K-polystability of a polarized variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a ‘most destabilizing’ degeneration. In this note we show that if such a degeneration exists, then the limiting scheme is itself relatively K-semistable.
{"title":"K-SEMISTABILITY OF OPTIMAL DEGENERATIONS","authors":"Ruadhaí Dervan","doi":"10.1093/qmathj/haaa012","DOIUrl":"https://doi.org/10.1093/qmathj/haaa012","url":null,"abstract":"K-polystability of a polarized variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a ‘most destabilizing’ degeneration. In this note we show that if such a degeneration exists, then the limiting scheme is itself relatively K-semistable.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"989-995"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49966561","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}
By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}sharp (S^ntimes S^n)^{sharp g}$ , relative to the boundary, are for $2nge 6$ independent of $g$ in degrees $*le (g-6)/2$ . In this note, we explain how this range can be improved to $*le g-2$ using cohomological vanishing results due to Borel and the classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.
{"title":"A NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS","authors":"Manuel Krannich","doi":"10.1093/qmathj/haaa017","DOIUrl":"https://doi.org/10.1093/qmathj/haaa017","url":null,"abstract":"By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre \u0000<tex>$D^{2n}sharp (S^ntimes S^n)^{sharp g}$</tex>\u0000, relative to the boundary, are for \u0000<tex>$2nge 6$</tex>\u0000 independent of \u0000<tex>$g$</tex>\u0000 in degrees \u0000<tex>$*le (g-6)/2$</tex>\u0000. In this note, we explain how this range can be improved to \u0000<tex>$*le g-2$</tex>\u0000 using cohomological vanishing results due to Borel and the classical invariant theory. This implies that the analogous ring for smooth bundles is independent of \u0000<tex>$g$</tex>\u0000 in the same range, provided the degree is small compared to the dimension.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"1069-1079"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49967015","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 discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.
{"title":"The Composition Operation on Spaces of Holomorphic Mappings","authors":"María D Acosta;Pablo Galindo;Luiza A Moraes","doi":"10.1093/qmathj/haz035","DOIUrl":"https://doi.org/10.1093/qmathj/haz035","url":null,"abstract":"We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"557-572"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haz035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49989716","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 different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.
{"title":"Flat ring epimorphisms and universal localizations of commutative rings","authors":"Lidia Angeleri Hügel;Frederik Marks;Jan Št’ovíček;Ryo Takahashi;Jorge Vitória","doi":"10.1093/qmath/haaa041","DOIUrl":"https://doi.org/10.1093/qmath/haaa041","url":null,"abstract":"We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"1489-1520"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49966432","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 deformations of the Chow group of zerocycles of the special fibre of a smooth scheme over a Henselian discrete valuation ring. Our main tools are Bloch's formula and differential forms. As a corollary we get an algebraization theorem for thickened zero cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.
{"title":"DEFORMATION THEORY OF THE CHOW GROUP OF ZERO CYCLES","authors":"Morten Lüders","doi":"10.1093/qmathj/haaa004","DOIUrl":"https://doi.org/10.1093/qmathj/haaa004","url":null,"abstract":"We study the deformations of the Chow group of zerocycles of the special fibre of a smooth scheme over a Henselian discrete valuation ring. Our main tools are Bloch's formula and differential forms. As a corollary we get an algebraization theorem for thickened zero cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"677-676"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49966581","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 produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class- $2$ -nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.
在M. N. Berman, B. Klopsch和U. Onn (A族2类幂零群，它们的自同构和亲同构zeta函数，数学)中，我们给出了与无限族-2类幂零李环的成员相关的各种理想zeta函数的显式公式。Z. 290(2018)， 909935)，根据Igusa函数。作为推论，我们得到了关于全局理想zeta函数、局部泛函方程、拓扑、约化和分级理想zeta函数的解析性质，以及与李环相关的单幂群格式的表示zeta函数。
{"title":"IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS","authors":"Christopher Voll","doi":"10.1093/qmathj/haaa010","DOIUrl":"https://doi.org/10.1093/qmathj/haaa010","url":null,"abstract":"We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-\u0000<tex>$2$</tex>\u0000-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"959-980"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49966587","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 establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood–Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.
{"title":"Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions","authors":"Shuichi Sato","doi":"10.1093/qmath/haz049","DOIUrl":"https://doi.org/10.1093/qmath/haz049","url":null,"abstract":"We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood–Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"295-320"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haz049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49982464","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}
{"title":"Erratum to: Sums of Linear Transformations in Higher Dimensions","authors":"","doi":"10.1093/qmath/haz023","DOIUrl":"https://doi.org/10.1093/qmath/haz023","url":null,"abstract":"","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"1169-1169"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haz023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49982468","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 in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to $$begin{equation*} max_{1 leq t leq q} left|sum_{n leq t} chi(n)right| = o_{q rightarrow infty}(sqrt{q}log q) end{equation*}$$ then for any ɛ > 0 one may exhibit cancellation in partial sums of χ on the interval [1, t] whenever $t gt q^{varepsilon}$ , i.e., $$begin{equation*} sum_{n leq t} chi(n) = o_{q rightarrow infty}(t) text{for all } t gt q^{varepsilon}. end{equation*}$$ We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.
{"title":"Short Character Sums and the Pólya–Vinogradov Inequality","authors":"Alexander P Mangerel","doi":"10.1093/qmath/haaa031","DOIUrl":"https://doi.org/10.1093/qmath/haaa031","url":null,"abstract":"We show in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to \u0000<tex>$$begin{equation*} max_{1 leq t leq q} left|sum_{n leq t} chi(n)right| = o_{q rightarrow infty}(sqrt{q}log q) end{equation*}$$</tex>\u0000 then for any ɛ > 0 one may exhibit cancellation in partial sums of χ on the interval [1, t] whenever \u0000<tex>$t gt q^{varepsilon}$</tex>\u0000, i.e., \u0000<tex>$$begin{equation*} sum_{n leq t} chi(n) = o_{q rightarrow infty}(t) text{for all } t gt q^{varepsilon}. end{equation*}$$</tex>\u0000 We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"1281-1308"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49982471","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}
Using a cap product, we construct an explicit Poincaré duality isomorphism between the blown-up intersection cohomology and the Borel–Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseudomanifolds.
{"title":"POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY","authors":"Martintxo Saralegi-Aranguren;Daniel Tanré","doi":"10.1093/qmathj/haaa009","DOIUrl":"10.1093/qmathj/haaa009","url":null,"abstract":"Using a cap product, we construct an explicit Poincaré duality isomorphism between the blown-up intersection cohomology and the Borel–Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseudomanifolds.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"71 1","pages":"943-958"},"PeriodicalIF":0.7,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46931832","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：