Pub Date : 2020-05-07DOI: 10.4007/annals.2023.197.1.3
G. Baldi, E. Ullmo
Let $Gamma subset operatorname{PU}(1,n)$ be a lattice, and $S_Gamma$ the associated ball quotient. We prove that, if $S_Gamma$ contains infinitely many maximal totally geodesic subvarieties, then $Gamma$ is arithmetic. We also prove an Ax-Schanuel Conjecture for $S_Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_Gamma$ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.
{"title":"Special subvarieties of non-arithmetic ball quotients and Hodge theory","authors":"G. Baldi, E. Ullmo","doi":"10.4007/annals.2023.197.1.3","DOIUrl":"https://doi.org/10.4007/annals.2023.197.1.3","url":null,"abstract":"Let $Gamma subset operatorname{PU}(1,n)$ be a lattice, and $S_Gamma$ the associated ball quotient. We prove that, if $S_Gamma$ contains infinitely many maximal totally geodesic subvarieties, then $Gamma$ is arithmetic. We also prove an Ax-Schanuel Conjecture for $S_Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_Gamma$ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"19 3","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41275591","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 : 2020-04-15DOI: 10.4007/annals.2023.197.1.1
S. Schröer
We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite etale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on the Weil Conjectures, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and Shepherd-Barron, some recent results on the base of their versal deformation, Shioda's theory of Mordell--Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.
{"title":"There is no Enriques surface over the integers","authors":"S. Schröer","doi":"10.4007/annals.2023.197.1.1","DOIUrl":"https://doi.org/10.4007/annals.2023.197.1.1","url":null,"abstract":"We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite etale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on the Weil Conjectures, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and Shepherd-Barron, some recent results on the base of their versal deformation, Shioda's theory of Mordell--Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46944521","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 : 2020-03-13DOI: 10.4007/annals.2020.192.2.8
P. Ivanisvili, R. Handel, A. Volberg
A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube.
{"title":"Rademacher type and Enflo type coincide","authors":"P. Ivanisvili, R. Handel, A. Volberg","doi":"10.4007/annals.2020.192.2.8","DOIUrl":"https://doi.org/10.4007/annals.2020.192.2.8","url":null,"abstract":"A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46680999","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 : 2020-02-27DOI: 10.4007/annals.2022.195.2.4
J. Witaszek
We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel's base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollar's theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and we provide a new proof of the existence of quotients by affine group schemes.
{"title":"Keel's base point free theorem and quotients in mixed characteristic","authors":"J. Witaszek","doi":"10.4007/annals.2022.195.2.4","DOIUrl":"https://doi.org/10.4007/annals.2022.195.2.4","url":null,"abstract":"We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel's base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollar's theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and we provide a new proof of the existence of quotients by affine group schemes.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42773230","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 : 2020-01-28DOI: 10.4007/annals.2021.194.1.4
V. Dimitrov, Ziyang Gao, P. Habegger
Consider a smooth, geometrically irreducible, projective curve of genus $g ge 2$ defined over a number field of degree $d ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for $1$-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.
{"title":"Uniformity in Mordell–Lang for curves","authors":"V. Dimitrov, Ziyang Gao, P. Habegger","doi":"10.4007/annals.2021.194.1.4","DOIUrl":"https://doi.org/10.4007/annals.2021.194.1.4","url":null,"abstract":"Consider a smooth, geometrically irreducible, projective curve of genus $g ge 2$ defined over a number field of degree $d ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for $1$-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49131665","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 : 2020-01-05DOI: 10.4007/annals.2022.195.1.3
Roger Casals, Honghao Gao
We prove that all maximal-tb Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m),(3,3),(3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances which induce faithful actions of the modular group PSL(2,Z) and the mapping class group M(0,4) into the coordinate rings of algebraic varieties associated to Legendrian links. Our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.
{"title":"Infinitely many Lagrangian fillings","authors":"Roger Casals, Honghao Gao","doi":"10.4007/annals.2022.195.1.3","DOIUrl":"https://doi.org/10.4007/annals.2022.195.1.3","url":null,"abstract":"We prove that all maximal-tb Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m),(3,3),(3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances which induce faithful actions of the modular group PSL(2,Z) and the mapping class group M(0,4) into the coordinate rings of algebraic varieties associated to Legendrian links. Our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47726981","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 : 2020-01-01DOI: 10.4007/annals.2020.191.2.7
D. Masser, U. Zannier
{"title":"Abelian varieties isogenous to no Jacobian","authors":"D. Masser, U. Zannier","doi":"10.4007/annals.2020.191.2.7","DOIUrl":"https://doi.org/10.4007/annals.2020.191.2.7","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"191 1","pages":"635-674"},"PeriodicalIF":4.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70185876","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 : 2020-01-01DOI: 10.4007/annamath.191.3.1031
{"title":"Index","authors":"","doi":"10.4007/annamath.191.3.1031","DOIUrl":"https://doi.org/10.4007/annamath.191.3.1031","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70187325","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 : 2020-01-01DOI: 10.4007/annals.2020.191.3.3
S. Goette, M. Kerin, K. Shankar
Summary: In this article, a six-parameter family of highly connected 7 -manifolds which admit an SO (3) invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO (3) -invariant metric of non-negative curvature.
{"title":"Highly connected 7-manifolds and non-negative sectional curvature","authors":"S. Goette, M. Kerin, K. Shankar","doi":"10.4007/annals.2020.191.3.3","DOIUrl":"https://doi.org/10.4007/annals.2020.191.3.3","url":null,"abstract":"Summary: In this article, a six-parameter family of highly connected 7 -manifolds which admit an SO (3) invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO (3) -invariant metric of non-negative curvature.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70186241","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 : 2020-01-01DOI: 10.4007/annamath.192.3.1069
{"title":"Index","authors":"","doi":"10.4007/annamath.192.3.1069","DOIUrl":"https://doi.org/10.4007/annamath.192.3.1069","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70187442","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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