Pub Date : 2021-08-12DOI: 10.1515/crelle-2023-0037
Lauren Cote, Yusuf Barış Kartal
Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.
{"title":"Categorical action filtrations via localization and the growth as a symplectic invariant","authors":"Lauren Cote, Yusuf Barış Kartal","doi":"10.1515/crelle-2023-0037","DOIUrl":"https://doi.org/10.1515/crelle-2023-0037","url":null,"abstract":"Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88307616","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 : 2021-08-09DOI: 10.1515/crelle-2022-0084
P. Koroteev, A. Zeitlin
Abstract In this paper, we describe a certain kind of q-connections on a projective line, namely Z-twisted ( G , q ) {(G,q)} -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q-connections and 𝑄𝑄 mathit{QQ} -systems/Bethe Ansatz equations. Here we associate to a Z-twisted ( G , q ) {(G,q)} -oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as ( G , q ) {(G,q)} -Wronskians. Among other things, we show that the 𝑄𝑄 mathit{QQ} -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.
{"title":"q-opers, QQ-systems, and Bethe Ansatz II: Generalized minors","authors":"P. Koroteev, A. Zeitlin","doi":"10.1515/crelle-2022-0084","DOIUrl":"https://doi.org/10.1515/crelle-2022-0084","url":null,"abstract":"Abstract In this paper, we describe a certain kind of q-connections on a projective line, namely Z-twisted ( G , q ) {(G,q)} -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q-connections and 𝑄𝑄 mathit{QQ} -systems/Bethe Ansatz equations. Here we associate to a Z-twisted ( G , q ) {(G,q)} -oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as ( G , q ) {(G,q)} -Wronskians. Among other things, we show that the 𝑄𝑄 mathit{QQ} -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83521011","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 : 2021-08-01DOI: 10.1515/crelle-2021-frontmatter777
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2021-frontmatter777","DOIUrl":"https://doi.org/10.1515/crelle-2021-frontmatter777","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87461788","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 : 2021-07-29DOI: 10.1515/crelle-2022-0065
Jean-Philippe Furter, Isac Hed'en
Abstract It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ( 𝔸 2 ) {{mathrm{Aut}}({mathbb{A}}^{2})} of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ( ℙ 2 ) {{rm Bir}({mathbb{P}}^{2})} up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ( ℙ 2 ) {{rm Bir}({mathbb{P}}^{2})} admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } {rin{0,1,2}} and that all Borel subgroups of rank r ∈ { 1 , 2 } {rin{1,2}} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 {mathcal{g}geq 1} . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ {mathcal{g}} , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ {mathcal{g}} . This moduli space is of dimension 2 ℊ - 1 {2mathcal{g}-1} .
{"title":"Borel subgroups of the plane Cremona group","authors":"Jean-Philippe Furter, Isac Hed'en","doi":"10.1515/crelle-2022-0065","DOIUrl":"https://doi.org/10.1515/crelle-2022-0065","url":null,"abstract":"Abstract It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ( 𝔸 2 ) {{mathrm{Aut}}({mathbb{A}}^{2})} of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ( ℙ 2 ) {{rm Bir}({mathbb{P}}^{2})} up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ( ℙ 2 ) {{rm Bir}({mathbb{P}}^{2})} admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } {rin{0,1,2}} and that all Borel subgroups of rank r ∈ { 1 , 2 } {rin{1,2}} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 {mathcal{g}geq 1} . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ {mathcal{g}} , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ {mathcal{g}} . This moduli space is of dimension 2 ℊ - 1 {2mathcal{g}-1} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74486121","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 : 2021-07-27DOI: 10.1515/crelle-2023-0044
P. Nagy, U. Semmelmann
Abstract We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.
{"title":"Eigenvalue estimates for 3-Sasaki structures","authors":"P. Nagy, U. Semmelmann","doi":"10.1515/crelle-2023-0044","DOIUrl":"https://doi.org/10.1515/crelle-2023-0044","url":null,"abstract":"Abstract We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77859534","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 : 2021-07-23DOI: 10.1515/crelle-2023-0008
Q. Ding
Abstract In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i {N_{i}} be a sequence of smooth manifolds with Ricci curvature ≥ - n κ 2 {geq-nkappa^{2}} on B 1 + κ ′ ( p i ) {B_{1+kappa^{prime}}(p_{i})} for constants κ ≥ 0 {kappageq 0} , κ ′ > 0 {kappa^{prime}>0} , and volume of B 1 ( p i ) {B_{1}(p_{i})} has a positive uniformly lower bound. Assume B 1 ( p i ) {B_{1}(p_{i})} converges to a metric ball B 1 ( p ∞ ) {B_{1}(p_{infty})} in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i {M_{i}} in B 1 ( p i ) {B_{1}(p_{i})} with ∂ M i ⊂ ∂ B 1 ( p i ) {partial M_{i}subsetpartial B_{1}(p_{i})} , we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit M ∞ {M_{infty}} of M i {M_{i}} is area-minimizing in B 1 ( p ∞ ) {B_{1}(p_{infty})} provided B 1 ( p ∞ ) {B_{1}(p_{infty})} is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ {M_{infty}} in ℛ {mathcal{R}} , and 𝒮 ∩ M ∞ {mathcal{S}cap M_{infty}} . Here, ℛ {mathcal{R}} and 𝒮 {mathcal{S}} are the regular and singular parts of B 1 ( p ∞ ) {B_{1}(p_{infty})} , respectively.
{"title":"Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below","authors":"Q. Ding","doi":"10.1515/crelle-2023-0008","DOIUrl":"https://doi.org/10.1515/crelle-2023-0008","url":null,"abstract":"Abstract In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i {N_{i}} be a sequence of smooth manifolds with Ricci curvature ≥ - n κ 2 {geq-nkappa^{2}} on B 1 + κ ′ ( p i ) {B_{1+kappa^{prime}}(p_{i})} for constants κ ≥ 0 {kappageq 0} , κ ′ > 0 {kappa^{prime}>0} , and volume of B 1 ( p i ) {B_{1}(p_{i})} has a positive uniformly lower bound. Assume B 1 ( p i ) {B_{1}(p_{i})} converges to a metric ball B 1 ( p ∞ ) {B_{1}(p_{infty})} in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i {M_{i}} in B 1 ( p i ) {B_{1}(p_{i})} with ∂ M i ⊂ ∂ B 1 ( p i ) {partial M_{i}subsetpartial B_{1}(p_{i})} , we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit M ∞ {M_{infty}} of M i {M_{i}} is area-minimizing in B 1 ( p ∞ ) {B_{1}(p_{infty})} provided B 1 ( p ∞ ) {B_{1}(p_{infty})} is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ {M_{infty}} in ℛ {mathcal{R}} , and 𝒮 ∩ M ∞ {mathcal{S}cap M_{infty}} . Here, ℛ {mathcal{R}} and 𝒮 {mathcal{S}} are the regular and singular parts of B 1 ( p ∞ ) {B_{1}(p_{infty})} , respectively.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74102239","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 : 2021-07-08DOI: 10.1515/crelle-2022-0093
Junyi Xie
Abstract In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.
{"title":"Remarks on algebraic dynamics in positive characteristic","authors":"Junyi Xie","doi":"10.1515/crelle-2022-0093","DOIUrl":"https://doi.org/10.1515/crelle-2022-0093","url":null,"abstract":"Abstract In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75251731","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 : 2021-07-05DOI: 10.1515/crelle-2022-0089
N. Bergeron, Pierre Charollois, Luis E. García
Abstract We study the arithmetic of degree N - 1 {N-1} Eisenstein cohomology classes for the locally symmetric spaces attached to GL N {mathrm{GL}_{N}} over an imaginary quadratic field k. Under natural conditions we evaluate these classes on ( N - 1 ) {(N-1)} -cycles associated to degree N extensions L / k {L/k} as linear combinations of generalized Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of L-functions attached to Hecke characters of L as polynomials in Kronecker–Eisenstein series evaluated at torsion points on elliptic curves with complex multiplication by k. We recover in particular the algebraicity of these critical values.
摘要研究了虚二次域k上局部对称空间GL N { mathm {GL}_{N}}的N-1 {N-1}次Eisenstein上同调类的算法。在自然条件下，我们将这些类在N次扩展L/k {L/k}相关的(N-1) {(N-1)}环上作为广义Dedekind和的线性组合求值。因此，我们证明了schzech和Colmez的一个重要猜想，将L的Hecke特征上的L函数的临界值表示为Kronecker-Eisenstein级数的多项式，在椭圆曲线的扭转点上用复数乘以k来求值。我们特别恢复了这些临界值的代数性。
{"title":"Eisenstein cohomology classes for GL N over imaginary quadratic fields","authors":"N. Bergeron, Pierre Charollois, Luis E. García","doi":"10.1515/crelle-2022-0089","DOIUrl":"https://doi.org/10.1515/crelle-2022-0089","url":null,"abstract":"Abstract We study the arithmetic of degree N - 1 {N-1} Eisenstein cohomology classes for the locally symmetric spaces attached to GL N {mathrm{GL}_{N}} over an imaginary quadratic field k. Under natural conditions we evaluate these classes on ( N - 1 ) {(N-1)} -cycles associated to degree N extensions L / k {L/k} as linear combinations of generalized Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of L-functions attached to Hecke characters of L as polynomials in Kronecker–Eisenstein series evaluated at torsion points on elliptic curves with complex multiplication by k. We recover in particular the algebraicity of these critical values.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91033910","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 : 2021-07-05DOI: 10.1515/crelle-2022-0028
Borys Kadets, Daniel Litt
Abstract Let ℓ{ell} be a prime, k a finitely generated field of characteristic different from ℓ{ell}, and X a smooth geometrically connected curve over k. Say a semisimple representation of π1ét(Xk¯){pi_{1}^{{text{'{e}t}}}(X_{bar{k}})} is arithmetic if it extends to a finite index subgroup of π1ét(X){pi_{1}^{{text{'{e}t}}}(X)}. We show that there exists an effective constant N=N(X,ℓ){N=N(X,ell)} such that any semisimple arithmetic representation of π1ét(Xk¯){pi_{1}^{{text{'{e}t}}}(X_{bar{k}})} into GLn(ℤℓ¯){operatorname{GL}_{n}(overline{mathbb{Z}_{ell}})}, which is trivial mod ℓN{ell^{N}}, is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel’s linearization theorem and the ℓ{ell}-adic form of Baker’s theorem on linear forms in logarithms.
{"title":"Level structure, arithmetic representations, and noncommutative Siegel linearization","authors":"Borys Kadets, Daniel Litt","doi":"10.1515/crelle-2022-0028","DOIUrl":"https://doi.org/10.1515/crelle-2022-0028","url":null,"abstract":"Abstract Let ℓ{ell} be a prime, k a finitely generated field of characteristic different from ℓ{ell}, and X a smooth geometrically connected curve over k. Say a semisimple representation of π1ét(Xk¯){pi_{1}^{{text{'{e}t}}}(X_{bar{k}})} is arithmetic if it extends to a finite index subgroup of π1ét(X){pi_{1}^{{text{'{e}t}}}(X)}. We show that there exists an effective constant N=N(X,ℓ){N=N(X,ell)} such that any semisimple arithmetic representation of π1ét(Xk¯){pi_{1}^{{text{'{e}t}}}(X_{bar{k}})} into GLn(ℤℓ¯){operatorname{GL}_{n}(overline{mathbb{Z}_{ell}})}, which is trivial mod ℓN{ell^{N}}, is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel’s linearization theorem and the ℓ{ell}-adic form of Baker’s theorem on linear forms in logarithms.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74955852","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 : 2021-07-05DOI: 10.1515/crelle-2023-0030
V. Guedj, C. H. Lu
Abstract We develop a new approach to L ∞ L^{infty} -a priori estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273], we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In [Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), 6, 688–713], we have studied the behavior of Monge–Ampère volumes on hermitian manifolds. We extend here the techniques of the former to the hermitian setting and use the bounds established in the latter, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge–Ampère equations on compact hermitian manifolds.
{"title":"Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds","authors":"V. Guedj, C. H. Lu","doi":"10.1515/crelle-2023-0030","DOIUrl":"https://doi.org/10.1515/crelle-2023-0030","url":null,"abstract":"Abstract We develop a new approach to L ∞ L^{infty} -a priori estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273], we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In [Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), 6, 688–713], we have studied the behavior of Monge–Ampère volumes on hermitian manifolds. We extend here the techniques of the former to the hermitian setting and use the bounds established in the latter, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge–Ampère equations on compact hermitian manifolds.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77747576","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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