Advances in Mathematics最新文献

{"title":"Topological dynamical systems induced by polynomials and combinatorial consequences","authors":"Wen Huang,&nbsp;Song Shao,&nbsp;Xiangdong Ye","doi":"10.1016/j.aim.2025.110440","DOIUrl":"10.1016/j.aim.2025.110440","url":null,"abstract":"<div><div>Let <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> be an integral polynomial with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math></span>. It is shown that if <em>S</em> is piecewise syndetic in <span><math><mi>Z</mi></math></span>, then<span><span><span><math><mo>{</mo><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>m</mi><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>∈</mo><mi>S</mi><mo>}</mo></math></span></span></span> is piecewise syndetic in <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, which extends the result by Glasner and Furstenberg for linear polynomials. Our result is obtained by showing the density of minimal points of a dynamical system of a <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> action associated with the piecewise syndetic set <em>S</em> and the polynomials <span><math><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>}</mo></math></span>.</div><div>Moreover, it is proved that if <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span> is minimal, then for each non-empty open subset <em>U</em> of <em>X</em>, there is <span><math><mi>x</mi><mo>∈</mo><mi>U</mi></math></span> with <span><math><mo>{</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mi>x</mi><mo>∈</mo><mi>U</mi><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mi>x</mi><mo>∈</mo><mi>U</mi><mo>}</mo></math></span> piecewise syndetic.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110440"},"PeriodicalIF":1.5,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687569","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":"Poissonian actions of Polish groups","authors":"Nachi Avraham-Re'em ,&nbsp;Emmanuel Roy","doi":"10.1016/j.aim.2025.110437","DOIUrl":"10.1016/j.aim.2025.110437","url":null,"abstract":"<div><div>We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct <em>spatial</em> Poissonian actions, answering partially a question of Glasner, Tsirelson &amp; Weiss about Lévy groups. We also construct for every diffeomorphism group a weakly mixing free spatial probability-preserving action. This constitutes a new class of Polish groups admitting non-essentially countable orbit equivalence relations, obtaining progress on a problem of Kechris.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110437"},"PeriodicalIF":1.5,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687568","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":"Sharp quantitative stability of Struwe's decomposition of the Poincaré-Sobolev inequalities on the hyperbolic space: Part I","authors":"Mousomi Bhakta ,&nbsp;Debdip Ganguly ,&nbsp;Debabrata Karmakar ,&nbsp;Saikat Mazumdar","doi":"10.1016/j.aim.2025.110447","DOIUrl":"10.1016/j.aim.2025.110447","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this article, we study the quantitative stability of the Poincaré-Sobolev equation on the hyperbolic space&lt;span&gt;&lt;span&gt;&lt;span&gt;(&lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;)&lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;. If we consider the upper half space model for the hyperbolic space, then the solutions to &lt;span&gt;&lt;span&gt;(&lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;)&lt;/span&gt;&lt;/span&gt; have certain equivalence with the &lt;em&gt;cylindrically symmetric solutions&lt;/em&gt; to the &lt;em&gt;Hardy-Sobolev-Mazy'a&lt;/em&gt; equation on the Euclidean space.&lt;/div&gt;&lt;div&gt;A classical result owing to Mancini and Sandeep &lt;span&gt;&lt;span&gt;[43]&lt;/span&gt;&lt;/span&gt; asserts that all positive solutions to &lt;span&gt;&lt;span&gt;(&lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;)&lt;/span&gt;&lt;/span&gt; are unique up to hyperbolic isometries, which henceforth will be called the &lt;em&gt;hyperbolic bubbles&lt;/em&gt;. In the spirit of Struwe, Bhakta-Sandeep &lt;span&gt;&lt;span&gt;[6]&lt;/span&gt;&lt;/span&gt; proved the following non-quantitative stability result: if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mtext&gt;dist&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mtext&gt;dist&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110447"},"PeriodicalIF":1.5,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144678810","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":"Finitely generated infinite torsion groups that are residually finite simple","authors":"Eduard Schesler","doi":"10.1016/j.aim.2025.110441","DOIUrl":"10.1016/j.aim.2025.110441","url":null,"abstract":"<div><div>We show that every countable residually finite torsion group <em>G</em> embeds in a finitely generated torsion group Γ that is residually <em>finite simple</em>. In particular we show the existence of finitely generated infinite torsion groups that are residually finite simple, which answers a question of Olshanskii and Osin.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110441"},"PeriodicalIF":1.5,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144670683","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":"Morse properties in convex projective geometry","authors":"Mitul Islam ,&nbsp;Theodore Weisman","doi":"10.1016/j.aim.2025.110430","DOIUrl":"10.1016/j.aim.2025.110430","url":null,"abstract":"<div><div>We study properties of “hyperbolic directions” in groups acting cocompactly on properly convex domains in real projective space, from three different perspectives simultaneously: the (coarse) metric geometry of the Hilbert metric, the projective geometry of the boundary of the domain, and the singular value gaps of projective automorphisms. We describe the relationship between different definitions of “Morse” and “regular” quasi-geodesics arising in these three different contexts. This generalizes several results of Benoist and Guichard to the non-Gromov hyperbolic setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110430"},"PeriodicalIF":1.5,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654949","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":"Geometry and periods of G2-moduli spaces","authors":"Thibault Langlais","doi":"10.1016/j.aim.2025.110435","DOIUrl":"10.1016/j.aim.2025.110435","url":null,"abstract":"<div><div>This paper is concerned with the geometry of the moduli space <span><math><mi>M</mi></math></span> of torsion-free <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structures on a compact <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-manifold <em>M</em>, equipped with the volume-normalised <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-metric <span><math><mi>G</mi></math></span>. When <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, this metric is known to be of Hessian type and to admit a global potential. Here we give a new description of the geometry of <span><math><mi>M</mi></math></span>, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space <span><math><mi>D</mi></math></span> diffeomorphic to <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mo>{</mo><mo>±</mo><mn>1</mn><mo>}</mo><mo>×</mo><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>n</mi><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>. We point out that the formal properties of this immersion <span><math><mi>Φ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>D</mi></math></span> are very similar to those of the period map defined on the moduli spaces of Calabi–Yau threefolds. With a view to understand the curvatures of <span><math><mi>G</mi></math></span>, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of <span><math><mi>Φ</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>⊂</mo><mi>D</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110435"},"PeriodicalIF":1.5,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654840","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":"Virtual localization revisited","authors":"Dhyan Aranha ,&nbsp;Adeel A. Khan ,&nbsp;Alexei Latyntsev ,&nbsp;Hyeonjun Park ,&nbsp;Charanya Ravi","doi":"10.1016/j.aim.2025.110434","DOIUrl":"10.1016/j.aim.2025.110434","url":null,"abstract":"<div><div>Let <em>T</em> be a split torus acting on an algebraic scheme <em>X</em> with fixed locus <em>Z</em>. Edidin and Graham showed that on localized <em>T</em>-equivariant Chow groups, (a) push-forward <span><math><msub><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> along <span><math><mi>i</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></math></span> is an isomorphism, and (b) when <em>X</em> is smooth the inverse <span><math><msup><mrow><mo>(</mo><msub><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> can be described via Gysin pullback <span><math><msup><mrow><mi>i</mi></mrow><mrow><mo>!</mo></mrow></msup></math></span> and cap product with <span><math><mi>e</mi><msup><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, the inverse of the Euler class of the normal bundle <em>N</em>. In this paper we show that (b) still holds when <em>X</em> is a quasi-smooth derived scheme (or Deligne–Mumford stack), using virtual versions of the operations <span><math><msup><mrow><mi>i</mi></mrow><mrow><mo>!</mo></mrow></msup></math></span> and <span><math><mo>(</mo><mo>−</mo><mo>)</mo><mo>∩</mo><mi>e</mi><msup><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. As a corollary we prove the virtual localization formula <span><math><msup><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mrow><mi>vir</mi></mrow></msup><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><msup><mrow><mo>[</mo><mi>Z</mi><mo>]</mo></mrow><mrow><mi>vir</mi></mrow></msup><mo>∩</mo><mi>e</mi><msup><mrow><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>vir</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> of Graber–Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110434"},"PeriodicalIF":1.5,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633177","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":"Exact structures and purity","authors":"Kevin Schlegel","doi":"10.1016/j.aim.2025.110433","DOIUrl":"10.1016/j.aim.2025.110433","url":null,"abstract":"<div><div>We relate the theory of purity of a locally finitely presented category with products to the study of exact structures on the full subcategory of finitely presented objects. Properties in the context of purity are translated to properties about exact structures. We specialize to the case of a module category over an Artin algebra and show that generic modules are in one to one correspondence with particular maximal exact structures.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110433"},"PeriodicalIF":1.5,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596904","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":"Blow-ups and the quantum spectrum of surfaces","authors":"Ádám Gyenge ,&nbsp;Szilárd Szabó","doi":"10.1016/j.aim.2025.110432","DOIUrl":"10.1016/j.aim.2025.110432","url":null,"abstract":"<div><div>We investigate the behavior of the spectrum of the quantum (or Dubrovin) connection of smooth projective surfaces under blow-ups. Our main result is that for small values of the parameters, the quantum spectrum of such a surface is asymptotically the union of the quantum spectrum of a minimal model of the surface and a finite number of additional points located “close to infinity”, that correspond bijectively to the exceptional divisors. This proves a conjecture of Kontsevich in the surface case.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110432"},"PeriodicalIF":1.5,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144587582","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 Rees algebra and analytic spread of a divisorial filtration","authors":"Steven Dale Cutkosky","doi":"10.1016/j.aim.2025.110428","DOIUrl":"10.1016/j.aim.2025.110428","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper we investigate some properties of Rees algebras of divisorial filtrations and their analytic spread. A classical theorem of McAdam shows that the analytic spread of an ideal &lt;em&gt;I&lt;/em&gt; in a formally equidimensional local ring is equal to the dimension of the ring if and only if the maximal ideal is an associated prime of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; for some &lt;em&gt;n&lt;/em&gt;. We show in Theorem 1.5 that McAdam's theorem holds for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtrations in an equidimensional local ring which is essentially of finite type over an excellent local ring of dimension less than or equal to 3. This generalizes an earlier result for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtrations in an equicharacteristic zero excellent local domain by the author. This theorem does not hold for more general filtrations.&lt;/div&gt;&lt;div&gt;We consider the question of the asymptotic behavior of the function &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;↦&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtration &lt;span&gt;&lt;math&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-primary ideals on a &lt;em&gt;d&lt;/em&gt;-dimensional normal excellent local ring. It is known from earlier work of the author that the multiplicity&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;munder&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; can be irrational. We show in Lemma 4.1 that the limsup of the first difference function&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;sup&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is always finite for a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtration. We then give an example in Section 4 showing that this limsup may not exist as a limit.&lt;/div&gt;&lt;div&gt;In the final section, we","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110428"},"PeriodicalIF":1.5,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144581248","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}