Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a7
Dominik Gutwein
This article constructs coassociative submanifolds in $mathrm{G}_2$-manifolds arising from Joyce’s generalised Kummer construction. The novelty compared to previous constructions is that these submanifolds all lie within the critical region of the $mathrm{G}_2$-manifold in which the metric degenerates. This forces the volume of the coassociatives to shrink to zero when the orbifold-limit is approached.
{"title":"Coassociative submanifolds in Joyce's generalised Kummer constructions","authors":"Dominik Gutwein","doi":"10.4310/pamq.2024.v20.n2.a7","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a7","url":null,"abstract":"This article constructs coassociative submanifolds in $mathrm{G}_2$-manifolds arising from Joyce’s generalised Kummer construction. The novelty compared to previous constructions is that these submanifolds all lie within the critical region of the $mathrm{G}_2$-manifold in which the metric degenerates. This forces the volume of the coassociatives to shrink to zero when the orbifold-limit is approached.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a8
Shoji Yokura
A bi-variant theory $mathbb{B}(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton–Mac Pherson’s bivariant theory $mathbb{B}(X xrightarrow{f} Y)$ defined for a morphism $f : X rightarrow Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $Omega^{ast,sharp} (X, Y )$ such that $Omega^{ast,sharp}(X, pt)$ is isomorphic to Lee–Pandharipande’s algebraic cobordism of vector bundles $Omega underline{}_{ast,sharp} (X)$. In particular, $Omega^ast (X, pt) = Omega^{ast,0} (X, pt)$ is isomorphic to Levine–Morel’s algebraic cobordism $Omega underline{}_{ast} (X)$. Namely, $Omega^{ast,sharp} (X,Y)$ is a bi-variant version of Lee–Pandharipande’s algebraic cobordism of bundles $Omega_{ast,sharp} (X)$.
{"title":"A bi-variant algebraic cobordism via correspondences","authors":"Shoji Yokura","doi":"10.4310/pamq.2024.v20.n2.a8","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a8","url":null,"abstract":"A bi-variant theory $mathbb{B}(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton–Mac Pherson’s bivariant theory $mathbb{B}(X xrightarrow{f} Y)$ defined for a morphism $f : X rightarrow Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $Omega^{ast,sharp} (X, Y )$ such that $Omega^{ast,sharp}(X, pt)$ is isomorphic to Lee–Pandharipande’s algebraic cobordism of vector bundles $Omega underline{}_{ast,sharp} (X)$. In particular, $Omega^ast (X, pt) = Omega^{ast,0} (X, pt)$ is isomorphic to Levine–Morel’s algebraic cobordism $Omega underline{}_{ast} (X)$. Namely, $Omega^{ast,sharp} (X,Y)$ is a <i>bi-variant version</i> of Lee–Pandharipande’s algebraic cobordism of bundles $Omega_{ast,sharp} (X)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a5
Yude Liu, Xinbao Lu, Qiang Sun, Ge Xiong
A necessary condition for the existence of solutions to the logarithmic Minkowski problem in $mathbb{R}^2$, which turns to be stronger than the celebrated subspace concentration condition, is given. The sufficient and necessary conditions for the existence of solutions to the logarithmic problem for quadrilaterals, as well as the number of solutions, are fully characterized.
{"title":"The logarithmic Minkowski problem in $R^2$","authors":"Yude Liu, Xinbao Lu, Qiang Sun, Ge Xiong","doi":"10.4310/pamq.2024.v20.n2.a5","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a5","url":null,"abstract":"A necessary condition for the existence of solutions to the logarithmic Minkowski problem in $mathbb{R}^2$, which turns to be stronger than the celebrated subspace concentration condition, is given. The sufficient and necessary conditions for the existence of solutions to the logarithmic problem for quadrilaterals, as well as the number of solutions, are fully characterized.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a2
Karim Adiprasito
We discuss mixed faces of Minkowski sums of polytopes, and show that any stable complete intersection of pointed hypersurfaces is homotopy Cohen–Macaulay, generalizing a result of Hacking, and answering the topological (or weak) version of a question of Markwig and Yu. In particular, the complete intersection has the homotopy type of a wedge of spheres of the same dimension.
{"title":"A note on the topology of Minkowski sums and complete intersections","authors":"Karim Adiprasito","doi":"10.4310/pamq.2024.v20.n1.a2","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a2","url":null,"abstract":"We discuss mixed faces of Minkowski sums of polytopes, and show that any stable complete intersection of pointed hypersurfaces is homotopy Cohen–Macaulay, generalizing a result of Hacking, and answering the topological (or weak) version of a question of Markwig and Yu. In particular, the complete intersection has the homotopy type of a wedge of spheres of the same dimension.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a8
Pavel Etingof, Edward Frenkel, David Kazhdan
We discuss a general framework for the analytic Langlands correspondence over an arbitrary local field F introduced and studied in our works [$href{http://arxiv.org/abs/1908.09677}{EFK1}$, $href{http://arxiv.org/abs/2103.01509}{EFK2}$, $href{http://arxiv.org/abs/2106.05243}{EFK3}$], in particular including non-split and twisted settings. Then we specialize to the archimedean cases ($F = mathbb{C}$ and $F = mathbb{R}$) and give a (mostly conjectural) description of the spectrum of the Hecke operators in various cases in terms of opers satisfying suitable reality conditions, as predicted in part in [$href{http://arxiv.org/abs/2103.01509}{EFK2}$, $href{http://arxiv.org/abs/2106.05243}{EFK3}$] and [$href{http://arxiv.org/abs/2107.01732}{GW}$]. We also describe an analogue of the Langlands functoriality principle in the analytic Langlands correspondence over $mathbb{C}$ and show that it is compatible with the results and conjectures of [$href{http://arxiv.org/abs/2103.01509}{EFK2}$]. Finally, we apply the tools of the analytic Langlands correspondence over archimedean fields in genus zero to the Gaudin model and its generalizations, as well as their $q$-deformations.
{"title":"A general framework and examples of the analytic Langlands correspondence","authors":"Pavel Etingof, Edward Frenkel, David Kazhdan","doi":"10.4310/pamq.2024.v20.n1.a8","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a8","url":null,"abstract":"We discuss a general framework for the analytic Langlands correspondence over an arbitrary local field F introduced and studied in our works [$href{http://arxiv.org/abs/1908.09677}{EFK1}$, $href{http://arxiv.org/abs/2103.01509}{EFK2}$, $href{http://arxiv.org/abs/2106.05243}{EFK3}$], in particular including non-split and twisted settings. Then we specialize to the archimedean cases ($F = mathbb{C}$ and $F = mathbb{R}$) and give a (mostly conjectural) description of the spectrum of the Hecke operators in various cases in terms of opers satisfying suitable reality conditions, as predicted in part in [$href{http://arxiv.org/abs/2103.01509}{EFK2}$, $href{http://arxiv.org/abs/2106.05243}{EFK3}$] and [$href{http://arxiv.org/abs/2107.01732}{GW}$]. We also describe an analogue of the Langlands functoriality principle in the analytic Langlands correspondence over $mathbb{C}$ and show that it is compatible with the results and conjectures of [$href{http://arxiv.org/abs/2103.01509}{EFK2}$]. Finally, we apply the tools of the analytic Langlands correspondence over archimedean fields in genus zero to the Gaudin model and its generalizations, as well as their $q$-deformations.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a10
Friedrich Knop
A quasi-Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are $0$-dimensional. In this paper, we classify compact, multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected, compact Lie groups. Thereby, we recover old and find new examples of these structures.
{"title":"Classification of multiplicity free quasi-Hamiltonian manifolds","authors":"Friedrich Knop","doi":"10.4310/pamq.2024.v20.n1.a10","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a10","url":null,"abstract":"A quasi-Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are $0$-dimensional. In this paper, we classify compact, multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected, compact Lie groups. Thereby, we recover old and find new examples of these structures.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a3
Andrea Appel, Valerio Toledano Laredo
Let $mathfrak{g}$ be a symmetrisable Kac–Moody algebra and $U_hbar mathfrak{g}$ its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of $U_hbar mathfrak{g}$ give rise to a canonical action of the pure braid group of $mathfrak{g}$ on any category $mathcal{O}$ (not necessarily integrable) $U_hbar mathfrak{g}$-module $mathcal{V}$. By relying on our recent results $href{http://arxiv.org/abs/1512.03041}{[textrm{ATL15}]}$, we show that this action describes the monodromy of the rational Casimir connection on the $mathfrak{g}$-module $V$ corresponding to $mathcal{V}$. We also extend these results to yield equivalent representations of parabolic pure braid groups on parabolic category $mathcal{O}$ for $U_hbar mathfrak{g}$ and $mathfrak{g}$.
{"title":"Pure braid group actions on category $mathcal{O}$ modules","authors":"Andrea Appel, Valerio Toledano Laredo","doi":"10.4310/pamq.2024.v20.n1.a3","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a3","url":null,"abstract":"Let $mathfrak{g}$ be a symmetrisable Kac–Moody algebra and $U_hbar mathfrak{g}$ its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of $U_hbar mathfrak{g}$ give rise to a canonical action of the pure braid group of $mathfrak{g}$ on any category $mathcal{O}$ (not necessarily integrable) $U_hbar mathfrak{g}$-module $mathcal{V}$. By relying on our recent results $href{http://arxiv.org/abs/1512.03041}{[textrm{ATL15}]}$, we show that this action describes the monodromy of the rational Casimir connection on the $mathfrak{g}$-module $V$ corresponding to $mathcal{V}$. We also extend these results to yield equivalent representations of parabolic pure braid groups on parabolic category $mathcal{O}$ for $U_hbar mathfrak{g}$ and $mathfrak{g}$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a11
Vladimir L. Popov
We prove that every orbit of the adjoint representation of any connected reductive algebraic group $G$ is a rational algebraic variety. For complex simply connected semisimple $G$, this implies rationality of homogeneous affine Hamiltonian $G$-varieties (which we classify).
{"title":"Rationality of adjoint orbits","authors":"Vladimir L. Popov","doi":"10.4310/pamq.2024.v20.n1.a11","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a11","url":null,"abstract":"We prove that every orbit of the adjoint representation of any connected reductive algebraic group $G$ is a rational algebraic variety. For complex simply connected semisimple $G$, this implies rationality of homogeneous affine Hamiltonian $G$-varieties (which we classify).","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a4
Roman Bezrukavnikov, Victor Kac, Vasily Krylov
Let $mathfrak{g}$ be a simple finite dimensional complex Lie algebra and let $widehat{mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a simple highest weight $widehat{mathfrak{g}}$-module are either bounded or are given by a linear function of the weight. We explain and generalize this observation using Kazhdan–Lusztig theory, by computing values at $q = 1$ of certain (parabolic) affine inverse Kazhdan–Lusztig polynomials. In particular, we obtain explicit character formulas for some $widehat{mathfrak{g}}$-modules of negative integer level $k$ when $mathfrak{g}$ is of type $D_n$, $E_6$, $E_7$, $E_8$ and $k geqslant -2,-3,-4,-6$ respectively, as conjectured by Kac and Wakimoto. The calculation relies on the explicit description of the canonical basis in the cell quotient of the anti-spherical module over the affine Hecke algebra corresponding to the subregular cell.We also present an explicit description of the corresponding objects in the derived category of equivariant coherent sheaves on the Springer resolution, they correspond to irreducible objects in the heart of a certain $t$-structure related to the so called non-commutative Springer resolution.
{"title":"Subregular nilpotent orbits and explicit character formulas for modules over affine Lie algebras","authors":"Roman Bezrukavnikov, Victor Kac, Vasily Krylov","doi":"10.4310/pamq.2024.v20.n1.a4","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a4","url":null,"abstract":"Let $mathfrak{g}$ be a simple finite dimensional complex Lie algebra and let $widehat{mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a simple highest weight $widehat{mathfrak{g}}$-module are either bounded or are given by a linear function of the weight. We explain and generalize this observation using Kazhdan–Lusztig theory, by computing values at $q = 1$ of certain (parabolic) affine inverse Kazhdan–Lusztig polynomials. In particular, we obtain explicit character formulas for some $widehat{mathfrak{g}}$-modules of negative integer level $k$ when $mathfrak{g}$ is of type $D_n$, $E_6$, $E_7$, $E_8$ and $k geqslant -2,-3,-4,-6$ respectively, as conjectured by Kac and Wakimoto. The calculation relies on the explicit description of the canonical basis in the cell quotient of the anti-spherical module over the affine Hecke algebra corresponding to the subregular cell.We also present an explicit description of the corresponding objects in the derived category of equivariant coherent sheaves on the Springer resolution, they correspond to irreducible objects in the heart of a certain $t$-structure related to the so called non-commutative Springer resolution.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a9
Giovanni Gaiffi, Oscar Papini
In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type $A$. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group.
{"title":"A basis for the cohomology of compact models of toric arrangements","authors":"Giovanni Gaiffi, Oscar Papini","doi":"10.4310/pamq.2024.v20.n1.a9","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a9","url":null,"abstract":"In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type $A$. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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