Inventiones mathematicae最新文献

{"title":"CAT(0) spaces of higher rank II","authors":"Stephan Stadler","doi":"10.1007/s00222-023-01230-4","DOIUrl":"https://doi.org/10.1007/s00222-023-01230-4","url":null,"abstract":"<p>This belongs to a series of papers motivated by Ballmann’s Higher Rank Rigidity Conjecture. We prove the following. Let <span>(X)</span> be a CAT(0) space with a geometric group action <span>(Gamma curvearrowright X)</span>. Suppose that every geodesic in <span>(X)</span> lies in an <span>(n)</span>-flat, <span>(ngeq 2)</span>. If <span>(X)</span> contains a periodic <span>(n)</span>-flat which does not bound a flat <span>((n+1))</span>-half-space, then <span>(X)</span> is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"68 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138553364","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}

{"title":"Effective equidistribution for multiplicative Diophantine approximation on lines","authors":"Sam Chow, Lei Yang","doi":"10.1007/s00222-023-01233-1","DOIUrl":"https://doi.org/10.1007/s00222-023-01233-1","url":null,"abstract":"<p>Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in <span>({mathrm{SL}}(3, {mathbb{R}})/{mathrm{SL}}(3,{mathbb{Z}}))</span>. We also provide a complementary convergence statement, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock’s from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"154 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885484","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}

{"title":"The time-like minimal surface equation in Minkowski space: low regularity solutions","authors":"Albert Ai, Mihaela Ifrim, Daniel Tataru","doi":"10.1007/s00222-023-01231-3","DOIUrl":"https://doi.org/10.1007/s00222-023-01231-3","url":null,"abstract":"<p>It has long been conjectured that for nonlinear wave equations that satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for the generic case. The aim of this article is to prove the first result in this direction, namely for the time-like minimal surface equation in the Minkowski space-time. Further, our improvement is substantial, namely by <span>(3/8)</span> derivatives in two space dimensions and by <span>(1/4)</span> derivatives in higher dimensions.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"15 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138559679","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}

{"title":"Scalar curvature rigidity of convex polytopes","authors":"Simon Brendle","doi":"10.1007/s00222-023-01229-x","DOIUrl":"https://doi.org/10.1007/s00222-023-01229-x","url":null,"abstract":"<p>We prove a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary. A variant of a theorem of Fefferman and Phong plays a central role in our analysis.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"39 2","pages":""},"PeriodicalIF":3.1,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508493","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}

{"title":"Derivation of Kubo’s formula for disordered systems at zero temperature","authors":"Wojciech De Roeck, Alexander Elgart, Martin Fraas","doi":"10.1007/s00222-023-01227-z","DOIUrl":"https://doi.org/10.1007/s00222-023-01227-z","url":null,"abstract":"<p>This work justifies the linear response formula for the Hall conductance of a two-dimensional disordered system. The proof rests on controlling the dynamics associated with a random time-dependent Hamiltonian.</p><p>The principal challenge is related to the fact that spectral and dynamical localization are intrinsically unstable under perturbation, and the exact spectral flow - the tool used previously to control the dynamics in this context - does not exist. We resolve this problem by proving a local adiabatic theorem: With high probability, the physical evolution of a localized eigenstate <span>(psi )</span> associated with a random system remains close to the spectral flow for a restriction of the instantaneous Hamiltonian to a region <span>(R)</span> where the bulk of <span>(psi )</span> is supported. Allowing <span>(R)</span> to grow at most logarithmically in time ensures that the deviation of the physical evolution from this spectral flow is small.</p><p>To substantiate our claim on the failure of the global spectral flow in disordered systems, we prove eigenvector hybridization in a one-dimensional Anderson model at all scales.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"50 4","pages":""},"PeriodicalIF":3.1,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508484","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}

{"title":"Higher Siegel–Weil formula for unitary groups: the non-singular terms","authors":"Tony Feng, Zhiwei Yun, Wei Zhang","doi":"10.1007/s00222-023-01228-y","DOIUrl":"https://doi.org/10.1007/s00222-023-01228-y","url":null,"abstract":"<p>We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the <span>(r^{mathrm{th}})</span> central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with <span>(r)</span> legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"51 4","pages":""},"PeriodicalIF":3.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508483","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}

{"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":"35 3","pages":"0"},"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}

{"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":"243 6","pages":"0"},"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}

{"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":"238 4","pages":"0"},"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}

{"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":"33 1","pages":"0"},"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}