Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.
{"title":"Basic kirwan injectivity and its applications","authors":"Yi Lin, Xiangdong Yang","doi":"10.1093/qmath/haac038","DOIUrl":"https://doi.org/10.1093/qmath/haac038","url":null,"abstract":"Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"199 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537728","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}
We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1 unicode{x003C} p,qunicode{x003C}infty$ be such that $1/p+1/qgeq 1$. Let X (resp., Y) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower p-estimate (resp., q-estimate). If X and Y are strongly weakly compactly generated, then so is their projective tensor product $X {widehat{otimes}_pi} Y$.
{"title":"On weak compactness in projective tensor products","authors":"José Rodríguez","doi":"10.1093/qmath/haac036","DOIUrl":"https://doi.org/10.1093/qmath/haac036","url":null,"abstract":"We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1 unicode{x003C} p,qunicode{x003C}infty$ be such that $1/p+1/qgeq 1$. Let X (resp., Y) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower p-estimate (resp., q-estimate). If X and Y are strongly weakly compactly generated, then so is their projective tensor product $X {widehat{otimes}_pi} Y$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"63 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537722","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}
We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $mathcal{C}$-conjugacy separability of a wreath product $A wr B$ in terms of the $mathcal{C}$-conjugacy separability of A and B, the growth of $mathcal{C}$-cyclic subgroup separability of B and the $mathcal{C}$-residual girth of $B.$ As an application, we provide a characterization of when $A wr B$ is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.
研究了群的受限环积共轭可分性的推广。利用a和B的$mathcal{C}$-共轭可分性、B的$mathcal{C}$-循环子群可分性的增长和$B的$mathcal{C}$-残围，给出环积$ a wr $的$mathcal{C}$-共轭可分性的有效上界。作为应用，我们给出了a wr B$是p共轭可分的一个表征。我们利用这一性质为每个素数p提供了一个具有无限基群的环积是p共轭可分的例子。给出了包括lamplighter群在内的幂零群环积共轭可分性的渐近上界，并给出了自由亚丫群共轭可分性的渐近上界。
{"title":"Quantifying conjugacy separability in wreath products of groups","authors":"Michal Ferov, Mark Pengitore","doi":"10.1093/qmath/haac031","DOIUrl":"https://doi.org/10.1093/qmath/haac031","url":null,"abstract":"We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $mathcal{C}$-conjugacy separability of a wreath product $A wr B$ in terms of the $mathcal{C}$-conjugacy separability of A and B, the growth of $mathcal{C}$-cyclic subgroup separability of B and the $mathcal{C}$-residual girth of $B.$ As an application, we provide a characterization of when $A wr B$ is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"74 23","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505990","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}
We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.
{"title":"Twists of rational Cherednik algebras","authors":"Y Bazlov, E Jones-Healey, A Mcgaw, A Berenstein","doi":"10.1093/qmath/haac033","DOIUrl":"https://doi.org/10.1093/qmath/haac033","url":null,"abstract":"We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"78 ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505986","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}
A. Schepper, J. Schillewaert, Hendrik Van Maldeghemmagali Victoor, Magali Victoor
{"title":"A Geometric Characterization of the Hjelmslev–Moufang Planes","authors":"A. Schepper, J. Schillewaert, Hendrik Van Maldeghemmagali Victoor, Magali Victoor","doi":"10.1093/QMATH/HAAB043","DOIUrl":"https://doi.org/10.1093/QMATH/HAAB043","url":null,"abstract":"","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44686002","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}
The Banana manifold $X_{{text{Ban}}}$ is a compact Calabi–Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, which was first studied by Bryan. We compute Katz’s genus 0 Gopakumar–Vafa invariants of fiber curve classes on the Banana manifold $X_{{text{Ban}}}to mathbf{P} ^1$. The weak Jacobi form of weight −2 and index 1 is the associated generating function for these genus 0 Gopakumar–Vafa invariants. The invariants are shown to be an actual count of structure sheaves of certain possibly non-reduced genus 0 curves on the universal cover of the singular fibers of $X_{{text{Ban}}}tomathbf{P}^1$.
Banana流形$X_｛text｛Ban｝｝$是一个紧的Calabi–Yau三重，由Bryan首次研究，它被构造为具有自身的一般有理椭圆表面的纤维乘积的针叶树分辨率。我们计算Banana流形$X_｛text｛Ban｝｝ to mathbf｛P｝^1$上纤维曲线类的Katz亏格0 Gopakumar–Vafa不变量。权重−2和索引1的弱Jacobi形式是这些亏格0 Gopakumar–Vafa不变量的相关生成函数。不变量被证明是$X_｛text｛Ban｝｝ to mathbf｛P｝^1$的奇异纤维的泛覆盖上某些可能非约化亏格0曲线的结构簇的实际计数。
{"title":"Genus 0 Gopakumar–Vafa Invariants of the Banana Manifold","authors":"Nina Morishige","doi":"10.1093/QMATH/HAAB026","DOIUrl":"https://doi.org/10.1093/QMATH/HAAB026","url":null,"abstract":"\u0000 The Banana manifold $X_{{text{Ban}}}$ is a compact Calabi–Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, which was first studied by Bryan. We compute Katz’s genus 0 Gopakumar–Vafa invariants of fiber curve classes on the Banana manifold $X_{{text{Ban}}}to mathbf{P} ^1$. The weak Jacobi form of weight −2 and index 1 is the associated generating function for these genus 0 Gopakumar–Vafa invariants. The invariants are shown to be an actual count of structure sheaves of certain possibly non-reduced genus 0 curves on the universal cover of the singular fibers of $X_{{text{Ban}}}tomathbf{P}^1$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47983126","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}
We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $mathbb{R}$.
{"title":"Two-Dimensional Locally Nash Groups","authors":"E. Baro, J. Vicente, M. Otero","doi":"10.1093/QMATH/HAAB021","DOIUrl":"https://doi.org/10.1093/QMATH/HAAB021","url":null,"abstract":"\u0000 We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $mathbb{R}$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QMATH/HAAB021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43866201","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}
We give a very concise proof of Ornstein’s L1 non-inequality for first- and second-order operators in two dimensions. The proof just needs a two-dimensional laminate supported on three points.
{"title":"Remarks On Ornstein’s Non-Inequality In ℝ2×2","authors":"D. Faraco, André Guerra","doi":"10.1093/QMATH/HAAB016","DOIUrl":"https://doi.org/10.1093/QMATH/HAAB016","url":null,"abstract":"\u0000 We give a very concise proof of Ornstein’s L1 non-inequality for first- and second-order operators in two dimensions. The proof just needs a two-dimensional laminate supported on three points.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QMATH/HAAB016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47954022","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}
We are concerned with an optimal regularity for ω-minimizers to double phase variational problems with variable exponents where the associated energy density is allowed to be discontinuous. We identify basic structure assumptions on the density for the absence of Lavrentiev phenomenon and higher integrability. Moreover, we establish a local Calderón–Zygmund theory for such generalized minimizers under minimal regularity requirements regarding such double phase functionals to the frame of Lebesgue spaces with variable exponents.
{"title":"Gradient Estimates of ω-Minimizers to Double Phase Variational Problems with Variable Exponents","authors":"Sun-Sig Byun, Ho-Sik Lee","doi":"10.1093/QMATH/HAAA067","DOIUrl":"https://doi.org/10.1093/QMATH/HAAA067","url":null,"abstract":"\u0000 We are concerned with an optimal regularity for ω-minimizers to double phase variational problems with variable exponents where the associated energy density is allowed to be discontinuous. We identify basic structure assumptions on the density for the absence of Lavrentiev phenomenon and higher integrability. Moreover, we establish a local Calderón–Zygmund theory for such generalized minimizers under minimal regularity requirements regarding such double phase functionals to the frame of Lebesgue spaces with variable exponents.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QMATH/HAAA067","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61271830","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}
We study automatic injectivity of surjective algebra homomorphisms from $mathscr{B}(X)$ , the algebra of (bounded, linear) operators on X, to $mathscr{B}(Y)$ , where X is one of the following long sequence spaces: c 0 (λ), $ell_{infty}^c(lambda)$ , and $ell_p(lambda)$ ( $1 leqslant p lt infty$ ) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.
我们研究了满射代数同态的自动注入性，从$mathscr{B}(X)$，(有界的，线性的)算子在X上的代数，到$mathscr{B}(Y)$，其中X是下列长序列空间之一:c0(λ)， $ell_{infty}^c(lambda)$和$ell_p(lambda)$ ($1 leqslant p lt infty$)， Y是任意的。在证明这些空间确实具有这样的性质的过程中，我们对任何上述的Banach空间的算子代数的双面理想进行了分类，这些空间相对于“顺序强算子拓扑”是封闭的。
{"title":"Surjective Homomorphisms from Algebras of Operators on Long Sequence Spaces are Automatically Injective","authors":"Bence HorvÁth;Tomasz Kania","doi":"10.1093/qmath/haaa066","DOIUrl":"https://doi.org/10.1093/qmath/haaa066","url":null,"abstract":"We study automatic injectivity of surjective algebra homomorphisms from \u0000<tex>$mathscr{B}(X)$</tex>\u0000, the algebra of (bounded, linear) operators on X, to \u0000<tex>$mathscr{B}(Y)$</tex>\u0000, where X is one of the following long sequence spaces: c\u0000<inf>0</inf>\u0000(λ), \u0000<tex>$ell_{infty}^c(lambda)$</tex>\u0000, and \u0000<tex>$ell_p(lambda)$</tex>\u0000 (\u0000<tex>$1 leqslant p lt infty$</tex>\u0000) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1167-1189"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa066","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978801","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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