Pub Date : 2026-03-01Epub Date: 2026-01-08DOI: 10.1016/j.aim.2025.110776
Sigrid Grepstad , Mihail N. Kolountzakis
We prove that for any two lattices of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set such that E tiles when translated by L or by M. A consequence of this is that the indicator function of E forms a Weyl–Heisenberg (Gabor) orthogonal basis of when translated by L and modulated by , the dual lattice of M.
{"title":"Bounded common fundamental domains for two lattices","authors":"Sigrid Grepstad , Mihail N. Kolountzakis","doi":"10.1016/j.aim.2025.110776","DOIUrl":"10.1016/j.aim.2025.110776","url":null,"abstract":"<div><div>We prove that for any two lattices <span><math><mi>L</mi><mo>,</mo><mi>M</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set <span><math><mi>E</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that <em>E</em> tiles <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> when translated by <em>L</em> or by <em>M</em>. A consequence of this is that the indicator function of <em>E</em> forms a Weyl–Heisenberg (Gabor) orthogonal basis of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> when translated by <em>L</em> and modulated by <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the dual lattice of <em>M</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110776"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928073","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}
Pub Date : 2026-03-01Epub Date: 2026-01-07DOI: 10.1016/j.aim.2025.110767
Tariq Syed
Cyclic coverings produce many examples of topologically contractible smooth affine complex varieties. In this paper, we study the motivic cohomology groups of cyclic coverings over algebraically closed fields of characteristic 0. In particular, we prove that in many situations Chow groups of cyclic coverings become trivial after tensoring with . Furthermore, we can prove that the Chow groups of certain bicyclic coverings are trivial even without tensoring with .
{"title":"Motivic cohomology of cyclic coverings","authors":"Tariq Syed","doi":"10.1016/j.aim.2025.110767","DOIUrl":"10.1016/j.aim.2025.110767","url":null,"abstract":"<div><div>Cyclic coverings produce many examples of topologically contractible smooth affine complex varieties. In this paper, we study the motivic cohomology groups of cyclic coverings over algebraically closed fields of characteristic 0. In particular, we prove that in many situations Chow groups of cyclic coverings become trivial after tensoring with <span><math><mi>Q</mi></math></span>. Furthermore, we can prove that the Chow groups of certain bicyclic coverings are trivial even without tensoring with <span><math><mi>Q</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110767"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928072","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}
Pub Date : 2026-03-01Epub Date: 2025-12-29DOI: 10.1016/j.aim.2025.110739
Henrik Bachmann , Jan-Willem van Ittersum
We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl–Kaneko–Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an -algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms.
{"title":"Formal multiple Eisenstein series and their derivations","authors":"Henrik Bachmann , Jan-Willem van Ittersum","doi":"10.1016/j.aim.2025.110739","DOIUrl":"10.1016/j.aim.2025.110739","url":null,"abstract":"<div><div>We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl–Kaneko–Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110739"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145847578","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}
Pub Date : 2026-03-01Epub Date: 2026-01-13DOI: 10.1016/j.aim.2025.110768
Caleb Eckhardt , Jianchao Wu
We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on -algebras.
{"title":"Nuclear dimension and virtually polycyclic groups","authors":"Caleb Eckhardt , Jianchao Wu","doi":"10.1016/j.aim.2025.110768","DOIUrl":"10.1016/j.aim.2025.110768","url":null,"abstract":"<div><div>We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>-algebras.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110768"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981119","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}
Pub Date : 2026-03-01Epub Date: 2026-01-22DOI: 10.1016/j.aim.2026.110797
Jin Gao , Meng Yang
We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński carpet cable system, resolving an open problem left by Devyver et al. (2023) [26]. Second, we prove that two-sided heat kernel estimates alone imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on strongly recurrent metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds.
{"title":"Hölder regularity of harmonic functions on metric measure spaces","authors":"Jin Gao , Meng Yang","doi":"10.1016/j.aim.2026.110797","DOIUrl":"10.1016/j.aim.2026.110797","url":null,"abstract":"<div><div>We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a <em>slow</em> volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński <em>carpet</em> cable system, resolving an open problem left by Devyver et al. (2023) <span><span>[26]</span></span>. Second, we prove that two-sided heat kernel estimates <em>alone</em> imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on <em>strongly recurrent</em> metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110797"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039202","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}
Pub Date : 2026-03-01Epub Date: 2025-12-31DOI: 10.1016/j.aim.2025.110758
Gefei Cai , Wen-Bo Li, Tim Mesikepp
We investigate several naturally-arising random fractals from the perspective of quasisymmetric geometry, and show that they fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with Brownian motion and various forms of the Schramm-Loewner evolution for , showing that a.s. neither is a quasisymmetric to a straight line. We also study the conformal loop ensemble for , and show that the collection of all points outside the loops is a.s. homeomorphic to the standard Sierpiński carpet, but not quasisymmetrically equivalent to a round carpet.
{"title":"Quasisymmetric geometry of low-dimensional random spaces","authors":"Gefei Cai , Wen-Bo Li, Tim Mesikepp","doi":"10.1016/j.aim.2025.110758","DOIUrl":"10.1016/j.aim.2025.110758","url":null,"abstract":"<div><div>We investigate several naturally-arising random fractals from the perspective of quasisymmetric geometry, and show that they fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with Brownian motion and various forms of the Schramm-Loewner evolution <span><math><msub><mrow><mi>SLE</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span> for <span><math><mi>κ</mi><mo>></mo><mn>0</mn></math></span>, showing that a.s. neither is a quasisymmetric to a straight line. We also study the conformal loop ensemble <span><math><msub><mrow><mi>CLE</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span> for <span><math><mi>κ</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>4</mn><mo>]</mo></math></span>, and show that the collection of all points outside the loops is a.s. homeomorphic to the standard Sierpiński carpet, but not quasisymmetrically equivalent to a round carpet.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110758"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886099","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}
Pub Date : 2026-03-01Epub Date: 2026-01-19DOI: 10.1016/j.aim.2026.110790
Pak Tung Ho
Cheng, Yang, and Zhang have studied two invariant surface area functionals in 3-dimensional CR manifolds. They deduced the Euler–Lagrange equations of the associated energy functionals when the 3-dimensional CR manifold has constant Webster curvature and vanishing torsion. In this paper, we deduce the Euler–Lagrange equations of the energy functionals in a more general 3-dimensional CR manifold. Moreover, we study the invariant area functionals on the disk bundle, on the Rossi sphere, and on 3-dimensional tori. In particular, we show that the Clifford torus is a minimizer for on the Rossi sphere when . Also, by computing the second variation formula, we show that the Clifford torus is not a minimizer for on the Rossi sphere when .
{"title":"On the invariant surface area functionals in 3-dimensional CR geometry","authors":"Pak Tung Ho","doi":"10.1016/j.aim.2026.110790","DOIUrl":"10.1016/j.aim.2026.110790","url":null,"abstract":"<div><div>Cheng, Yang, and Zhang have studied two invariant surface area functionals in 3-dimensional CR manifolds. They deduced the Euler–Lagrange equations of the associated energy functionals when the 3-dimensional CR manifold has constant Webster curvature and vanishing torsion. In this paper, we deduce the Euler–Lagrange equations of the energy functionals in a more general 3-dimensional CR manifold. Moreover, we study the invariant area functionals on the disk bundle, on the Rossi sphere, and on 3-dimensional tori. In particular, we show that the Clifford torus is a minimizer for <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> on the Rossi sphere <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> when <span><math><mi>t</mi><mo>=</mo><mo>−</mo><mn>4</mn><mo>+</mo><msqrt><mrow><mn>15</mn></mrow></msqrt></math></span>. Also, by computing the second variation formula, we show that the Clifford torus is not a minimizer for <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> on the Rossi sphere <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> when <span><math><mi>t</mi><mo>></mo><mo>−</mo><mn>4</mn><mo>+</mo><msqrt><mrow><mn>15</mn></mrow></msqrt></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110790"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039198","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}
Pub Date : 2026-03-01Epub Date: 2026-01-19DOI: 10.1016/j.aim.2025.110773
Pedro Abdalla , Afonso S. Bandeira , Martin Kassabov , Victor Souza , Steven H. Strogatz , Alex Townsend
The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any and , the homogeneous Kuramoto model on the Erdős–Rényi random graph is globally synchronizing with probability tending to one as n goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the Kuramoto model is globally synchronizing on any d-regular Ramanujan graph, and on typical d-regular graphs, for .
{"title":"Expander graphs are globally synchronizing","authors":"Pedro Abdalla , Afonso S. Bandeira , Martin Kassabov , Victor Souza , Steven H. Strogatz , Alex Townsend","doi":"10.1016/j.aim.2025.110773","DOIUrl":"10.1016/j.aim.2025.110773","url":null,"abstract":"<div><div>The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>p</mi><mo>⩾</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo><mo>/</mo><mi>n</mi></math></span>, the homogeneous Kuramoto model on the Erdős–Rényi random graph <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is globally synchronizing with probability tending to one as <em>n</em> goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the Kuramoto model is globally synchronizing on any <em>d</em>-regular Ramanujan graph, and on typical <em>d</em>-regular graphs, for <span><math><mi>d</mi><mo>⩾</mo><mn>600</mn></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110773"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039200","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}
Pub Date : 2026-03-01Epub Date: 2026-01-19DOI: 10.1016/j.aim.2026.110787
Ayush Khaitan
We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the Ricci flow by constructing infinite families of fully non-linear analogues of Perelman's and functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a “Ricci flow vector field” in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.
{"title":"The weighted ambient metric for manifolds with density","authors":"Ayush Khaitan","doi":"10.1016/j.aim.2026.110787","DOIUrl":"10.1016/j.aim.2026.110787","url":null,"abstract":"<div><div>We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the Ricci flow by constructing infinite families of fully non-linear analogues of Perelman's <span><math><mi>F</mi></math></span> and <span><math><mi>W</mi></math></span> functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a “Ricci flow vector field” in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110787"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039257","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}
Pub Date : 2026-03-01Epub Date: 2026-01-08DOI: 10.1016/j.aim.2025.110775
Matej Filip
We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair , defined by a polytope P, and mutations of a Laurent polynomial f with Newton polytope . For a Laurent polynomial f in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair over , where is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if f is 0-mutable. The Kodaira–Spencer map of the constructed deformation is injective, and if f is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space.
{"title":"Laurent polynomials and deformations of non-isolated Gorenstein toric singularities","authors":"Matej Filip","doi":"10.1016/j.aim.2025.110775","DOIUrl":"10.1016/j.aim.2025.110775","url":null,"abstract":"<div><div>We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair <span><math><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>,</mo><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>)</mo></math></span>, defined by a polytope <em>P</em>, and mutations of a Laurent polynomial <em>f</em> with Newton polytope <span><math><mi>Δ</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>P</mi></math></span>. For a Laurent polynomial <em>f</em> in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair <span><math><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></msub><mo>,</mo><mo>∂</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></msub><mo>)</mo></math></span> over <span><math><mi>C</mi><mo>[</mo><mo>[</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>]</mo><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if <em>f</em> is 0-mutable. The Kodaira–Spencer map of the constructed deformation is injective, and if <em>f</em> is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110775"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928077","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号