{"title":"Geometric structures for the G2′-Hitchin component","authors":"Parker Evans","doi":"10.1016/j.aim.2024.110091","DOIUrl":"10.1016/j.aim.2024.110091","url":null,"abstract":"<div><div>We give an explicit geometric structures interpretation of the <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Hitchin component <span><math><mrow><mi>Hit</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo><mo>⊂</mo><mi>χ</mi><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo></math></span> of a closed oriented surface <em>S</em> of genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span>. In particular, we prove <span><math><mrow><mi>Hit</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo></math></span> is naturally homeomorphic to a moduli space <span><math><mi>M</mi></math></span> of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>-structures for <span><math><mi>G</mi><mo>=</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> and <span><math><mi>X</mi><mo>=</mo><msup><mrow><mi>Ein</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msup></math></span> on a fiber bundle <span><math><mi>C</mi></math></span> over <em>S</em> via the descended holonomy map. Explicitly, <span><math><mi>C</mi></math></span> is the direct sum of fiber bundles <figure><img></figure> with fiber <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>S</mi><mo>×</mo><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>S</mi><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, where <em>UTS</em> denotes the unit tangent bundle.</div><div>The geometric structure associated to a <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Hitchin representation <em>ρ</em> is explicitly constructed from the unique associated <em>ρ</em>-equivariant alternating almost-complex curve <span><math><mover><mrow><mi>ν</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>→</mo><msup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></msup></math></span>; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Fuchsian case and shown to be unrelated to the <span><math><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msup><mrow><mi>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110091"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A semi-pseudo-Kähler structure on the SL(3,R)-Hitchin component and the Goldman symplectic form","authors":"Nicholas Rungi , Andrea Tamburelli","doi":"10.1016/j.aim.2024.110066","DOIUrl":"10.1016/j.aim.2024.110066","url":null,"abstract":"<div><div>The aim of this paper is to show the existence and give an explicit description of a semi-pseudo-Riemannian metric and a symplectic form on the <span><math><mi>SL</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>R</mi><mo>)</mo></math></span>-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they are non-degenerate on a neighborhood of the Fuchsian locus, where they give rise to a mapping class group invariant pseudo-Kähler structure that restricts to a multiple of the Weil-Petersson metric on Teichmüller space. By comparing our symplectic form with Goldman's <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, we prove that the pair <span><math><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>,</mo><mi>I</mi><mo>)</mo></math></span> cannot define a Kähler structure on the Hitchin component.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110066"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hausdorff dimension of singular vectors in function fields","authors":"Noy Soffer Aranov , Taehyeong Kim","doi":"10.1016/j.aim.2024.110084","DOIUrl":"10.1016/j.aim.2024.110084","url":null,"abstract":"<div><div>We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of <em>ε</em>-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier (2016) <span><span>[9]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110084"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164294","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":"Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with L∞ coefficients","authors":"Mihalis Mourgoglou , Thanasis Zacharopoulos","doi":"10.1016/j.aim.2024.110054","DOIUrl":"10.1016/j.aim.2024.110054","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, be an open set with <em>s</em>-Ahlfors regular boundary ∂Ω, for some <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span>, such that either <span><math><mi>s</mi><mo>=</mo><mi>n</mi></math></span> and Ω is a corkscrew domain with the pointwise John condition, or <span><math><mi>s</mi><mo><</mo><mi>n</mi></math></span> and <span><math><mi>Ω</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>E</mi></math></span>, for some <em>s</em>-Ahlfors regular set <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. In this paper we provide a unifying method to construct Varopoulos type extensions of BMO and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> boundary functions. In particular, we show that a) if <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BMO</mi></mrow><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>, there exists <span><math><mi>F</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> such that <span><math><mtext>dist</mtext><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo><mo>|</mo><mi>∇</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math></span> is uniformly bounded in Ω and the Carleson functional of <span><math><mtext>dist</mtext><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi><mo>−</mo><mi>n</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math></span> as well the sharp non-tangential maximal function of <em>F</em> are uniformly bounded on ∂Ω with norms controlled by the BMO-norm of <em>f</em>, and <span><math><mi>F</mi><mo>→</mo><mi>f</mi></math></span> in a certain non-tangential sense <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mo>|</mo></mrow><mrow><mo>∂</mo><mi>Ω</mi></mrow></msub></math></span>-almost everywhere; b) if <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>, <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span>, there exists <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> such that the non-tangential maximal functions of <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> and <span><math><","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110054"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Analytic and topological nets","authors":"Kirill Lazebnik","doi":"10.1016/j.aim.2024.110042","DOIUrl":"10.1016/j.aim.2024.110042","url":null,"abstract":"<div><div>We characterize which planar graphs arise as the pullback, under a rational map <em>r</em>, of an analytic Jordan curve passing through the critical values of <em>r</em>. We also prove that such pullbacks are dense within the collection of <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span>, where <em>f</em> is a branched cover of the sphere and Σ is a Jordan curve passing through the branched values of <em>f</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110042"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164290","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":"Projective rigidity of circle packings","authors":"Francesco Bonsante, Michael Wolf","doi":"10.1016/j.aim.2024.110024","DOIUrl":"10.1016/j.aim.2024.110024","url":null,"abstract":"<div><div>We prove that the space of circle packings compatible with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110024"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The fake monster algebra and singular Borcherds products","authors":"Haowu Wang , Brandon Williams","doi":"10.1016/j.aim.2024.110083","DOIUrl":"10.1016/j.aim.2024.110083","url":null,"abstract":"<div><div>In this paper we consider several problems in the theory of automorphic products and generalized Kac–Moody algebras proposed by Borcherds in 1995. We show that the denominator of the fake monster algebra defines the unique holomorphic Borcherds product of singular weight on a maximal lattice. We give a full classification of symmetric holomorphic Borcherds products of singular weight on lattices of prime level. Finally we prove that all twisted denominator identities of the fake monster algebra arise as the Fourier expansions of Borcherds products of singular weight at a certain cusp. The proofs rely on an identification between modular forms for the Weil representation attached to lattices of type <span><math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>⊕</mo><mi>U</mi><mo>⊕</mo><mi>L</mi></math></span> and certain tuples of Jacobi forms of level <em>N</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110083"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164293","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":"New exotic examples of Ricci limit spaces","authors":"Xilun Li , Shengxuan Zhou","doi":"10.1016/j.aim.2024.110098","DOIUrl":"10.1016/j.aim.2024.110098","url":null,"abstract":"<div><div>For any integers <span><math><mi>m</mi><mo>⩾</mo><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>, we construct a Ricci limit space <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> such that for a fixed point, some tangent cones are <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and some are <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. This is an improvement of Menguy's example <span><span>[3]</span></span>. Moreover, we show that for any finite collection of closed Riemannian manifolds <span><math><mo>(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>Ric</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>⩾</mo><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>)</mo><mo>⩾</mo><mn>1</mn></math></span>, there exists a collapsed Ricci limit space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> such that each Riemannian cone <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is a tangent cone of <em>X</em> at <em>x</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110098"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150241","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}
Dennis Borisov , Ludmil Katzarkov , Artan Sheshmani
{"title":"Shifted symplectic structures on derived Quot-stacks II – derived Quot-schemes as dg manifolds","authors":"Dennis Borisov , Ludmil Katzarkov , Artan Sheshmani","doi":"10.1016/j.aim.2024.110092","DOIUrl":"10.1016/j.aim.2024.110092","url":null,"abstract":"<div><div>It is proved that derived <span><math><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi></math></span>-schemes, as defined by Ciocan-Fontanine and Kapranov, are represented by dg manifolds of finite type. This is the second part of a work aimed to analyze shifted symplectic structures on moduli spaces of coherent sheaves on Calabi–Yau manifolds. The first part related dg manifolds to derived schemes as defined by Toën and Vezzosi.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110092"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity","authors":"Tao Su","doi":"10.1016/j.aim.2024.110101","DOIUrl":"10.1016/j.aim.2024.110101","url":null,"abstract":"<div><div>The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson states that, a smooth Betti moduli space of complex dimension <em>d</em> over a punctured Riemann surface has the dual boundary complex homotopy equivalent to a sphere of dimension <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span>. Via a microlocal geometric perspective, we verify this conjecture for a class of rank <em>n</em> wild character varieties over the two-sphere with one puncture, associated with any Stokes Legendrian link defined by an <em>n</em>-strand positive braid.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110101"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150250","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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