Christian Böhning , Hans-Christian Graf von Bothmer , Yuri Tschinkel
{"title":"Equivariant birational geometry of cubic fourfolds and derived categories","authors":"Christian Böhning , Hans-Christian Graf von Bothmer , Yuri Tschinkel","doi":"10.1016/j.aim.2025.110249","DOIUrl":"10.1016/j.aim.2025.110249","url":null,"abstract":"<div><div>We study equivariant birationality from the perspective of derived categories. We produce examples of nonlinearizable but stably linearizable actions of finite groups on smooth cubic fourfolds.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110249"},"PeriodicalIF":1.5,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746669","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":"Local terms for the categorical trace","authors":"Dennis Gaitsgory , Yakov Varshavsky","doi":"10.1016/j.aim.2025.110223","DOIUrl":"10.1016/j.aim.2025.110223","url":null,"abstract":"<div><div>In this paper we introduce the categorical “true local terms” maps for Artin stacks and show that they are additive and commute with proper pushforwards, smooth pullbacks and specializations. In particular, we generalizing results of <span><span>[14]</span></span> to this setting.</div><div>As an application, we supply proofs of two theorems stated in <span><span>[1]</span></span>. Namely, we show that the “true local terms” of the Frobenius endomorphism coincide with the “naive local terms” and that the “naive local terms” commute with !-pushforwards. The latter result is a categorical version of the classical Grothendieck–Lefschetz trace formula.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110223"},"PeriodicalIF":1.5,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746788","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}
Debsoumya Chakraborti , Oliver Janzer , Abhishek Methuku , Richard Montgomery
{"title":"Edge-disjoint cycles with the same vertex set","authors":"Debsoumya Chakraborti , Oliver Janzer , Abhishek Methuku , Richard Montgomery","doi":"10.1016/j.aim.2025.110228","DOIUrl":"10.1016/j.aim.2025.110228","url":null,"abstract":"<div><div>In 1975, Erdős asked for the maximum number of edges that an <em>n</em>-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Turán-type results can be used to prove an upper bound of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. However, this approach cannot give an upper bound better than <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>. We show that there is an absolute constant <em>t</em> and some constant <span><math><mi>c</mi><mo>=</mo><mi>c</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> such that for each <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, every <em>n</em>-vertex graph with at least <span><math><mi>c</mi><mi>n</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>t</mi></mrow></msup></math></span> edges contains <em>k</em> pairwise edge-disjoint cycles with the same vertex set, resolving this old problem in a strong form up to a polylogarithmic factor. The well-known construction of Pyber, Rödl and Szemerédi of graphs without 4-regular subgraphs shows that there are <em>n</em>-vertex graphs with <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> edges which do not contain two cycles with the same vertex set, so the polylogarithmic term in our result cannot be completely removed.</div><div>Our proof combines a variety of techniques including sublinear expanders, absorption and a novel tool for regularisation, which is of independent interest. Among other applications, this tool can be used to regularise an expander while still preserving certain key expansion properties.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110228"},"PeriodicalIF":1.5,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739280","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":"Gibbs measures for geodesic flow on CAT(-1) spaces","authors":"Caleb Dilsavor , Daniel J. Thompson","doi":"10.1016/j.aim.2025.110231","DOIUrl":"10.1016/j.aim.2025.110231","url":null,"abstract":"<div><div>For a proper geodesically complete CAT(-1) space equipped with a discrete non-elementary action, and a bounded continuous potential with the Bowen property, we construct weighted quasi-conformal Patterson densities and use them to build a Gibbs measure on the space of geodesic lines. Our construction yields a Gibbs measure with local product structure for any potential in this class, which includes bounded Hölder continuous potentials. Furthermore, if the Gibbs measure is finite, then we prove that it is the unique equilibrium state. In contrast to previous results in this direction, we do not require any condition that the potential must take the same value on two geodesic lines which share a common segment.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110231"},"PeriodicalIF":1.5,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739282","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 geometric concentration theorem","authors":"Olivier Haution","doi":"10.1016/j.aim.2025.110237","DOIUrl":"10.1016/j.aim.2025.110237","url":null,"abstract":"<div><div>We establish a purely geometric form of the concentration theorem (also called localization theorem) for actions of a linearly reductive group <em>G</em> on an affine scheme <em>X</em> over an affine base scheme <em>S</em>. It asserts the existence of a <em>G</em>-representation without trivial summand over <em>S</em>, which acquires over <em>X</em> an equivariant section vanishing precisely at the fixed locus of <em>X</em>.</div><div>As a consequence, we show that the equivariant stable motivic homotopy theory of a scheme with an action of a linearly reductive group is equivalent to that of the fixed locus, upon inverting appropriate maps, namely the Euler classes of representations without trivial summands. We also discuss consequences for equivariant cohomology theories obtained using Borel's construction. This recovers most known forms of the concentration theorem in algebraic geometry, and yields generalizations valid beyond the setting of actions of diagonalizable groups on one hand, and that of oriented cohomology theories on the other hand.</div><div>Finally, we derive a version of Smith theory for motivic cohomology, following the approach of Dwyer–Wilkerson in topology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110237"},"PeriodicalIF":1.5,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739281","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":"Boundary behaviour of universal covering maps","authors":"Gustavo R. Ferreira , Anna Jové","doi":"10.1016/j.aim.2025.110232","DOIUrl":"10.1016/j.aim.2025.110232","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> be a multiply connected domain, and let <span><math><mi>π</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>Ω</mi></math></span> be a universal covering map. In this paper, we analyze the boundary behaviour of <em>π</em>, describing the interplay between radial limits and angular cluster sets, the tangential and non-tangential limit sets of the deck transformation group, and the geometry and the topology of the boundary of Ω.</div><div>As an application, we describe accesses to the boundary of Ω in terms of radial limits of points in the unit circle, establishing a correspondence, in the same spirit as in the simply connected case. We also develop a theory of prime ends for multiply connected domains which behaves properly under the universal covering, providing an extension of the Carathéodory–Torhorst Theorem to multiply connected domains.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110232"},"PeriodicalIF":1.5,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739208","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 point spaces of symbolic dynamical systems","authors":"Jason P. Bell , Be'eri Greenfeld","doi":"10.1016/j.aim.2025.110211","DOIUrl":"10.1016/j.aim.2025.110211","url":null,"abstract":"<div><div>We study point modules of monomial algebras associated with symbolic dynamical systems, parametrized by proalgebraic varieties which ‘linearize’ the underlying dynamical systems. Faithful point modules correspond to transitive sub-systems, equivalently, to monomial algebras associated with infinite words. In particular, we prove that the space of point modules of every prime monomial algebra with Hilbert series <span><math><mn>1</mn><mo>/</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>—which is thus thought of as a ‘monomial <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>’—is isomorphic to a union of a classical projective line with a Cantor set. While there is a continuum of monomial <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>'s with non-equivalent graded module categories, they all share isomorphic parametrizing spaces of point modules. In contrast, free algebras are geometrically rigid, and are characterized up to isomorphism from their spaces of point modules.</div><div>Furthermore, we derive enumerative and ring-theoretic consequences from our analysis. In particular, we show that the formal power series counting the irreducible components of the parametrizing spaces of truncated point modules of finitely presented monomial algebras are rational functions, and classify isomorphisms and automorphisms of projectively simple monomial algebras.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110211"},"PeriodicalIF":1.5,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705006","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":"Corrigendum to “A Rockafellar-type theorem for non-traditional costs” [Adv. Math. 395 (2022) 108157]","authors":"S. Artstein-Avidan, S. Sadovsky, K. Wyczesany","doi":"10.1016/j.aim.2025.110227","DOIUrl":"10.1016/j.aim.2025.110227","url":null,"abstract":"<div><div>This note describes corrections to an error in the published version of the paper “A Rockafellar-type theorem for non-traditional costs” regarding the solvability of an uncountable family of inequalities. In this note, we describe the mathematical error and show that one must add an extra assumption – either countability of the family or an assumption on the coefficients not allowing the existence of what we call an infinite “black hole”.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110227"},"PeriodicalIF":1.5,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705007","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 capillary Minkowski problem","authors":"Xinqun Mei , Guofang Wang , Liangjun Weng","doi":"10.1016/j.aim.2025.110230","DOIUrl":"10.1016/j.aim.2025.110230","url":null,"abstract":"<div><div>In this article, we introduce a capillary Minkowski problem, which asks for the existence of a strictly convex capillary hypersurface <span><math><mi>Σ</mi><mo>⊂</mo><mover><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><mo>‾</mo></mover></math></span> supported on <span><math><mover><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><mo>‾</mo></mover></math></span> with a prescribed Gauss-Kronecker curvature on a spherical cap <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span>. We reduce it to a Monge-Ampère type equation with a Robin boundary value problem and then obtain a necessary and sufficient condition for solving this problem provided <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></math></span>. The restriction <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></math></span> comes from the difficult part of the proof, <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-estimation. We manage to prove <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-estimates by using this restriction and leave the problem open if <span><math><mi>θ</mi><mo>></mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. This is a natural Robin boundary version of the classical Minkowski problem.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110230"},"PeriodicalIF":1.5,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704919","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 orbifold DT/PT vertex correspondence","authors":"Yijie Lin","doi":"10.1016/j.aim.2025.110222","DOIUrl":"10.1016/j.aim.2025.110222","url":null,"abstract":"<div><div>We present an orbifold topological vertex formalism for PT invariants of toric Calabi-Yau 3-orbifolds with transverse <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> singularities. We give a proof of the orbifold DT/PT Calabi-Yau topological vertex correspondence. As an application, we derive an explicit formula for the PT <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-vertex in terms of loop Schur functions and prove the multi-regular orbifold DT/PT correspondence.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110222"},"PeriodicalIF":1.5,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705005","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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