Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a $G$-linearized vector bundle on an abelian variety, quotient of $G$. This generalizes a classical result of Sumihiro for actions of smooth connected affine algebraic groups.
{"title":"Algebraic group actions on normal varieties","authors":"M. Brion","doi":"10.1090/MOSC/263","DOIUrl":"https://doi.org/10.1090/MOSC/263","url":null,"abstract":"Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a $G$-linearized vector bundle on an abelian variety, quotient of $G$. This generalizes a classical result of Sumihiro for actions of smooth connected affine algebraic groups.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"78 1","pages":"85-107"},"PeriodicalIF":0.0,"publicationDate":"2017-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/263","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43726634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ be a spherical variety for a connected reductive group $G$. Work of Gaitsgory-Nadler strongly suggests that the Langlands dual group $G^vee$ of $G$ has a subgroup whose Weyl group is the little Weyl group of $X$. Sakellaridis-Venkatesh defined a refined dual group $G^vee_X$ and verified in many cases that there exists an isogeny $phi$ from $G^vee_X$ to $G^vee$. In this paper, we establish the existence of $phi$ in full generality. Our approach is purely combinatorial and works (despite the title) for arbitrary $G$-varieties.
{"title":"The dual group of a spherical variety","authors":"F. Knop, B. Schalke","doi":"10.1090/mosc/270","DOIUrl":"https://doi.org/10.1090/mosc/270","url":null,"abstract":"Let $X$ be a spherical variety for a connected reductive group $G$. Work of Gaitsgory-Nadler strongly suggests that the Langlands dual group $G^vee$ of $G$ has a subgroup whose Weyl group is the little Weyl group of $X$. Sakellaridis-Venkatesh defined a refined dual group $G^vee_X$ and verified in many cases that there exists an isogeny $phi$ from $G^vee_X$ to $G^vee$. In this paper, we establish the existence of $phi$ in full generality. Our approach is purely combinatorial and works (despite the title) for arbitrary $G$-varieties.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"78 1","pages":"187-216"},"PeriodicalIF":0.0,"publicationDate":"2017-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/mosc/270","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43795747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.
{"title":"Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras","authors":"V. Gritsenko, V. Nikulin","doi":"10.1090/MOSC/265","DOIUrl":"https://doi.org/10.1090/MOSC/265","url":null,"abstract":"Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"78 1","pages":"75-83"},"PeriodicalIF":0.0,"publicationDate":"2017-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/265","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46519290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $hat{frak{g}}$ span an $SL_2(mathbf{Z})$-invariant space. This result extends to admissible $hat{frak{g}}$-modules, where $frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $hat{frak{g}}$-modules when $frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. Another instance of modular invariance occurs for boundary level admissible modules, including when $frak{g}$ is a basic Lie superalgebra. For example, if $frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant families of $hat{frak{g}}$-modules, whose QHR produces the minimal series modules for the $N=2$ superconformal algebras (resp. a modular invariant family of $N=3$ superconformal algebra modules). However, in the case when $frak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized supercharacters of admissible $hat{frak{g}}$-modules holds outside of boundary levels only after their modification in the spirit of Zwegers' modification of mock theta functions. Applying the QHR, we obtain families of representations of $N=2,3,4$ and big $N=4$ superconformal algebras, whose modified (super)characters span an $SL_2(mathbf{Z})$-invariant space.
{"title":"Representations of superconformal algebras and mock theta functions","authors":"V. Kac, M. Wakimoto","doi":"10.1090/MOSC/268","DOIUrl":"https://doi.org/10.1090/MOSC/268","url":null,"abstract":"It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $hat{frak{g}}$ span an $SL_2(mathbf{Z})$-invariant space. This result extends to admissible $hat{frak{g}}$-modules, where $frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $hat{frak{g}}$-modules when $frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. \u0000Another instance of modular invariance occurs for boundary level admissible modules, including when $frak{g}$ is a basic Lie superalgebra. For example, if $frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant families of $hat{frak{g}}$-modules, whose QHR produces the minimal series modules for the $N=2$ superconformal algebras (resp. a modular invariant family of $N=3$ superconformal algebra modules). \u0000However, in the case when $frak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized supercharacters of admissible $hat{frak{g}}$-modules holds outside of boundary levels only after their modification in the spirit of Zwegers' modification of mock theta functions. Applying the QHR, we obtain families of representations of $N=2,3,4$ and big $N=4$ superconformal algebras, whose modified (super)characters span an $SL_2(mathbf{Z})$-invariant space.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"78 1","pages":"9-74"},"PeriodicalIF":0.0,"publicationDate":"2017-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/268","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47514998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver varieties. The physical origin of the correspondence is the 6d little string theory. The quantum Langlands correspondence emerges in the limit in which the 6d string theory becomes the 6d conformal field theory with (2,0) supersymmetry.
{"title":"Quantum $q$-Langlands Correspondence","authors":"Mina Aganagic, E. Frenkel, A. Okounkov","doi":"10.1090/MOSC/278","DOIUrl":"https://doi.org/10.1090/MOSC/278","url":null,"abstract":"We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver varieties. The physical origin of the correspondence is the 6d little string theory. The quantum Langlands correspondence emerges in the limit in which the 6d string theory becomes the 6d conformal field theory with (2,0) supersymmetry.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/278","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41370485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matrix divisors are introduced in the work by A.Weil (1938) which is considered as a starting point of the theory of holomorphic vector bundles on Riemann surfaces. In this theory matrix divisors play the role similar to the role of usual divisors in the theory of line bundles. Moreover, they provide explicit coordinates (Tyurin parameters) in an open subset of the moduli space of stable vector bundles. These coordinates turned out to be helpful in integration of soliton equations. We would like to gain attention to one more relationship between matrix divisors of vector G-bundles (where G is a complex semi-simple Lie group) and the theory of integrable systems, namely to the relationship with Lax operator algebras. The result we obtain can be briefly formulated as follows: the moduli space of matrix divisors with certain discrete invariants and fixed support is a homogeneous space. Its tangent space at the unit is naturally isomorphic to the quotient space of M-operators by L-operators, both spaces essentially defined by the same invariants (the result goes back to Krichever, 2001). We give one more description of the same space in terms of root systems.
{"title":"Matrix divisors on Riemann surfaces and Lax operator algebras","authors":"O. Sheinman","doi":"10.1090/mosc/267","DOIUrl":"https://doi.org/10.1090/mosc/267","url":null,"abstract":"Matrix divisors are introduced in the work by A.Weil (1938) which is considered as a starting point of the theory of holomorphic vector bundles on Riemann surfaces. In this theory matrix divisors play the role similar to the role of usual divisors in the theory of line bundles. Moreover, they provide explicit coordinates (Tyurin parameters) in an open subset of the moduli space of stable vector bundles. These coordinates turned out to be helpful in integration of soliton equations. \u0000We would like to gain attention to one more relationship between matrix divisors of vector G-bundles (where G is a complex semi-simple Lie group) and the theory of integrable systems, namely to the relationship with Lax operator algebras. The result we obtain can be briefly formulated as follows: the moduli space of matrix divisors with certain discrete invariants and fixed support is a homogeneous space. Its tangent space at the unit is naturally isomorphic to the quotient space of M-operators by L-operators, both spaces essentially defined by the same invariants (the result goes back to Krichever, 2001). We give one more description of the same space in terms of root systems.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"78 1","pages":"109-121"},"PeriodicalIF":0.0,"publicationDate":"2017-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/mosc/267","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45369862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some problems concerning the solvability of the nonlinear stationary Boltzmann equation in the framework of the BGK model","authors":"A. Khachatryan, K. Khachatryan","doi":"10.1090/MOSC/255","DOIUrl":"https://doi.org/10.1090/MOSC/255","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"77 1","pages":"87-106"},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/255","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Moscow Mathematical Society — the oldest mathematical society in Russia, and one of the oldest in the world — was established one hundred and fifty years ago, in the Autumn of 1864. Its foundation was one of the most significant events in the development of mathematics in Russia, testifying to the creation in the country of a mathematical community that needed special forms of organization for its activities. It should be noted that this was not the first attempt to organize a mathematical society in Moscow: already in 1810, a group of teachers and students at Moscow State University had tried to establish a similar society; see [1, p. 316] and [2]. However, it existed only very briefly: a sufficiently large community of active professional mathematicians had not yet formed in the ancient capital to maintain its regular activities. Until the mid-1830s, Moscow was, in relation to mathematics, profoundly provincial, significantly inferior to St. Petersburg, in which the Imperial Academy of Sciences was located, and Kazan’, workplace of N. I. Lobachevskĭı (see [1]. But by as early as the middle of the nineteenth century, the works of N. D. Brashman and N. E. Zernova of Moscow had become notable points on the mathematical map of Europe.
{"title":"The Moscow Mathematical Society and the development of mathematics in Russia (on the 150th anniversary of the Society’s creation)","authors":"S. Demidov, V. Tikhomirov, T. A. Tokareva","doi":"10.1090/MOSC/260","DOIUrl":"https://doi.org/10.1090/MOSC/260","url":null,"abstract":"The Moscow Mathematical Society — the oldest mathematical society in Russia, and one of the oldest in the world — was established one hundred and fifty years ago, in the Autumn of 1864. Its foundation was one of the most significant events in the development of mathematics in Russia, testifying to the creation in the country of a mathematical community that needed special forms of organization for its activities. It should be noted that this was not the first attempt to organize a mathematical society in Moscow: already in 1810, a group of teachers and students at Moscow State University had tried to establish a similar society; see [1, p. 316] and [2]. However, it existed only very briefly: a sufficiently large community of active professional mathematicians had not yet formed in the ancient capital to maintain its regular activities. Until the mid-1830s, Moscow was, in relation to mathematics, profoundly provincial, significantly inferior to St. Petersburg, in which the Imperial Academy of Sciences was located, and Kazan’, workplace of N. I. Lobachevskĭı (see [1]. But by as early as the middle of the nineteenth century, the works of N. D. Brashman and N. E. Zernova of Moscow had become notable points on the mathematical map of Europe.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"77 1","pages":"127-148"},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/260","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local dynamics of two-component singularly perturbed parabolic systems","authors":"I. Kashchenko, S. A. Kashchenko","doi":"10.1090/MOSC/252","DOIUrl":"https://doi.org/10.1090/MOSC/252","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"77 1","pages":"55-68"},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/252","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies","authors":"T. M. Mitryakova, O. Pochinka","doi":"10.1090/MOSC/253","DOIUrl":"https://doi.org/10.1090/MOSC/253","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"77 1","pages":"69-86"},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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