We compute the behaviour of Hodge data under additive middle convolution for irreducible variations of polarized complex Hodge structures on punctured complex affine lines.
我们计算了极化复Hodge结构在删截复仿射线上的不可约变化在加性中间卷积下的Hodge数据的行为。
{"title":"On the Hodge Theory of the Additive Middle Convolution","authors":"M. Dettweiler, Stefan Reiter","doi":"10.4171/prims/56-3-3","DOIUrl":"https://doi.org/10.4171/prims/56-3-3","url":null,"abstract":"We compute the behaviour of Hodge data under additive middle convolution for irreducible variations of polarized complex Hodge structures on punctured complex affine lines.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-3-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49567696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the moduli of G-local systems on smooth but not necessarily proper complex algebraic varieties. We show that, when suitably considered as derived algebraic stacks, they carry natural Poisson structures, generalizing the well known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has non-trivial crossings.
{"title":"Poisson Geometry of the Moduli of Local Systems on Smooth Varieties","authors":"T. Pantev, B. Toen","doi":"10.4171/prims/57-3-8","DOIUrl":"https://doi.org/10.4171/prims/57-3-8","url":null,"abstract":"We study the moduli of G-local systems on smooth but not necessarily proper complex algebraic varieties. We show that, when suitably considered as derived algebraic stacks, they carry natural Poisson structures, generalizing the well known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has non-trivial crossings.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46750433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ihara, Kaneko, and Zagier defined two regularizations of multiple zeta values and proved the regularization theorem that describes the relation between those regularizations. We show that the regularization theorem can be generalized to polynomials whose coefficients are regularizations of multiple zeta values and that specialize to symmetric multiple zeta values defined by Kaneko and Zagier.
{"title":"Polynomial Generalization of the Regularization Theorem for Multiple Zeta Values","authors":"M. Hirose, H. Murahara, Shingo Saito","doi":"10.4171/prims/56-1-9","DOIUrl":"https://doi.org/10.4171/prims/56-1-9","url":null,"abstract":"Ihara, Kaneko, and Zagier defined two regularizations of multiple zeta values and proved the regularization theorem that describes the relation between those regularizations. We show that the regularization theorem can be generalized to polynomials whose coefficients are regularizations of multiple zeta values and that specialize to symmetric multiple zeta values defined by Kaneko and Zagier.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-1-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45070986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of hypergeometric type is obtained with respect to volume density.
{"title":"Harmonic Hadamard Manifolds and Gauss Hypergeometric Differential Equations","authors":"M. Itoh, H. Satoh","doi":"10.4171/PRIMS/55-3-3","DOIUrl":"https://doi.org/10.4171/PRIMS/55-3-3","url":null,"abstract":"A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of hypergeometric type is obtained with respect to volume density.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/55-3-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42403587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal of this paper is to make a surprising connection between several central conjectures in algebraic geometry: the Nonvanishing Conjecture, the Abundance Conjecture, and the Semiampleness Conjecture for nef line bundles on K-trivial varieties.
{"title":"On Generalised Abundance, I","authors":"Vladimir Lazi'c, T. Peternell","doi":"10.4171/PRIMS/56-2-3","DOIUrl":"https://doi.org/10.4171/PRIMS/56-2-3","url":null,"abstract":"The goal of this paper is to make a surprising connection between several central conjectures in algebraic geometry: the Nonvanishing Conjecture, the Abundance Conjecture, and the Semiampleness Conjecture for nef line bundles on K-trivial varieties.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/56-2-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43728171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we study the anabelian geometry of pointed stable curves over algebraically closed fields of positive characteristic. We prove that the semigraph of anabelioids of PSC-type arising from a pointed stable curve over an algebraically closed field of positive characteristic can be reconstructed group-theoretically from its fundamental group. This result may be regarded as a version of the combinatorial Grothendieck conjecture in positive characteristic. As an application, we prove that, if a pointed stable curve over an algebraic closure of a finite field satisfies certain conditions, then the isomorphism class of the admissible fundamental group of the pointed stable curve completely determines the isomorphism class of the pointed stable curve as a scheme. This result generalizes a result of A. Tamagawa to the case of (possibly singular) pointed stable curves.
{"title":"On the Admissible Fundamental Groups of Curves over Algebraically Closed Fields of Characteristic $p > 0$","authors":"Yu Yang","doi":"10.4171/PRIMS/54-3-4","DOIUrl":"https://doi.org/10.4171/PRIMS/54-3-4","url":null,"abstract":"In the present paper, we study the anabelian geometry of pointed stable curves over algebraically closed fields of positive characteristic. We prove that the semigraph of anabelioids of PSC-type arising from a pointed stable curve over an algebraically closed field of positive characteristic can be reconstructed group-theoretically from its fundamental group. This result may be regarded as a version of the combinatorial Grothendieck conjecture in positive characteristic. As an application, we prove that, if a pointed stable curve over an algebraic closure of a finite field satisfies certain conditions, then the isomorphism class of the admissible fundamental group of the pointed stable curve completely determines the isomorphism class of the pointed stable curve as a scheme. This result generalizes a result of A. Tamagawa to the case of (possibly singular) pointed stable curves.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-3-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47897064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of basic slc-trivial fibrations. It is a generalization of that of Ambro's lc-trivial fibrations. Then we study fundamental properties of basic slc-trivial fibrations by using the theory of variations of mixed Hodge structure on cohomology with compact support. More precisely, we prove that the moduli part of a basic slc-trivial fibration is b-strongly nef. Note that the notion of basic slc-trivial fibrations is closely related to that of normal irreducible quasi-log canonical pairs. So the results obtained in this paper will play an important role in the theory of quasi-log schemes. Here we give a structure theorem for normal irreducible quasi-log canonical pairs as an application of the main theorem. This result makes the theory of quasi-log schemes more powerful and more flexible.
{"title":"Fundamental Properties of Basic Slc-Trivial Fibrations I","authors":"O. Fujino, T. Fujisawa, Haidong Liu","doi":"10.4171/prims/58-3-2","DOIUrl":"https://doi.org/10.4171/prims/58-3-2","url":null,"abstract":"We introduce the notion of basic slc-trivial fibrations. It is a generalization of that of Ambro's lc-trivial fibrations. Then we study fundamental properties of basic slc-trivial fibrations by using the theory of variations of mixed Hodge structure on cohomology with compact support. More precisely, we prove that the moduli part of a basic slc-trivial fibration is b-strongly nef. Note that the notion of basic slc-trivial fibrations is closely related to that of normal irreducible quasi-log canonical pairs. So the results obtained in this paper will play an important role in the theory of quasi-log schemes. Here we give a structure theorem for normal irreducible quasi-log canonical pairs as an application of the main theorem. This result makes the theory of quasi-log schemes more powerful and more flexible.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47465555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The generalized quantum group of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang-Baxter equation. In this paper, we develop the crystal base theory for finite-dimensional representations of generalized quantum group of type $A$. As a main result, we construct Kirillov-Reshetikhin modules, that is, a family of irreducible modules which have crystal bases. We also give an explicit combinatorial description of the crystal structure of Kirillov-Reshetikhin modules, the combinatorial $R$ matrix, and energy function on their tensor products.
{"title":"Kirillov–Reshetikhin Modules of Generalized Quantum Groups of Type $A$","authors":"Jae-Hoon Kwon, M. Okado","doi":"10.4171/prims/57-3-9","DOIUrl":"https://doi.org/10.4171/prims/57-3-9","url":null,"abstract":"The generalized quantum group of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang-Baxter equation. In this paper, we develop the crystal base theory for finite-dimensional representations of generalized quantum group of type $A$. As a main result, we construct Kirillov-Reshetikhin modules, that is, a family of irreducible modules which have crystal bases. We also give an explicit combinatorial description of the crystal structure of Kirillov-Reshetikhin modules, the combinatorial $R$ matrix, and energy function on their tensor products.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41591856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassmannians. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this paper we undertake a comprehensive study of a certain semi-local transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing recent work of Inoue-Lam-Pylyavskyy. Special cases of transformations we consider include geometric R-matrices appearing in Berenstein-Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov-Shen.
{"title":"Plabic R-Matrices","authors":"Sunita Chepuri","doi":"10.4171/prims/56-2-2","DOIUrl":"https://doi.org/10.4171/prims/56-2-2","url":null,"abstract":"Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassmannians. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this paper we undertake a comprehensive study of a certain semi-local transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing recent work of Inoue-Lam-Pylyavskyy. Special cases of transformations we consider include geometric R-matrices appearing in Berenstein-Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov-Shen.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-2-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45862710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to Hodge Theory of the Middle Convolution","authors":"M. Dettweiler, C. Sabbah","doi":"10.4171/PRIMS/54-2-8","DOIUrl":"https://doi.org/10.4171/PRIMS/54-2-8","url":null,"abstract":"We give a correction to the statement of Theorem 3.2.3 of [2]. 2010 Mathematics Subject Classification: 14D07, 32G20, 32S40, 34M99.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-2-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44345806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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