Pub 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":"2026-01-19","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 : 2026-01-19DOI: 10.1016/j.aim.2026.110791
Kan Jiang , Junjie Miao , Lifeng Xi
It is well-known that fractal dimensions are invariant under bi-Lipschitz mappings on Euclidean spaces, and therefore, bi-Lipschitz mappings are important in the classification of fractal sets. On locally finite discrete metric spaces, bi-Lipschitz mappings are a class of special quasi-isometries which constitute a fundamental concept in geometric group theory.
In this paper, we extend discrete fractal dimensions to locally finite discrete metric spaces, establishing their quasi-isometric invariance. For discrete self-similar sets with integer digits, we prove a complete classification of bi-Lipschitz and quasi-isometric equivalences, providing a discrete analogue to Falconer and Marsh's seminal results on Lipschitz equivalence of self-similar fractals. Our main theorem shows that two non-trivial such sets are quasi-isometric if and only if the logarithm of their scaling ratios and digit set cardinalities are rationally proportional. Furthermore, the bi-Lipschitz equivalence of these structures is strictly determined by the inclusion of zero in their digit sets, distinguishing them from standard self-similar fractals.
{"title":"Discrete fractals: Dimensions, quasi-isometric invariance and self-similarity","authors":"Kan Jiang , Junjie Miao , Lifeng Xi","doi":"10.1016/j.aim.2026.110791","DOIUrl":"10.1016/j.aim.2026.110791","url":null,"abstract":"<div><div>It is well-known that fractal dimensions are invariant under bi-Lipschitz mappings on Euclidean spaces, and therefore, bi-Lipschitz mappings are important in the classification of fractal sets. On locally finite discrete metric spaces, bi-Lipschitz mappings are a class of special quasi-isometries which constitute a fundamental concept in geometric group theory.</div><div>In this paper, we extend discrete fractal dimensions to locally finite discrete metric spaces, establishing their quasi-isometric invariance. For discrete self-similar sets with integer digits, we prove a complete classification of bi-Lipschitz and quasi-isometric equivalences, providing a discrete analogue to Falconer and Marsh's seminal results on Lipschitz equivalence of self-similar fractals. Our main theorem shows that two non-trivial such sets are quasi-isometric if and only if the logarithm of their scaling ratios and digit set cardinalities are rationally proportional. Furthermore, the bi-Lipschitz equivalence of these structures is strictly determined by the inclusion of zero in their digit sets, distinguishing them from standard self-similar fractals.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110791"},"PeriodicalIF":1.5,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039199","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 : 2026-01-15DOI: 10.1016/j.aim.2025.110762
Dimitri Ara , Léonard Guetta
Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict ω-categories. To state the most general functorialities, we use the language of Gray ω-categories, that is, categories enriched in the category of strict ω-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict ω-categories defines a Gray ω-functor, that is, a morphism of Gray ω-categories. To makes sense of this statement, we prove that slices of Gray ω-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict ω-categories defines a Gray ω-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray ω-functors, which we plan to investigate in future work.
{"title":"Lax functorialities of the comma construction for ω-categories","authors":"Dimitri Ara , Léonard Guetta","doi":"10.1016/j.aim.2025.110762","DOIUrl":"10.1016/j.aim.2025.110762","url":null,"abstract":"<div><div>Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict <em>ω</em>-categories. To state the most general functorialities, we use the language of Gray <em>ω</em>-categories, that is, categories enriched in the category of strict <em>ω</em>-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict <em>ω</em>-categories defines a Gray <em>ω</em>-functor, that is, a morphism of Gray <em>ω</em>-categories. To makes sense of this statement, we prove that slices of Gray <em>ω</em>-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict <em>ω</em>-categories defines a Gray <em>ω</em>-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray <em>ω</em>-functors, which we plan to investigate in future work.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110762"},"PeriodicalIF":1.5,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981068","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 : 2026-01-15DOI: 10.1016/j.aim.2026.110785
Tal Gottesman
We prove that the bounded derived category of the lattice of order ideals of the product of two ordered chains is fractionally Calabi–Yau. We also show that these lattices are derived equivalent to higher Auslander algebras of type A. The proofs involve the study of intervals of the poset that have resolutions described with antichains having rigid properties. These two results combined corroborate a conjecture by Chapoton linking posets to Fukaya–Seidel Categories.
{"title":"Fractionally Calabi–Yau lattices that tilt to higher Auslander algebras of type A","authors":"Tal Gottesman","doi":"10.1016/j.aim.2026.110785","DOIUrl":"10.1016/j.aim.2026.110785","url":null,"abstract":"<div><div>We prove that the bounded derived category of the lattice of order ideals of the product of two ordered chains is fractionally Calabi–Yau. We also show that these lattices are derived equivalent to higher Auslander algebras of type A. The proofs involve the study of intervals of the poset that have resolutions described with antichains having rigid properties. These two results combined corroborate a conjecture by Chapoton linking posets to Fukaya–Seidel Categories.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110785"},"PeriodicalIF":1.5,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981070","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 : 2026-01-15DOI: 10.1016/j.aim.2026.110786
Zhu Ye
Let M be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of M is always greater than or equal to the dimension of some (or every) asymptotic cone of M.
Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of M equals k, then there exists an asymptotic cone of M whose upper box dimension is at most k. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature.
In the subsequent part of the paper, we extend or partially extend Sormani's results concerning M with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with and linear volume growth.
Finally, we construct an example of an open n-manifold M with whose volume growth order oscillates between 1 and n.
{"title":"Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature","authors":"Zhu Ye","doi":"10.1016/j.aim.2026.110786","DOIUrl":"10.1016/j.aim.2026.110786","url":null,"abstract":"<div><div>Let <em>M</em> be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of <em>M</em> is always greater than or equal to the dimension of some (or every) asymptotic cone of <em>M</em>.</div><div>Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of <em>M</em> equals <em>k</em>, then there exists an asymptotic cone of <em>M</em> whose upper box dimension is at most <em>k</em>. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature.</div><div>In the subsequent part of the paper, we extend or partially extend Sormani's results concerning <em>M</em> with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with <span><math><mrow><mi>Ric</mi></mrow><mo>≥</mo><mn>0</mn></math></span> and linear volume growth.</div><div>Finally, we construct an example of an open <em>n</em>-manifold <em>M</em> with <span><math><msub><mrow><mi>sec</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span> whose volume growth order oscillates between 1 and <em>n</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110786"},"PeriodicalIF":1.5,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981226","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 : 2026-01-14DOI: 10.1016/j.aim.2025.110774
Avy Soffer , Xiaoxu Wu
For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in an adapted way the Free Channel Wave Operator, and further tools from the recent works [21], [22], [35]. This work generalizes the results of the first part of [21], [22] to arbitrary dimension, and non-radial data.
{"title":"On the large time asymptotics of Schrödinger type equations with general data","authors":"Avy Soffer , Xiaoxu Wu","doi":"10.1016/j.aim.2025.110774","DOIUrl":"10.1016/j.aim.2025.110774","url":null,"abstract":"<div><div>For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in an adapted way the Free Channel Wave Operator, and further tools from the recent works <span><span>[21]</span></span>, <span><span>[22]</span></span>, <span><span>[35]</span></span>. This work generalizes the results of the first part of <span><span>[21]</span></span>, <span><span>[22]</span></span> to arbitrary dimension, and non-radial data.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110774"},"PeriodicalIF":1.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981067","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 : 2026-01-14DOI: 10.1016/j.aim.2025.110770
Lukas Müller , Christoph Schweigert , Lukas Woike , Yang Yang
The Levin-Wen string-nets of a spherical fusion category describe, by results of Kirillov and Bartlett, the representations of mapping class groups of closed surfaces obtained from the Turaev-Viro construction applied to . We provide a far-reaching generalization of this statement to arbitrary pivotal finite tensor categories, including non-semisimple or non-spherical ones: We show that the finitely cocompleted string-net modular functor built from the projective objects of a pivotal finite tensor category is equivalent to Lyubashenko's modular functor built from the Drinfeld center .
{"title":"The Lyubashenko modular functor for Drinfeld centers via non-semisimple string-nets","authors":"Lukas Müller , Christoph Schweigert , Lukas Woike , Yang Yang","doi":"10.1016/j.aim.2025.110770","DOIUrl":"10.1016/j.aim.2025.110770","url":null,"abstract":"<div><div>The Levin-Wen string-nets of a spherical fusion category <span><math><mi>C</mi></math></span> describe, by results of Kirillov and Bartlett, the representations of mapping class groups of closed surfaces obtained from the Turaev-Viro construction applied to <span><math><mi>C</mi></math></span>. We provide a far-reaching generalization of this statement to arbitrary pivotal finite tensor categories, including non-semisimple or non-spherical ones: We show that the finitely cocompleted string-net modular functor built from the projective objects of a pivotal finite tensor category is equivalent to Lyubashenko's modular functor built from the Drinfeld center <span><math><mi>Z</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110770"},"PeriodicalIF":1.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981069","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 : 2026-01-13DOI: 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":"2026-01-13","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 : 2026-01-13DOI: 10.1016/j.aim.2025.110771
Laurenţiu Maxim , Jörg Schürmann
<div><div>We give a <em>K</em>-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope <em>P</em>, depending on a given homogeneous polynomial function <em>φ</em> on <em>P</em>, and with Laurent polynomial weights <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span> associated to the faces <span><math><mi>Q</mi><mo>⪯</mo><mi>P</mi></math></span> of the polytope. For this purpose, we calculate equivariant <em>K</em>-theoretic Hodge–Chern classes of a torus-equivariant mixed Hodge module <span><math><mi>M</mi></math></span> on the toric variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> associated to <em>P</em> (defined via an equivariant embedding of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> into an ambient smooth variety). For any integer <em>ℓ</em>, we introduce a corresponding equivariant Hodge <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span>-polynomial <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>,</mo><mi>ℓ</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>;</mo><mo>[</mo><mi>M</mi><mo>]</mo><mo>)</mo></math></span>, with <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> the corresponding ample Cartier divisor on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> (defined by the facet presentation of <em>P</em>). Motivic properties of the Hodge–Chern classes are used to express this equivariant Hodge polynomial in terms of weighted character sums fitting with a generalized weighted Ehrhart theory. The equivariant Hodge polynomials are shown to satisfy a reciprocity and purity formula fitting with the duality for equivariant mixed Hodge modules, and implying the corresponding properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function corresponding to Stanley's <em>g</em>-function of the polar polytope of <em>P</em>, we recover in geometric terms a recent combinatorial formula of Beck–Gunnells–Materov. More generally, motivated by the analogy to the Kazhdan–Lusztig theory, we introduce a duality involution on the free <span><math><mi>Z</mi><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span>-module of weight functions corresponding to the duality of equivariant mixed Hodge modules, and prove a new reciprocity formula in terms of this duality. This unifies and generalizes the classical reciprocity formula of Brion–Vergne in Ehrhart theory as well as t
{"title":"Weighted Ehrhart theory via equivariant toric geometry","authors":"Laurenţiu Maxim , Jörg Schürmann","doi":"10.1016/j.aim.2025.110771","DOIUrl":"10.1016/j.aim.2025.110771","url":null,"abstract":"<div><div>We give a <em>K</em>-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope <em>P</em>, depending on a given homogeneous polynomial function <em>φ</em> on <em>P</em>, and with Laurent polynomial weights <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span> associated to the faces <span><math><mi>Q</mi><mo>⪯</mo><mi>P</mi></math></span> of the polytope. For this purpose, we calculate equivariant <em>K</em>-theoretic Hodge–Chern classes of a torus-equivariant mixed Hodge module <span><math><mi>M</mi></math></span> on the toric variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> associated to <em>P</em> (defined via an equivariant embedding of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> into an ambient smooth variety). For any integer <em>ℓ</em>, we introduce a corresponding equivariant Hodge <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span>-polynomial <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>,</mo><mi>ℓ</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>;</mo><mo>[</mo><mi>M</mi><mo>]</mo><mo>)</mo></math></span>, with <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> the corresponding ample Cartier divisor on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> (defined by the facet presentation of <em>P</em>). Motivic properties of the Hodge–Chern classes are used to express this equivariant Hodge polynomial in terms of weighted character sums fitting with a generalized weighted Ehrhart theory. The equivariant Hodge polynomials are shown to satisfy a reciprocity and purity formula fitting with the duality for equivariant mixed Hodge modules, and implying the corresponding properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function corresponding to Stanley's <em>g</em>-function of the polar polytope of <em>P</em>, we recover in geometric terms a recent combinatorial formula of Beck–Gunnells–Materov. More generally, motivated by the analogy to the Kazhdan–Lusztig theory, we introduce a duality involution on the free <span><math><mi>Z</mi><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span>-module of weight functions corresponding to the duality of equivariant mixed Hodge modules, and prove a new reciprocity formula in terms of this duality. This unifies and generalizes the classical reciprocity formula of Brion–Vergne in Ehrhart theory as well as t","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110771"},"PeriodicalIF":1.5,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981072","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 : 2026-01-12DOI: 10.1016/j.aim.2025.110765
Tyler Arant , Alexander S. Kechris , Patrick Lutz
This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of , the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides , the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypergraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.
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