber bundles","authors":"N. Yang","doi":"10.4310/cjm.2022.v10.n4.a5","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n4.a5","url":null,"abstract":". We study the Milnor-Witt motives which are a finite direct sum of Z ( q )[ p ] and Z /η ( q )[ p ]. We show that for MW-motives of this type, we can determine an MW-motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroups of Witt cohomology, if the MW-motive splits as above. As an application, we give the splitting formula of Milnor-Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real complete flags has only 2-torsions.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44146428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-09DOI: 10.4310/CJM.2022.v10.n3.a3
H. Gimperlein, Bernhard Krotz, Job J. Kuit, H. Schlichtkrull
We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.
{"title":"A Paley–Wiener theorem for Harish–Chandra modules","authors":"H. Gimperlein, Bernhard Krotz, Job J. Kuit, H. Schlichtkrull","doi":"10.4310/CJM.2022.v10.n3.a3","DOIUrl":"https://doi.org/10.4310/CJM.2022.v10.n3.a3","url":null,"abstract":"We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47547649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-17DOI: 10.4310/cjm.2021.v9.n2.a2
Jessica Fintzen, S. Shin
Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(mathbb A_F)$ and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of $G$. We illustrate how such congruences can be applied in the construction of Galois representations. Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton--Paskūnas on density of supercuspidal points from definite unitary groups to general $G$ as above.
{"title":"Congruences of algebraic automorphic forms and supercuspidal representations","authors":"Jessica Fintzen, S. Shin","doi":"10.4310/cjm.2021.v9.n2.a2","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n2.a2","url":null,"abstract":"Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(mathbb A_F)$ and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of $G$. We illustrate how such congruences can be applied in the construction of Galois representations. \u0000Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton--Paskūnas on density of supercuspidal points from definite unitary groups to general $G$ as above.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46293216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-16DOI: 10.4310/cjm.2022.v10.n4.a1
Chieh-Yu Chang, Yen-Tsung Chen, Yoshinori Mishiba
This paper is a culmination of [CM20] on the study of multiple zeta values (MZV's) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$-adic MZV's satisfy the same $bar{k}$-algebraic relations that their corresponding $infty$-adic MZV's satisfy. Equivalently, we show that the $v$-adic MZV's form an algebra with multiplication law given by the $q$-shuffle product which comes from the $infty$-adic MZV's, and there is a well-defined $bar{k}$-algebra homomorphism from the $infty$-adic MZV's to the $v$-adic MZV's.
{"title":"Algebra structure of multiple zeta values in positive characteristic","authors":"Chieh-Yu Chang, Yen-Tsung Chen, Yoshinori Mishiba","doi":"10.4310/cjm.2022.v10.n4.a1","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n4.a1","url":null,"abstract":"This paper is a culmination of [CM20] on the study of multiple zeta values (MZV's) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$-adic MZV's satisfy the same $bar{k}$-algebraic relations that their corresponding $infty$-adic MZV's satisfy. Equivalently, we show that the $v$-adic MZV's form an algebra with multiplication law given by the $q$-shuffle product which comes from the $infty$-adic MZV's, and there is a well-defined $bar{k}$-algebra homomorphism from the $infty$-adic MZV's to the $v$-adic MZV's.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45976440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-14DOI: 10.4310/cjm.2022.v10.n3.a2
J. M. Manzano, Francisco Torralbo
We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $mathbb{S}^2timesmathbb{R}$ and $mathbb{H}^2timesmathbb{R}$, being the mean curvature larger than $frac{1}{2}$ in the latter case. These surfaces are not equivariant but singly periodic, lie at bounded distance from a horizontal geodesic, and complete the family of horizontal unduloids previously given by the authors. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $mathbb H^2timesmathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $mathbb S^2timesmathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. In particular, we find the first non-equivariant examples of embedded tori in $mathbb{S}^2timesmathbb{R}$, which have constant mean curvature $H>frac12$. Finally, we prove that there are no properly immersed surface with constant mean curvature $Hleqfrac{1}{2}$ at bounded distance from a horizontal geodesic in $mathbb{H}^2timesmathbb{R}$.
{"title":"Horizontal Delaunay surfaces with constant mean curvature in $mathbb{S}^2 times mathbb{R}$ and $mathbb{H}^2 times mathbb{R}$","authors":"J. M. Manzano, Francisco Torralbo","doi":"10.4310/cjm.2022.v10.n3.a2","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n3.a2","url":null,"abstract":"We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $mathbb{S}^2timesmathbb{R}$ and $mathbb{H}^2timesmathbb{R}$, being the mean curvature larger than $frac{1}{2}$ in the latter case. These surfaces are not equivariant but singly periodic, lie at bounded distance from a horizontal geodesic, and complete the family of horizontal unduloids previously given by the authors. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $mathbb H^2timesmathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $mathbb S^2timesmathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. In particular, we find the first non-equivariant examples of embedded tori in $mathbb{S}^2timesmathbb{R}$, which have constant mean curvature $H>frac12$. Finally, we prove that there are no properly immersed surface with constant mean curvature $Hleqfrac{1}{2}$ at bounded distance from a horizontal geodesic in $mathbb{H}^2timesmathbb{R}$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49255291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-20DOI: 10.4310/CJM.2022.v10.n1.a2
T. Creutzig, A. Linshaw
We prove the conjecture of Gaiotto and Rapcak that the $Y$-algebras $Y_{L,M,N}[psi]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $mathcal{W}_{1+infty}$-algebra, and satisfy a symmetry known as triality. These $Y$-algebras are defined as the cosets of certain non-principal $mathcal{W}$-algebras and $mathcal{W}$-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on $mathcal{W}$-algebras of type $A$. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal $mathcal{W}$-algebras due to Arakawa and us, (3) Feigin and Semikhatov's conjectured triality between subregular $mathcal{W}$-algebras, principal $mathcal{W}$-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular $mathcal{W}$-algebras due to Arakawa and van Ekeren, (5) the identification of Heisenberg cosets of subregular $mathcal{W}$-algebras with principal rational $mathcal{W}$-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Prochazka and Rapcak on the explicit truncation curves realizing the simple $Y$-algebras as $mathcal{W}_{1+infty}$-quotients, and on their minimal strong generating types.
{"title":"Trialities of $mathcal{W}$-algebras","authors":"T. Creutzig, A. Linshaw","doi":"10.4310/CJM.2022.v10.n1.a2","DOIUrl":"https://doi.org/10.4310/CJM.2022.v10.n1.a2","url":null,"abstract":"We prove the conjecture of Gaiotto and Rapcak that the $Y$-algebras $Y_{L,M,N}[psi]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $mathcal{W}_{1+infty}$-algebra, and satisfy a symmetry known as triality. These $Y$-algebras are defined as the cosets of certain non-principal $mathcal{W}$-algebras and $mathcal{W}$-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on $mathcal{W}$-algebras of type $A$. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal $mathcal{W}$-algebras due to Arakawa and us, (3) Feigin and Semikhatov's conjectured triality between subregular $mathcal{W}$-algebras, principal $mathcal{W}$-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular $mathcal{W}$-algebras due to Arakawa and van Ekeren, (5) the identification of Heisenberg cosets of subregular $mathcal{W}$-algebras with principal rational $mathcal{W}$-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Prochazka and Rapcak on the explicit truncation curves realizing the simple $Y$-algebras as $mathcal{W}_{1+infty}$-quotients, and on their minimal strong generating types.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44711646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-17DOI: 10.4310/cjm.2021.v9.n1.a2
Chenyang Xu, Ziquan Zhuang
We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a valuation is a minimizer if and only if it is K-semistable, and that K-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of klt singularities and an effective bound of the local fundamental group of a klt singularity.
{"title":"Uniqueness of the minimizer of the normalized volume function","authors":"Chenyang Xu, Ziquan Zhuang","doi":"10.4310/cjm.2021.v9.n1.a2","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n1.a2","url":null,"abstract":"We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a valuation is a minimizer if and only if it is K-semistable, and that K-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of klt singularities and an effective bound of the local fundamental group of a klt singularity.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44536390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-14DOI: 10.4310/CJM.2022.v10.n4.a3
A. Carlotto, Giada Franz, Mario B. Schulz
We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
{"title":"Free boundary minimal surfaces with connected boundary and arbitrary genus","authors":"A. Carlotto, Giada Franz, Mario B. Schulz","doi":"10.4310/CJM.2022.v10.n4.a3","DOIUrl":"https://doi.org/10.4310/CJM.2022.v10.n4.a3","url":null,"abstract":"We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47334496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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