Pub Date : 2023-11-06DOI: 10.1007/s00222-023-01224-2
David Ben-Zvi, Harrison Chen, David Helm, David Nadler
Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant $K$ K -group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ F with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$ K -theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf . As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $mathrm{GL}_{n}(F)$ GLn(F) into coherent sheaves on the stack of Langlands parameters.
{"title":"Coherent Springer theory and the categorical Deligne-Langlands correspondence","authors":"David Ben-Zvi, Harrison Chen, David Helm, David Nadler","doi":"10.1007/s00222-023-01224-2","DOIUrl":"https://doi.org/10.1007/s00222-023-01224-2","url":null,"abstract":"Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant $K$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>K</mml:mi> </mml:math> -group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>F</mml:mi> </mml:math> with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>K</mml:mi> </mml:math> -theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf . As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $mathrm{GL}_{n}(F)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:math> into coherent sheaves on the stack of Langlands parameters.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135544860","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 : 2023-11-03DOI: 10.1007/s00222-023-01225-1
Laure Dumaz, Cyril Labbé
We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $mathcal{H}_{L}$ , with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order 1 (Bulk regime) or of order $1ll E ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper (Dumaz and Labbé in Ann. Probab. 51(3):805–839, 2023), this identifies completely the transition between the localized and delocalized phases of the spectrum of $mathcal{H}_{L}$ . The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.
我们研究了具有白噪声势的连续安德森哈密顿函数$mathcal{H}_{L}$的谱在一个长度$L$被发送到无穷长的段上的行为。我们放大1阶能级$E$(散装能级)或1阶能级$ L$(交叉能级)。我们证明了特征值和质心的点过程收敛于泊松点过程。我们还以显式的速率证明了本征函数的指数局域化。此外，我们证明了特征函数收敛于良好识别的极限:在交叉区域，这些极限是普遍的。结合我们的同伴论文(Dumaz and labb in Ann)的结果。概率。51(3):805 - 839,2023)，这完全确定了$mathcal{H}_{L}$谱的局域相和非局域相之间的跃迁。两个主要的技术挑战是两点估计或Minami估计的证明，以及对亚椭圆扩散收敛到平衡的估计，其证明依赖于Malliavin演算和亚矫顽力理论。
{"title":"Localization crossover for the continuous Anderson Hamiltonian in 1-d","authors":"Laure Dumaz, Cyril Labbé","doi":"10.1007/s00222-023-01225-1","DOIUrl":"https://doi.org/10.1007/s00222-023-01225-1","url":null,"abstract":"We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $mathcal{H}_{L}$ , with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order 1 (Bulk regime) or of order $1ll E ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper (Dumaz and Labbé in Ann. Probab. 51(3):805–839, 2023), this identifies completely the transition between the localized and delocalized phases of the spectrum of $mathcal{H}_{L}$ . The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775117","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 : 2023-11-03DOI: 10.1007/s00222-023-01226-0
Gregorio Baldi, Bruno Klingler, Emmanuel Ullmo
Abstract Given a polarizable ℤ-variation of Hodge structures $mathbb{V}$ V over a complex smooth quasi-projective base $S$ S , a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$ S , called the special subvarieties for $mathbb{V}$ V . Our main result in this paper is that, if the level of ${mathbb{V}}$ V is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree $d$ d smooth hypersurfaces in $mathbf{P}^{n+1}_{mathbb{C}}$ PCn+1 , $ngeq 3$ n≥3 , $dgeq 5$ d≥5 and $(n,d)neq (4,5)$ (n,d)≠(4,5) , is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in $S^{mbox{an}}$ San as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ S of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.
摘要给定复杂光滑拟射光基$S$ S上的Hodge结构$mathbb{V}$ V的一个可极化变分，Cattani, Deligne和Kaplan的经典结果表明，它的Hodge轨迹(即例外Hodge张量出现的轨迹)是$S$ S的不可约代数子变种的可数并，称为$mathbb{V}$ V的特殊子变种。本文的主要结果是，如果${mathbb{V}}$ V的阶数至少为3，那么这个Hodge轨迹实际上是这类特殊子变量的有限并(因此是代数的)，至少如果我们将Hodge轨迹限制为正周期维数(定理1.5)。例如，在$mathbf{P}^{n+1}_{mathbb{C}}$ P C n + 1, $ngeq 3$ n≥3,$dgeq 5$ d≥5和$(n,d)neq (4,5)$ (n, d)≠(4,5)条件下，阶次为$d$ d的光滑超曲面泛族的正周期维的Hodge轨迹是代数的。另一方面，我们证明了在一级或二级，Hodge基因座在$S^{mbox{an}}$ S an中只要包含一个典型的特殊亚种就是解析密集的。这些结果来自于对$S$ S中除零周期维的非典型特殊子变种外的典型/非典型交集的分布的完整阐明。
{"title":"On the distribution of the Hodge locus","authors":"Gregorio Baldi, Bruno Klingler, Emmanuel Ullmo","doi":"10.1007/s00222-023-01226-0","DOIUrl":"https://doi.org/10.1007/s00222-023-01226-0","url":null,"abstract":"Abstract Given a polarizable ℤ-variation of Hodge structures $mathbb{V}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> </mml:math> over a complex smooth quasi-projective base $S$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>S</mml:mi> </mml:math> , a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>S</mml:mi> </mml:math> , called the special subvarieties for $mathbb{V}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> </mml:math> . Our main result in this paper is that, if the level of ${mathbb{V}}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> </mml:math> is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree $d$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>d</mml:mi> </mml:math> smooth hypersurfaces in $mathbf{P}^{n+1}_{mathbb{C}}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> , $ngeq 3$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:math> , $dgeq 5$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:math> and $(n,d)neq (4,5)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:math> , is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in $S^{mbox{an}}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>S</mml:mi> <mml:mtext>an</mml:mtext> </mml:msup> </mml:math> as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>S</mml:mi> </mml:math> of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775119","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 : 2023-10-12DOI: 10.1007/s00222-023-01223-3
Claudio Llosa Isenrich, Pierre Py
Abstract We prove that in a cocompact complex hyperbolic arithmetic lattice $Gamma < {mathrm{PU}}(m,1)$ Γ<PU(m,1) of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type $mathscr{F}_{m-1}$ Fm−1 but not of type $mathscr{F}_{m}$ Fm . This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.
摘要证明了紧复双曲算术格$Gamma <{ mathm {PU}}(m,1)$ Γ <最简单类型的PU (m, 1)，足够深的有限索引子群承认具有核类型为$mathscr{F}_{m-1}$ F m-1但不具有核类型为$mathscr{F}_{m}$ F m的大量同态。这提供了许多有限表示的双曲群的非双曲子群，并回答了Brady的一个老问题。我们的方法也证明了非球面Kähler流形的辛格猜想的一个特例。
{"title":"Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices","authors":"Claudio Llosa Isenrich, Pierre Py","doi":"10.1007/s00222-023-01223-3","DOIUrl":"https://doi.org/10.1007/s00222-023-01223-3","url":null,"abstract":"Abstract We prove that in a cocompact complex hyperbolic arithmetic lattice $Gamma < {mathrm{PU}}(m,1)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>PU</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:math> of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type $mathscr{F}_{m-1}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> but not of type $mathscr{F}_{m}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> . This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923152","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 : 2023-10-04DOI: 10.1007/s00222-023-01222-4
Connor Mooney, Yang Yang
Abstract We construct nonlinear entire anisotropic minimal graphs over $mathbb{R}^{4}$ R4 , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over $mathbb{R}^{n},, n geq 8$ Rn,n≥8 .
在$mathbb{R}^{4}$ r4上构造了非线性全各向异性极小图，完成了各向异性Bernstein问题的求解。我们构建的示例具有各种增长率，并且我们的方法既可以推广到高维，也可以恢复和阐明$mathbb{R}^{n},, n geq 8$ R n, n≥8上的非线性完整最小图的已知示例。
{"title":"The anisotropic Bernstein problem","authors":"Connor Mooney, Yang Yang","doi":"10.1007/s00222-023-01222-4","DOIUrl":"https://doi.org/10.1007/s00222-023-01222-4","url":null,"abstract":"Abstract We construct nonlinear entire anisotropic minimal graphs over $mathbb{R}^{4}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over $mathbb{R}^{n},, n geq 8$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>8</mml:mn> </mml:math> .","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135592232","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 : 2023-09-29DOI: 10.1007/s00222-023-01221-5
Robert M. Guralnick, Michael Larsen, Pham Huu Tiep
Abstract The main results of the paper develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to $|G|$ |G| in a logarithmic sense.
{"title":"Character levels and character bounds for finite classical groups","authors":"Robert M. Guralnick, Michael Larsen, Pham Huu Tiep","doi":"10.1007/s00222-023-01221-5","DOIUrl":"https://doi.org/10.1007/s00222-023-01221-5","url":null,"abstract":"Abstract The main results of the paper develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to $|G|$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mo>|</mml:mo> </mml:math> in a logarithmic sense.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135132129","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 : 2023-09-27DOI: 10.1007/s00222-023-01216-2
Cristiana De Filippis, Giuseppe Mingione
Abstract Local Schauder theory holds in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic problems are locally Hölder continuous if so are their coefficients.
{"title":"Nonuniformly elliptic Schauder theory","authors":"Cristiana De Filippis, Giuseppe Mingione","doi":"10.1007/s00222-023-01216-2","DOIUrl":"https://doi.org/10.1007/s00222-023-01216-2","url":null,"abstract":"Abstract Local Schauder theory holds in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic problems are locally Hölder continuous if so are their coefficients.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135536574","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 : 2023-09-26DOI: 10.1007/s00222-023-01219-z
Gergely Bérczi, Frances Kirwan
Abstract The Green–Griffiths–Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.
{"title":"Non-reductive geometric invariant theory and hyperbolicity","authors":"Gergely Bérczi, Frances Kirwan","doi":"10.1007/s00222-023-01219-z","DOIUrl":"https://doi.org/10.1007/s00222-023-01219-z","url":null,"abstract":"Abstract The Green–Griffiths–Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904364","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 : 2023-09-26DOI: 10.1007/s00222-023-01220-6
Giulio Bresciani
Abstract Let $X$ X be a curve over a field $k$ k finitely generated over ℚ and $t$ t an indeterminate. We prove that, if $s$ s is a section of $pi _{1}(X)to operatorname{Gal}(k)$ π1(X)→Gal(k) such that the base change $s_{k(t)}$ sk(t) is birationally liftable, then $s$ s comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.
{"title":"On the birational section conjecture with strong birationality assumptions","authors":"Giulio Bresciani","doi":"10.1007/s00222-023-01220-6","DOIUrl":"https://doi.org/10.1007/s00222-023-01220-6","url":null,"abstract":"Abstract Let $X$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> </mml:math> be a curve over a field $k$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>k</mml:mi> </mml:math> finitely generated over ℚ and $t$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> an indeterminate. We prove that, if $s$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>s</mml:mi> </mml:math> is a section of $pi _{1}(X)to operatorname{Gal}(k)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mo>Gal</mml:mo> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:math> such that the base change $s_{k(t)}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>s</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> </mml:math> is birationally liftable, then $s$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>s</mml:mi> </mml:math> comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904248","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 : 2023-09-26DOI: 10.1007/s00222-023-01218-0
Gergely Bérczi, Frances Kirwan
Abstract Let $H$ H be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety $X$ X . Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup $Q$ Q of $H$ H , we define a notion of moment map for the action of $H$ H , and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient $X/!/H$ X//H introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of $X/!/H$ X//H and express the rational cohomology ring of $X/!/H$ X//H in terms of the rational cohomology ring of the GIT quotient $X/!/T^{H}$ X//TH , where $T^{H}$ TH is a maximal torus in $H$ H . We relate intersection pairings on $X/!/H$ X//H to intersection pairings on $X/!/T^{H}$ X//TH , obtaining a residue formula for these pairings on $X/!/H$ X//H analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.
摘要设$H$ H是作用于一个复射影变量$X$ X上具有内阶单幂根的复线性代数群。给定作用的充分线性化和相关的Fubini-Study Kähler形式，该形式对于$H$ H的最大紧子群$Q$ Q是不变的，我们定义了$H$ H的作用的矩映射概念，并在适当的条件下(线性化适应良好且半稳定性与稳定性一致)我们描述了(非约化)GIT商$X/!/H$ X / /H引入了(b2013.2013.12)等人在J. Topol. 11(3): 826-855, 2018)。利用这一描述，我们推导出$X/!/H$ X/ /H并表示$X/!/H$ X/ /关于GIT商$X/!/T^{H}$ X / /T H，其中$T^{H}$ T H是$H$ H中的最大环面。我们将$X/!/H$ X/ /H到$X/!/T^{H}$ X/ /T H，得到$X/!/H$ X / /H类似于(Jeffrey and Kirwan在拓扑34(2):291-327,1995)的残差公式。作为应用，我们给出了多项式次射影超曲面的Green-Griffiths-Lang猜想和Kobayashi猜想的证明。
{"title":"Moment maps and cohomology of non-reductive quotients","authors":"Gergely Bérczi, Frances Kirwan","doi":"10.1007/s00222-023-01218-0","DOIUrl":"https://doi.org/10.1007/s00222-023-01218-0","url":null,"abstract":"Abstract Let $H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>H</mml:mi> </mml:math> be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety $X$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> </mml:math> . Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup $Q$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Q</mml:mi> </mml:math> of $H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>H</mml:mi> </mml:math> , we define a notion of moment map for the action of $H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>H</mml:mi> </mml:math> , and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient $X/!/H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> </mml:math> introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of $X/!/H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> </mml:math> and express the rational cohomology ring of $X/!/H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> </mml:math> in terms of the rational cohomology ring of the GIT quotient $X/!/T^{H}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> </mml:msup> </mml:math> , where $T^{H}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> </mml:msup> </mml:math> is a maximal torus in $H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>H</mml:mi> </mml:math> . We relate intersection pairings on $X/!/H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> </mml:math> to intersection pairings on $X/!/T^{H}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> </mml:msup> </mml:math> , obtaining a residue formula for these pairings on $X/!/H$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> </mml:math> analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904363","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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