{"title":"Moduli spaces of filtered G-local systems on curves","authors":"Pengfei Huang , Hao Sun","doi":"10.1016/j.aim.2025.110420","DOIUrl":"10.1016/j.aim.2025.110420","url":null,"abstract":"<div><div>In this paper, we construct the moduli spaces of filtered <em>G</em>-local systems on curves for an arbitrary reductive group <em>G</em> over an algebraically closed field of characteristic zero. This provides an algebraic construction for the Betti moduli spaces in the tame nonabelian Hodge correspondence for vector bundles/principal bundles on noncompact curves. As a direct application, the tame nonabelian Hodge correspondence on noncompact curves holds not only for the relevant categories, but also for the moduli spaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110420"},"PeriodicalIF":1.5,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549852","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":"Spectral quantization for ancient asymptotically cylindrical flows","authors":"Wenkui Du, Jingze Zhu","doi":"10.1016/j.aim.2025.110422","DOIUrl":"10.1016/j.aim.2025.110422","url":null,"abstract":"<div><div>We study ancient mean curvature flows in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> whose tangent flow at −∞ is a shrinking cylinder <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>(</mo><msqrt><mrow><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>|</mo><mi>t</mi><mo>|</mo></mrow></msqrt><mo>)</mo></math></span>, where <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. We prove that the cylindrical profile function <em>u</em> of these flows have the asymptotics <span><math><mi>u</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>ω</mi><mo>,</mo><mi>τ</mi><mo>)</mo><mo>=</mo><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>⊤</mo></mrow></msup><mi>Q</mi><mi>y</mi><mo>−</mo><mn>2</mn><mtext>tr</mtext><mo>(</mo><mi>Q</mi><mo>)</mo><mo>)</mo><mo>/</mo><mo>|</mo><mi>τ</mi><mo>|</mo><mo>+</mo><mi>o</mi><mo>(</mo><mo>|</mo><mi>τ</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>τ</mi><mo>→</mo><mo>−</mo><mo>∞</mo></math></span>, where the cylindrical matrix <em>Q</em> is a constant symmetric <span><math><mi>k</mi><mo>×</mo><mi>k</mi></math></span> matrix whose eigenvalues are quantized to be either 0 or <span><math><mo>−</mo><mfrac><mrow><msqrt><mrow><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo></mrow></msqrt></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>. Compared with the bubble-sheet quantization theorem in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> obtained by Haslhofer and the first author, this theorem has full generality in the sense of removing noncollapsing condition and being valid for all dimensions. In addition, we establish symmetry improvement theorem which generalizes the corresponding results of Brendle-Choi and the second author to all dimensions. Finally, we give some geometric applications of the two theorems. In particular, we obtain the asymptotics, compactness and <span><math><mtext>O</mtext><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> symmetry of <em>k</em>-ovals in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> which are ancient noncollapsed flows in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> satisfying full rank condition that <span><math><mtext>rk</mtext><mo>(</mo><mi>Q</mi><mo>)</mo><mo>=</mo><mi>k</mi></math></span>, and we also obtain the classification of ancient noncollapsed flows in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> satisfying vanishing rank condition that <span><math><mtext>rk</mtext><mo>(</mo><mi>Q</mi><mo>)</mo><mo>=</mo><mn>0<","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110422"},"PeriodicalIF":1.5,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534866","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":"Fractional distortion in hyperbolic groups","authors":"Pallavi Dani , Timothy Riley","doi":"10.1016/j.aim.2025.110418","DOIUrl":"10.1016/j.aim.2025.110418","url":null,"abstract":"<div><div>For all integers <span><math><mi>p</mi><mo>></mo><mi>q</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>k</mi><mo>></mo><mn>0</mn></math></span>, and all non-elementary torsion-free hyperbolic groups <em>H</em>, we construct a hyperbolic group <em>G</em> in which <em>H</em> is a subgroup, such that the distortion function of <em>H</em> in <em>G</em> grows like <span><math><msup><mrow><mi>exp</mi></mrow><mrow><mi>k</mi></mrow></msup><mo></mo><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>p</mi><mo>/</mo><mi>q</mi></mrow></msup><mo>)</mo></math></span>. Here, <span><math><msup><mrow><mi>exp</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> denotes the <em>k</em>-fold-iterated exponential function.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110418"},"PeriodicalIF":1.5,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534867","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":"Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows","authors":"Jérôme Buzzi , Sylvain Crovisier , Yuri Lima","doi":"10.1016/j.aim.2025.110410","DOIUrl":"10.1016/j.aim.2025.110410","url":null,"abstract":"<div><div>We construct symbolic dynamics for three dimensional flows with positive speed. More precisely, for each <span><math><mi>χ</mi><mo>></mo><mn>0</mn></math></span>, we code a set of full measure for every invariant probability measure which is <em>χ</em>–hyperbolic. These include all ergodic measures with entropy bigger than <em>χ</em> as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of <span><math><mo>[</mo><mo>−</mo><mi>χ</mi><mo>,</mo><mi>χ</mi><mo>]</mo></math></span>. This contrasts with a previous work of Lima & Sarig which built a coding associated to a given invariant probability measure <span><span>[28]</span></span>. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110410"},"PeriodicalIF":1.5,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523559","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":"Noncommutative resolution of SUC(2)","authors":"Elias Sink, Jenia Tevelev","doi":"10.1016/j.aim.2025.110421","DOIUrl":"10.1016/j.aim.2025.110421","url":null,"abstract":"<div><div>We study the derived category of the moduli space <span><math><mi>S</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo></math></span> of rank 2 vector bundles on a smooth projective curve <em>C</em> of genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span> with trivial determinant. This generalizes the recent work by Tevelev and Torres on the case with fixed odd determinant. Since <span><math><mi>S</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo></math></span> is singular, we work with its noncommutative resolution of singularities constructed by Pădurariu and Špenko–Van den Bergh (in the more general setting of symmetric stacks). We show that this noncommutative resolution admits a semiorthogonal decomposition into derived categories of symmetric powers <span><math><msup><mrow><mi>Sym</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mi>C</mi></math></span> for <span><math><mn>2</mn><mi>k</mi><mo>≤</mo><mi>g</mi><mo>−</mo><mn>1</mn></math></span>. In the case of even genus, each block appears four times. This is also true in the case of odd genus, except that the top symmetric power <span><math><msup><mrow><mi>Sym</mi></mrow><mrow><mi>g</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>C</mi></math></span> appears twice. In the case of even genus, the noncommutative resolution is strongly crepant in the sense of Kuznetsov and categorifies the intersection cohomology of <span><math><mi>S</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo></math></span>. Since all of its components are “geometric,” our semiorthogonal decomposition provides evidence for the expectation, which dates back to the work of Newstead and Tyurin, that <span><math><mi>S</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo></math></span> is a rational variety. Finally, we study mutations of semiorthogonal decompositions on the Hecke correspondence, answering a question of Pădurariu and Toda.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110421"},"PeriodicalIF":1.5,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517546","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":"Kummer surfaces and quadratic line complexes in characteristic two","authors":"Toshiyuki Katsura , Shigeyuki Kondō","doi":"10.1016/j.aim.2025.110416","DOIUrl":"10.1016/j.aim.2025.110416","url":null,"abstract":"<div><div>In this paper, we study the classical theory of quadratic line complexes and Kummer surfaces. A quadratic line complex is the intersection of the Grassmannian <span><math><mi>G</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span> and a quadric hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>, and a Kummer surface is the quotient of the Jacobian of a curve of genus 2 by the inversion. F. Klein discovered a relationship between a quadratic line complex and a curve of genus 2, its Jacobian and the associated Kummer surface. This theory holds in any characteristic not equal to two. However the situation in characteristic two is entirely different. The purpose of this paper is to give an analogue in characteristic 2 of this classical theory.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110416"},"PeriodicalIF":1.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490328","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":"Theta liftings for (GLn,GLn) type dual pairs of loop groups","authors":"Yanze Chen , Yongchang Zhu","doi":"10.1016/j.aim.2025.110417","DOIUrl":"10.1016/j.aim.2025.110417","url":null,"abstract":"<div><div>In this article we prove the theta liftings of a cusp form on the loop <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> group over <span><math><mi>Q</mi></math></span> induced from a classical cusp form for the loop group version of the “dual pair” <span><math><mo>(</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> is an Eisenstein series.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110417"},"PeriodicalIF":1.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491074","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 Chow ring of the universal Picard stack over the hyperelliptic locus","authors":"Hannah Larson","doi":"10.1016/j.aim.2025.110412","DOIUrl":"10.1016/j.aim.2025.110412","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>J</mi></mrow><mrow><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>→</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> be the universal Picard stack parametrizing degree <em>d</em> line bundles on genus <em>g</em> curves, and let <span><math><msubsup><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> be its restriction to locus of hyperelliptic curves <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>g</mi></mrow></msub><mo>⊂</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>. We determine the rational Chow ring of <span><math><msubsup><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> for all <em>d</em> and <em>g</em>. In particular, we prove it is generated by restrictions of tautological classes on <span><math><msubsup><mrow><mi>J</mi></mrow><mrow><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> and we determine all relations among the restrictions of such classes. We also compute the integral Picard group of <span><math><msubsup><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, completing (and extending the <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-equivariant case) prior work of Erman and Wood. As a corollary, we prove that <span><math><msubsup><mrow><mi>J</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>g</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> is either a trivial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>-gerbe over its rigidification, or has Brauer class of order 2, depending on the parity of <span><math><mi>d</mi><mo>−</mo><mi>g</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110412"},"PeriodicalIF":1.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490329","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":"Intersection probabilities for flats in hyperbolic space","authors":"Ercan Sönmez, Panagiotis Spanos, Christoph Thäle","doi":"10.1016/j.aim.2025.110415","DOIUrl":"10.1016/j.aim.2025.110415","url":null,"abstract":"<div><div>Consider the <em>d</em>-dimensional hyperbolic space <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> of constant curvature <span><math><mi>K</mi><mo><</mo><mn>0</mn></math></span> and fix a point <em>o</em> playing the role of an origin. Let <strong>L</strong> be a uniform random <em>q</em>-dimensional totally geodesic submanifold (called <em>q</em>-flat) in <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> passing through <em>o</em> and, independently of <strong>L</strong>, let <strong>E</strong> be a random <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mi>q</mi><mo>+</mo><mi>γ</mi><mo>)</mo></math></span>-flat in <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> which is uniformly distributed in the set of all <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mi>q</mi><mo>+</mo><mi>γ</mi><mo>)</mo></math></span>-flats intersecting a hyperbolic ball of radius <span><math><mi>u</mi><mo>></mo><mn>0</mn></math></span> around <em>o</em>. We are interested in the distribution of the random <em>γ</em>-flat arising as the intersection of <strong>E</strong> with <strong>L</strong>. In contrast to the Euclidean case, the intersection <span><math><mi>E</mi><mo>∩</mo><mi>L</mi></math></span> can be empty with strictly positive probability. We determine this probability and the full distribution of <span><math><mi>E</mi><mo>∩</mo><mi>L</mi></math></span>. Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behavior as <span><math><mi>d</mi><mo>↑</mo><mo>∞</mo></math></span> and also <span><math><mi>K</mi><mo>↑</mo><mn>0</mn></math></span>. Thereby we obtain a phase transition with three different phases which we completely characterize, including a critical phase with distinctive behavior and a phase recovering the Euclidean results. In the background are methods from hyperbolic integral geometry.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110415"},"PeriodicalIF":1.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490327","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":"Riesz-type calculus for Lorentz-Morrey spaces","authors":"Liguang Liu , Jie Xiao","doi":"10.1016/j.aim.2025.110414","DOIUrl":"10.1016/j.aim.2025.110414","url":null,"abstract":"<div><div>This paper is devoted to a highly nontrivial study of <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo><mo>∋</mo><mi>α</mi></math></span>-order Fourier transform-based Riesz integral-differential calculus for Lorentz-Morrey spaces (covering Lebesgue spaces): (i) Riesz integral traces of Lorentz-Morrey spaces; (ii) Riesz differential equations in dual Lorentz-Morrey spaces; (iii) Riesz variational capacities for Lorentz-Morrey spaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110414"},"PeriodicalIF":1.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491075","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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