Pub Date : 2024-07-18DOI: 10.4310/cjm.2024.v12.n2.a3
Fabio Cavalletti, Andrea Mondino
The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of “timelike Ricci curvature bounded below and dimension bounded above” for a measured Lorentzian pre-length space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. This notion is proved to be stable under a suitable weak convergence of measured Lorentzian pre-length spaces, giving a glimpse on the strength of the approach we propose. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting. The framework of Lorentzian pre-length spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity.
{"title":"Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications","authors":"Fabio Cavalletti, Andrea Mondino","doi":"10.4310/cjm.2024.v12.n2.a3","DOIUrl":"https://doi.org/10.4310/cjm.2024.v12.n2.a3","url":null,"abstract":"The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of “timelike Ricci curvature bounded below and dimension bounded above” for a measured Lorentzian pre-length space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. This notion is proved to be stable under a suitable weak convergence of measured Lorentzian pre-length spaces, giving a glimpse on the strength of the approach we propose. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting. The framework of Lorentzian pre-length spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745795","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 : 2024-07-18DOI: 10.4310/cjm.2024.v12.n2.a1
Ruadhaí Dervan, John Benjamin McCarthy, Lars Martin Sektnan
We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang–Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang–Mills equation, this provides the first examples of solutions in higher rank.
{"title":"$Z$-critical connections and Bridgeland stability conditions","authors":"Ruadhaí Dervan, John Benjamin McCarthy, Lars Martin Sektnan","doi":"10.4310/cjm.2024.v12.n2.a1","DOIUrl":"https://doi.org/10.4310/cjm.2024.v12.n2.a1","url":null,"abstract":"We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang–Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang–Mills equation, this provides the first examples of solutions in higher rank.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"68 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737662","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 : 2024-07-18DOI: 10.4310/cjm.2024.v12.n2.a2
Ashay Burungale, Matthias Flach
Let $E/F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$. We give a complete proof of the conjecture of Birch and Swinnerton-Dyer for $E/F$, as well as its equivariant refinement formulated by Gross $href{https://doi.org/10.1007/978-1-4899-6699-5_14}{[39]}$, under the assumption that $L(E/F, 1) neq 0$ and that $F(E_{tors})/K$ is abelian. We also prove analogous results for CM abelian varieties $A/K$.
{"title":"The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication","authors":"Ashay Burungale, Matthias Flach","doi":"10.4310/cjm.2024.v12.n2.a2","DOIUrl":"https://doi.org/10.4310/cjm.2024.v12.n2.a2","url":null,"abstract":"Let $E/F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$. We give a complete proof of the conjecture of Birch and Swinnerton-Dyer for $E/F$, as well as its equivariant refinement formulated by Gross $href{https://doi.org/10.1007/978-1-4899-6699-5_14}{[39]}$, under the assumption that $L(E/F, 1) neq 0$ and that $F(E_{tors})/K$ is abelian. We also prove analogous results for CM abelian varieties $A/K$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"29 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737663","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 : 2024-01-30DOI: 10.4310/cjm.2024.v12.n1.a3
Yang Li
We prove the metric version of the SYZ conjecture for a class of Calabi–Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge–Ampère equation on certain polytopes. This does not rely on discrete symmetry.
{"title":"Metric SYZ conjecture for certain toric Fano hypersurfaces","authors":"Yang Li","doi":"10.4310/cjm.2024.v12.n1.a3","DOIUrl":"https://doi.org/10.4310/cjm.2024.v12.n1.a3","url":null,"abstract":"We prove the metric version of the SYZ conjecture for a class of Calabi–Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge–Ampère equation on certain polytopes. This does not rely on discrete symmetry.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"87 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646470","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 : 2024-01-30DOI: 10.4310/cjm.2024.v12.n1.a1
Georgios Pappas, Michael Rapoport
We establish basic results on p-adic shtukas and apply them to the theory of local and global Shimura varieties, and on their interrelation. We construct canonical integral models for (local, and global) Shimura varieties of Hodge type with parahoric level structure.
{"title":"$p$-adic shtukas and the theory of global and local Shimura varieties","authors":"Georgios Pappas, Michael Rapoport","doi":"10.4310/cjm.2024.v12.n1.a1","DOIUrl":"https://doi.org/10.4310/cjm.2024.v12.n1.a1","url":null,"abstract":"We establish basic results on <i>p</i>-adic shtukas and apply them to the theory of local and global Shimura varieties, and on their interrelation. We construct canonical integral models for (local, and global) Shimura varieties of Hodge type with parahoric level structure.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"13 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646534","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 : 2024-01-30DOI: 10.4310/cjm.2024.v12.n1.a2
Yannick Guedes-Bonthonneau, Colin Guillarmou, Malo Jézéquel
For analytic negatively curved Riemannian manifolds with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular, one recovers both the topology and the metric. More generally our result holds in the analytic category under the no conjugate point and hyperbolic trapped set assumptions.
{"title":"Scattering rigidity for analytic metrics","authors":"Yannick Guedes-Bonthonneau, Colin Guillarmou, Malo Jézéquel","doi":"10.4310/cjm.2024.v12.n1.a2","DOIUrl":"https://doi.org/10.4310/cjm.2024.v12.n1.a2","url":null,"abstract":"For analytic negatively curved Riemannian manifolds with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular, one recovers both the topology and the metric. More generally our result holds in the analytic category under the no conjugate point and hyperbolic trapped set assumptions.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"11 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646923","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 : 2023-09-29DOI: 10.4310/cjm.2023.v11.n4.a3
Nguyen Viet Dang, Matthieu Léautaud, Gabriel Rivière
In analogy with the study of Pollicott–Ruelle resonances on negatively curved manifolds, we define anisotropic Sobolev spaces that are well-adapted to the analysis of the geodesic vector field associated with any translation invariant Finsler metric on the torus $mathbb{T}^d$. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex subsets of $mathbb{T}^d$ (i.e. projection of the boundaries of strictly convex bodies of $mathbb{R}^d$). Associated with the set of lengths of such orthogeodesics, we define a geometric Epstein function and prove its meromorphic continuation. We compute its residues in terms of intrinsic volumes of the convex sets. We also prove Poisson-type summation formulae relating the set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.
{"title":"Length orthospectrum of convex bodies on flat tori","authors":"Nguyen Viet Dang, Matthieu Léautaud, Gabriel Rivière","doi":"10.4310/cjm.2023.v11.n4.a3","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n4.a3","url":null,"abstract":"In analogy with the study of Pollicott–Ruelle resonances on negatively curved manifolds, we define anisotropic Sobolev spaces that are well-adapted to the analysis of the geodesic vector field associated with any translation invariant Finsler metric on the torus $mathbb{T}^d$. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex subsets of $mathbb{T}^d$ (i.e. projection of the boundaries of strictly convex bodies of $mathbb{R}^d$). Associated with the set of lengths of such orthogeodesics, we define a geometric Epstein function and prove its meromorphic continuation. We compute its residues in terms of intrinsic volumes of the convex sets. We also prove Poisson-type summation formulae relating the set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"194 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538148","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 : 2023-01-01DOI: 10.4310/cjm.2023.v11.n4.a2
Tobias Barthel, Drew Heard, Beren Sanders
We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-to-global principle and establish that the Balmer-Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson-Iyengar-Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite etale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height $0$ and height $infty$ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of $E(n)$-local spectral Mackey functors noting that it bijects onto the height $le n$ chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.
{"title":"Stratification in tensor triangular geometry with applications to spectral Mackey functors","authors":"Tobias Barthel, Drew Heard, Beren Sanders","doi":"10.4310/cjm.2023.v11.n4.a2","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n4.a2","url":null,"abstract":"We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-to-global principle and establish that the Balmer-Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson-Iyengar-Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite etale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height $0$ and height $infty$ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of $E(n)$-local spectral Mackey functors noting that it bijects onto the height $le n$ chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181013","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 : 2023-01-01DOI: 10.4310/cjm.2023.v11.n1.a1
P. Allen, Chandrashekhar B. Khare, J. Thorne
{"title":"Modularity of $mathrm{GL}_2 (mathbb{F}_p)$-representations over CM fields","authors":"P. Allen, Chandrashekhar B. Khare, J. Thorne","doi":"10.4310/cjm.2023.v11.n1.a1","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n1.a1","url":null,"abstract":"","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404584","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 : 2023-01-01DOI: 10.4310/cjm.2023.v11.n4.a1
Christopher D. Hacon, Jihao Liu
{"title":"Existence of flips for generalized $operatorname{lc}$ pairs","authors":"Christopher D. Hacon, Jihao Liu","doi":"10.4310/cjm.2023.v11.n4.a1","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n4.a1","url":null,"abstract":"","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784075","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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