Pub Date : 2022-06-25DOI: 10.1515/crelle-2022-0031
Fabien Trihan, Olivier Brinon
Abstract We express the order of the pole and the leading coefficient of the L-function of a (large class of) ℓ{ell}-adic coefficients (ℓ{ell} any prime) over a quasi-projective variety over a finite field of characteristic p. We use the technique of [13] with coefficients with, as new ingredient, the use of F-gauges and their equivalence, in the derived category, with Raynaud modules proved by Ekedahl.
{"title":"On the zeroes and poles of L-functions over varieties in positive characteristic","authors":"Fabien Trihan, Olivier Brinon","doi":"10.1515/crelle-2022-0031","DOIUrl":"https://doi.org/10.1515/crelle-2022-0031","url":null,"abstract":"Abstract We express the order of the pole and the leading coefficient of the L-function of a (large class of) ℓ{ell}-adic coefficients (ℓ{ell} any prime) over a quasi-projective variety over a finite field of characteristic p. We use the technique of [13] with coefficients with, as new ingredient, the use of F-gauges and their equivalence, in the derived category, with Raynaud modules proved by Ekedahl.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83031348","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 : 2022-06-25DOI: 10.1515/crelle-2022-0029
Ming Xiao
Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F=(F1,…,Fm){F=(F_{1},dots,F_{m})} from the complex unit ball 𝔹n{mathbb{B}^{n}}, n≥2{ngeq 2}, to a bounded symmetric domain Ω=Ω1×⋯×Ωm{Omega=Omega_{1}timescdotstimesOmega_{m}} up to constant conformal factors, where Ωi′{Omega_{i}^{prime}}s are irreducible factors of Ω. We prove every non-constant component Fi{F_{i}} must map generic boundary points of 𝔹n{mathbb{B}^{n}} to the boundary of Ωi{Omega_{i}}. In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls.
{"title":"Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains","authors":"Ming Xiao","doi":"10.1515/crelle-2022-0029","DOIUrl":"https://doi.org/10.1515/crelle-2022-0029","url":null,"abstract":"Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F=(F1,…,Fm){F=(F_{1},dots,F_{m})} from the complex unit ball 𝔹n{mathbb{B}^{n}}, n≥2{ngeq 2}, to a bounded symmetric domain Ω=Ω1×⋯×Ωm{Omega=Omega_{1}timescdotstimesOmega_{m}} up to constant conformal factors, where Ωi′{Omega_{i}^{prime}}s are irreducible factors of Ω. We prove every non-constant component Fi{F_{i}} must map generic boundary points of 𝔹n{mathbb{B}^{n}} to the boundary of Ωi{Omega_{i}}. In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75736241","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 : 2022-06-21DOI: 10.1515/crelle-2023-0043
Hülya Argüz, Pierrick Bousseau
Abstract Cluster varieties come in pairs: for any 𝒳 {mathcal{X}} cluster variety there is an associated Fock–Goncharov dual 𝒜 {mathcal{A}} cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the 𝒳 {mathcal{X}} cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 {mathcal{A}} cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.
{"title":"Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors","authors":"Hülya Argüz, Pierrick Bousseau","doi":"10.1515/crelle-2023-0043","DOIUrl":"https://doi.org/10.1515/crelle-2023-0043","url":null,"abstract":"Abstract Cluster varieties come in pairs: for any 𝒳 {mathcal{X}} cluster variety there is an associated Fock–Goncharov dual 𝒜 {mathcal{A}} cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the 𝒳 {mathcal{X}} cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 {mathcal{A}} cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81723502","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 : 2022-06-01DOI: 10.1515/crelle-2022-2001
{"title":"Erratum to Table of Contents (J. reine angew. Math. 785 (2022), i–iv)","authors":"","doi":"10.1515/crelle-2022-2001","DOIUrl":"https://doi.org/10.1515/crelle-2022-2001","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75118494","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 : 2022-06-01DOI: 10.1515/crelle-2023-0034
Jianchun Chu, Man-Chun Lee
Abstract In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824].
{"title":"Hypercritical deformed Hermitian-Yang–Mills equation revisited","authors":"Jianchun Chu, Man-Chun Lee","doi":"10.1515/crelle-2023-0034","DOIUrl":"https://doi.org/10.1515/crelle-2023-0034","url":null,"abstract":"Abstract In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824].","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90679355","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 : 2022-05-20DOI: 10.1515/crelle-2023-0054
S. Brendle, M. Eichmair
Abstract We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2.
{"title":"Proof of the Michael–Simon–Sobolev inequality using optimal transport","authors":"S. Brendle, M. Eichmair","doi":"10.1515/crelle-2023-0054","DOIUrl":"https://doi.org/10.1515/crelle-2023-0054","url":null,"abstract":"Abstract We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90996593","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 : 2022-05-18DOI: 10.1515/crelle-2023-0028
Gregory R. Chambers, Yevgeny Liokumovich, A. Nabutovsky, R. Rotman
Abstract A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.
{"title":"Geodesic nets on non-compact Riemannian manifolds","authors":"Gregory R. Chambers, Yevgeny Liokumovich, A. Nabutovsky, R. Rotman","doi":"10.1515/crelle-2023-0028","DOIUrl":"https://doi.org/10.1515/crelle-2023-0028","url":null,"abstract":"Abstract A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85308539","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 : 2022-05-01DOI: 10.1515/crelle-2022-frontmatter786
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2022-frontmatter786","DOIUrl":"https://doi.org/10.1515/crelle-2022-frontmatter786","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81768593","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 : 2022-04-29DOI: 10.1515/crelle-2022-0090
Qintao Deng, Xinrui Zhao
Abstract Unique continuation of harmonic functions on RCD {operatorname{RCD}} space is a long-standing open problem, with little known even in the setting of Alexandrov spaces. In this paper, we establish the weak unique continuation theorem for harmonic functions on RCD ( K , 2 ) {operatorname{RCD}(K,2)} spaces and give a counterexample for strong unique continuation in the setting of RCD ( K , N ) {operatorname{RCD}(K,N)} space for any N ≥ 4 {Ngeq 4} and any K ∈ ℝ {Kinmathbb{R}} .
{"title":"Failure of strong unique continuation for harmonic functions on RCD spaces","authors":"Qintao Deng, Xinrui Zhao","doi":"10.1515/crelle-2022-0090","DOIUrl":"https://doi.org/10.1515/crelle-2022-0090","url":null,"abstract":"Abstract Unique continuation of harmonic functions on RCD {operatorname{RCD}} space is a long-standing open problem, with little known even in the setting of Alexandrov spaces. In this paper, we establish the weak unique continuation theorem for harmonic functions on RCD ( K , 2 ) {operatorname{RCD}(K,2)} spaces and give a counterexample for strong unique continuation in the setting of RCD ( K , N ) {operatorname{RCD}(K,N)} space for any N ≥ 4 {Ngeq 4} and any K ∈ ℝ {Kinmathbb{R}} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77969018","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 : 2022-04-11DOI: 10.1515/crelle-2023-0005
Paul Apisa, Matt Bainbridge, Jane Wang
Abstract We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 1993, 3, 589–689], we show that the moduli space 𝒜 g , n ( 𝒎 ) {{mathcal{A}}_{g,n}(boldsymbol{m})} of genus g affine surfaces with cone points of complex order 𝒎 = ( m 1 … , m n ) {boldsymbol{m}=(m_{1}ldots,m_{n})} is a holomorphic affine bundle over ℳ g , n {mathcal{M}_{g,n}} , and the moduli space 𝒟 g , n ( 𝒎 ) {{mathcal{D}}_{g,n}(boldsymbol{m})} of dilation surfaces is a covering space of ℳ g , n {mathcal{M}_{g,n}} . We then classify the connected components of 𝒟 g , n ( 𝒎 ) {{mathcal{D}}_{g,n}(boldsymbol{m})} and show that it is an orbifold- K ( G , 1 ) {K(G,1)} , where G is the framed mapping class group of [A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials, preprint 2020].
{"title":"Moduli spaces of complex affine and dilation surfaces","authors":"Paul Apisa, Matt Bainbridge, Jane Wang","doi":"10.1515/crelle-2023-0005","DOIUrl":"https://doi.org/10.1515/crelle-2023-0005","url":null,"abstract":"Abstract We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 1993, 3, 589–689], we show that the moduli space 𝒜 g , n ( 𝒎 ) {{mathcal{A}}_{g,n}(boldsymbol{m})} of genus g affine surfaces with cone points of complex order 𝒎 = ( m 1 … , m n ) {boldsymbol{m}=(m_{1}ldots,m_{n})} is a holomorphic affine bundle over ℳ g , n {mathcal{M}_{g,n}} , and the moduli space 𝒟 g , n ( 𝒎 ) {{mathcal{D}}_{g,n}(boldsymbol{m})} of dilation surfaces is a covering space of ℳ g , n {mathcal{M}_{g,n}} . We then classify the connected components of 𝒟 g , n ( 𝒎 ) {{mathcal{D}}_{g,n}(boldsymbol{m})} and show that it is an orbifold- K ( G , 1 ) {K(G,1)} , where G is the framed mapping class group of [A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials, preprint 2020].","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74260321","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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