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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given an algorithm for resolution of singularities that satisfies certain conditions (``a good algorithm''), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme \(T\)) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced \(T\), this parameter scheme can be naturally expressed as a disjoint union of locally closed sets \(T_j\), such that the induced family on each part \(T_j\) is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families. resolution of singularities; algorithmic resolution; simultaneous resolution; Hilbert schemes Encinas, S., Nobile, A. and Villamayor, O.: On algorithmic equi-resolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), no. 3, 607-648. Global theory and resolution of singularities (algebro-geometric aspects) On algorithmic equi-resolution and stratification of Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Artin-Schelter regular algebras of global dimension 3 have been completely classified. Much has been written about the associated geometry in the case where \(A\) is a regular algebra generated by elements of degree 1. In this paper, the author continues a study of the geometry of \(\text{Tails }A=\text{GrMod }A/\text{Fdim }A\) when \(A\) is a regular algebra of global dimension 3 that is not generated by elements of degree 1. Here \(\text{GrMod }A\) is the category of graded right \(A\)-modules and \(\text{Fdim }A\) is the full subcategory consisting of direct limits of finite-dimensional modules.
This paper starts by sharpening the classification of regular algebras of weight \((1,1,n)\) with \(n>1\). \textit{D. R. Stephenson} [J. Algebra 183, No. 1, 55-73 (1996; Zbl 0868.16027)] proved that a regular algebra \(A\) of weight \((1,1,n)\) is isomorphic to an Ore extension \(A\cong R[z;\sigma,\delta]\) where \(R\) is regular of global dimensional 2, \(\sigma\) is a graded automorphism of \(R\), and \(\delta\) is a graded left \(\sigma\)-derivation on \(R\). In Section 1 of this paper, by direct computation, the author determines all possibilities for \(\sigma\) and \(\delta\) in each case. In Section 3, a point module over \(A\) is defined to be a graded, normalized right \(A\)-module which is 1-critical and has multiplicity at most 1. In Section 5, it is shown that the set of isomorphism classes of point modules over \(A\) has the structure of the graph of an automorphism \(\tau=\tau_A\) of a subscheme \(D_A\subseteq P(1,1,n)\). If \(R\cong k\{x,y\}/(yx-qxy)\) and \(A\cong R[z;\sigma,\delta]\) with \(\sigma\) generic, it is proved that \(D_A\) breaks down into three components: two lines and a nonsingular curve. The automorphism \(\tau\) stabilizes these components. Other possibilities exist for the decomposition of \(D_A\). Some examples are given in Section 5 and 6. noncommutative schemes; quantum planes; Artin-Schelter regular algebras; categories of graded right modules; finite-dimensional modules; Ore extensions; graded automorphisms; point modules Darin R. Stephenson, Quantum planes of weight (1,1,\?), J. Algebra 225 (2000), no. 1, 70 -- 92. , Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Homological dimension in associative algebras, Ordinary and skew polynomial rings and semigroup rings, Rings arising from noncommutative algebraic geometry Quantum planes of weight \((1,1,n)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well-known that the Hilbert function \(H_{\mathbb{X}}\) of a finite set of points \(\mathbb{X}\) in the projective space \(\mathbb{P}^n\) over a field \(k\) contains important geometric information about the embedding \(\mathbb{X}\subset \mathbb{P}^n\). In this paper, the authors construct a new invariant, called the \(n\)-type vector of \(\mathbb{X}\), which encodes the information of \(H_{\mathbb{X}}\) in a different way. In their main theorem, they prove that there is a 1-1 correspondence between \(n\)-type vectors and Hilbert functions of finite point sets in \(\mathbb{P}^n\).
For point sets in \(\mathbb{P}^2\), \textit{L. Gruson} and \textit{C. Peskine} introduced the numerical character of \(H_\mathbb{X}\). [in: Algebraic Geometry, Proc., Trans. Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14001)]. The authors show that their 2-type vector is equivalent to this numerical character and conclude that one can characterize the Hilbert functions of points in \(\mathbb{P}^2\) having the uniform position property in terms of their 2-type vector. Similarly, the 2-type vectors of complete intersections in \(\mathbb{P}^2\) are easy to describe. Furthermore, the authors use the \(n\)-type vector to give a bound for the largest Hilbert function of a subset of \(\mathbb{X}\) which lies on a hyperplane, and they show that \(k\)-configurations are extremal with respect to this bound.
The \(n\)-type vector has proven to be a useful and information-rich invariant of the embedding \(\mathbb{X}\subset \mathbb{P}^n\). The authors were able to extend their results about it in their subsequent paper [\textit{A. V. Geramita, T. Harima} and \textit{Y. S. Shin}, in: The curves seminar at Queen's, Kingston 1998, Queen's Pap. Pure Appl. Math. 114, 97-140, Exposé II D (1998; see the following review Zbl 0943.13013)]. \(n\)-type vector; Hilbert functions of finite point sets; numerical character; complete intersections A. V. Geramita, T. Harima, and, Y. S. Shin, An alternative to the Hilbert function for the ideal of a finite set of points in Pn, Illinois J. Math, in press. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Configurations and arrangements of linear subspaces An alternative to the Hilbert function for the ideal of a finite set of points in \(\mathbb{P}^n\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A very beautiful connection was found by Brieskorn and Slodowy between simple singularities (rational double points) and simple Lie algebras. In the paper under review the author seeks an analogue of this connection in the class of 1-dimensional simple singularities which are complete intersection in \({\mathbb{C}}^ 3.\) These singularities are classified by \textit{M. Giusti} [Singularities, Summer Inst., Arcata/Calif. 1987, Proc. Symp. Pure Math. 40, Part 1, 457-494 (1983; Zbl 0525.32006)] and they are labelled \(S_{\mu} (\mu =5,6,7,...)\), \(T_ 7,T_ 8,T_ 9,U_ 7,U_ 8,U_ 9,W_ 9,W_{10},Z_ 9,Z_{10}\). The author associate to \(S_{\mu}^ a \)diagram called \(D=D_ k[*]\), \(k=\mu -1\) which is constructed from the Dynkin diagram \(D_ k\) by adding one distinguished vertex. They are connected as follows: The diagram determines a torus embedding \({\mathcal X}(D)\) and the parameter space of the semi-universal deformation of the singularity is identified with \({\mathcal X}(D)/W_ 2(D)\), where \(W_ 2(D)\) is an analogue of the Weyl group, and this identification respects the discriminants of the both spaces. A similar result is proved for \(T_{\mu}\) and \(E_{\mu}[*]\) \((\mu =6,7,8)\). extended Dynkin diagram; 1-dimensional simple singularities; complete intersection; torus embedding; semi-universal deformation K. Wirthmüller , Torus embeddings and deformations of simple singularities of space curves , Acta Math. 157 (1986), 159-241. Singularities of curves, local rings, Deformations of complex singularities; vanishing cycles, Deformations of singularities, Projective techniques in algebraic geometry, Embeddings in algebraic geometry Torus embeddings and deformation of simple singularities of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author calculates in an explicit form bases of invariant rings of indecomposable real finite reflection groups. Note that a similar problem is considered earlier by \textit{V. F. Ignatenko} [J. Sov. Math. 33, 933-953 (1986); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 16, 195-229 (1984; Zbl 0592.51008)] but the author's treatment is shorter. bases of invariant rings; indecomposable real finite reflection groups M. L. Mehta, Basic sets of invariant polynomials for finite reflection groups, Comm. Algebra 16 (1988), 1083-1098. Other geometric groups, including crystallographic groups, Vector and tensor algebra, theory of invariants, Reflection groups, reflection geometries, Geometric invariant theory, Other algebraic groups (geometric aspects) Basic sets of invariant polynomials for finite reflection groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities simple algebras; Brauer groups; Kronecker product of quaternion algebras Tignol, J. -P.: Corps à involution neutralisés par une extension abélienne elémentaire. Lecture notes in math. 844 (1981) Finite rings and finite-dimensional associative algebras, Rings with involution; Lie, Jordan and other nonassociative structures, Brauer groups (algebraic aspects), Generalizations (algebraic spaces, stacks) Corps à involution neutralisés par une extension abélienne élémentaire | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Seit vor nun fast hundert Jahren Hilbert im Zusammenhang mit dem 12. Hilbertschen Problem das nähere Studium der Hilbertschen Modulgruppen \(\Gamma\) total-reeller algebraischer Zahlkörper \(K\) und ihrer Modulfunktionen als direkte Analoga zur klassischen Modulgruppe anregte, hat sich, ausgehend von den Arbeiten von Blumenthal und Hecke, eine umfangreiche Theorie entwickelt. Es gab jedoch keine einführende Darstellung, so daß jeder Interessierte von Anfang an auf die ständig wachsende Flut von Originalarbeiten angewiesen war. Anfang der siebziger Jahre gelang es F. Hirzebruch, die Singularitäten des kompaktifizierten Quotientenraumes von \(\Gamma\) im Fall \([K:\mathbb Q]=2\) aufzulösen, wodurch eine singularitätenfreie projektive algebraische Fläche entstand. Damit war die unmittelbare Verbindung zur algebraischen Geometrie hergestellt, für die diese Flächen unter anderem eine Menge interessanten Beispiele und Sonderfälle lieferten.
Der Verf. des vorliegenden Buches schrieb 1981 zusammen mit \textit{F. Hirzebruch} eine erste Einfüh\-rung in diesen Teil der Theorie der Hilbertschen Modulfunktionen (d.h., \(\Gamma\) für \([K:\mathbb Q]=2\), zugehörige Quotientenflächen und ihre algebraischen Untervarietäten) unter dem Titel ``Lectures on Hilbert modular surfaces'' [Séminaire de Mathématiques Supérieures, Séminaire Scientifique OTAN (NATO Advanced Study Institute), Dep. Math. Stat., Univ. Montréal 77 (1981; Zbl 0483.14009)]. Die neue Darstellung entspricht im Aufbau bis auf einige Umstellungen im wesentlichen dem vom Verf. stammenden zweiten Teil der ursprünglichen Einführung einschließlich des Stoffes des Kapitels über Hilbertsche Modulflächen des ersten, damals von Hirzebruch verfaßten Teiles [für den Inhalt sei auf die ausführliche Besprechung von \textit{D. Zagier} im Zbl 0483.14009 verwiesen], ist aber im einzelnen bedeutend ausführlicher. Am Anfang steht jetzt ein zusätzliches Kapitel mit einer kurzen Beschreibung der Grundlagen (``Spitzen, Fundamentalbereich, elliptische Fixpunkte, Modulformen'') und am Schluß ein Kapitel über die Tate-Vermutungen für Hilbertsche Modulflächen, in dem über die neuen Ergebnisse von \textit{G. Harder}, \textit{R. P. Langlands} und \textit{M. Rapoport} [J. Reine Angew. Math. 366, 53--120 (1986; Zbl 0575.14004)] und \textit{C. Klingenberg} [Invent. Math. 89, 291--318 (1987; Zbl 0601.14007)] berichtet wird. Hervorzuheben ist das ebenfalls neue Kapitel VIII mit einer Zusammenstellung vieler interessanter Beispiele.
Die übrigen Kapitel (II.\ Auflösung der Spitzensingularitäten, III.\ Lokale Invarianten, IV.\ Globale Invarianten, V.\ Modulkurven auf Modulflächen, VI.\ Kohomologie, VII.\ Klassifikation der Hilbertschen Modulflächen, IX.\ Humbertsche Flächen, X.\ Moduln abelscher Flächen mit reeller Multipikation') bringen jeweils eine gut lesbare Einführung und verleihen durch eine Übersicht über weitere Ergebnisse mit ausführlichen Literaturhinweisen dem Buch den Charakter eines Handbuchs. Angefügt ist eine Tabelle numerischer Invarianten für die Hilbertschen Modulgruppen reell-quadratischer Zahlkörper mit den Diskriminanten \(D<500\).
Das Buch dürfte für längere Zeit das Standardwerk über dieses Gebiet sein. classification of Hilbert modular surfaces; Hilbert-Blumenthal surfaces; resolution of cusp singularities; Humbert surfaces; moduli Van Der Geer, G., \textit{Hilbert modular surfaces}, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in mathematics and related areas (3)], Vol. 16, (1988), Springer, Berlin Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic ground fields for surfaces or higher-dimensional varieties, Modular and Shimura varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to number theory, Global theory and resolution of singularities (algebro-geometric aspects) Hilbert modular surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors give an introduction to the theory of finite sets of points in affine and projective spaces. While providing the necessary theoretical background, they emphasize the computational methods which have been developed to deal with such sets of points.
First they describe the Buchberger-Möller algorithm for computing a Gröbner basis of the defining ideal of a set of \(K\)-rational points in affine space \(\mathbb{A}^n_K\), if one is given their coordinates. They present the analysis of \textit{M. G. Marinari}, \textit{H. M. Möller} and \textit{T. Mora} [Appl. Algebra Eng. Commun. Comput. 4, No. 2 103-145 (1993; Zbl 0785.13009)] which shows that this algorithm is of polynomial complexity in \(n\) and the number of points. Then they treat the algorithm from the same paper which allows one to find a minimal system of generators of the homogeneous ideal of a set of points in projective space. -- In the second part of the paper, the authors give an overview of more advanced recent developments in the theory of point sets such as computation of their Hilbert-Poincaré series, separators, Cayley-Bacharach schemes, \(G\)-symmetric and \(G\)-transitive sets of points. Here they are mainly citing the results of the papers by \textit{A. V. Geramita}, \textit{M. Kreuzer} and \textit{L. Robbiano} [Trans. Am. Math. Soc. 339, No. 1, 163-189 (1993; Zbl 0793.14002)], and by \textit{G. Niesi} and \textit{L. Robbiano} [in: Computational algebraic geometry, Pap. Conf. Nice 1992, Prog. Math. 109, 195-201 (1993; Zbl 0806.14046)].
For anyone interested in computational aspects of finite set of points in affine or projective spaces, the paper under review presents an ideal starting point. Beginning with the first elementary observations and leading up to recent developments, the authors manage to give a clear and comprehensible overview allowing the reader to start implementing the described algorithms right away. rational points; Gröbner basis of the defining ideal; Hilbert-Poincaré series; separators; Cayley-Bacharach schemes; computational aspects of finite set of points Mora, T.; Robbiano, L.: Points in affine and projective spaces. Computational algebraic geometry and commutative algebra, cortona-91, 106-150 (1993) Computational aspects in algebraic geometry, Rational points, Relevant commutative algebra, Symbolic computation and algebraic computation, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Analysis of algorithms and problem complexity Points in affine and projective spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In [Invent. Math. 3, 75-135 (1967; Zbl 0219.14024)] \textit{D. Mumford} gave a satisfactory answer to the possibility to embed the moduli space \(\Gamma_g (n, 2n) \setminus \mathbb{H}_g\) of principally polarized abelian varieties with level \(n\)-structure in some projective space through theta-nullwerte in case \(n\geq 4\). In a recent paper the author settled the case \(n=2\) showing the map to be generically injective. Now in the present paper it is first shown by using the Gauß map as before that the map \(\Gamma_g (3, 6)\setminus \mathbb{H}_g\to \mathbb{P}^{3^g-1}\) given by the theta-nullwerte \(\tau\mapsto \vartheta {m' \brack 0} (3\tau, 0)\), \(m'\in ({1\over 3} \mathbb{Z}/ \mathbb{Z})^g\) is an embedding, too. Moreover, evaluating the gradients at the origin of the analogous theta-functions \(\vartheta {m' \brack 0} (n\tau, nz)\), \(m'\in (n^{-1} \mathbb{Z}/ \mathbb{Z})^g\), which are modular forms with respect to the representation \(\text{det}^{1/2} \otimes \text{St}\), St being the tautological one, and obvious multiplier system \(v_n\), yields for \(n\geq 3\) a mapping \(\Phi_n: \Gamma_g (n, 2n) \setminus \mathbb{H}_g\to G(g, n^g)\) into the Grassmannian of \(g\)-planes, the differential of which is always injective in case \(n\geq 5\) and with well-understood ramification locus in case \(n=4\), \(g\geq 2\). level theta structure; Siegel modular groups; Siegel modular forms; moduli space of principally polarized abelian varieties; theta-functions; differential Salvati Manni R.: On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure. Math. Z. 221, 231--241 (1996) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Theta functions and abelian varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0707.16006. pfaffian; Azumaya algebra; quadratic form; alternating elements; discriminant module; tensor products of quaternion algebras; involution of orthogonal type; algebra with involution; involutions on central simple algebras; group of special similitudes Knus, M.-A., Parimala, R., Sridharan, R.: Pfaffians, central simple algebras and similitudes. Math. Z.206, 589-604 (1991) Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Rings with involution; Lie, Jordan and other nonassociative structures, Algebraic theory of quadratic forms; Witt groups and rings, Finite-dimensional division rings, Quadratic forms over general fields, Quadratic spaces; Clifford algebras, Clifford algebras, spinors, Quadratic and bilinear forms, inner products, Brauer groups of schemes, Linear algebraic groups over arbitrary fields Pfaffians, central simple algebras and similitudes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Lambda\) be a canonical algebra with \(n\) simple modules and \({\mathcal D}^ b(\text{mod}(\Lambda))\) the derived category of right \(\Lambda\)- modules. We show that the semidirect product of \(\mathbb{Z}^ n\) and the braid group on \(n\) strings acts transitively on the set of exceptional sequences in \({\mathcal D}^ b(\text{mod}(\Lambda))\). Similar results are proved for coherent sheaves on weighted projective lines. canonical algebras; simple modules; derived category of right \(\Lambda\)- modules; semidirect products; braid groups; exceptional sequences; coherent sheaves on weighted projective lines Meltzer, H.: Exceptional sequences for canonical algebras. Arch. Math. 64, 304--312 (1995) Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of orders, lattices, algebras over commutative rings, Module categories in associative algebras, Simple and semisimple modules, primitive rings and ideals in associative algebras Exceptional sequences for canonical algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The major concern of the book under review, to whom the author modestly refers to as ``Lecture Notes'', is the algebraic enumerative geometry understood in the most modern of the possible modern senses. This, however, without neglecting a constant and respectful eye to the classical roots of the subject, never left aside but enhanced by that deeper understanding gained from new visions and perspectives. Self-defined as ``Lecture Notes'', this collection of ten dense chapters form, to say the truth, an impressive monumental book, 375 pages long, whose title ``\textsl{K-theoretic computations in Enumerative Geometry}'', that seems very focused, looks more quite a pretest for an amazing interdisciplinary carousel of mathematical fireworks than a way to confine the subject between artificial walls.
The treatise, that's the way we think more appropriate to refer to the ``Lecture Notes'', is written with an excited and exciting mood. It is a vehicle through which the author transmits enthusiasm and arouses the interests of many many readers, regardless of the harsh reality, namely, that only a few elected and very dedicated people will be really able to grasp the deepness and the subtleties of most of the contents, which is what one needs to feel breathtaking emotions.
In the very first lines of his work, the author declares that the lectures are intended ``\textsl{for graduate students who want to learn how to do the computations from the title}''. That gave the reviewer, mostly unaware of the material developed in the notes, the pleasant sensation to feel younger, a little bit like a graduate student, although certainly not among the best.
The author clearly emphasizes that the lectures are focused on computations. As he puts it, it is very important to keep a ``\textsl{connection between abstract notions of algebraic geometry, which can be very abstract indeed, and something we can see, feel, or test with code}''. If the journey is promising, however, certainly is not effortless: everyone is warned that the exciting but tiring reading is demanding, that everyone is asked to work himself and is exhorted to read with an active, and not passive, attitude.
Who wish to be initiated to the mysteries of the enumerative \(K\)-theory through these notes, is invited, nearly every half a page, to think her/himself to work out an exercise, to propose creative examples. To say it with the same words of the author, in fact, while ``\textsl{it is a challenge to adequately illustrate a modern algebraic geometry narrative, one can become familiar with main characters of the notes by working out examples}'' and his hope is ``\textsl{that these notes will be placed alongside a pad of paper, a pencil, an eraser, and a symbolic computation interface.}''
One difficulty in reviewing this book is that it is not just an expository work, although it is also that, and is not just a research book, although it is also that. In fact it covers a wide range of interrelated subjects that even now, while the reviewer is writing the present report, or the reader is reading the inadequate reviewer's words, is in rapidly expansion. This treatise forms the body not of a static but of a dynamic book. It sounds like the foundation of a new literary genre, that of mathematical snapshots taken by experienced photographers from the window of a running train, while the landscape rapidly changes before the eyes. The titanic flavor of the work, in spite of the prudent warning of the author (\textsl{These notes are meant to be a partial sample of basic techniques and results, and this is not an attempt to write an A to Z technical manual on the subject, nor to present a panorama of geometric applications that these techniques have}), comes from the (successful) attempt to draw the state of art of the subject as photographed in the exactly the same instant the notes were going to be written.
Stopping by listing too easy and obvious praises, both to the text and the fresh style it is written with, the best way to end this review is probably that of giving the reader the idea of how its content is organized. The book is divided into ten chapter, that the author calls Sections: hard work already begins in Section 2, despite is entitled ``\textsl{Before we begin}''. This is a kind of warning, like the game level \(0\) where one must gains points to access to the next one. The first section ``\textsl{Aim and Scopes}'' is dramatically intriguing and transmits a lot of excitement to the potential reader who will tend to forget, due to the beauty of the exposition, how much the book is demanding.
The third section is dedicated to the study of the still mysterious land of Hilbert schemes of points in threefolds: by contrast with the case of points on a surface, even its dimension is not known. This topic is used as a pretest to introduce the traveler into the realm of the Donaldson-Thomas (DT) invariants. A central result of this section is the Nekrasov formula related with the DT theory, originally born in the context of string theory. A remarkable family of equivariant symplectic resolutions are provided by Nakajima quiver varieties, that have found many geometric and representation-theoretic applications and whose construction is recalled in Section 4.
Next level: Section 5. Devoted to the DT theory on symmetric powers of curves, instead of working with stable maps, whose first appearance dates back to the milestone papers by Kontsevich and Manin [\textsl{Gromov-Witten classes, quantum cohomology, and enumerative geometry}. Mirror symmetry, II, 607--653, AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, RI, 1997], the author deals with the more general \textsl{quasimaps}, extensively treated in Section 4.3. These are not the same as the stable ones, but the author implicitly suggests a research direction to check his guess that the methods used by \textit{A. Givental} and \textit{V. Tonita} [Math. Sci. Res. Inst. Publ. 62, 43--91 (2014; Zbl 1335.19002)] may be possibly adapted to the situation considered in Section 6, where quasimaps and related material are extensively treated.
\textsl{Vertices and edges} (certain kind of tensors associated to fixed points and invariant curves under the torus action of a toric threefold) have direct analog in \(K\)-theory, and are the ``\textit{Nuts and bolts}'' of the subject, widely treated in Section 7, through which the mathematical performer. according to the Master, may measure his own degree of professional skill.
Section 8 is concerned with equivariant difference equations: key words for this chapter are \textsl{difference equations for vertices}. The last two chapters are entitled respectively \textsl{Stable envelops} and \textsl{Quantum Knizhnik-Zamolodchikov equations}. These are rather technical sections, and one will not even attempt to popularize them within the few lines left in this already too long review. The best recommendation is to read over and over the very first section, ``\textsl{Aim and Scope}'', to arouse once more in the beginner the desire to deeply penetrate, and become intimate with, the content of the notes.
The notes, the book, the paper, or whatever this mathematical exposition be, concludes itself with a huge list of related literature to whom the reader is really invited to refer to. As if the reviewed masterpiece were, more than a usual report, an Arianna thread, a sort of compass necessary to avoid getting lost in the labyrinth of a unusually spectacular mathematics that the young brave reader might like to venture out. enumerative geometry; computations in K-Theory; Donaldson-Thomas theory; moduli spaces; Hilbert schemes of points in threefolds; Nakajima varieties; stable envelops and quantum groups; quantum Knizhnik-Zamolodchikov equations A. Okounkov, \textit{Lectures on K-theoretic computations in enumerative geometry}, arXiv:1512.07363 [INSPIRE]. Research exposition (monographs, survey articles) pertaining to \(K\)-theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Projective and enumerative algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Parametrization (Chow and Hilbert schemes) Lectures on \(K\)-theoretic computations in enumerative geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities rational curves; multiple points; one-to-one correspondence between lattices of points; fundamental period parallelograms of elliptic functions; absolute invariant Algebraic geometry Curve razionali che esprimono le relazioni modulari | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper under review, the authors follow relations of the triality with theory of simple singularities in the sense of V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko [\textit{V. I. Arnol'd} et al., Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, ed. by V. I. Arnol'd. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001)].
The authors recall É. Cartan's construction (from 1925) of the triality automorphism of the Lie algebra \(\mathfrak{so(8)}\), give a matrix representation for the real form \(\mathfrak{so(4,4)}\), describe how the cohomology homomorphism induced by the triality automorphism operates on the characteristic classes in the cohomology of the classifying space \(B\text{Spin}(8)\), and study the triality automorphism of the singularity \(D_4\).
The reviewer remarks that the following paper (not listed in the references) is closely related to the cohomology aspect of this work: [\textit{A. Gray} and \textit{P. Green}, Pac. J. Math. 34, 83--96 (1970; Zbl 0194.22804)]. triality; characteristic class; Lie algebra; singularities of smooth functions Homology of classifying spaces and characteristic classes in algebraic topology, Exceptional (super)algebras, Singularities of surfaces or higher-dimensional varieties, Local complex singularities Triality, characteristic classes, \(D_4\) and \(G_2\) singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a field and \(G\) a quasi-simple subgroup of the Chevalley group \(F_4(K)\). We assume that \(G\) is generated by a class \(\Sigma\) of abstract root subgroups such that there are \(A,C\in\Sigma\) with \([A,C]\in\Sigma\) and any \(A\in\Sigma\) is contained in a long root subgroup of \(F_4(K)\). We determine the possibilities for \(G\) and describe the embedding of \(G\) in \(F_4(K)\). groups of Lie type; Chevalley groups; long root elements; root subgroups; orthogonal groups; buildings; polar spaces; Moufang hexagons Steinbach, A.: Groups of Lie type generated by long root elements in \(F4(K)\), J. algebra 255, 463-488 (2002) Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Simple groups: alternating groups and groups of Lie type, Generators, relations, and presentations of groups, Groups with a \(BN\)-pair; buildings Groups of Lie type generated by long root elements in \(F_4(K)\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Basic concepts of the theory of formal groups are introduced and applications to groups and Lie algebras of dynamical systems are briefly summarized. formal groups; Lie algebras of dynamical systems General properties and structure of complex Lie groups, Formal groups, \(p\)-divisible groups, Dynamical systems and ergodic theory Certain applications of formal groups in the theory of dynamical systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The concept of initial ideals for ideals of a polynomial ring in Gröbner bases theory is generalized in a natural way for subalgebras of a polynomial ring, and they are called initial algebras. A set of generators of a subalgebra is called a SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis if their initial monomials generate the initial algebra. The main difference between the initial ideal and the initial algebra is that the former always is finitely generated while the latter does not. Hence it is an important problem to find a criterion for the finite generation of initial algebras.
\textit{M. Göbel} [J. Symb. Comput. 26, 261--272 (1998; Zbl 0916.13011)], studied this problem for the subalgebra of invariants of a permutation group \(G\). He proved that, with respect to the lexicographic order, the initial algebra of \(K[V]^G\) is finitely generated if and only if \(G\) is a direct product of symmetric groups.
In the present paper, the same result is proved for any multiplicative order, i.e., a monomial order which does not require the minimality of the unit \(1\), and also for the action on the Laurent polynomial ring.
A principal role in the proof plays the topological structure on the set of multiplicative orders. Using this structure and the Baire theorem, the author proves that there exist uncountable cardinality of distinct initial algebras for each invariant ring, when \(G\) is not a direct product of symmetric groups. (Note that in the case of initial ideals, there exist only a finite cardinality of distinct initial ideals for an ideal under a certain condition, although there exist infinitely many orders.) If \(G\) is a product of symmetric groups, the number of distinct initial algebras of the subalgebra of invariants is finite and is equal to the order of \(G\). subalgebra analogue to Gröbner bases for ideals; SAGBI; Laurent polynomial; polynomial invariants; finite permutation groups; initial terms; multiplicative orders; finite generation of initial algebras; subalgebra of invariants of a permutation group Kuroda, O.: The infiniteness of the SAGBI bases for certain invariant rings. Osaka J. Math. 39, 665--680 (2002) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Group actions on affine varieties, Commutative rings and modules of finite generation or presentation; number of generators, Actions of groups on commutative rings; invariant theory, Polynomial rings and ideals; rings of integer-valued polynomials The infiniteness of the SAGBI bases for certain invariant rings. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a complex \(2\)-dimensional vector space and let \(\Gamma\) be a finite subgroup of \(\mathrm{SL}(V)\). Up to conjugacy, such subgroups are classified by the Dynkin diagrams of type \(A\), \(D\) and \(E\). Suppose \(\Delta\) is the Dynkin diagram of \(\Gamma\). The quotient \(V/\Gamma\) embeds as a hypersurface in \(\mathbb{A}^3\) and is a Kleinian singularity or rational double point of type \(\Delta\). Identifying \(V/\Gamma\) with the set of zeros of a weight homogeneous polynomial in \(\mathbb{C}^3\), it follows that there is a Possion bracket on the coordinate ring \(\mathbb{C}[V]^\Gamma\) and an associated Poisson structure on the polynomial ring \(\mathbb{C}[x,y,z]\).
In this paper, the author constructs all possible noncommutative deformations of \(V/\Gamma\) of type \(D\) in terms of generators and relations. He also proves that the moduli space of isomorphism classes of noncommutative deformations in type \(D_n\) is isomorphic to a vector space of dimension \(n\).
Earlier, \textit{P. Boddington} already constructed all possible noncommutative deformations of type \(D\) Kleinian singularities in his Ph.D. thesis [``No-cycle algebras and representation theory'', Ph.D. thesis, University of Warwick (2004)]. Boddington's parametrization of the noncommutative deformations is more closely related to the construction by \textit{W. Crawley-Boevey} and \textit{M. P. Holland}, [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)], than the construction in the article under review which uses methods similar to those of \textit{V. V. Bavula} and \textit{D. A. Jordan}, [Trans. Am. Math. Soc. 353, No. 2, 769-794 (2001; Zbl 0961.16016)]. Kleinian singularities of type \(D\); noncommutative deformations; simply laced Dynkin diagrams; coordinate rings; Poisson brackets; generators and relations; moduli spaces Levy, P., Isomorphism problems of noncommutative deformations of type \textit{D} Kleinian singularities, Trans. Amer. Math. Soc., 361, 5, 2351-2375, (2009) Deformations of associative rings, Rings arising from noncommutative algebraic geometry, Deformations of singularities, Noncommutative algebraic geometry Isomorphism problems of noncommutative deformations of type \(D\) Kleinian singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0516.00012.]
This paper addresses the problem of attaching a ''motif'' \(M_ f\) to a suitable modular form f on an indefinite quaternion group over a totally real number field F [\textit{R. P. Langlands} in Math. Developm. Hilbert Probl., Proc. Sympos. Pure Math. 28, De Kalb 1974, 401-418 (1976; Zbl 0345.14006)].
For \([F:{\mathbb{Q}}]=n\) odd (for simplicity), we describe \(M_ f\). Take \(\Gamma\) to be a \(\Gamma_ 0\)-type congruence subgroup, f a holomorphic weight-2 eigenform. Let \(X_{\Gamma}\) be the associated Hilbert modular variety, and \(H^ n_{sp}(X_{\Gamma})\) the weight-n part of \(H^ n(X_{\Gamma})\) [the latter is the ''n-th motif of \(X_{\Gamma}''\); see \textit{P. Deligne} in Automorphic forms, representations and L-functions, Proc. Symp. Pure Math. 33, No.2, 313-346 (1979; Zbl 0449.10022)]. Let \(H^ n_{sp}(X_{\Gamma})^{new}\) be the orthogonal complement of the part coming from smaller \(\Gamma\) 's. Let e be the idempotent in the Hecke algebra for f. Put \(M_ f=eH^ n_{sp}(X_{\Gamma})^{new}.\)
The author conjectures that: (1) \(M_ f\) is ''realizable'' by an abelian variety \(A_ f\) (plus structure...) defined over f, and \((2)\quad the\) ''functor'' \(M_ f\mapsto A_ f\) turns the Eichler-Shimizu correspondence into an isogeny over F (with structure). - With an assumption that L(1,f,X)\(\neq 0\) for certain quadratic characters X, the author proves the first conjecture for \([F:{\mathbb{Q}}]=odd\), and, for \([F:{\mathbb{Q}}]=even\), finds \(A_ f\) defined over \({\mathbb{C}}\). For \(f_ 1\) and \(f_ 2\) both satisfying this ''nonvanishing criterion'', it is shown that if \(f_ 1\) and \(f_ 2\) correspond under Eichler-Shimizu, then \(A_{f_ 1}\) and \(A_{f_ 2}\) are isogenous over \({\mathbb{C}}\). realization of motif of modular form by abelian variety; Hilbert; modular cusp form; totally real number field; Hilbert modular variety; Hecke algebra; Eichler-Shimizu correspondence T. Oda : Hodge structures of Shimura varieties attached to the unit groups of quaternion algebras , in Advanced Studies in Pare Mathematics 2: Galois Groups and Their Representations , 15-36 Tokyo: Kinokuniya Press (1983). Generalizations (algebraic spaces, stacks), Algebraic theory of abelian varieties, Holomorphic modular forms of integral weight, Totally real fields Hodge structures of Shimura varieties attached to the unit groups of quaternion algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the ramification sets of finite analytic mappings and applications of his results and methods to punctual Hilbert schemes and to finite analytic maps. He uses essentially the technique of ``chaining'' which consists in associating to a finite map a sequence (or chain) of sets, which are components of ramification loci of increasing multiplicity, and then in controlling their dimensions. For that purpose he uses a theorem of Grothendieck about the order of connexity of subgerms of an irreducible analytic germ, and also in the projective case the Fulton-Hansen theorem and a theorem of Deligne.
In {\S} 1, the author introduces three different notions of multiplicity. The topological multiplicity, the stable multiplicity and the algebraic one. In {\S} 2, he gives a fairly general lower bound to the dimension of the ramification set \(T^{d+1}(f)\), the set of points at which the multiplicity is at least \(d+1\). For the topological case he needs a hypothesis about f, called weak multitransversality which guarantees the additivity of multiplicity under deformation. This theorem is proved by a complicated induction involving multiproducts of ramification sets and the theorem of Grothendieck.
In {\S} 3, the author gives applications of {\S} 2, and of the chaining technique to the punctual Hilbert scheme \(Hilb'{\mathcal O}_{X,x}\) which parametrizes in \(Hilb'(X)\) the punctual schemes concentrated at \(x\in X\). The idea consists in identifying the germ of \(Hilb'({\mathcal O}_{X,x})\) at a smoothable element z with the ramification loci an appropriate map obtained by unfolding the equation of z. He thus obtains a lower bound for the local dimension at z of the open set U of smoothable points in X. This bound is (n-1)(\(\ell -1)\) with \(n=\dim (X)\) in the easiest case (X everywhere irreducible). Various, and more complicated results are obtained when we drop the irreducibility hypothesis or consider instead of U the open set of weakly smoothable (i.e. smoothable in a smooth ambient space) element.
Finally in {\S} 4, the author proves similar results for a finite projective morphism \(f:\quad X^ n\to P^ p.\) He generalizes a previous joint result of himself with Lazarsfeld (case \(n=p)\). This consists again in giving cases of non-emptiness for \(T^{d+1}(f)\) under some complicated numerical conditions. ramification sets of finite analytic mappings; punctual Hilbert schemes; ramification loci of increasing multiplicity T. Gaffney, ''Multiple points, chaining and Hilbert schemes,'' Amer. J. Math., vol. 110, iss. 4, pp. 595-628, 1988. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Multiple points, chaining and Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let the group \(\mu_m\) of \(m\)th roots of unity act on the complex line by multiplication. This gives a \(\mu_m\)-action on Diff, the algebra of polynomial differential operators on the line. Following \textit{W. Crawley-Boevey} and \textit{M. P. Holland} [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)], we introduce a multiparameter deformation \(D_\tau\) of the smash product \(\text{Diff}\#\mu_m\). Our main result provides natural bijections between (roughly speaking) the following spaces:
(1) \(\mu_m\)-equivariant version of Wilson's adelic Grassmannian of rank \(r\);
(2) rank \(r\) projective \(D_\tau\)-modules (with generic trivialization data);
(3) rank \(r\) torsion-free sheaves on a ``noncommutative quadric'' \(\mathbb{P}^1\times_\tau\mathbb{P}^1\);
(4) disjoint union of Nakajima quiver varieties for the cyclic quiver with \(m\) vertices.
The bijection between (1) and (2) is provided by a version of Riemann-Hilbert correspondence between \(\mathcal D\)-modules and sheaves. The bijections between (2), (3), and (4) were motivated by our previous work [Compos. Math. 134, No. 3, 283-318 (2002; Zbl 1048.14001)]. The resulting bijection between (1) and (4) reduces, in the very special case: \(r=1\) and \(\mu_m=\{1\}\), to the partition of (rank 1) adelic Grassmannian into a union of Calogero-Moser spaces discovered by Wilson. This gives, in particular, a natural and purely algebraic approach to \textit{G. Wilson}'s result [Invent. Math. 133, No. 1, 1-41 (1998; Zbl 0906.35089)]. algebras of polynomial differential operators; multiparameter deformations; smash products; adelic Grassmannians; projective \(D\)-modules; sheaves; quadrics; Riemann-Hilbert correspondence; quiver varieties; Calogero-Moser spaces V. Baranovsky, V. Ginzburg, and, A. Kuznetsov, Wilson's Grassmannian and a non-commutative quadric, arXiv:math.AG/0203116. Group structures and generalizations on infinite-dimensional manifolds, Rings of differential operators (associative algebraic aspects), Sheaves of differential operators and their modules, \(D\)-modules, Noncommutative algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Wilson's Grassmannian and a noncommutative quadric. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a field of characteristic \(\neq 2\). Two important invariants of a quadratic form over \(K\) are its determinant (belonging to the square class group \(K^\times/K^{\times^2})\) and its Hasse invariant (which is an element of the subgroup \(Br_2(K)\) of elements of order at most 2 in the Brauer group of \(K)\). Simple formulas are derived to compute these invariants for trace forms of central simple algebras over \(K\). Galois cohomology; orthogonal group; quadratic form; determinant; Hasse invariant; Brauer group; trace forms of central simple algebras D. Lewis, J. Morales, The Hasse invariant of the trace form of a central simple algebra, Pub. Math. Fac. Sci. Besançon, 92/93--93/94, Univ. Franche-Comté, Besançon Quadratic forms over general fields, Galois cohomology, Brauer groups of schemes The Hasse invariant of the trace form of a central simple algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Over perfect fields, the study of connected linear algebraic groups reduces to that of smooth connected unipotent groups and of reductive groups. In particular, this aspect is crucial in proving results like the finiteness of Tamagawa number, finiteness of Tate-Shafarevich sets etc. However, there are a plethora of imperfect fields arising as local and global function fields over which the earlier methods fail and none of the above-mentioned results were proved until very recently. The recent proofs of such results due to Brian Conrad were made possible due to the development of a theory of pseudo-reductive groups. The subject was initiated by Tits and the present authors in collaboration with Gabber developed a comprehensive structure theory. The book under review takes off from their important work and carries out a complete classification of pseudo-reductive groups. In order to review the present book, it is necessary to recall various aspects covered in the book of \textit{B. Conrad, O. Gabber} and \textit{G. Prasad} [Pseudo-reductive groups. Cambridge: Cambridge University Press (2010; Zbl 1216.20038)]. Both the books will indubitably be standard references for a long time to come. Let us describe precisely what the objects of study are, what natural questions arise and which ones among them are answered and what their impact has been.
A smooth, connected, affine \(k\)-group \(G\) is said to be pseudo-reductive if its \(k\)-unipotent radical \(R_{u,k}(G)\) is trivial. This is different from \(G\) being a connected, reductive \(k\)-group when the unipotent radical over an algebraic closure is not trivial (this can happen only when \(k\) is imperfect). Already, Tits observed years ago that for a finite, inseparable extension \(l/k\) and a nontrivial reductive \(l\)-group \(G\), the Weil restriction \(R_{l/k}(G)\) is a pseudo-reductive \(k\)-group which is \textit{not} reductive. For instance, if \(G=GL_1\), then \(R_{l/k}(GL_1)\) is the Zariski-open subspace complementary to the hypersurface defined by the vanishing of the norm map associated to the \(k\)-vector space \(l\). In general, for a smooth, connected, affine \(k\)-group \(G\), the \(k\)-group \(G/R_{u,k}(G)\) is pseudo-reductive. A pseudo-reductive \(k\)-group is said to be pseudo-semisimple if it equals its derived subgroup. A solvable pseudo-reductive \(k\)-group is necessarily commutative. However, a classification of the latter groups seems an intractable problem. Nevertheless, Conrad, Gabber and Prasad demonstrated that this is not an obstacle to the study of pseudo-reductive groups. A nontrivial pseudo-semisimple group \(G\) which does not contain any nontrivial smooth connected normal \(k\)-subgroup is said to be pseudo-simple. Conrad-Gabber-Prasad showed that if a pseudo-reductive \(k\)-group \(G\) contains a split maximal \(k\)-torus, then all such tori are \(G(k)\)-conjugate -- such groups are called pseudo-split over \(k\). Some basic questions remained after the work of Conrad-Gabber-Prasad on the structure theory and, the present text addresses a number of them.
The main results proved in this book can be listed as:{\parindent=6mm\begin{itemize}\item[(a)] existence, uniqueness and isogeny theorems for pseudo-reductive groups,\item[(b)] structure theorem over arbitrary imperfect fields,\item[(c)] uniqueness and optimal existence results for pseudo-split and quasi-split \(k_{sep}/k\)-forms for imperfect \(k\),\item[(d)] examples in every positive characteristic where pseudo-split and quasi-split \(k_{sep}/k\)-forms do not exist,\item[(e)] a Tits-style classification of pseudo-semisimple \(k\)-groups \(G\) in terms of the Dynkin diagram of \(G_{k_{sep}}\) with \(*\)-action of \(\mathrm{Gal}(k_{sep}/k)\) and in terms of the embedded anisotropic kernel.
\end{itemize}} The authors of the book under review do not only build on Conrad-Gabber-Prasad's work, they also simplify some of the earlier proofs. Conrad-Gabber-Prasad gave a so-called standard construction of pseudo-reductive groups.
If \(k'\) is a finite, reduced nonzero \(k\)-algebra and \(G'\to\mathrm{Spec}(k')\) is a smooth affine group scheme with connected reductive fibers and \(T'\) is a maximal \(k'\)-torus of \(G'\), consider the maximal \(k'\)-torus \(T_0:=T'/Z_{G'}\) of \(G_0:=G'/Z_{G'}\) where \(Z_{G'}\) denotes the scheme-theoretic center of \(G'\). Suppose there is a commutative pseudo-reductive \(k\)-group \(C\) (whose structure is treated as a black box) and a factorization in \(k\)-homomorphisms
\[
R_{k'/k}(T')\to C\to R_{k'/k}(T_0).
\]
The natural action of \(G_0\) on \(G'\) over \(k'\) gives a \(R_{k'/k}(G_0)\)-action on \(R_{k'/k}(G')\) over \(k\) and hence an action of \(C\) via
\[
C\to R_{k'/k}(T_0)\to R_{k'/k}(G_0).
\]
The central quotient \((C\propto R_{k'/k}(G'))/R_{k'/k}(T')\) is pseudo-reductive, and a group produced by such a construction is said to be standard.
For \(\mathrm{char\,}k\neq 2,3\), the connected \(k\)-groups which are pseudo-simple over the separable closure of \(k\) are standard.
Over every imperfect field of characteristic \(2\) or \(3\), there exist non-standard pseudo-split absolutely pseudo-simple groups. This prompted Conrad-Gabber-Prasad to refine the notion of standard construction to a generalized standard construction. In the present book, the authors further fine-tune this construction.
If \(G\) is a pseudo-reductive \(k\)-group and \(K\) is the minimal field extension of \(k\) which is a field of definition of the geometric unipotent radical -- \(K\) is necessarily a purely inseparable finite extension of \(k\) -- then there is a natural map \(i_G\) from \(G\) to \(R_{K/k}(G_K/R_{u,K}(G_K))\). One calls \(G\) to be of minimal type if the intersection of the unipotent \(k\)-group scheme \(\ker(I_G)\) with some (and hence any) Cartan \(k\)-subgroup \(Z_G(T)\) of \(G\) is trivial. This notion is important in classification and structure theorems as the central pseudo-reductive quotient group is of minimal type and has the same associated \(K\). For instance, simplifiying to a large extent the proof in the Conrad-Gabber-Prasad book, the authors use this idea to prove standardness away from characteristics \(2\) and \(3\). Further, ``minimal type'' behaves well with respect to passage to normal \(k\)-subgroups and to centralizers of subgroups of multiplicative type (we note that a central quotient of a pseudo-reductive group need not be pseudo-reductive).
Over any imperfect field \(k\) of characteristic \(p\), with \(a\in k-k^p\), consider the fields \(k_1=k(a^{1/p})\), \(k_2=k(a^{1/p^2})\). Then, the \(k\)-group \(R_{k_2/k}(SL_p)/R_{k_1/k}(\mu_p)\) is a standard pseudo-simple \(k\)-group which is not of minimal type.
When \(k\) is any imperfect field, standard absolutely pseudo-simple \(k\)-groups can fail to be of minimal type. So, the authors define a weaker notion called locally of minimal type. A pseudo-reductive \(k\)-group \(G\) is locally of minimal type if the \(k\)-subgroup of \(G_{k_{sep}}\) generated by any pair of opposite root groups relative to a maximal \(k_{sep}\)-torus is a central quotient of an absolutely pseudo-simple \(k_{sep}\)-group of minimal type. Groups locally of minimal type encompass most of our groups of interest -- the generalized standard pseudo-reductive groups are easily shown to be locally of minimal type.
One of the main structure theorems proved in the book asserts: A pseudo-reductive group which is locally of minimal type is generalized standard. In particular, for any pseudo-reductive group \(G\), the group \(G/Z_G\) is a pseudo-reductive group which is generalized standard.
We note that when characteristic of \(k\) is \(2\) and \([k:k^2]>2\), there exist for each \(n\geq 1\), pseudo-split absolutely pseudo-simple \(k\)-groups \(G\) with non-reduced root system \(BC_n\) such that \(G\) is \textit{not locally of minimal type}.
Remarkably, many variants of classical results are either false or require completely different arguments. For instance, the proof of the pseudo-isogeny theorems given by the authors builds on a pseudo-reductive variant of an idea of Steinberg and is completely different from the traditional proof of the isogeny theorem for connected semisimple groups.
A natural question is whether a pseudo-reductive \(k\)-group \(G\) admits a pseudo-split \(k_{sep}/k\)-form. In other words, is there a pseudo-reductive \(k\)-group \(H\) such that \(G_{k_{sep}}\cong H_{k_{sep}}\) and \(H\) admits a split maximal \(k\)-torus? Amazingly, in every positive characteristic, the authors give examples of pseudo-semisimple groups which do not admit pseudo-split forms. Further, when \(\mathrm{char\,}k=2\), \([k:k^2]>4\) and \(k\) admits a quadratic Galois extension \(k'\) such that \(\ker(Br(k)\to Br(k'))\neq\{1\}\), then the authors prove the existence, for every \(n\geq 1\), of non-standard absolutely pseudo-simple \(k\)-groups of types \(B_n\), \(C_n\) and \(BC_n\) over \(k_{sep}\) which do \textit{not} admit a quasi-split \(k_{sep}/k\)-form.
For a pseudo-semisimple \(k\)-group \(G\), the authors study the structure of the maximal smooth closed \(k\)-subgroup of \(\Aut_{G/k}\) (the latter is not \(k\)-smooth in general) and, prove a Tits-style classification theorem for general pseudo-semisimple groups. This recovers Tits's result for the semisimple case. However, when \(k\) is imperfect of characteristic \(2\), then absolutely pseudo-simple \(k\)-groups of type \(F_4\) that are not pseudo-split, cannot have \(k\)-rank \(3\) whereas they can have \(k\)-rank \(2\) -- a situation unlike the semisimple case.
The book under review carries a treasury of information regarding affine algebraic groups. There is a very readable exposition by these two authors which encompasses results from both books -- this is published in Proc. Symp. Pure Math. The interested reader will benefit by looking through that exposition before delving into the present book. Further, it is worthwhile reading the scholarly review of the previous book in the Mathematical Reviews by Bertrand Remy. The notions and techniques developed to prove the classification theorems in this book and the structure theorems in the earlier book are tours-de-force. The authors of the present book have done a wonderful job of carefully recalling the results in the previous book at appropriate places and putting them in perspective while proving new results. For several years to come, both these books can be expected to constitute essential reading for graduate students and researchers interested in applications of affine algebraic groups. linear algebraic groups; pseudo-reductive groups; generalized standard construction; groups locally of minimal type; structure and classification of pseudo-reductive groups; imperfect fields; pseudo-split groups; central extensions; affine group schemes Conrad, B.; Prasad, G., Classification of pseudo-reductive groups, Annals of Mathematics Studies, (2015), Princeton University Press Linear algebraic groups over arbitrary fields, Structure theory for linear algebraic groups, Affine algebraic groups, hyperalgebra constructions, Linear algebraic groups over local fields and their integers Classification of pseudo-reductive groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is devoted to the study of the theory of finite dimensional modules over a finitely generated associative algebra. In the first three sections basic definitions and recent results concerning degenerations of modules and related topics are discussed. Most of the material is contained in previous works of the author [see \textit{K. Bongartz}, Adv. Math. 121, No. 2, 245-287 (1996; Zbl 0862.16007), Comment. Math. Helv. 69, No. 4, 575-611 (1994; Zbl 0832.16008)] and his followers [see e.g. \textit{G. Zwara}, Colloq. Math. 72, No. 2, 281-303 (1997; Zbl 0890.16006)]. The next section is an account of well-known results from the theory of geometric quotients of varieties endowed with actions of algebraic groups. As an application in the final section the author considers in some detail the classical wild problem of pairs of \(n\times n\)-matrices. In particular, he classifies geometrically the orbits in the \(\text{PGL}_2\)-variety of pairs of \(2\times 2\)-matrices. In addition some pathologies occurring in the case of \(3\times 3\)-matrices are described. It should be noted that the paper under review contains many references (the list of references contains 67 citations), non-formal remarks and interesting examples. finite dimensional modules; finitely generated algebras; \(G\)-varieties; geometric quotients; cancellation; preprojective modules; Auslander-Reiten quivers; tame quivers; wild quivers; pairs of matrices Klaus Bongartz, Some geometric aspects of representation theory, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 1 -- 27. Representations of quivers and partially ordered sets, Finite rings and finite-dimensional associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Group actions on varieties or schemes (quotients), Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras), Representation type (finite, tame, wild, etc.) of associative algebras, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Some geometric aspects of representation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide a group-theoretic realization of two-parameter quantum toroidal algebras using finite subgroups of \(\text{SL}_2(\mathbb{C})\) via McKay correspondence. In particular our construction contains the vertex representation of the two-parameter quantum affine algebras of ADE types as special subalgebras. two-parameter quantum affine algebra; finite groups; wreath products; McKay correspondence DOI: 10.1090/S0002-9947-2011-05284-0 Quantum groups (quantized enveloping algebras) and related deformations, McKay correspondence, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Two-parameter quantum vertex representations via finite groups and the McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities \(\mathbb{C}^n/G\), where \(G\) is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity \(A_n\) leads to a remarkable formula. theta function; McKay correspondence; elliptic class of singular varieties; quotient singularities Local complex singularities, Singularities in algebraic geometry Elliptic classes, McKay correspondence and theta identities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the last years there have been several new constructions of surfaces of general type with \(p_{g} = 0\), and important progress on their classification. The present paper presents the status of the art on surfaces of general type with \(p_{g} = 0\), and gives an updated list of the existing surfaces, in the case where \(K^{2} = 1,\dots,7\). It also focuses on certain important aspects of this classification. surfaces of general type with genus \(0\); Godeaux surfaces; Campedelli surfaces; Burniat surfaces; Bloch conjecture; actions of finite groups I. Bauer, F. Catanese and R. Pignatelli, Surfaces of general type with geometric genus zero: A survey, Complex and Differential Geometry, Springer Proc. Math. 8, Springer, Heidelberg (2011), 1-48. Surfaces of general type, Families, moduli, classification: algebraic theory, Coverings of curves, fundamental group, Generators, relations, and presentations of groups Surfaces of general type with geometric genus zero: a survey | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Possible ramification of finite dimensional central simple algebras of exponent two over \(\mathbb{R}(\!(t)\!)(x)\) which are trivial in all real closures of their centers is described. The proof of the local Pfister conjecture is given in cases where either points of ramification are defined by quadratic polynomials or polynomials with roots which are not squares in the corresponding extensions of the field of constants. finite-dimensional central simple algebras; real closures of centers; local Pfister conjecture Finite-dimensional division rings, Brauer groups (algebraic aspects), Algebraic functions and function fields in algebraic geometry On algebras of exponent two that are trivial in all real closures of their centers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers three problems of an algebro-geometric nature connected with the 27-dimensional tube domain \({T}_e\) on which a certain real form of the exceptional Lie group \(E_7\) acts. Not much attention seems to have been given to arithmetic and moduli problems connected with this domain, and it seemed worthwile to propose for consideration certain specific problems connected with it which might have answers somewhat different in appearance to what is in many cases well known for classical domains.
To explain the problems, the following notation is used: Let \(V \cong \mathbb{R}^n\) be an \(n\)-dimensional real vector space and \({K}\) be a self-adjoint homogeneous convex cone such that the tube domain \({T} = V + i {K}\) is a complex hermitian symmetric domain whose group \(G\) of holomorphic automorphisms is a semisimple real algebraic Lie group defined over \(\mathbb{Q}\). Let \(\Gamma\) be an arithmetic subgroup of \(G(\mathbb{Q})\), let \(R\) be any subring of \(\mathbb{C}\) and \({Q} (\Gamma, R)\) be the graded \(R\)-algebra of modular forms \(f\) on \({T}\) with respect to \(\Gamma\) such that all the coefficients of the Fourier expansion of \(f\) are in \(R\), and let \({Q} (\Gamma) = {Q} (\Gamma, \mathbb{C})\).
Problem I. Assume that \({Q} (\Gamma) = {Q} (\Gamma, \mathbb{Q}) \otimes \mathbb{C}\). The first problem is to decide whether \({Q} (\Gamma, \mathbb{Z})\) is finitely generated as a graded algebra over \(\mathbb{Z} \). This is known to be the case when \({T}\) is a Siegel upper half-plane \({H}_n\) of degree \(n\). If we let \({T}_e\) be the exceptional 27-dimensional tube domain \({I} + {K}\), where \({I}\) is the exceptional 27-dimensional real Jordan division algebra and \({K}\) is its set of positive squares, and if \(\Gamma_e\) is a certain nice maximal arithmetic subgroup of \(G(\mathbb{Q})\), described in an earlier paper of the author, one may ask the same question, and this is the first problem.
Problem II. With the same notation as above, one may ask for the specific nature of the reciprocity laws of the reflex field at special points of \({T}_e\). This question has been solved by J. Milne in principal, and as a second problem, the author asks what specific, possible interesting form it may take in this case.
Problem III. With notation as above, if \({H}_n\) is the Siegel upper half-space of degree \(n\) and \(\Gamma = \Gamma_n\), the Siegel modular group of degree \(n\), then the orbits of \(\Gamma_n\) in \({H}_n\) correspond one-to-one to the isomorphism classes of normally polarized abelian varieties of dimension \(n\). Moreover, there is a \(\Gamma_n\)-invariant complex analytic closed subset \({I}_n\) of \({H}_n\), of which a Zariski-open subset \({I}^0_n\) is in one-to-one correspondence with canonically polarized Jacobian varieties of curves of genus \(n\), hence, via Torelli's theorem with the isomorphism classes of nonsingular curves of genus \(n\). If \(n = 3\), then \({I}_3 = {H}_3\), and this special case is important for the considerations which follow.
There is no known interpretation of the space of orbits of \(\Gamma_e\) in \({T}_e\) as the space of moduli of some family of polarized algebraic varieties. Problem 3 is to seek such a family. The author approaches this problem by considering the four Severi varieties \(S_n\) of Zak, where \(\dim (S_n) = 2^n\), \(n = 1,2,3,4\). These are given in their ``natural'' projective embeddings as follows: \(S_1 = \mathbb{P}^2 \hookrightarrow \mathbb{P}^5\) (Veronese embedding); \(S_2 = \mathbb{P}^2 \times \mathbb{P}^2 \hookrightarrow \mathbb{P}^8\) (Segre embedding); \(S_3 = G(2,6) \hookrightarrow \mathbb{P}^{14}\) (embedding by Plücker coordinates), where \(G(2,6)\) is the Grassmannian of planes in 6-space; and \(S_4 = {\mathfrak C} \mathbb{P}^2 \hookrightarrow \mathbb{P}^{26}\), where \({\mathfrak C}\) is the complex Cayley numbers, and \({\mathfrak C} \mathbb{P}^2\) is the complex projective Cayley plane.
Now observe that the generic quadric hypersurface section \({Q} \cap S_1\) of \(S_1\) as imbedded in \(\mathbb{P}^5\) is a nonsingular, non-hyperelliptic plane quartic curve \(C\) of genus 3 in \(\mathbb{P}^2\). (In the text of this article, \(S_2\) was mistakenly put for \(S_1\).) The moduli of such curves are essentially given via Torelli's theorem by the orbits of \(\Gamma_3\) in \({H}_3\). Thus we are led to consider the four irreducible symmetric hermitian domains of \((\mathbb{R}-)\) rank 3 of which \({H}_3\) is that of lowest dimension, and of which \({T}_e\) is of the highest dimension, 27. One associates these in the order of increasing dimension with the four Severi varieties, and considers for each certain geometric configurations associated with a generic quadric section of each Severi variety in its natural projective imbedding, thus attempting to gain insight into possibly associated moduli problems, such as they may be.
Progress is reported on the case of the second Severi variety, and further advances were realized in a report on the topic ``A Problem on an Exceptional Domain'' which appeared in the proceedings of the Chebotarev conference in Kazan, Russia, held in 1994. Subsequently the author has developed fairly specific ideas in connection with the case of \(S_3\) by considering the trigonal construction of Recillas and the tetragonal construction of Donagi, as related in part by the Prym map. exceptional domain; action of exceptional Lie group; finite generation of algebra of modular forms; tube domain; moduli problems; reciprocity laws; Siegel modular group; Severi varieties; embeddings Complex Lie groups, group actions on complex spaces, Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube), Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Three problems on an exceptional domain | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose \(G\) is a connected reductive algebraic group, \(P\) is a parabolic subgroup of \(G\), \(L\) is a Levi factor of \(P\), and \(e\) is a regular nilpotent element in \(\text{Lie }L\). We assume that the characteristic of the underlying field is good for \(G\). Choose a maximal torus, \(T\), and a Borel subgroup, \(B\), of \(G\), so that \(T\subseteq B\cap L\), \(B\subseteq P\) and \(e\in\text{Lie} B\). Let \(\mathfrak B\) be the variety of Borel subgroups of \(G\) and let \({\mathfrak B}_e\) be the subset of \(\mathfrak B\) consisting of Borel subgroups whose Lie algebras contain \(e\). Finally, let \(W\) be the Weyl group of \(G\) with respect to \(T\). For \(w\in W\) let \({\mathcal O}_w\) be the \(B\)-orbit in \(\mathfrak B\) containing \(^wB\). We consider the intersections \({\mathcal O}_w\cap{\mathfrak B}_e\). The main result is that if \(\dim{\mathcal O}_w\cap{\mathfrak B}_e=\dim{\mathfrak B}_e\), then \({\mathcal O}_w\cap{\mathfrak B}_e\) is an affine space. Thus, the irreducible components of \({\mathfrak B}_e\) are indexed by Weyl group elements. It is also shown that if \(G\) is of type \(A\), then this set of Weyl group elements is a right cell in \(W\). connected reductive algebraic groups; parabolic subgroups; Levi factors; regular nilpotent elements; maximal torus; varieties of Borel subgroups; Lie algebras; Weyl groups; affine spaces; irreducible components; right cells Douglass, JM, Irreducible components of fixed point subvarieties of flag varieties, Math. Nachr., 189, 107-120, (1998) Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Lie algebras of linear algebraic groups Irreducible components of fixed point subvarieties of flag varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Mit G ist in der Arbeit stets eine reduktive komplexe lineare algebraische Gruppe gemeint, und \({\mathbb{C}}[\phi]\) bezeichnet den Koordinatenring des Darstellungsraumes einer endlichdimensionalen komplexen Darstellung \(\phi\) von G. Wie üblich ist \({\mathbb{C}}[\phi]^ G\) die \({\mathbb{C}}\)-Unteralgebra aller G-invarianten Polynome in \({\mathbb{C}}[\phi]\) unter dieser Wirkung von G. Die Darstellung (\(\phi\),G) heißt koregulär, wenn \({\mathbb{C}}[\phi]^ G\) ein Polynomring über \({\mathbb{C}}\) ist, und sie ist ein vollständiger Kodurchschnitt, wenn \({\mathbb{C}}[\phi]^ G\) ein vollständiger Durchschnitt ist. In den ersten beiden Sätzen der Arbeit gibt der Verf. notwendige, sich auf die Dimension und die Anzahl der Gewichtsräume eines maximalen Torus beziehende Bedingungen an, unter denen die Darstellung \(\phi\) einer Gruppe G des Ranges 1 vollständiger Kodurchschnitt ist. Der Rest der Arbeit zielt darauf ab, die gefundenen Kriterien auf einfache, zusammenhängende und einfach zusammenhängende algebraische Gruppen anzuwenden.
Ist G eine solche Gruppe vom Rang r, so seien \(\phi_ i\), \(i=1,...,r\), die nach \textit{N. Bourbaki} [Groupes et algèbres de Lie, Chaps. 7 et 8 (1975; Zbl 0329.17002)] numerierten fundamentalen Darstellungen von G. Ist \(\phi\) eine Darstellung von G und k eine natürliche Zahl, so bezeichnet \(k\phi\) die direkte Summe von k Kopien von \(\phi\) ; sind \(\phi\) und \(\psi\) irreduzible Darstellungen, so ist \(\phi\) \(\psi\) die irreduzible Komponente des höchsten Gewichtes von \(\phi\) \(\otimes \psi\). Der Verf. zeigt: Sei \(\phi\) eine nichttriviale irreduzible Darstellung von G. Ist (\(\phi\),G) vollständiger Kodurchschnitt, aber nicht koregulär, so kann die Darstellung (\(\phi\),G) oder ihr Dual mit einer der Darstellungen \(\phi^ 5_ 1(A_ 1)\), \(\phi^ 6_ 1(A_ 1)\), \(\phi^ 3_ 1(A_ 3)\) oder \(\phi_ 1\phi_ 2(A_ 3)\) identifiziert werden.
Daraus ergibt sich als Korollar: Enthält die Darstellung \(\phi\) von G keine von Null verschiedene triviale Subdarstellung und ist sie vollständiger Kodurchschnitt, so ist \(\phi\) irreduzibel, wenn nur eine nicht koreguläre irreduzible Subdarstellung von \(\phi\) existiert. Außerdem sei noch der folgende, am Schluß der Arbeit placierte Satz erwähnt: Für jede koreguläre irreduzible Darstellung (\(\phi\),G) kann man effektiv eine natürliche Zahl \(n_{\phi}\) angeben, so daß die Darstellung \((n_{\phi}\phi,G)\) kein vollständiger Kodurchschnitt ist. Für Gruppen vom Typ \(A_ n\) (n\(\geq 2)\) etwa ist \((m\phi_ 1,A_ n)\) genau dann vollständiger Kodurchschnitt, wenn \(m\leq n+2\) gilt.
Ein Teil der Ergebnisse dieser Arbeit ist auch in einem Beitrag des Autors in ''Algebraic groups and related topics'' [Proc. Symp. Res. Inst. Math. Sci., Kyoto Univ., Kyoto 1983, RIMS Kôkyûroku 512, 137-147 (1984)] enthalten. Lie groups of rank one; reductive complex linear algebraic group; finite- dimensional complex representation; coregular representation; invariant polynomials; complete intersection; maximal torus; complete cointersection; fundamental representations; irreducible representations; highest weights Haruhisa Nakajima, ?Invariants of reductive Lie groups of rank one and their applications,? Proc. Japan Acad.,A60, No. 6, 221?224 (1984). Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Group actions on varieties or schemes (quotients), Geometric invariant theory Invariants of reductive Lie groups of rank one and their applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(A\) is a differential graded (DG) algebra and \(D(A)\) its derived category, a categorical resolution of \(A\) is a pair \((B,M)\) consisting of a smooth DG algebra \(B\) and a DG \(A^{\text{op}}\otimes B\)-module \(M\) such that the derived tensor functor from \(D(A)\) to \(D(B)\) associated with \(M\) is full and faithful on the subcategory of perfect DG \(A\)-modules. Here, a DG algebra \(B\) is smooth if \(B\) is perfect as a DG \(B^{\text{op}}\otimes B\)-module. In particular, since the derived category \(D(X)\) of quasi-coherent sheaves on a quasi-compact and separated scheme \(X\) is equivalent to \(D(A)\) for some DG algebra \(A\), it makes sense to speak of a categorical resolution of \(X\). Note that \(X\) is smooth if and only if \(D(X)\) is smooth in the above sense.
The main result of the paper under review states that if \(X\) is a reduced separated scheme of finite type over a field of characteristic zero, then it has a categorical resolution by a smooth poset scheme if and only if \(X\) has Du Bois singularities.
A poset scheme is, roughly speaking, defined as follows. Fix a finite partially ordered set (poset) \(S\). An \(S\)-scheme \(\mathcal{X}\) is the datum of a scheme \(X_\alpha\) for every \(\alpha\in S\) and morphisms \(f_{\alpha\beta}: X_\alpha \rightarrow X_\beta\) for \(\alpha\geq \beta\) satisfying the obvious compatibility condition. The usual notions for schemes, for example, regularity, are defined componentwise. Similarly, a quasi-coherent sheaf on a poset scheme is a collection of sheaves \(F_\alpha\) on the \(X_\alpha\) and maps \(\phi_{\alpha\beta}: f^*_{\alpha\beta}F_\beta\rightarrow F_\alpha\) satisfying the usual cocycle conditions. Quasi-coherent sheaves on a poset scheme \(\mathcal{X}\) form an abelian Grothendieck category and one can consider its derived category \(D(\mathcal{X})\). Among many other properties relevant in this context the author shows that \(D(\mathcal{X})\) admits a semi-orthogonal decomposition into the derived categories of the \(X_\alpha\), that \(D(\mathcal{X})\) admits a compact generator and that it is smooth if \(\mathcal{X}\) is a regular \(S\)-scheme essentially of finite type over a perfect field.
If \(S\) and \(S'\) are posets and \(\tau\) is an order preserving map, one can define a map from an \(S\)-scheme \(\mathcal{X}\) to an \(S'\)-scheme \(\mathcal{X}'\). Note that any scheme \(Y\) can be considered as an \(S'\)-scheme for \(S'\) a point. Given a smooth \(S\)-scheme \(\mathcal{X}\), a map \(\pi: \mathcal{X}\rightarrow Y\) is a categorical resolution if the derived pullback functor is one. In Section 7 the author establishes, in particular, that \(\pi\) is a categorical resolution if and only if the adjunction morphism \(\mathcal{O}_Y\rightarrow R\pi_*\mathcal{O}_\mathcal{X}\) is a quasi-isomorphism. Now, in characteristic zero any reduced separated scheme of finite type \(X\) has a so-called cubical hyperresolution \(\mathcal{Z}\) and if \(X\) has Du Bois singularities, the result from Section 7 precisely gives that \(\mathcal{Z}\) provides a categorical resolution. For the other direction the author shows that the adjunction map \(\mathcal{O}_X\rightarrow R\pi_*\mathcal{O}_\mathcal{Z}\) of a hyperresolution \(\pi: \mathcal{Z}\rightarrow X\) has a left inverse, which is sufficient to conclude that \(X\) is Du Bois.
The author also proves three results on degeneration of spectral sequences for smooth projective poset schemes. These are, in particular, used to prove that the de Rham-Du Bois complex can be defined by means of any smooth projective poset scheme which satisfies descent in the classical topology. derived categories; differential graded algebras; resolutions of singularities; poset schemes; categorical resolution; Du Bois singularities; cubical hyperresolution; degeneration of spectral sequence Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck categories, Spectral sequences, hypercohomology Categorical resolutions, poset schemes, and Du Bois singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Algebraic geometry may be viewed as a global version of commutative algebra and, alternatively, commutative algebra as a local version of algebraic geometry. This is essentially due to the fact that schemes are locally affine and that affine schemes (with their associated category of quasicoherent sheaves) and commutative rings (with their associated category of modules) are just two ways of looking at the same thing, due to Serre's theorem.
The last couple of decades, several types of ``noncommutative algebraic geometry'' have been proposed, more or less successfully, mainly in order to be able to apply geometric methods (and intuition) to tackle problems originating within noncommutative algebra. However, even now, it remains at least unclear what a noncommutative algebra should be -- much depending on what one really aims to do with it and how strong the noncommutative analog of the dictionary commutative algebra -- algebraic geometry should become in the noncommutative set-up.
For example, one of the many reasons for the success of the classical Grothendieck approach of associating to any commutative ring \(R\) its affine scheme \((\text{Spec}(R),{\mathcal O}_R)\) is that the stalks of the structure sheaf \({\mathcal O}_R\) are local rings, due to the very fact that \({\mathcal O}_{R,P}=R_P\) at every prime \(P\) of \(R\). This obvious fact allows the study of a general ring \(R\) essentially to be reduced to that of (much simpler) local rings, which geometrically amounts to study \(\text{Spec}(R)\) locally.
When one tries to generalize this to the noncommutative case, one is inevitably confronted with the problem that noncommutative localization techniques hardly ever produce local rings. In this way, it is clear that exactly this lack of a ``good'' localization theory complicates matters if one wants to generalize the construction of structure sheaves to the noncommutative case. In particular, we should point out that the essential fact that schemes are not only covered by affines but possess a basis of affine subsets is due to this: only the nice behaviour of commutative localization guarantees that for \(f\in R\) the open set \(D(f)\) may be identified as a scheme with \(\text{Spec}(R_f)\). The fact that the intersection of two affines is affine again is also due to this type of argument.
Although the ``classical'' approach to algebraic geometry (affine schemes are defined as ``spaces'' endowed with a structure sheaf constructed through localization) does not appear to work in general, there still remains the possibility of trying to generalize this set-up at least for a restricted class of rings.
The present book should be considered from this point-of-view: the author introduces the notion of so-called schematic algebras and associates to these ``schemes'' with a sufficiently nice behaviour. The book is very nicely written and presents an excellent overview of the author's ideas concerning noncommutative algebraic geometry as well as the basic geometric and algebraic behaviour of these schematic algebras.
Roughly speaking, schematic algebras are graded algebras with sufficiently many Ore sets, thus guaranteeing that the associated ``projective scheme'' is at least covered by ``affine schemes''. Although this does not completely generalizes the commutative case (the associated structure sheaf does not give rise to local rings, affine open sets do not form a basis, \dots) nevertheless the methods developed here are sufficiently strong to allow for a generalization of Serre's theorem (the projective version, relating the behaviour of quasicoherent sheaves and graded modules) or basic cohomology theory, for example. Moreover, the class of schematic algebras is sufficiently vast to motivate the restriction of the present construction to these. Actually, although ``classical'' rings like general group rings, enveloping algebras or PI algebras hardly ever are schematic, many examples of algebras naturally occurring within the context of quantum groups are. It is thus mainly in this context that schematic algebras and the present version of noncommutative algebraic geometry should be considered.
The book contains a wealth of material ranging from the purely geometric set-up to more specialized topics like applications within the framework of regular algebras, noncommutative valuation theory and braided categories, just to mention a few. One particular aspect which this reviewer liked was the often open-ended presentation of many topics, inviting the reader to further develop the subject and add new insights. (Also submitted to MR). affine schemes; categories of quasicoherent sheaves; Serre's theorem; noncommutative localizations; structure sheaves; schematic algebras; noncommutative algebraic geometry; graded algebras; Ore sets; quantum groups; braided categories F. van Oystaeyen. \textit{Algebraic geometry for associative algebras}. Series ''Lect. Notes in Pure and Appl. Mathem.'' \textbf{232} (Marcel Dekker: New York, 2000). Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Research exposition (monographs, survey articles) pertaining to associative rings and algebras, , Associative rings of functions, subdirect products, sheaves of rings, Graded rings and modules (associative rings and algebras), Ore rings, multiplicative sets, Ore localization Algebraic geometry for associative algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The 21 open problems presented here were collected on the occasion of a workshop of Arithmetic Geometry at the University of Utrecht, 26-30 June, 2000.
The subjects involved are: curves, Abelian varieties, moduli spaces, \(p\)-divisible groups, Hilbert modular forms, rigid geometry, group schemes, Neron model, \(K3\) surfaces, Calabi-Yau varieties, sporadic groups, knot-invariants. curves; Abelian varieties; moduli spaces; \(p\)-divisible group; Hilbert modular forms; rigid geometry; group schemes; Neron model; \(K3\) surfaces; Calabi-Yau varieties; sporadic groups; knot-invariants Edixhoven, S. J.; Moonen, B. J. J.; Oort, F., Open problems in algebraic geometry, Bulletin des Sciences Mathématiques, 125, 1-22, (2001) Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Abelian varieties of dimension \(> 1\), Arithmetic aspects of modular and Shimura varieties, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects) Open problems in algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves \textit{P. Deligne}'s rationality conjecture \([= 2.8\) of Proc. Symp. Pure Math. 33, No.2, 313-346 (1979; Zbl 0449.10022)] for all critical values of all algebraic Hecke characters of CM-fields. Thus he establishes the fact that - up to an algebraic number whose Galois behaviour is that of the Hecke character \(\chi\) in question - the values L(\(\chi\),k), for \(k\in {\mathbb{Z}}\) such that the \(\Gamma\)-factors on either side of the functional equation have no pole at k, are (either zero or) certain periods derived from abelian varieties with complex multiplication.
The idea of proof is, in the first place, a refinement of the method of Eisenstein-Damerell-Shimura: write the L-value as a linear combination of (values of in general non-holomorphic derivatives of) Eisenstein- Kronecker-Lerch-Hecke-Kloosterman series which are Hilbert modular forms relative to the totally real subfield of the CM-field for which \(\chi\) is an algebraic Hecke character [see \textit{R. M. Damerell}, Acta Arith. 17, 287-301 (1970; Zbl 0209.246); ibid. 19, 311-317 (1971; Zbl 0229.12015); \textit{A. Weil}, Elliptic functions according to Eisenstein and Kronecker (1976; Zbl 0318.33004); \textit{G. Shimura}, Ann. Math., II. Ser. 91, 144- 222 (1970; Zbl 0237.14009)].
However, in order to refine Shimura's result (which is up to an unspecified factor in \({\bar {\mathbb{Q}}})\) to the precise rationality statement of Deligne's conjecture, the author had to overcome the notorious difficulty that Deligne's construction of periods looks \textit{prima facie} quite incompatible with the period that comes naturally with Shimura's proof [see \textit{P. Deligne}, loc. cit., 8.14-8.21; \textit{G. Harder} and the reviewer, Lect. Notes Math. 1111, 17-49 (1985; Zbl 0561.10012)].
The author does this by brilliantly combining both Shimura's reciprocity laws for special values of Hilbert modular forms and the recent generalization of the Shimura-Taniyama reciprocity law for CM-abelian varieties due to Tate and Deligne [see \textit{S. Lang}, Complex multiplication (1983; Zbl 0536.14029), chap. 7] with his own remodelling of Deligne's period definition inside the category of motives (for absolute Hodge cycles constructed from CM-abelian varieties).
This seems to be the first publication where an existing and manageable theory of motives [specifically: the one derived from \textit{P. Deligne}'s theorem on absolute Hodge cycles on abelian varieties; see Lect. Notes Math. 900 (1982), chap. I, p. 9-100 (Zbl 0537.14006)and chap. IV, p. 261- 279 (Zbl 0499.16001)] is essentially used to prove a theorem about values of L-functions.
G. Harder has announced (in Harder-Schappacher, loc. cit.), but not yet published in detail, a method which would allow to carry over the author's result to Hecke characters of arbitrary number fields. Hecke L-series; critical values; algebraic Hecke characters; CM-fields; \(\Gamma \)-factors; functional equation; periods; abelian varieties with complex multiplication; Kloosterman series; Hilbert modular forms; Shimura's reciprocity laws; category of motives; absolute Hodge cycles; values of L-functions D. Blasius, On the critical values of Hecke \(L\)-series, Ann. of Math. (2) 124 (1986), 23--63. Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Complex multiplication and moduli of abelian varieties, Generalizations (algebraic spaces, stacks), Complex multiplication and abelian varieties, Zeta functions and \(L\)-functions of number fields, Zeta functions and \(L\)-functions, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols On the critical values of Hecke L-series | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies Fourier transforms of relatively invariant functions on the square matrix spaces and Hermitian matrix spaces over the finite field \(\mathbb{F}_q\), which are typical prehomogeneous vector spaces over a finite field. In the general setting, the formula of Fourier transform of a relatively invariant function on a prehomogeneous vector space was already obtained by the author provided that the characteristic of \(\mathbb{F}_q\) is sufficiently large and that we consider only the open orbit of the prehomogeneous vector space. One purpose of this paper is to exclude the constraint in these special examples and give an explicit formula of the Fourier transform. Though in this paper we only consider two examples, this is the first step to the study of prehomogeneous vector spaces using intersection cohomology theory. Gauss sum; character sum; Fourier transforms of relatively invariant functions; square matrix spaces; Hermitian matrix spaces; finite field; prehomogeneous vector spaces; intersection cohomology theory Gyoja, A.: Character sums and intersection cohomology complexes associated to the space of square matrices. Indag. math., N.S. 8, No. 3, 371-385 (1997) Other character sums and Gauss sums, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Homogeneous spaces and generalizations Character sums and intersection cohomology complexes associated to the space of square matrices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Cluster algebras, invented by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups, are a class of commutative algebras endowed with a distinguished set of generators, the cluster variables. The cluster variables are grouped into finite subsets, called clusters, and are defined recursively from initial variables through mutation on the clusters. The seed includes the cluster variables and the iterative process or mutation is codified in an exchange matrix. Cluster algebras are defined via matrices and mutations and also exchange patterns via matrices and polynomials. The book under review serves as an introductory survey on cluster algebras. It contains nine chapters and is well written.
Chapter 1 is of introductory nature and contains motivations for cluster algebras. Some combinatorial examples involving recurrences and some notations are provided. In Chapter 2, one learns about the definition of a cluster algebra. In these notes, the author focus on cluster algebras of geometric type, although more general versions have been defined. All necessary ingredients involved in the definition such as skew-symmetrizable matrices, quiver notations and exchange graphs are introduced. The first definition of a cluster algebra involves the polynomials more directly, with mutation being defined by substitutions rather than via sign-skew-symmetric matrices. In Chapter 3, the author describes this approach in detail and explains how it corresponds to the version with matrices given in Chapter 2.
Recall that a cluster algebra is said to be of finite type if it has finitely many seeds. The classification of cluster algebras of finite type is done in terms of Cartan matrices of finite type. The corresponding reflection groups and their root system play a key role in describing the cluster algebras. The author follows J.E. Humphreys' book on reflection groups and Coxeter groups to give a brief introduction to the theory of reflection groups and root system in Chapter 4. In Chapter 5, the finite type cluster algebras in terms of Dynkin diagrams are classified.
Associated to each cluster algebra of finite type is a corresponding abstract simplicial complex on the clusters known as the generalized associahedron. An overview of these results are provided in Chapter 6. Chapter 7 discusses two aspects of periodicity in cluster algebras. One is the periodicity of quivers with respect to mutations and the other is the categorical periodicity considered by \textit{B. Keller} which led to his proof in [Ann. Math. (2) 177, No. 1, 111--170 (2013; Zbl 1320.17007)].
Recall that a cluster algebra is said to be of finite mutation type if the set of principal parts of mutation matrices is finite. The classification of all cluster algebras of finite mutation type associated to skew-symmetric matrices is addressed. Some of these cluster algebras are given in terms of the quivers associated to marked Riemann surfaces.
In the last chapter, the author illustrates one important example of cluster algebras. The beautiful result is that the homogeneous coordinate ring of the Grassmannian of \(k\)-subspaces of an \(n\)-dimensional space admits a cluster algebra structure. associahedron; cluster algebra; cluster complex; Dynkin diagram; finite mutation type; Grassmannian; Laurent phenomenon; reflection group; periodicity; quiver mutation; root system; surface Marsh, Robert J., Lecture notes on cluster algebras, Zurich Lectures in Advanced Mathematics, ii+117 pp., (2013), European Mathematical Society (EMS), Zürich Research exposition (monographs, survey articles) pertaining to commutative algebra, Cluster algebras, Combinatorial aspects of commutative algebra, Grassmannians, Schubert varieties, flag manifolds, Poisson algebras, Root systems, \(n\)-dimensional polytopes Lecture notes on cluster algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is Volume II of a study in derived algebraic geometry, and depends heavily on Volume I [Zbl 1408.14001]. In short, Volume I defines the different categories and the functors on them, needed for derived algebraic geometry. Then it tells which properties these functors have, and how to work with them. Volume I contains explanations of why we need the higher geometry: It is needed so that the functors have adjoint functors. It considers Ind-coherent sheaves and categories of correspondences. So Volume I gives the basic theory of derived algebraic geometry based on $(\infty,2)$-categories, and Volume II gives essential results from this theory, solving algebraic geometric problems by representation theory extended to derived theory. \par The geometric objects of interest in Volume II are the class of geometric objects named inf-schemes, with their corresponding categories of ind-coherent sheaves. These are the categories that appear in representation theoretic situations. \par Volume II consists of two parts. Part I has four chapters, and starts with a recalling (from Volume I) of inf-schemes, and a slight functorial interpretation of deformation theory. Also, the theory of Ind-coherent sheaves on inf-schemes are recalled. The introduction ends with an interesting discussion linking Crystals and $D$-modules. Chapter 1 give a proper definition of deformation theory, using representation theory, (or even higher category theory): Push-out of (derived) schemes, (pro)-cotangent and tangent spaces are defined on their infinitesimal objects, their properties are given, infinitesimal cohesiveness, and the the higher category definition of deformation theory. After this, chapter 1 gives the consequences of admitting deformation theory; categories with deformation theory is a concept influencing the rest of Volume II. For instance, a subsection on isomorphism properties is given; the property of having deformation theory can be used to show that certain maps between prestacks are isomorphisms, and the same technique is used to prove a criterion for being locally almost of finite type. The final section of chapter 1 considers square-zero extensions of prestacks. Chapter 2 start with a functorial definition of ind-schemes connected to inductive limits. The basic results of this category is then summed up and proved, and the different concepts are combined to (Ind)-inf-schemes and nil-closed embedding (to mention its link to deformation theory). Chapter three grows on the previous chapter. It considers Ind-coherent sheaves on ind-schemes involving proper base change for Ind-Schemes, and the concept corresponding to coherent sheaves, InCoh on (ind)-inf-schemes and the direct image functor for ind-inf-schemes. The formalism of correspondences is extended to inf-schemes and the self-duality and multiplicative structure of IndCoh ond ind-inf-schemes are considered. Chapter 3 looks deeply into Ind-coherent sheaves on ind-inf-schemes: Their proper base-change, their inductive extension IndCoh on (ind)-inf schemes, and the direct image functor for ind-inf-schemes. The formalism of correspondences is extended to inf-schemes, and the authors apply self-duality and the multiplicative structure of IndCoh on ind-inf-schemes. The final chapter of part I, chapter 4, is an application of crystals: First, the book defines crystals on prestacks and inf-schemes. The category of crystals on a prestack $\mathcal X$ is defined to be IndCoh on the corresponding prestack $\mathcal X_{dR}.$ This is also seen as a functor out of the category of correspondences, and the introduction of crystals means that the forgetful functor $\text{Crys}(\mathcal Z)\rightarrow\text{IndCoh}(Z)$ has a left adjoint. This is provided that $\mathcal Z$ is a prestack admitting deformation theory. Finally, in this chapter this is compared to the classical theory of $D$-modules. \par Part II is named formal geometry. The introduction to this chapter more or less indicates that in geometry, formal can be thought of as convergent sequences, thereby as inductive limits, and so completing up to analytic geometry. Going from geometry to algebra, Lie algebras comes up as the kernel of the exponential map. From this it also follows that a Lie algebroid corresponds to a functor of Lie algebras, leading to infinitesimal differential geometry. Chapter 5 is short and concise, and defines formal moduli and formal moduli problems, by introducing functors to groupoids. From this reviewer's viewpoint, this is a vital organ of the book, and readers should pay attention. Chapter 6 on Lie algebras and co-commutative co-algebras considers algebras over operads, Koszul duality, Associative algebras, universal enveloping algebras and ways to construct it. The chapter Introduces modules and proves the Poincare-Birkhoff-Witt (PBW) theorem, and defines commutative co-algebras and bialgebras. Chapter 7 goes into formal groups and formal Lie algebras. It tells how formal moduli problems leads to co-algebras, it confirms that inf-affines is what one would think it is, and it introduces modules over formal groups and formal Lie algebras. These modules leads to actions of formal groups on prestacks. Chapter 8 is dedicated to the study of Lie algebroids, in derived geometry: The inertial group, the basic structure, examples, results about modules on Lie algebroids and the universal enveloping algebra, square-zero extensions and Lie algebroids, IndCoh of a square-zero extension. Also there are results about global sections of a Lie algebroid, and Lie algebras as modules over a monad. Chapter 8 ends with a comment on the relation to classical Lie algebroids and gives an application of ind-coherent sheaves on push-outs. Chapter 9 makes sense of (infinitesimal) differential geometry in the context of derived algebraic geometry. The notions include deformations to the normal cone of a closed embedding, the notion of the $n$-th infinitesimal neighbourhood of a scheme embedded in another, the PBW filtration on the universal enveloping algebra of a Lie algebroid over a smooth scheme and the Hodge filtration (a.k.a. de Rham resolution) of the dualizing $D$-module. \par The two books together, and in particular this volume II, is written in a totally categorical language. This says that there are no explicit definitions, and the objects under study are given by their functorial properties. This makes the two books rather hard to read, especially as the notation must be stored by imagination, but then after all it illustrates advanced material in a relatively compact way. The ideas presented by the volumes are truly magnificent, and it is proved that the properties we want from a derived algebraic geometry holds. Thus it is magnificent both as an introduction to the power of derived algebraic geometry, and as a reference work. derived scheme; connective pro-cotangent space; connective deformation theory; almost of finite type (pro-)quasicoherent sheaf; anchor map; Chevalley functor; ind scheme; cocommutative Hopf algebra; cocommutative bi-algebra; co-operad; composition monoidal structure; crystal; de Rham prestack; differential of $x$; exponential map; filtered object; formal moduli problem; formally smooth; Hodge filtration; ind-inf-scheme; inertia object; Lie operad; left-Lax equivariance; $n$-coconnective ind-scheme; pro-cotangent complex; reduced indscheme; shifted anchor map; smooth of relative dimension $n$; splitting of a Lie algebroid; universal envelope of a Lie algebra; Verdier duality; Weil restriction; zero Lie algebroid Research exposition (monographs, survey articles) pertaining to algebraic geometry, Generalizations (algebraic spaces, stacks), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Double categories, \(2\)-categories, bicategories, hypercategories, Nonabelian homotopical algebra A study in derived algebraic geometry. Volume II: Deformations, Lie theory and formal geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let G be a finite group, k an algebraically closed field in which \(| G|\) is nonzero, and \(V_ 1,...,V_ n\) the irreducible kG-modules. For a finite k-dimensional kG-module V, there is the McKay quiver, which has \(V_ 1,...,V_ n\) as vertices, and \(t_{ij}\) arrows from \(V_ j\) to \(V_ i\), where \(t_{ij}\) is the multiplicity of \(V_ j\) in \(V\otimes_ kV_ i.\)
The main result is that if \([V:k]=2\), then the underlying graph of the separated version of the McKay quiver is a disjoint union of extended Dynkin diagrams, but it is never of this form for \([V:k]>2\). The first part generalizes an observation of \textit{J. McKay} [Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026)]. This result is connected with work of \textit{D. Happel}, \textit{U. Preiser} and \textit{C. M. Ringel} [Manuscr. Math. 31, 317-329 (1980; Zbl 0436.20005)] and with some work of \textit{M. Auslander} on skew group rings and reflexive modules over rings of group- invariant power series [Trans. Am. Math. Soc. 293, 511-532 (1986)]. Several interesting examples of McKay quivers are given. finite group; irreducible kG-modules; McKay quiver; disjoint union of extended Dynkin diagrams Auslander, M., Reiten, I.: McKay quiver and extended Dynkin diagrams. Trans. Amer. Math. Soc., 293, 193--301 (1986) Group rings of finite groups and their modules (group-theoretic aspects), Representation theory of associative rings and algebras, Singularities in algebraic geometry, Ordinary representations and characters McKay quivers and extended Dynkin diagrams | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A surface isogenous to a product is a complex algebraic surface that is isomorphic to \((C_1\times C_2)/G\), where the \(C_i\) are curves of genus \(g_i\geq2\), and \(G\) is a finite group acting freely on \(C_1\times C_2\). A Beauville surface is a surface isogenous to a product which satisfies the following additional conditions: \(G\) does not interchange the two curves, the two quotients \(C_i/G\) are both isomorphic to \(\mathbb{P}^1\), and the covers \(C_i\to C_i/G\) are both ramified in exactly three points.
The existence of a surface isogenous to a product with given group \(G\) can be reduced to a purely group theoretical question, via the Riemann existence theorem. Moreover, counting connected components of the moduli space of such surfaces corresponds to counting orbits of certain group actions.
The article under review gives a clear and readable account of this group-theoretic approach, and applies it to give the asymptotic growth of the number of components of the moduli space of certain surfaces of general type that are isogenous to a product. Results are obtained for alternating and symmetric groups, and \((\mathbb{Z}/p)^r\) where \(p\) is prime. The Beauville surfaces are treated as a special case, and further results are obtained for Beauville surfaces with group \(\mathrm{PSL}(2,p)\) . Beauville surfaces; finite groups; moduli spaces; surfaces of general type Garion, S., Penegini, M.: Beauville surfaces, moduli spaces and finite groups. Commun. Algebra \textbf{42}, 2126-2155 (2014) Surfaces of general type, Families, moduli, classification: algebraic theory, Simple groups: alternating groups and groups of Lie type, Fuchsian groups and their generalizations (group-theoretic aspects), Riemann surfaces Beauville surfaces, moduli spaces and finite groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Simple space curve singularities were classified by the first author [Commun. Algebra 27, No.8, 3993--4013 (1999; Zbl 0963.14011)]. The same methods are used to determine the simple isolated determinantal codimension two singularities in other dimensions. By a counting argument simple higher dimensional ones can only exist for dimension at most 4, and only \(2\times 3\) matrices occur. In high dimensions even rigid determinantal singularities exist, but they are not isolated. For the case of fat points in the plane the counting argument does not work, but it turns out that there is only one family, again given by a \(2\times3\) matrix.
In the surface case the simple codimension two determinantal singularities are exactly the rational triple points. All singularities on the lists are quasi-homogeneous, but in dimension four there are examples where one variable has a negative weight.
For simple fat points and space curves the complete list of adjacencies is computed, with computer assistance. simple singularities; space curves; fat points; Hilbert-Burch theorem; classification of singularities; adjacencies Frühbis-Krüger, A.; Neumer, A., Simple Cohen-Macaulay codimension 2 singularities, Commun. Algebra, 38, 454-495, (2010) Local complex singularities, Complex surface and hypersurface singularities, Deformations of singularities, Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties Simple Cohen-Macaulay codimension 2 singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth. moduli spaces; representations of quivers; extended Dynkin quivers; smooth representations; tame representation type; invariants Mátyás Domokos, On singularities of quiver moduli, Glasg. Math. J. 53 (2011), no. 1, 131 -- 139. Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Representation type (finite, tame, wild, etc.) of associative algebras On singularities of quiver moduli. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a classification of simple hypersurface singularities by the classification of irreducible Weyl groups not using the normal forms. He proves: For any singularity the following conditions are equivalent:
(1) the singularity is simple;
(2) the singularity is elliptic;
(3) the monodromy group of the singularity is finite;
(4) the monodromy group of the singularity is isomorphic to a Weyl group of type \(A_K\), \(D_K\), \(E_6\), \(E_7\), \(E_8\);
(5) the mixed Hodge structure in the vanishing cohomologies of the singularity is trivial;
(6) the length of the spectrum of the singularity is less than one.
Furthermore, if two simple singularities have isomorphic monodromy groups then they are stably equivalent. simple hypersurface singularities; classification of irreducible Weyl groups; monodromy group Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Singularities in algebraic geometry On the A-D-E classification of the simple singularities of functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Classification problems of vector bundles on smooth projective varieties over an algebraically closed field \(k\) and classification problems of modules over finite dimensional \(k\)-algebras are closely related via derived equivalences of the corresponding bounded derived categories. We review known results and provide an introduction to the concept of moduli spaces for a certain class of representations of quivers. vector bundles; smooth projective varieties; finite dimensional algebras; derived equivalences; bounded derived categories; moduli spaces; representations of quivers Lutz Hille, Tilting line bundles and moduli of thin sincere representations of quivers, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 4 (1996), no. 2, 76 -- 82. Representation theory of groups, algebras, and orders (Constanţa, 1995). Representations of quivers and partially ordered sets, Representations of associative Artinian rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Free, projective, and flat modules and ideals in associative algebras Tilting line bundles and moduli of thin sincere representations of quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{GL}(n,\mathbb C)\) be a finite group and consider the corresponding quotient singularity \(X=\mathbb A^n/G\). Inspired by the classical McKay correspondence, many interesting connections between resolutions \(Y\to X\) of the quotient singularity and the representation theory of \(G\) have been discovered. In particular, for \(G\subset \text{SL}(3,\mathbb C)\) abelian, every projective crepant resolution is given by a moduli space of \(G\)-constellations or, equivalently, a moduli space of representations of the McKay quiver of \(G\) of dimension vector \((1,\dots,1)\); see \textit{A. Craw} and \textit{A. Ishii} [Duke Math.\ J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)]. Given a resolution \(Y\to \mathbb A^3/G\) with \(G\not\subset \text{SL}(3,\mathbb C)\) one may ask whether \(Y\) can be identified with a moduli space of representations of the McKay quiver too.
The author studies this question for the Danilov resolution (also known as the economic resolution) of the terminal quotient singularity of type \(\frac 1r(1,a,r-a)\) for coprime numbers \(a\), \(r\). This means that \(G\subset \text{GL}(3,\mathbb C)\) is the cyclic group of order \(r\) acting diagonally with eigenvalues \(\zeta\), \(\zeta^a\), and \(\zeta^{r-a}\) where \(\zeta\) is a primitive \(r\)-th root of unity. The Danilov resolution \(Y\to X=\mathbb A^3/G\) is a toric resolution given by a series of weighted blow-ups.
For \(G\subset \text{GL}(n,\mathbb C)\), a \(G\)-constellation is defined as a \(G\)-equivariant sheaf \(F\) on \(\mathbb A^n\) whose global sections \(\Gamma(F)\) are given by the regular representation \(R\) of \(G\). A stability parameter for \(G\)-constellations is given by a \(\mathbb Z\)-linear map \(\theta: \text R(G)\to \mathbb Q\) from the representation ring to the rational numbers such that \(\theta(R)=0\). A \(G\)-constellation \(F\) is called \(\theta\)-stable if for every non-trivial \(G\)-subsheaf \(E\subset F\) we have \(\theta(\Gamma(F))>0\). Due to more general results of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)], there is a fine moduli space \(\mathcal M_\theta\) of \(\theta\)-stable \(G\)-constellations. As structure sheaves of free \(G\)-orbits do not have any non-trivial \(G\)-subsheaves, they are \(\theta\)-stable for every parameter \(\theta\). The irreducible component \(Y_\theta\) of \(\mathcal M_\theta\) containing the structure sheaves of free orbits is called the coherent component. Let \(f: Y\to X\) be a resolution of the singularities. A family \(\mathcal F\) of \(G\)-clusters over \(Y\) is called a gnat-family (short for \(G\)-natural) if \(\mathcal F_y\) is supported on the \(G\)-orbit \(f(y)\) for every \(y\in Y\). If \(\mathcal F\) is in addition \(\theta\)-stable, this means that the classifying morphism \(Y\to\mathcal M_\theta\) factorises over the coherent component and commutes with \(f\) and the canonical morphism \(Y_\theta\to X=\mathbb A^n/G\) given by sending a \(G\)-constellation to the \(G\)-orbit supporting it. For \(G\) abelian, \textit{T. Logvinenko} [Doc. Math., J. DMV 13, 803--823 (2008; Zbl 1160.14025)] gave a characterization of all gnat-families on \(Y\) in terms of \(G\)-Weyl divisors.
In the paper under review, toric divisors on the Danilov resolution \(Y\) are used in order to construct a gnat-family \(\mathcal F\). Then the cone of stability parameters \(\theta\) for which \(\mathcal F\) is \(\theta\)-stable is computed. Furthermore, it is shown that the fibres of \(\mathcal F\) are pairwise non-isomorphic. It follows that the classifying morphism \(Y\to Y_\theta\) is bijective and consequently that \(Y\) is the normalisation of \(Y_\theta\). It is conjectured that \(Y_\theta\) is normal so that \(Y\cong Y_\theta\). McKay correspondence; resolutions of terminal quotient singularities; Danilov resolution; moduli of quiver representations Kȩdzierski, O.: Danilov's resolution and representations of the mckay quiver. Tohoku math. J. (2) 66, No. 3, 355-375 (2014) McKay correspondence, Representations of quivers and partially ordered sets, Geometric invariant theory Danilov's resolution and representations of the McKay quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let k be an imaginary quadratic field in which 3 splits as \({\mathfrak P}\cdot {\mathfrak P}'\) and H be the Hilbert class field of k. Denote by \(A_ k\) the ring of integers of k so that \({\mathbb{C}}/A_ k\) gives an elliptic curve. Let f be a primitive 3-division point of \({\mathbb{C}}/A_ k\). Then \(H=k(9)=k(\alpha^ 3(f))\) where \(\alpha^ 3(f)\) (see equation 2) is given in terms of the Weber function and the j-invariant associated with \({\mathbb{C}}/A_ k.\)
The main result (theorem 1.3) states that if \({\mathfrak I}\) is an integral ideal of k prime to 6 then the ring of integers of the field k(9\(\cdot {\mathfrak I})\) (respectively of the field k(9)\(\cdot k({\mathfrak I}))\) is a monogeneous extension over the ring of integers \(A_{k(9)}\) generated by \(F(\beta)\) (respectively \(F(\lambda)\)) where \(\beta =\lambda +\phi\), \(\phi\) being a primitive 9 division point and \(\lambda\) a primitive \({\mathfrak I}\)-division point of \({\mathbb{C}}/A_ k.\)
The discussions on elliptic functions, modular forms, Deuring's model of elliptic curve and relevant results on complex multiplication make the paper self contained. monogenity of rings of integers; imaginary quadratic field; Hilbert class field; elliptic functions; modular forms; complex multiplication V. Fleckinger , Monogénéité de certains anneaux d'entiers , Sém. de Théorie des Nombres de Bordeaux ( 1986 - 1987 ), Exposé 7 , 7-01 - 7-11 . Article | Zbl 0662.12003 Algebraic numbers; rings of algebraic integers, Complex multiplication and abelian varieties, Holomorphic modular forms of integral weight Monogénéité de certains anneaux d'entiers. (Monogenity of certain rings of integers) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X/\mathbb{F}_q\) be a smooth projective variety of even dimension \(2r\). The author proves that the order of the non-divisible part of the higher Brauer groups \(\mathrm{Br}^r(X)(\ell)_{\mathrm{nd}}\) is a square number for every \(\ell \neq \mathrm{char}(\mathbb{F}_q)\). Here, \(A(\ell) := \bigcup_nA_{\ell^n}\) denotes the \(\ell\)-primary torsion subgroup of an Abelian group \(A\). The higher Brauer groups are defined as \(\mathrm{Br}^r(X) = \mathrm{H}_{\mathrm{L}}^{2r+1}(X,\mathbb{Z}(r))\) with \(\mathbb{Z}(r)\) Bloch's cycle complex on the small étale site of \(X\). Note that because of \(\mathbb{Z}(1) = \mathbb{G}_m[-1]\), one has \(\mathrm{Br}^1(X) = \mathrm{H}^2(X,\mathbb{G}_m) =\mathrm{Br}(X)\), the cohomological Brauer group.
Sketch of the proof. First, identify
\[
\mathrm{Br}^r(X)(\ell)_{\mathrm{nd}} = \mathrm{H}_{\mathrm{et}}^{2r+1}(X,\mathbb{Z}_\ell(r))_{\mathrm{tor}},
\]
\(A_{\mathrm{tor}}\) the torsion subgroup of an Abelian group \(A\). Then, for \(\ell \neq 2\), construct a non-degenerate skew-symmetric bilinear form
\[
\mathrm{Br}^r(X)(\ell)_{\mathrm{nd}} \times \mathrm{Br}^r(X)(\ell)_{\mathrm{nd}} \to \mathbb{Q}_\ell/\mathbb{Z}_\ell
\]
from the pairings
\[
\mathrm{H}_{\mathrm{et}}^{2r}(X,\mu_{\ell^m}^{\otimes r}) \times \mathrm{H}_{\mathrm{et}}^{2r+1}(X,\mu_{\ell^m}^{\otimes r}) \to \mathrm{H}_{\mathrm{et}}^{4r+1}(X,\mu_{\ell^m}^{\otimes 2r})
\]
and
\[
\mathrm{H}_{\mathrm{et}}^{2r}(X,\mathbb{Q}_\ell/\mathbb{Z}_\ell(r))_{\mathrm{nd}} \times \mathrm{H}_{\mathrm{et}}^{2r+1}(X,\mathbb{Z}_\ell(r))_{\mathrm{tor}} \to \mathbb{Q}_\ell/\mathbb{Z}_\ell.
\]
For \(\ell = 2\), more work has to be done. Brauer groups of schemes; finite ground fields; arithmetic ground fields; motivic cohomology; motivic homotopy theory Brauer groups of schemes, Finite ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Motivic cohomology; motivic homotopy theory The order of higher Brauer groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Q\) be a tame connected quiver (i.e. the underlying graph \(|Q|\) of \(Q\) is a Dynkin or extended Dynkin diagram), and let \(d\) be a prehomogeneous dimension vector (i.e. the space of representations of \(Q\) with dimension vector \(d\) contains a Zariski open \(\text{GL}(d)\)-orbit). Denote by \(Z_{Q,d}\) the closed subscheme in the representation space of the common zeros of the basic semi-invariant polynomial functions \(f_1,\dots,f_s\) (the null-cone); recall that the zero sets \(Z(f_1),\dots,Z(f_s)\) are the codimension \(1\) irreducible components of the complement of the open orbit.
In an earlier paper, \textit{Ch. Riedtmann} and \textit{G. Zwara} [Comment. Math. Helv. 79, No. 2, 350-361 (2004; Zbl 1063.14052)] showed that \(Z_{Q,d}\) is a complete intersection if \(|Q|=A_n\) or \(\widetilde A_n\).
The main result of the present paper is that \(Z_{Q,d}\) is not too far from being a complete intersection for all tame prehomogeneous quiver settings. More precisely, it is proved that \(s-\text{codim}(Z_{Q,d})\leq\gamma(|Q|)\), where \(\gamma(|Q|)\in\{0,1,2,3,4\}\) is explicitly given for each (extended) Dynkin diagram, and the bound is sharp with the possible exception of the case of \(\widetilde E_8\). The proof uses the Auslander-Reiten theory for representations of tame quivers. (Also submitted to MR.) tame quivers; prehomogeneous dimension vectors; open orbits; common zero loci of semi-invariants; extended Dynkin diagrams; semi-invariant polynomials Riedtmann, Ch, Tame quivers, semi-invariants, and complete intersections, J. Algebra, 279, 362-382, (2004) Representations of quivers and partially ordered sets, Complete intersections, Geometric invariant theory, Vector and tensor algebra, theory of invariants Tame quivers, semi-invariants, and complete intersections. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The representation theory of a connected smooth affine group scheme over a field \(k\) of characteristic \(p>0\) is faithfully captured by that of its family of Frobenius kernels. Such Frobenius kernels are examples of infinitesimal group schemes, affine group schemes \(G\) whose coordinate (Hopf) algebra \(k[G]\) is a finite-dimensional local \(k\)-algebra. This paper presents a study of the cohomology algebra \(H^* (G,k)\) of an arbitrary infinitesimal group seheme over \(k\).
We provide a geometric determination of the ``cohomological support variety''
\[
|G|\equiv\text{Spec} H^{ev} (G,k)
\]
analogous to that given by \textit{D. Quillen} for the cohomology of finite groups [Ann. Math., II. Ser. 94, 549-572, 573-602 (1971; Zbl 0247.57013)]. We further study finite-dimensional rational \(G\)-modules \(M\) for arbitrary infinitesimal group schemes \(G\) over \(k\). In a manner initiated by \textit{J. L. Alperin} and \textit{L. Evens} [J. Pure Appl. Algebra 22, 1-9 (1981; Zbl 0469.20008)] and \textit{J. F. Carlson} [J. Algebra 85, 104-143 (1983; Zbl 0526.20040)] for finite groups, we consider the variety \(|G|_M\subset |G|\) of the ideal \(I_M=\text{ker} \{H^{ev}(G,K) \to\text{Ext}^*_G (M, M)\}\) and provide a geometric description of this variety which is analogous to that given by \textit{G. S. Avrunin} and \textit{L. L. Scott} for finite-dimensional modules for finite groups [Invent. Math. 66, 277-286 (1982; Zbl 0489.20042)].
This paper is a continuation of our recent work establishing the finite generation of \(H^*(G,k)\) [\textit{E. M. Friedlander} and \textit{A. Suslin}, Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] and investigating the infinitesimal 1-parameter subgroups of \(G\) [\textit{A. Suslin}, \textit{E. M. Friedlander} and \textit{C. P. Bendel}, J. Am. Math. Soc. 10, No. 3, 693-728 (1997; see the preceding review Zbl 0960.14023)]. Earlier work by E. Friedlander and B. Parshall and J. Jantzen concerning the cohomology of restricted Lie algebras are forerunners of the results presented here: Finite-dimensional restricted Lie algebras are in 1-1 correspondence with infinitesimal group schemes of height \(\leq 1\). Our main theorems (theorems 5.2 and 6.7 below) when restricted to infinitesimal group schemes of height \(\leq 1\) significantly strengthen previously known cohomological information for restricted Lie algebras.
An interesting aspect of our work is the extent to which infinitesimal 1-parameter subgroups \(\nu:\mathbb{G}_{a (r)}\to G\) for infinitesimal group schemes \(G\) of height \(\leq r\) play the role of elementary abelian \(p\)-subgroups (and their generalizations, shifted subgroups) for finite groups. Indeed, much of our effort is dedicated to proving that cohomology classes are detected (modulo nilpotence) by such 1-parameter subgroups.
The proof of the detection theorem for arbitrary infinitesimal group schemes over \(k\) relies upon a generalization of a spectral sequence introduced by \textit{H. H. Andersen} and \textit{J. C. Jantzen} [Math. Ann. 269, 487-525 (1984; Zbl 0529.20027)]. Our generalized spectral sequence is presented in \S 3, enabling the proof in \S 4 of the general detection theorem (theorem 4.3). -- The detection theorem demonstrates the essential injectivity of the natural map considered in the preceding paper cited above
\[
\psi:H^{ev} (G,k)\to k\bigl[V_r (G)\bigr],
\]
where \(V_r(G)\) is the scheme of infinitesimal 1-parameter subgroup schemes \(\nu: \mathbb{G}_{a(r)}\to G\) of an infinitesimal group scheme \(G\) over \(k\) of height \(\leq r\). The essential surjectivity of \(\psi\) (more precisely, surjective onto \(p^r\)-th powers) is a main result of our preceding paper reviewed above. This is formalized in theorem 5.2 which presents a geometric, non-cohomological description of the cohomological support variety \(|G|\) of \(G\). Corollary 6.8 gives a similarly geometric, non-cohomological identification of \(|G|_M\subset |G|\) for any finite-dimensional rational \(G\)-module \(M\). We conclude in \S 7 with a few applications of these descriptions, applications analogous to results obtained previously for the cohomology of finite groups. infinitesimal group schemes; cohomological support variety; characteristic \(p\); representation theory of a connected smooth affine group scheme; Frobenius kernels; cohomology algebra; restricted Lie algebras; infinitesimal 1-parameter subgroups A. Suslin, E. M. Friedlander and C. P. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), no. 3, 729-759. Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters, Homological methods in Lie (super)algebras, Cohomology of Lie (super)algebras Support varieties for infinitesimal group schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This book gives the foundations of a theory of differential algebraic groups. It is intended that such a theory bears to algebraic groups the same relation that the theory of differential equations bears to the theory of algebraic equations. Now algebraic groups can be viewed as groups in the category of algebraic varieties, where the latter are taken to be locally given as sets of simultaneous solutions of algebraic equations. But in fact the foundational aspects of the theory are better served by an abstract view of algebraic varieties as sets with an additional structure, where the latter is usually a sheaf of integral domains, but could be taken as a collection of fields with specialization relations (the ring of sections over a set is replaced by its quotient field and the restriction maps indue the specializations). To lay the foundations of differential algebraic groups, the author adopts a similar formulation: the objects are sets with attached collections of fields and specialization maps, except now the fields are extensions, inside some fixed universal differential field U, of a fixed subfield F with set of derivation operators D, and the specializations must preserve the D-F structure.
With these objects the theory is developed. This requires defining the objects, which are termed D-F groups (Chapter I), describing what happens when F, D, and U change (Chapter II), endowing the objects with a topology (Chapter II) and rational functions (Chapter V), how the associated objects like D-F Lie algebras should be defined (Chapter VI), and what the differential analogue of Galois cohomology should be (Chapter VII). In this framework, the basic theory is produced: subgroups and quotients make sense (Chapter IV), and there is a suitable Lie theory (Chapter VIII).
This is a demanding book. The author requires familiarity with differential algebra as presented in his ''Differential algebra and algebraic groups'' (1973; Zbl 0264.12102) and in his ''Constrained extensions of differential fields'' [Adv. Math. 12, 141-170 (1974; Zbl 0279.12103)]. Although the reader familiar with the theory of algebraic groups will find that many of the results of the theory of differential algebraic groups are analogous to the algebraic theory, he should be warned that not all are: for example, D-F groups may not be locally affine, and even affine ones need not be linear. In other words, this is a genuinely different subject. By his careful presentation of the subject's foundations in this volume, the author has prepared the way for its future development and applications. differential algebraic groups; differential field; D-F groups; rational functions; D-F Lie algebras; Galois cohomology E. Kolchin, \textit{Differential Algebraic Groups, Pure and Applied Mathematics}, Vol. 114, Academic Press, Orlando, FL, 1985. Differential algebra, Research exposition (monographs, survey articles) pertaining to field theory, Other algebraic groups (geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Modules of differentials, Lie algebras of linear algebraic groups Differential algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We just quote the authors' abstract: The congruence subgroup \(\Gamma_ 2(2,4,8)\) of the group \(\Gamma_ 2\) of \(4\times 4\) integral symplectic matrices is contained in \(\Gamma_ 2(4)\) and contains \(\Gamma_ 2(8)\), with \(\Gamma_ 2(n)\) the principal congruence subgroup of level \(n\). The Satake compactification of the quotient of the three-dimensional Siegel upper half space by \(\Gamma_ 2(2,4,8)\) is shown to be a complete intersection of ten quadrics in \(\mathbb{P}^{13}\). We determine the space of global holomorphic three forms on this space, which coincides with the space of cusp forms of weight 3 on \(\Gamma_ 2(2,4,8)\); it has dimension 2283. Finally, we study the action of the Hecke operators on this space and consider the Andrianov \(L\)-functions of some eigenforms. Siegel modular groups; Siegel modular forms; space of global holomorphic three forms; principal congruence subgroup; Satake compactification; complete intersections; action of Hecke operators; Andrianov \(L\)- functions Van Geemen, B.; Van Straten, D.: The cuspform of weight 3 on \(\Gamma 2\)(2,4,8). Math. comp. 61, No. 204, 849-872 (1993) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Hecke-Petersson operators, differential operators (several variables), Complete intersections, Langlands \(L\)-functions; one variable Dirichlet series and functional equations The cusp forms of weight 3 on \(\Gamma_ 2(2,4,8)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author classifies simple hypersurface K3 singularities (X,x) defined by a non-degenerated polynomial \(f(z_ 0,z_ 1,z_ 2,z_ 3)\) and he studies the minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) and the singularities on the exceptional divisor E.
A three-dimensional singularity (X,x) is a simple K3 singularity if (X,x) is quasi-Gorenstein and if the exceptional divisor E of any minimal resolution is a normal K3 surface, where a minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) is a proper morphism with only terminal singularities on \(\tilde X,\) with \(\tilde X\simeq X\setminus \{x\}\) and with \(K_{\tilde X}\) nef with respect to \(\pi\).
- The simple K3 singularities could be regarded as three-dimensional generalizations of simple elliptic singularities.
If the simple K3 singularity (X,x) is defined by a non-degenerated polynomial f(z), then \((1,1,1,1)\in \Gamma(f)\). The weight \(\alpha =\alpha(f)=(\alpha_ 1,\alpha_ 2,\alpha_ 3,\alpha_ 4)\) of the quasi-homogeneous polynomial \(f_{\Delta_ 0}\) associated to the face \(\Delta_ 0\) containing (1,1,1,1) verifies \(\sum^{4}_{i=1}\alpha_ i =1\). - Then to classify the simple K3 singularities we need to study the set \(W_ 4\) of weights: \(W_ 4=\{\alpha \in {\mathbb{Q}}^ 4_+| \quad \sum^{4}_{i=1}\alpha_ i =1,\alpha_ 1\geq...\geq \alpha_ 4\) and \((1,1,1,1)\in Int(C(\alpha))\},\) where \(C(\alpha\)) is the closed cone in \({\mathbb{R}}^ 4\) generated by the set \(T(\alpha)=\{\nu \in {\mathbb{Z}}^ 4_ 0| \alpha.\nu =1\}.\)
The author shows that the cardinality of \(W_ 4\) is 95, and for each weight \(\alpha\) he gives a quasi-homogeneous f of weight \(\alpha\) which defines a simple K3 singularity and such that \(\Delta_ 0=\Gamma (f)\) is the convex hull of \(T(\alpha\)). Then he constructs a minimal resolution \(\pi: \tilde X\to X\) using torus embedding: if the weight \(\alpha(f)=(p_ 1/p,...,p_ 4/p)\), where \(p_ 1,...,p_ 4\) are relatively prime integers, the filtered blow-up with weight \((p_ 1,...,p_ 4)\), \(\Pi: (V,F)\to ({\mathbb{C}}^ 4,0)\), induces a minimal resolution of (X,x). In the last part the author shows that under some conditions on the nondegenerate polynomial f defining the simple K3 singularity (X,x), the type and the number of the singularities on E are determined by the weight \(\alpha(f)\), independently of f. type of singularities; simple hypersurface K3 singularities; minimal resolution; exceptional divisor; number of the singularities; weight Yonemura, T., Hypersurface simple \textit{K}3 singularities, Tohoku Math. J. (2), 42, 3, 351-380, (1990), MR 1066667 Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Hypersurface simple K3 singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I see ibid. 38, 321-341 (1986; Zbl 0618.14004), for part II see the preceding review.]
In this last part of the paper the author applies the concept of universally one-equicodimensional schemes (defined and studied in the second part of the paper) to the problem of finite generation of k- subalgebras of a k-algebra of finite type, with k a field. This approach is different from the methods developed by Zariski and Nagata in connection with the Hilbert's fourteenth problem. The problem can be translated into more geometric terms as follows: if X is an algebraic variety over k and if Y is a k-scheme dominated by X, find natural conditions ensuring that Y is also an algebraic variety. For example, if Y is quasi-compact and integral, then one of the main results of the paper says that Y is an algebraic variety iff Y is a universally one- equicodimensional scheme. As applications, the author considers the low- dimensional cases and gives new proofs to some results due to Zariski and Nagata. finite generation of subalgebras; universally one-equicodimensional schemes; algebra of finite type Adrian Constantinescu. Proper morphisms and finite generation of subalgebras. III. Schemes dominated by algebraic varieties. Stud. Cerc. Mat., 38(6):477--510, 1986. Schemes and morphisms, Commutative Artinian rings and modules, finite-dimensional algebras Proper morphisms and finite generation of subalgebras. III: Schemes dominated by algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is a sequel of [\textit{M. Domokos, H. Lenzing}, J. Algebra 228, No. 2, 738-762 (2000; Zbl 0955.16015)] where the authors investigated the relative invariants and moduli spaces introduced by \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515-530 (1994; Zbl 0837.16005)]. Let \(k\) be an algebraically closed field, let \(\Sigma\) be a finite-dimensional \(k\)-algebra, and let \(\text{mod }\Sigma\) be the category of finite-dimensional right \(\Sigma\)-modules. The authors study in detail the module category when \(\Sigma\) is a concealed-canonical algebra. In fact the class of such algebras consists of representation-infinite algebras of tame or wild representation type. They prove that for any admissible weight all the corresponding moduli spaces are isomorphic to a certain projective space. As a consequence the authors show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental. In particular, a generalization of a result of \textit{C. M. Ringel} [Invent. Math. 58, 217-239 (1980; Zbl 0433.15009)] for the rational invariants of extended Dynkin quivers is obtained. finite-dimensional algebras; representations; concealed-canonical algebras; tame concealed algebras; relative invariants; admissible weights; semistability; coarse moduli spaces; extended Dynkin quivers; perpendicularity DOI: 10.1006/jabr.2001.9117 Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Module categories in associative algebras, Group actions on varieties or schemes (quotients) Moduli spaces for representations of concealed-canonical algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a group scheme. The notion of a diagram of schemes was introduced by [\textit{M. Hashimoto}, Foundations of Grothendieck duality for diagrams of schemes. Lecture Notes in Mathematics 1960. Berlin: Springer. (2009; Zbl 1163.14001)]. His purpose is to study \(G\)-linearized quasi-coherent sheaves over \(G\)-schemes and \((G, A)\)-modules where \(A\) is a \(G\)-algebra.
In the present paper, the authors established the theory of local cohomology for diagrams of schemes. As a consequence, they generalized the theorem of \textit{M. Hochster} and \textit{J. A. Eagon} [Am. J. Math. 93, 1020--1058 (1971; Zbl 0244.13012)]. diagram of schemes; group scheme; invariant theory; local cohomology DOI: 10.1307/mmj/1220879415 Generalizations (algebraic spaces, stacks), Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients) Local cohomology on diagrams of schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a normal projective surface \(S\), there exists a function \(R_S\) on the group of Weil divisors on \(S\) modulo Cartier divisors such that
\[
\chi(\mathcal O_S(D))=\chi(\mathcal O_S)+\frac{1}{2}D\cdot (D-K_S)+R_S(D),
\]
where \(D\) is any Weil divisor. The invariant \(R_S(D)\) is the sum of the local invariants \(R_{S,x}(D_x)\) of the germ \(D_x \subset (S,x)\) at singular points \(x\).
For quotient singularity germs, \textit{R. Blache} [Abh. Math. Semin. Univ. Hamb. 65, 307--340 (1995; Zbl 0877.14007)] gave a description of this invariant including the delta invariant of curve singularities.
This paper deals with the invariant \(R_X\) of cyclic quotient singularities \((X,x)\) and shows an explicit computation providing a new interpretation for the delta invariant.
For the proof, the authors deeply analyze invariants of eigenfunctions with respect to the cyclic group action on \(\mathcal O_{\mathbb C^2, 0}\).
As an application, for the \(\mathcal O_X\)-modules of eigenfunctions, they provide explicitly the McKay decomposition of those into special indecomposable reflexive modules.
Furthermore, they give answers two questions posed by Blache [loc. cit.] on the behavior of the function \(R_X(m K_X)\) of \(m\). Riemann-Roch; delta invariant; cyclic quotient singularities; McKay correspondence; reflexive modules; curvettes Local complex singularities, Plane and space curves, Complex surface and hypersurface singularities, Topological properties in algebraic geometry The correction term for the Riemann-Roch formula of cyclic quotient singularities and associated invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the structure of singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities.
Assuming that the boundary \(\partial\Omega\) of \(\Omega\in\mathbb C^{n+1}\) is \(C^\infty\)-smooth, the Bergman kernel \(B(z)\) of \(\Omega\) takes the form near a boundary point \(p\):
\[
B(z) =\frac{\Phi(w,p)}{\rho^{2+2/d_F}(\log(1/\rho))^{m_F-1}},
\]
where \((w,p)\) is some polar coordinates on a nontangential cone \(\Lambda\) with apex at \(p\) and \(\rho\) means the distance from the boundary. Here \(\Phi\) admits some asymptotic expansion with respect to the variables \(\rho^{1/m}\) and \(\log(1/\rho)\) as \(\rho\to 0\) on \(\Lambda\). The values of \(d_F > 0\), \(m_F\in\mathbb Z_+\) and \(m\in\mathbb N\) are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of \(\Phi\) as \(\rho\to 0\) on \(\Lambda\) is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. The analysis is based on some integral formula for the Bergman kernel due to Haslinger.
The Fourier transform gives a clear connection between the Bergman space of \(\Omega\) and an associated weighted Bergman space \(H_\tau\) in \(\mathbb C^n\) with a parameter \(\tau\). As a result, the Bergman kernel of \(\Omega\) is expressed as a superposition of the reproducing kernel of \(H_\tau\) by using the Laplace transform with respect to \(\tau\).
From the Haslinger formula, the problem is reduced to the analysis of a Laplace integral of the following form: \(I(\tau)=\int_{\mathbb R_+^n}e^{-\tau f(x)}\varphi(x)\,dx,\) \(\tau>0\), where the phase \(f\) is associated with the defining function of \(\Omega\). In order to investigate the boundary behavior of the Bergman kernel, the author analyzes the behavior of the integral \(I(\tau)\) at infinity. The essential idea of the analysis is due to the work of Varchenko on the analysis of an oscillatory integral: \(\tilde I(\tau)=\int_{\mathbb R^n}e^{i\tau f(x)}\varphi(x)\,dx,\) \(\tau>0\).
Assuming \(f(x) =\prod x_j^{p_j}\), \(p_j\geq0\), an asymptotic expansion of \(I(\tau)\) as \(\tau\to\infty\) is computed. The problem on the resolution of singularities of the phase \(f\) at a critical point \(x_0\) is considered by applying the theory of toric varieties. Toric resolutions are constructed in terms of the Newton polyhedron of the phase \(f\) at \(x_0\).
As a result, the principal term of the asymptotic expansion of \(I(\tau)\) can be seen directly by using the geometry of the Newton diagram of \(f\). Moreover the form of the asymptotic expansion of \(I(\tau)\) is determined by quantitative information on the toric resolutions. From this result, the essential structure of singularities of the Bergman kernel appears in the symbol of its Laplace integral representation. singularities of the Bergman kernel; pseudoconvex domains of finite type; Newton polyhedra; Szegö kernel; Laplace integral; asymptotic expansion; Fourier analysis; toric varieties; toric resolutions Kamimoto J. Newton polyhedra and the Bergman kernel. Math Z, 246: 405--440 (2004) Bergman spaces of functions in several complex variables, Finite-type domains, Toric varieties, Newton polyhedra, Okounkov bodies, Integral representations; canonical kernels (Szegő, Bergman, etc.) Newton polyhedra and the Bergman kernel | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let k be an algebraically closed field, Q a finite quiver with relations such that the corresponding k-algebra is finite-dimensional. Suppose X and Y are representations of the quiver with relations, and the finite vector dimensions of X and Y over k coincide. There is a virtual degeneration from X to Y if there is a degeneration from \(X\oplus Z\) for some representation Z. Obviously, the existence of a degeneration implies the existence of a virtual degeneration; the converse is not true. However, when the quiver with relations is of finite representation type, the author proves that the existence of a virtual degeneration does imply the existence of a degeneration if either the underlying non-oriented graph of Q is a Dynkin diagram \(A_ n\) or \(D_ n\), or the Auslander- Reiten quiver is simply connected and a certain numerical characteristic does not exceed 2. Another result is that the inequality \(\dim_ kHom(U,X)\leq \dim_ kHom(U,Y)\) for all representations U, which is necessary for the existence of a degeneration from X to Y, is both necessary and sufficient for the existence of a virtual degeneration provided the representation type is finite. finite quiver with relations; representations; virtual degeneration; finite representation type; Dynkin diagram; Auslander-Reiten quiver Riedtmann, Christine, Degenerations for representations of quivers with relations, Ann. Sci. École Norm. Sup. (4), 19, 2, 275-301, (1986) Representation theory of associative rings and algebras, Finite rings and finite-dimensional associative algebras, Group actions on varieties or schemes (quotients) Degenerations for representations of quivers with relations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From the authors's abstract: ``We investigate a class of Lie group actions on \({\mathbb{C}}^ N\), the so-called polar actions, that naturally generalize the standard \(T^ N\) actions. For a domain invariant under such an action (i.e., a generalized Reinhardt domain) we characterize the invariant plurisubharmonic functions and determine the envelope of holomorphy in geometric terms. For a generalized Reinhardt domain containing the origin of \({\mathbb{C}}^ N\) we also compute its automorphism group.'' Lie group actions; polar actions; generalized Reinhardt domain; invariant plurisubharmonic functions; envelope of holomorphy; automorphism group Eric Bedford and Jiri Dadok, Generalized Reinhardt domains, J. Geom. Anal. 1 (1991), no. 1, 1 -- 17. Complex Lie groups, group actions on complex spaces, Group actions on varieties or schemes (quotients), Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube), Plurisubharmonic functions and generalizations, Domains of holomorphy, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Generalized Reinhardt domains | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a \(F\)-algebra where \(F\) is a field, and let \(W\) be an \(A\)-module of finite presentation. We use the linear Lie-Rinehart algebra \(\mathbf V_W\) of \(W\) to define the first Chern-class \(c_1(W)\) in \(H^2(\mathbf V_{W|U}, \mathcal O_U)\), where \(U\) in \(\text{Spec}(A)\) is the open subset where \(W\) is locally free. We compute explicitly algebraic \(\mathbf V_W\)-connections on maximal Cohen-Macaulay modules \(W\) on the hypersurface-singularities \(B_{mn2} = x^m + y^n + z^2\), and show that these connections are integrable, hence the first Chern-class \(c_1(W)\) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanishes. Kodaira-Spencer maps; Lie-algebroids; connections; Chern-classes; Brieskorn singularities; Alexander-polynomials; quotient singularities; McKay correspondence Helge Maakestad, Chern classes and Lie-Rinehart algebras, Indag. Math. (N.S.) 18 (2007), no. 4, 589 -- 599. Homological methods in Lie (super)algebras, Cohen-Macaulay modules, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Chern classes and Lie-Rinehart algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum Hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [\textit{P. Etingof} and \textit{V. Ginzburg}, Invent. Math. 147, No. 2, 243-348 (2002; Zbl 1061.16032)].
We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products. spherical subalgebras; symplectic reflection algebras; wreath products; quantum Hamiltonian reductions; algebras of differential operators; representation spaces; extended Dynkin quivers; reflection functors; generalized preprojective algebras P. Etingof, W. L. Gan, V. Ginzburg and A. Oblomkov, \textit{Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products}, \textit{Publ. Math. IHES}\textbf{105} (2007) 91 [math/0511489]. Representations of quivers and partially ordered sets, Rings of differential operators (associative algebraic aspects), Representations of associative Artinian rings, Noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations, Deformations of associative rings, Representations of finite symmetric groups, Combinatorial aspects of representation theory Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book under review is an introduction to the theory of algebraic curves over finite fields with applications to coding theory. New results on error correcting codes on algebraic (modular) curves over finite fields are presented in book form for the first time.
The first two chapters are introductory, preparing the readers to the basic notions about algebraic curves, their function fields, and the Riemann-Roch theorem. In chapter 3, the zeta-functions and L-functions of algebraic curves are defined, and the functional equations and the Riemann hypothesis are discussed. Exponential sums and Kloosterman sums are extensively dealt with in chapter 4. The novelty of the book is chapter 5 where applications of the theory of algebraic curves over finite fields to algebraic codes are beautifully presented. Goppa initiated the use of algebraic geometry to the study of linear codes, for instance, the construction of linear codes from the rational points on an algebraic curve. A new proof on a theorem of Tsfasman-Vladut-Zink is presented employing more direct approach than the original proof, counting the number of the rational points on modular curves using the Eichler-Selberg trace formula.
Algebraic Goppa codes are defined as follows. Let C be a smooth projective absolutely irreducible algebraic curve defined over \({\mathbb{F}}_ q\) of genus g. Let \(P_ 1,...,P_ n\) be distinct points on C. Let \(D=P_ 1+...+P_ n\), and \(G=\sum_{Q}m_ QQ \) be two positive divisors on C with disjoint support. Let \(\Omega_ C(D-G)=\{\omega \in \Omega_ C| (\omega)+D-G\geq 0\}\) be the space of differentials which have zeros at the points Q in the support of G with multiplicities at least \(m_ Q\) and which are regular outside the set \(\{P_ 1,...,P_ n\}\) where they are allowed to have at most simple poles. Suppose that \(2g-2<\deg (G)<\deg (D)\). Then the image of the residue mapping \(\Omega_ C(D-G)\to {\mathbb{F}}^ n_ q,\quad \omega \to (res_{P_ 1}\omega,...,res_{P_ n}\omega)\) is defined to be an algebraic Goppa code and denoted by \(\Gamma_ C(D,G)\). The dimension of \(\Gamma_ C(D,G)\) is \(k:=g-1+\deg (D)-\deg (G)\) and its relative distance d satisfies \(d\cdot \max_{1\leq i\leq n}(\deg (P_ i))\geq \deg (G)-2g+2\). The dual definition of an algebraic Goppa code \(\Gamma_ C(D,G)\) is the following:
Let \(\phi_ 1,...,\phi_ s\) be a basis over \({\mathbb{F}}_ q\) of L(G). The matrix \(H=(\phi_ i(P_ j))\), \(1\leq i\leq s\), \(1\leq j\leq n\), is the parity check matrix of the code \(\Gamma_ C(D,G)\). - For an algebraic Goppa code, there are two basic parameters: the transmission rate \(R=k/n\) and the relative distance \(\delta:=d/n.\)
The theorem of Tsfasman, Vladut and Zink, and its generalization by the same authors, and independently by Ihara is formulated as follows: Let \(q=p^{2f}\geq 49\). For each prime \(\ell >p\), let \(C_{\ell}=X_ 0(\ell)\) be the modular curve which parametrizes elliptic curves with a subgroup of order \(\ell\). Let \(g_{\ell}\) be the genus of \(C_{\ell}\) and \(N_{\ell}\) the number of points on \(C_{\ell}\) rational over \({\mathbb{F}}_ q\). Let \(\gamma (C_{\ell})=(g_{\ell}-1)/(N_{\ell}-1)\). Then \(\lim_{\ell \to \infty}\gamma (C_{\ell}) =1/(\sqrt{q}-1)\). Furthermore, if \(\phi (\delta)=\delta \log_ q(q-1)-\delta \log_ q\delta -(1-\delta)\log_ q(1-\delta)\) denotes the entropy function, then the equation \(\phi (\delta)-\delta =1/(\sqrt{q}-1)\) has two distinct roots \(\delta_ 1, \delta_ 2\) and hence it is possible to construct algebraic Goppa codes with transmission rate and relative distance which are better than the Varshamov-Gilbert bound in the interval \(\delta_ 1, \delta_ 2.\)
This was proved for \(q=p^ 2\) or \(p^ 4\) by Tsfasman, Vladut and Zink. The author of this book gives a new proof using the Eichler-Selberg trace formula to count the number of rational points on modular curves.
The book is written in a very friendly manner, and the reader can get a fairly good picture on algebraic Goppa codes as an application of the theory of algebraic curves over finite fields. algebraic curves over finite fields; coding theory; error correcting codes; zeta-functions; L-functions; algebraic Goppa code; entropy; number of rational points on modular curves C. J. Moreno, \textit{Algebraic Curves Over Finite Fields} (Cambridge University Press, 1991). Computational aspects of algebraic curves, Arithmetic ground fields for curves, Arithmetic codes, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Finite ground fields in algebraic geometry Algebraic curves over finite fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the book under review is to provide details to one of the most important achievement of the mathematics of this century, namely the difficult proof of the Mordell conjecture for number fields found recently by \textit{G. Faltings} [Invent. Math. 73, 349-366 (1983; see the preceding review)], as well as to present some additional results about arithmetic surfaces. The book, based on a seminar conducted by the editors at Max-Plank-Institut für Mathematik Bonn, is divided into seven chapters. The first three chapters introduce the basic tools, in the next two the conjectures of Tate, Shafarevich and Mordell are proved, while in the last two ones some complements and basic facts about arithmetic surfaces are given.
Chapter one (by \textit{G. Faltings}) describes the necessary basic facts from the theory of moduli spaces and their compactifications (due to many people) and is written rather as a survey. Indeed, it is impossible to give all details, because such an attempt would require a whole book itself. However, it is an excellent introduction to moduli spaces, logarithmic singularities and the compactification of the moduli space of abelian varieties. Some examples are also presented. Chapter two (by \textit{G. Faltings}) uses chapter one in order to define the modular height of an abelian variety over a number field and to prove its main properties. The concept of modular height of an abelian variety, entirely due to Faltings, is the basic tool in proving the conjectures mentioned above.
Chapter three (by \textit{F. Grunewald}) discusses some results from the theory of p-divisible groups and finite flat group schemes. These developments are necessary in chapters 5 and 6. The details are in general omitted, although - when possible - some proofs or relevant examples are included.
Chapter four (by \textit{N. Schappacher}) proves the Tate conjecture. Namely, let K be a number field, \(\bar K\) an algebraic closure of K, \(\pi =Gal(\bar K/K)\), A an abelian variety over K, \(T_{\ell}(A)=\lim_{\overset \leftarrow n}A(\ell^ n)\) and \(V_{\ell}(A)=T_{\ell}(A)\otimes_{{\mathbb{Z}}_{\ell}}{\mathbb{Q}}_{\ell},\) where \(\ell\) is a prime number and \(A(\ell^ n)\) denotes the kernel of the multiplication by \(\ell^ n\) on A. Then \(\pi\) acts naturally on \(T_{\ell}(A)\) and on \(V_{\ell}(A)\). Then Tate's conjecture asserts that the action of \(\pi\) on \(V_{\ell}(A)\) is semi-simple and the natural map \(End_ K(A)\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_{\ell}\to End_{{\mathbb{Z}}_{\ell}[\pi]}(T_{\ell}(A))\) is an isomorphism.
Chapter five (by \textit{G. Wüstholz}) deals with the proof of Faltings' finiteness theorem. Using this one proves the Mordell conjecture which asserts that if X is a smooth projective curve over K of genus \(g\geq 2\), then the set of rational points X(K) of X over K is finite.
Chapter six (by \textit{G. Faltings}) extends the conjectures of Tate, Shafarevich and Mordell from the case of number fields to the case of finitely generated extensions over \({\mathbb{Q}}\). This is done by reducing the situation to number fields via complex Hodge theory.
The last chapter (by \textit{U. Stuhler}) is an introduction to the intersection theory on arithmetic surfaces, which has been initiated by \textit{S. Yu. Arakelov} [Math. USSR, Izv. 8 (1974), 1167-1180 (1976), translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002) and Proc. Int. Congr. Math., Vancouver 1974, Vol. 1, 405- 408 (1975; Zbl 0351.14003)] and further developed by \textit{G. Faltings} [Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005)]. Among other things one proves the Riemann-Roch theorem and the Hodge index theorem.
Contents:
Chapter I: \textit{Gerd Faltings}: ''Moduli spaces'' (p. 1-32). - Chapter II: \textit{Gerd Faltings}: ''Heights'' (p. 33-52). - Chapter III: \textit{Fritz Grunewald}: ''Some facts from the theory of group schemes'' (p. 53-113). - Chapter IV: \textit{Norbert Schappacher}: ''Tate's conjecture on the endomorphisms of abelian varieties'' (p. 114-153). - Chapter V: \textit{G. Wüstholz}: ''The finiteness theorems of Faltings'' (p. 154-202). - Chapter VI: \textit{Gerd Faltings}: ''Complements'' (p. 203-227). - Chapter VII: \textit{Ulrich Stuhler}: ''Intersection theory on arithmetic surfaces'' (p. 228-268). Rational points; Seminar; Bonn; Wuppertal; Mordell conjecture; proof of Tate conjecture; proof of Shafarevich conjecture; proof of the Mordell conjecture; logarithmic singularities; compactification of the moduli space of abelian varieties; modular height of an abelian variety; p-divisible groups; intersection theory on arithmetic surfaces; Riemann-Roch theorem; Hodge index theorem G. Faltings , G. Wüstholz , Rational Points, Aspects of Mathematics No. E6 , Vieweg, Braunschweig/Wiesbaden, 1984. Arithmetic ground fields for abelian varieties, Rational points, Special algebraic curves and curves of low genus, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Elliptic curves Rational points. Seminar Bonn/Wuppertal 1983/84 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive group over an algebraically closed field. We propose a definition of a partition of this variety into locally closed smooth subvarieties indexed by the unipotent classes in the corresponding group over the complex numbers. We obtain explicit results in types \(A\), \(C\) and \(D\).
Let \(\Bbbk\) be an algebraically closed field of characteristic exponent \(p\geq 1\). Let \(G\) be a connected reductive algebraic group over \(\Bbbk\) and let \(\mathfrak g\) be the Lie algebra of \(G\). Note that \(G\) acts on \(G\) and on \(\mathfrak g\) by the adjoint action and on \(\mathfrak g^*\) by the coadjoint action. (For any \(\Bbbk\)-vector space \(V\) we denote by \(V^*\) the dual vector space.) Let \(G_{\mathbb C}\) be the reductive group over \(\mathbb C\) of the same type as \(G\). Let \(\mathcal U_G\) be the variety of unipotent elements of \(G\). Let \(\mathcal N_{\mathfrak g}\) be the variety of nilpotent elements of \(\mathfrak g\). Let \(\mathcal N_{\mathfrak g^*}\) be the variety of nilpotent elements of \(\mathfrak g^*\).
In parts I-III [Transform. Groups 10, No. 3-4, 449-487 (2005; Zbl 1107.20036); ibid. 13, No. 3-4, 773-797 (2008; Zbl 1196.20056); J. Algebra 329, No. 1, 163-189 (2011; Zbl 1243.20061)] we have proposed a definition of a partition of \(\mathcal U_G\) and of \(\mathcal N_{\mathfrak g}\) into smooth locally closed \(G\)-stable pieces which are indexed by the unipotent classes in \(G_{\mathbb C}\) and which in many ways depend very smoothly on \(p\). In this paper we propose a definition of an analogous partition of \(\mathcal N_{\mathfrak g^*}\) into pieces which are indexed by the unipotent classes in \(G_{\mathbb C}\). (This definition is only of interest for \(p>1\), small; for \(p=1\) or \(p\) large we can identify \(\mathcal N_{\mathfrak g}\) with \(\mathcal N_{\mathfrak g^*}\) and the partition of \(\mathcal N_{\mathfrak g^*}\) is deduced from the partition of \(\mathcal N_{\mathfrak g}\).) We will illustrate this in the case where \(G\) is of type \(A\), \(C\) or \(D\) and \(p\) is arbitrary. unipotent elements; nilpotent elements; unipotent classes; unipotent pieces; reductive connected algebraic groups; varieties of Borel subgroups; Lie algebras; adjoint actions; partitions; locally closed smooth subvarieties G. Lusztig, Unipotent elements in small characteristic, IV, Transform. Groups 15 (2010), 921--936. Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions Unipotent elements in small characteristic. IV. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities \(\mathbb C^2/G\) where \(G=\frac{1}{r}(1,a)\leq\mathrm{GL}(2,\mathbb C)\).
This paper is organized as follows. In Section 2 we define the reconstruction algebra associated to a labelled Dynkin diagram of type \(A\) and describe some of its basic structure. In Section 3 we prove that it is isomorphic to the endomorphism ring of some Cohen-Macaulay modules. In Section 4 the minimal resolution of the singularity \(\mathbb C^2/\frac{1}{r}(1,a)\) is obtained via a certain moduli space of representations of the associated reconstruction algebra \(A_{r,a}\), and in Section 5 we produce a tilting bundle which gives us our derived equivalence. In Section 6 we prove that \(A_{r,a}\) is a prime ring and use this to show that the Azumaya locus of \(A_{r,a}\) coincides with the smooth locus of its centre \(\mathbb C[x,y]^{\frac{1}{r}(1,a)}\). This then gives a precise value for the global dimension of \(A_{r,a}\), which shows that the reconstruction algebra need not be homologically homogeneous. reconstruction algebras; Cohen-Macaulay singularities; labelled Dynkin diagrams; endomorphism rings of Cohen-Macaulay modules; resolutions of singularities; moduli spaces of representations; tilting bundles; derived equivalences; global dimension Wemyss, M, Reconstruction algebras of type \(A\), Trans. Am. Math. Soc., 363, 3101-3132, (2011) Rings arising from noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Cohen-Macaulay modules, Representations of quivers and partially ordered sets Reconstruction algebras of type \(A\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi--Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include:
(1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over \(\mathbb{Q}\) or number fields,
(2) the modularity of solutions of Picard-Fuchs differential equations of families of Calabi--Yau varieties, and mirror maps (mirror moonshine),
(3) the modularity of generating functions of invariants counting certain quantities on Calabi-Yau varieties, and
(4) the modularity of moduli for families of Calabi-Yau varieties. The topic (4) is commonly known as geometric modularity.{
}Discussions in this paper are centered around arithmetic modularity, namely on (1), and (2), with a brief excursion to (3). elliptic curves; \(K3\) surfaces; Calabi-Yau threefolds; CM type Calabi-Yau varieties; Galois representations; modular (cusp) forms; automorphic inductions; geometry and arithmetic of moduli spaces; Hilbert and Siegel modular forms; Families of Calabi-Yau varieties; mirror symmetry; mirror maps; Picard-Fuchs differential equations Yui, N.: Modularity of Calabi-Yau varieties: 2011 and beyond. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, Fields Institute Communications, vol. 67, pp. 101-139. Springer, New York (2013) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Arithmetic mirror symmetry, Galois representations, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Moduli, classification: analytic theory; relations with modular forms, Mirror symmetry (algebro-geometric aspects), History of algebraic geometry, History of mathematics in the 21st century Modularity of Calabi-Yau varieties: 2011 and beyond | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) denote the finite field \(\mathbb F_q\) with \(q\) elements for \(q\) a power of a prime \(p\), and let \(G\) denote the general linear group \(\text{GL}_2(F)\). Let \(V=F^2\) denote the natural two-dimensional \(FG\)-module. Further, for a non-negative integer \(k\), let \(V_k\) denote the \(k\)-th symmetric power of \(V\) as an \(FG\)-module. The author first considers a map \(e\otimes V_{k-(q+1)}\to V_k\) for \(k>q\) where \(e\) denotes the character determinant. It is shown that the cokernel of this map (having dimension \(q+1\)) is isomorphic to the reduction mod \(p\) of a principal series representation.
The main focus of the paper is on a map \(D\colon V_k\to V_{k+(q-1)}\) defined by Serre. The main result is an identification of the cokernel of \(D\) for \(q>2\), \(2\leq k\leq p-1\), \(k\neq\frac{q+1}{2}\). Precisely, it is shown that the cokernel of \(D\) is isomorphic to the reduction mod \(p\) of an integral model of a cuspidal representation for \(\overline{\mathbb Q}_pG\) (where \(\overline{\mathbb Q}_p\) is the algebraic closure of the \(p\)-adic field). The proof makes use of a short exact sequence involving the cokernel of \(D\). This short exact sequence is identified with a short exact sequence in crystalline cohomology for the projective curve \(XY^q-X^qY-Z^{q+1}=0\) due to \textit{B. Haastert} and \textit{J.~C. Jantzen} [J. Algebra 132, No. 1, 77-103 (1990; Zbl 0724.20030)].
Lastly, in the case \(q=p>3\), the author applies his results to modular forms over \(G\). The map \(D\) discussed above is used to extend a cohomological analogue of the Hasse invariant operator constructed by \textit{B. Edixhoven} and \textit{C. Khare} [Doc. Math., J. DMV 8, 43-50 (2003; Zbl 1044.11030)] on the cohomology of spaces of mod \(p\) modular forms for \(\text{GL}_2\). modular representations of finite groups; congruences for mod \(p\) modular forms; general linear groups; principal series representations; cuspidal representations; symmetric powers; crystalline cohomology Modular representations and characters, Congruences for modular and \(p\)-adic modular forms, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Representations of finite groups of Lie type, Representation theory for linear algebraic groups, Linear algebraic groups over finite fields Reduction mod \(p\) of cuspidal representations of \(\text{GL}_2(\mathbb F_{p^n})\) and symmetric powers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In {\S} 1 the definition of relative Lie groups \({\mathcal G}\) over algebraic curves is given with Lie algebras which correspond to the classical (not necessarily unitary) r-matrices, introduced by L. D. Faddeev and others. The action of the group of adèles of \({\mathcal G}\) on the quotient by the principal adèles induces the generalized Bäcklund transformations. In {\S} 3 the connection of Baker functions with the problem of description of ''maximal'' subtori of \({\mathcal G}\) is given. In {\S} 4 the definition of \(\tau\)-function is given in the spirit of \textit{V. G. Kac} and \textit{D. H. Peterson} [Proc. Natl. Acad. Sci. USA 78, 3308-3312 (1981; Zbl 0469.22016)] which is a grouplike version of the definition from a paper by the present author [Funkts. Anal. Prilozh. 17, No.3, 93-95 (1983; Zbl 0528.17004)]. Proofs are omitted. relative Lie groups; algebraic curves; group of adèles; principal adèles; generalized Bäcklund transformations; Baker functions; subtori; \(\tau \) -function I. V. Cherednik, ''Group interpretation of Baker functions and ?-functions,'' Usp. Mat. Nauk,38, No. 6, 133-134 (1983). Linear algebraic groups over adèles and other rings and schemes, Infinite-dimensional Lie groups and their Lie algebras: general properties, Classical groups (algebro-geometric aspects), Associated Lie structures for groups On the group-theoretical interpretation of Baker functions and \(\tau\)- functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Inspired by the Bloch-Beilinson conjectures, \textit{C. Voisin} [Lect. Notes Pure Appl. Math. 179, 265--285 (1996; Zbl 0912.14003)] has formulated a conjecture concerning the Chow group of 0-cycles on complex varieties of geometric genus one. This note presents some new examples of surfaces for which Voisin's conjecture is verified. algebraic cycles; Chow groups; motives; finite-dimensional motives; \(K3\) surfaces; surfaces of general type Laterveer, R., Some results on a conjecture of voisin for surfaces of geometric genus one, Boll. Unione Mat. Ital., 9, 435-452, (2016) (Equivariant) Chow groups and rings; motives, Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Surfaces of general type, Automorphisms of surfaces and higher-dimensional varieties Some results on a conjecture of Voisin for surfaces of geometric genus one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this brief note, we present our closedness theorem in geometry over Henselian valued fields with analytic structure. It enables, among others, application of resolution of singularities and of transformation to normal crossings by blowing up in much the same way as over locally compact ground fields. Also given are many applications which, at the same time, provide useful tools in geometry and topology of definable sets and functions. They include several versions of the Łojasiewicz inequality, Hölder continuity of definable functions continuous on closed bounded subsets of the affine space, piecewise continuity of definable functions or curve selection. We also present our most recent research concerning definable retractions and the extension of continuous definable functions. These results were established in several successive papers of ours, and their proofs made, in particular, use of the following fundamental tools: elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers-Lipshitz-Robinson, relative quantifier elimination for ordered abelian groups, due to Cluckers-Halupczok, the closedness theorem as well as canonical resolution of singularities and transformation to normal crossings by blowing up due to Bierstone-Milman. As for the last tool, our approach requires its definable version established in our most recent paper within a category of definable, strong analytic manifolds and maps. Henselian fields; closedness theorem; analytic structure; b-minimal cell decomposition; quantifier elimination; ordered abelian groups; fiber shrinking; Łojasiewicz inequalities; piecewise continuity; Hölder continuity; curve selection; transformation to normal crossings; resolution of singularities; definable retractions; extension of continuous definable functions Non-Archimedean analysis, Analytic algebras and generalizations, preparation theorems, Rigid analytic geometry, Applications of model theory, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects) A closedness theorem over Henselian fields with analytic structure and its applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of this work is the complete description of the degeneracy relations for conjugacy classes of nilpotent elements in the symplectic and orthogonal Lie algebras over an algebraically closed field of characteristic two. Here a class \(C\) is said to degenerate to a class \(C'\) if \(C'\) is contained in the Zariski closure of \(C\). Such results are well known for characteristics different from two [\textit{M. Gerstenhaber}, Ann. Math. (2) 70, 167--205 (1959; Zbl 0168.28103), ibid. 73, 324--348 (1961; Zbl 0168.28201), ibid. 74, 532--569 (1961; Zbl 0111.24502); \textit{W. Hesselink}, Trans. Am. Math. Soc. 222, 1--32 (1976; Zbl 0332.14017)]. The author's methods are based on the classification of nilpotent classes in characteristic two by partitions and index functions as established by \textit{W. Hesselink} [Math. Z. 166, 165--181 (1979; Zbl 0387.20038)]. He also uses a process of induction for nilpotent classes introduced by \textit{G. Lusztig} and \textit{N. Spaltenstein} [J. Lond. Math. Soc., II. Ser. 19, 41--52 (1979; Zbl 0407.20035)]. In the course of the developments a conjectured dimension formula is proved for the fixed point variety of a nilpotent element acting on the manifold of full flags. Other results concern algebro-geometric properties of the nilpotent variety (complete intersection, normality, differential criterion for regular elements) and the validity of the Chevalley restriction theorem for invariant polynomials on the Lie algebras. classical Lie algebras; degeneracy relations for conjugacy classes of nilpotent elements; symplectic and orthogonal Lie algebras; dimension formula; fixed point variety of nilpotent element; manifold of full flags; complete intersection; normality; differential criterion for regular elements; validity of Chevalley restriction theorem for invariant polynomials Lie algebras of linear algebraic groups, Coadjoint orbits; nilpotent varieties, Classical groups (algebro-geometric aspects) Degeneration behavior of nilpotent conjugacy classes of classical Lie algebras of characteristic 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In their landmark paper [Invent. Math. 93, No.~1, 185-223 (1988; Zbl 0651.20040)] \textit{F. J. Grunewald, D. Segal} and \textit{G. C. Smith} introduced the notion of the zeta function
\[
\zeta^\leq_G(s)=\sum_{H\leq G}|G:H|^{-s}=\sum_{n=1}^\infty a^\leq_n(G)\cdot n^{-s}
\]
for a group G. Here \(a^\leq_n(G)\) stands for the number of subgroups of index \(n\) in \(G\). (One can work in a similar way with normal subgroups.) Based on the available examples, and the analogy with the Dedekind zeta function of a number field, they were led to conjecture that if \(G\) is a finitely generated nilpotent group, then there exist finitely many rational functions \(W_1(X,Y),\dots,W_r(x,y)\in\mathbb{Q}(X,Y)\) such that for each prime \(p\) there is an \(i\) such that \(\zeta^\leq_{G,p}(s)=W_i(p,p^{-s})\). Here \(\zeta^\leq_{G,p}(s)=\sum_{n=1}^\infty a^\leq_{p^n}(G)\cdot p^{-ns}\) is the local zeta function. If this holds for a group \(G\), one says that the local zeta functions are finitely uniform. If one rational function suffices (that is, one can take \(r=1\) above), one says that the local zeta functions are uniform.
Recent work of \textit{M. du Sautoy} and \textit{F. Grunewald} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No.~5, 351-356 (1999; Zbl 1062.11503), Ann. Math. (2) (to appear)] shows that the behaviour of the local factors as one varies over the prime is not related to the behaviour of primes in number fields, as originally suspected, but rather to the number of points modulo \(p\) of a variety. This number is known to vary wildly with \(p\), so that it does not lend itself to a finitely uniform description. For instance, the number of points modulo \(p\) on the elliptic curve given by \(y^2-x^3+x=0\) is \(p\) if \(p\equiv 3\pmod 4\); when \(p\equiv 1\pmod 4\), though, the number is \(p-2a\), where \(p=a^2+b^2\), and \(a+ib\equiv 1\pmod{2+2i}\).
The purpose of the paper under review is to exhibit a nilpotent group \(G\) whose zeta function depends on the behaviour modulo \(p\) of the elliptic curve above, and thus does not lend itself to a finitely uniform description. This is a group of nilpotency class two and Hirsch length \(9\). The equation describing the elliptic curve is cleverly embedded in the commutator pattern. It is possible that this method can be extended to encode an arbitrary variety. Although it is well-known that nilpotent groups of class two can be quite wild taken as a whole, the new phenomena occurring within an individual group, brought to light by this paper, are striking. finitely generated nilpotent groups; zeta functions; elliptic curves; numbers of subgroups; subgroups of finite index M. du Sautoy, ''A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups,'' Israel J. Math., vol. 126, pp. 269-288, 2001. Nilpotent groups, Other Dirichlet series and zeta functions, Elliptic curves, Subgroup theorems; subgroup growth A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K_ n={\mathbb{Q}}(\alpha)\) be the totally real cubic number field determined by \(\alpha\), a root of \(\phi_ n(x)=x^ 3-(n+3)x^ 2+(n+4)x-1\), for \(n\geq 3\). Let \(M_ n={\mathbb{Z}}[\alpha]\) and let \(U_ n^+\) (respectively \(U^ 2_ n)\) denote the group of totally positive units (respectively squares of units) in \(M_ n\). The pairs \((M_ n,U_ n^+)\) and \((M_ n,U^ 2_ n)\) each determine a normal complex space with one cusp. This paper provides a formula for the defect series for each of these two families of cusps. As an application of the result, bounds are computed for the size of the plurigenera of Hilbert modular varieties over the fields \(K_ 3\) and \(K_ 4\). These bounds yield a specific example proving that unlike the case of Hilbert modular surfaces, a Hilbert modular threefold may have arithmetic genus equal to one and still not be rational. non-rational Hilbert modular threefold; defects of cusp singularities; totally real cubic number field; plurigenera of Hilbert modular varieties; arithmetic genus Modular and Shimura varieties, Global ground fields in algebraic geometry, \(3\)-folds, Singularities in algebraic geometry Defects of cusp singularities and the classification of Hilbert modular threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of Frobenius splitting, introduced by \textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27--40 (1985; Zbl 0601.14043)] and refined by \textit{S. Ramanan} and \textit{A. Ramanathan} [Invent. Math. 79, 217--224 (1985; Zbl 0553.14023)], is a powerful method in the study of flag varieties and representation theory of linear algebraic groups in positive characteristic. One can also recover from it many of the analogous results in characteristic 0 by reduction to positive characteristic. The present monography is the first exposition in book form of this theory and of its major applications. It addresses to mathematicians and graduate students having a basic knowledge of algebraic geometry and representation theory of algebraic groups as provided by standard texts.
The first chapter of the book is devoted to the general study of Frobenius split schemes. They are defined as follows : Let \(X\) be a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p > 0\). Consider the absolute Frobenius morphism \(F = (\text{id}_X, F^{\#}) : X\rightarrow X\), where \(F^{\#} : {\mathcal O}_X\rightarrow F_{\ast}{\mathcal O}_X (={\mathcal O}_X)\) maps \(f\) to \(f^p\). \(X\) is Frobenius split if there exists a morphism of \({\mathcal O}_X\)-modules \(\phi : F_{\ast}{\mathcal O}_X\rightarrow {\mathcal O}_X\) which is a left inverse of \(F^{\#}\). This has remarkable consequences: \(X\) is reduced (because \(F^{\#}\) is injective), weakly normal and, if \(X\) is projective, \(\text{H}^i(X,L) = 0\), \( \forall i > 0\), for every ample line bundle \(L\) on \(X\) (because \(F^{\ast}L\simeq L^{\otimes p}\), hence \(\text{H}^i(X,L)\) injects into \(\text{H}^i(X,L^{\otimes p})\)). Moreover, (variants of) the Kodaira, Grauert-Riemenschneider and Kawamata-Viehweg vanishing theorems are quick consequences of the existence of a Frobenius splitting.
On the other hand, the existence of a Frobenius splitting is a quite restrictive condition: if \(X\) is smooth one can show (using the Cartier operator) that the dual of the locally free \({\mathcal O}_X\)-module \(F_{\ast}{\mathcal O}_X\) (if \(t_1,\dots ,t_n\) is a local system of parameters for \(X\) at a point \(x\), then the monomials \(t_1^{a_1}\dots t_n^{a_n}\) with \(0\leq a_i\leq p-1\) form a local \({\mathcal O}_X\)-basis for \(F_{\ast}{\mathcal O}_X\) at \(x\)) is isomorphic to \(F_{\ast}({\omega}_X^{\otimes 1-p})\) hence, if \(X\) is Frobenius split, \(\text{H}^0(X,{\omega}_X^{\otimes 1-p})\neq 0\). For example, the only smooth projective curves which are Frobenius split are the projective line and the elliptic curves of Hasse invariant 1. However, many varieties where a linear algebraic group acts with a dense orbit turn out to be Frobenius split: the flag varieties \(G/P\) (\(G\) a connected, simply-connected, semisimple algebraic group and \(P\subset G\) a parabolic subgroup) and their Schubert subvarieties (chapter 2), the cotangent bundles of flag varieties (chapter 5) and the equivariant embeddings of connected reductive groups, e.g., toric varieties (chapter 6).
The proofs of these results use a (simple) criterion of Mehta and Ramanathan asserting that, for \(X\) projective and smooth, a section \(\phi \in \text{H}^0(X,{\omega}_X^{\otimes 1-p})\) defines a Frobenius splitting of \(X\) iff, in the local expansion of \(\phi \) at a point \(x\in X\), the ``monomial'' \(t_1^{p-1}\dots t_n^{p-1}(dt_1\wedge \cdots \wedge dt_n)^{1-p}\) occurs with coefficient 1. The verification of this condition in specific situations, is a matter of geometry or representation theory. For example, in the case of Schubert varieties, one firstly proves that their Bott-Samelson-Demazure-Hansen desingularizations are Frobenius split using the concrete description of the canonical bundle of these desingularizations. Also, if \(X = G/P\), then \(\text{H}^0(X,{\omega}_X^{\otimes 1-p})\) is a \(G\)-module.
Once the Frobenius splitting of the above mentioned varieties has been established, many important geometric consequences can be quickly drawn: Schubert varieties have rational singularities and are projectively Cohen-Macaulay and projectively normal in any projective embedding defined by a complete linear system (chapter 3), the full and subregular nilpotent cones have rational singularities (chapter 5), the equivariant embeddings of reductive groups have rational singularities (chapter 6), etc.
The book also contains some remarkable applications of Frobenius splitting in the representation theory of semi-simple groups : the Demazure character formula (chapter 3), a proof of the Parthasarathy-Ranga Rao-Varadarajan-Kostant conjecture on the existence of certain components in the tensor product of two dual Weyl modules and the existence of good filtrations for such tensor products and for the coordinate rings of semisimple groups in positive characteristic (chapter 4), etc.
The final chapter of the book (the 7th) contains a result of a different nature : the punctual Hilbert schemes of a smooth, projective, Frobenius split surface are Frobenius split, too. The authors provide all the necessary preliminary results on symmetric products, punctual Hilbert schemes and the Hilbert-Chow morphism. A key point is the fact that, for a smooth surface \(X\), the Hilbert-Chow morphism \(\text{Hilb}^nX\rightarrow \text{Sym}^nX\) is crepant.
Each section of the book is complemented with exercises; each chapter ends with useful comments, and open problems are suggested throughout. The book leads clearly and rapidly an interested reader from basic results to the research level. flag varieties; Schubert varieties; representations of algebraic groups; Hilbert schemes of points Brion, M.; Kumar, S., Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, vol. 231, (2005), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields, Representation theory for linear algebraic groups, Parametrization (Chow and Hilbert schemes), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Frobenius splitting methods in geometry and representation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{M. Orzech} proposed [in Lect. Notes Math. 917, 66-90 (1982; Zbl 0492.13003)] a generalization of the Brauer group construction for suitable categories of divisorial lattices over Krull domains, which includes as particular cases the reflexive Brauer group of the domain as well as the ordinary Brauer group. On the other hand, \textit{R. Parimala} and \textit{V. Srinivas} defined [in Duke Math. J. 66, 207-237 (1992; Zbl 0780.13002)] a Brauer group of schemes based on sheaves of Azumaya algebras with involution. The purpose of the present paper is to generalize the Parimala-Srinivas construction following Orzech's method. Along the way, the authors also define involutive versions of the Picard and class groups, and give exact sequences connecting these groups and relating them to the 2-torsion part of the (ordinary) Brauer group. In the last section, a geometric interpretation is given for affine normal domains over algebraically closed fields of characteristic zero in terms of étale cohomology groups of the spectrum. categories of divisorial lattices over Krull domains; reflexive Brauer groups; ordinary Brauer groups; Brauer group of schemes; sheaves of Azumaya algebras with involution; class groups; exact sequences; affine normal domains; étale cohomology groups Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Brauer groups of schemes, Dedekind, Prüfer, Krull and Mori rings and their generalizations, Associative rings of functions, subdirect products, sheaves of rings, Rings with involution; Lie, Jordan and other nonassociative structures Involutive Brauer groups of a Krull domain | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Completely reducible representations \(R,R'\) of Lie groups \(G,G'\) on a Hilbert space are said to be dual if their actions commute and the spectral decomposition of each uniquely determines that of the other. Duality is of common occurrence, Schur-Weyl duality between the representations of the symmetric and general linear group being the classical example. The author proves several straightforward but useful results about Casimir operators and about dual pairs \(H,H'\) with \(H \subset G\), \(H' \supset G'\). The theory is illustrated in detail in the case of pairs \((G,G') = (U(N), U(p,q))\) and \((H,H') = (U(N - 1), U(p,q) \ltimes_\tau H_n)\). dual representations; invariant polynomials; Lie groups; Hilbert space; Schur-Weyl duality; general linear group; Casimir operators Representations of Lie and linear algebraic groups over real fields: analytic methods, Vector and tensor algebra, theory of invariants, Classical groups (algebro-geometric aspects) Dual representations and invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The primitive cohomology of nonsingular cubic fourfold \(Y \subset {\mathbb P}^{5}\) is located in the middle dimension (four) and has as nonzero Hodge numbers \(h^{3,1} = h^{1,3} = 1\) and \(h^{2,2}_{0} = 20.\) The question that remained was the image of this period map. The main goal of this paper is to prove Hassett's conjecture. But proof yield more, such as a new proof of Voisin's injectivity theorem. The author shows that Vinberg's Dynkin diagram of an arithmetic reflection group of hyperbolic type of rank 20 gives an insightful picture of the boundary strata and their incidence relations.
Theorem 4.1. The period map for cubic fourfolds with at most simple singularities, \(P : \dot{M} \to X,\) is an open embedding with image \(\dot{X}.\) It identifies the automorphic line bundle restricted to \(\dot{X}\) with the line bundle \(O_{M^{o}}(2)\) so that we obtain an isomorphism of \({\mathbb C}\)-algebras
\[
\bigoplus_{k}H^{0}(\dot{{\mathbb D}}, A(k))^{\Gamma} \rightarrow {\mathbb C}[Sym^{3}V^{*}]^{SL(V)}
\]
which multiplies the degree by 2. The passage to \(Proj\) makes the above embedding extend to an isomorphism of the GIT completion of \(\dot{M}\) onto the Baily-Borel type compactification \(\dot{X}^{bb}\) of \(\dot{X}.\) primitive cohomology of nonsingular cubic fourfold; Hodge numbers; Vinberg's Dynkin diagram; period map for cubic fourfolds; Baily-Borel type compactification E. Looijenga, ''The period map for cubic fourfolds,'' Invent. Math., vol. 177, pp. 213-233, 2009. Period matrices, variation of Hodge structure; degenerations, \(4\)-folds, Automorphic functions in symmetric domains The period map for cubic fourfolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A=k\langle X_1,\dots, X_p\rangle\) be the free associative algebra over an algebraically closed field \(k\). For a fixed \(d\), the variety of \(d\)-dimensional \(A\)-modules \(V(p,d)\) is the set of \(p\)-tuples \((m_1,\dots,m_p)\) of \(d\times d\) matrices with entries from \(k\), and the orbits under simultaneous conjugation of \(\text{GL}_d\) correspond to the isomorphism classes. Although for arbitrary \(d\) the classification of these orbits is a ``wild'' problem, \textit{S. Friedland} [Adv. Math. 50, 189-265 (1983; Zbl 0532.15009)] constructed a disjoint decomposition of \(V(p,d)\) into finitely many locally closed \(\text{GL}_d\)-stable subvarieties and for each such variety found a finite number of \(\text{GL}_d\)-invariant regular functions whose values determine the orbit up to a finite number of choices.
The purpose of the paper under review is to modify the Friedland construction in order to obtain on each stratum a finite number of \(\text{GL}_d\)-invariant regular functions which separate the orbits. As a consequence of the method the author obtains a constructive proof of a result of \textit{M. Rosenlicht} [Am. J. Math. 78, 401-443 (1956; Zbl 0073.376)], where the functions determine the orbits under the action of an arbitrary algebraic group. isomorphism classes of modules; free associative algebras; varieties of \(n\)-dimensional modules; \(\text{GL}_ d\)-stable subvarieties; regular functions; actions; algebraic groups Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Group actions on varieties or schemes (quotients), Automorphisms and endomorphisms, Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras), Representation type (finite, tame, wild, etc.) of associative algebras A remark on Friedland's stratification of varieties of modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Quantum groups are non-cocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra while quantum cohomology is a commutative deformation of the algebra structure on the cohomology of an algebraic variety that takes into account the enumerative geometry of rational curves in the variety. That is, quantum cohomology is a commutative associative deformation of ordinary multiplication in equivariant cohomology \(H^\bullet_{\mathsf{G}}(X)\) defined by \((\gamma_1 * \gamma_2, \gamma_3) = \sum_{\beta>0} q^\beta \langle \gamma_1,\gamma_2,\gamma_3 \rangle_\beta\), where \((\gamma_1,\gamma_2)=\int_X \gamma_1 \cup \gamma_2\) is the standard bilinear form on \(H^\bullet_{\mathsf{G}}(X)\), \(\beta\) ranges over the cone of effective classes in \(H_2(X,\mathbb{Z})\), \(q^\beta\) denotes the corresponding element of the semigroup algebra of the effective cone, and \(\langle \gamma_1,\gamma_2,\gamma_3 \rangle_\beta \in H^\bullet_{\mathsf{G}}(\mathsf{pt},\mathbb{Q})\) is the virtual count of rational curves of degree \(\beta\) meeting cycles Poincaré dual to \(\gamma_1\), \(\gamma_2\), and \(\gamma_3\).
The authors study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver \(Q\), connecting equivariant quantum cohomology of symplectic resolutions with their quantizations and derived autoequivalences. Using a geometric \(R\)-matrix formalism (Section 1.2.1, page 3), which is related to the reflection operator in Liouville field theory, they construct the Yangian \(\mathsf{Y}_Q\) of \(Q\) acting on the cohomology \(H(\mathsf{w}) = \bigoplus_{\mathsf{v}} H^\bullet_{\mathsf{G}}\left(\mathcal{M}_{\theta,\zeta}(\mathsf{v},\mathsf{w})\right)\) of Nakajima quiver varieties. When \(Q\) does not have a loop, this construction is related to \textit{M. Varagnolo}'s [Lett. Math. Phys. 53, No. 4, 273--283 (2000; Zbl 0972.17010)] and \textit{H. Nakajima}'s [J. Am. Math. Soc. 14, No. 1, 145--238 (2001; Zbl 0981.17016)], who construct a certain subalgebra of \(\mathsf{Y}_Q\) via generators and relations.
The authors prove a formula for quantum multiplication by divisors in terms of the Yangian action, where the structure constants of quantum multiplication are formal power series in \(q^\beta\). The quantum connection through the commuting difference connection can be identified with the trigonometric Casimir connection for \(\mathsf{Y}_Q\), which is a generalization of the rational Casimir connection. Equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of \(\mathsf{Y}_Q\). A key role is played by geometric shift operators (Section 8.1.7, page 115) which can be identified with the quantum KZ difference connection.
In the second part of the book under review, the authors give an extended example of the general theory for moduli spaces of sheaves on \(\mathbb{C}^2\) framed at infinity. In Chapter 12 (page 141), the general framework of the quiver has been specialized to the one with one vertex and one loop. The Uhlenbeck compactification, which is the compactification of the moduli of framed instantons, as well as polarizations of Nakajima varieties to construct a Quot-scheme, Baranovsky operators, Fock spaces, and Virasoro algebras have been defined. The Yangian action is analyzed explicitly in terms of a free field realization, and they show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary, an action of the \(W\)-algebra \(\mathcal{W}\big(\mathfrak{gl}(r)\big)\) on the equivariant cohomology of rank \(r\) moduli spaces is obtained. quantum group; quantum cohomology; Nakajima quiver varieties; Yangians; quantum connections; Hilbert schemes; Baxter algebras; Fock bosons; instanton moduli; Baranovsky operators; Virasoro algebra; Gamma functions Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum groups and quantum cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of Kac-Moody Lie algebras has undergone a tremendous development since they were introduced more than 40 years ago. These algebras generalize the finite-dimensional semisimple Lie algebras, and they have close and important connections to several areas in mathematics, and also in mathematical physics. Today the topic has become a standard tool in mathematics. The purpose of the present book is to present the algebro-geometric and topological aspects of Kac-Moody theory in characteristic \(0\).
A lot of different topics are treated in this monumental work. Many of the topics are brought together for the first time in book form. The emphasis is on the study of Kac-Moody groups and their flag varieties. Among the main topics are, the Weyl-Kac character formula, a detailed construction of Kac-Moody groups and their flag varieties; the Demazure character formula, a study of the geometry of Schubert varieties, including normality and Cohen-Macaulayness; the Borel-Weil-Bott theorem; the Bernstein-Gelfand-Gelfand resolution; the Kempf resolution; conjugacy theorems for the Cartan subalgebras and invariance of the generalized Cartan matrix; determination of the defining ideal of the flag varieties via the Plücker relations; a study of the \(T\)-equivariant and singular cohomology of the flag varieties via the nil-Hecke ring, including positivity results for the cup product, and various criteria for the smoothness and rational smoothness of points on Schubert varieties. A particular chapter is devoted to the realization of affine Kac-Moody algebras and the corresponding groups, as well as their flag varieties.
There are several sections introducing the necessary background material, like ind-varieties; pro-groups, pro-Lie algebras; Tits systems; the nil-Hecke ring; Coxeter groups; the Bernstein-Gelfand-Gelfand \(\mathcal O\) category; Lie algebra homology; as well as appendices on results from algebraic geometry; local cohomology; results from topology; relative homological algebra; introduction to spectral sequences.
Although the book is devoted to the general Kac-Moody theory, many of the topics of the book will be useful for those only interested in the finite-dimensional case.
The book is self contained, but is on the level of advanced graduate students. Most statements, definitions and proofs are short, and there are few examples. Moreover many results are given as exercises, some of these are on the difficult side. For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. Chapter by chapter the contents is:
I. Kac-Moody algebras; Basic theory
This section contains the definition of Kac-Moody algebras, root space decompositions and the Weyl groups associated to Kac-Moody algebras. Moreover the dominant chamber and the Tits cone are introduced, and the invariant bilinear form and the Casimir operator are treated.
II. Representation theory of Kac-Moody algebras
Here the Bernstein-Gelfand-Gelfand \(\mathcal O\) category is introduced, the Weyl-Kac character formula is proved, and the important Shapovalov bilinear form is given.
III. Lie algebra homology and cohomology
Results of Konstant-Garland-Lepowsky are proved and the Laplacian is introduced and used in calculations.
IV. An introduction to ind-varieties and pro-groups
Ind-varieties, ind-groups and their Lie algebras are defined and the smoothness of ind-varieties discussed. Pro-groups and pro-Lie algebras are introduced.
V. Tits systems; Basic theory
The basic theory of Tits systems and refined Tits systems is presented.
VI. Kac-Moody groups; Basic theory
The definitions of Kac-Moody groups and the related parabolic subgroups are given. Moreover, the representations of Kac-Moody groups are treated.
VII. Generalized flag varieties of Kac-Moody groups
The ind-variety structure of generalized flag varieties is given, and line bundles on these varieties are studied. Moreover the group \(\mathcal G^{\text{min}}\), defined by Kac-Peterson, and the group \(\mathcal U^-\) are studied.
VIII. Demazure and Weyl-Kac character formulas
The normality of Schubert varieties is proved and the Demazure Character formula proved. Moreover extensions of the Weyl-Kac character formula and the Borel-Weil-Bott theorem are given.
IX. Bernstein-Gelfand-Gelfand and Kempf resolutions
The Bernstein-Gelfand-Gelfand resolution and the dual Kempf resolution are obtained in the general Kac-Moody situation.
X. Defining equations of \(\mathcal G/\mathcal P\) and conjugacy theorems
The quadratic relations defining \(\mathcal G/\mathcal P\) in the canonical projective embedding are given, and the conjugacy theorems for Lie algebras, and groups, are proved.
XI. Topology of Kac-Moody groups and their flag varieties
The nil-Hecke ring is introduced, and the \(T\)-equivariant cohomology of \(\mathcal G/\mathcal B\) is determined. Some positivity results of cup products are given, and the degeneracy of the Leray-Serre spectral sequence for the fibration \(\mathcal G^{\text{min}} \to \mathcal G^{\text{min}}/T\) is proved.
XII. Smoothness and rational smoothness of Schubert varieties
The singular locus and rational smoothness of Schubert varieties are studied.
XIII. An introduction to affine Kac-Moody Lie algebras and groups
The affine Kac-Moody Lie algebras and groups are introduced and studied. Kac-Moody Lie algebras; Kac-Moody groups; representation theory; flag varieties; semisimple simply-connected algebraic groups; parabolic subgroups; Weyl-Kac character formula; \(\mathfrak n\)-homology; ind-varieties; pro-groups; pro-Lie algebras; Tits systems; Demazure character formula; Schubert varieties; Cohen-Macaulay varieties; Borel-Weil-Bott theorem; Bernstein-Gelfand-Gelfand resolution; Kempf resolution; Cartan subalgebras; generalized Cartan matrix; Plücker relations; nil-Hecke rings; cup products; Leray-Serre spectral sequence; smoothness of Schubert varieties; affine Kac-Moody algebras Kumar, S., Kac-Moody groups, their flag varieties and representation theory, 204, (2002), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc., Boston, MA Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Loop groups and related constructions, group-theoretic treatment, Research exposition (monographs, survey articles) pertaining to topological groups, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Grassmannians, Schubert varieties, flag manifolds, Rational and unirational varieties Kac-Moody groups, their flag varieties and representation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0742.00065. This paper is a continuation of the second named author's paper in Math. Z. 191, 489-506 (1986; Zbl 0589.17012)].
The authors compute the generalized Cartan matrix (GCM) associated to the simple-elliptic surface singularities \(\tilde E_ 6\), \(\tilde E_ 7\), and \(\tilde E_ g\) (see the paper of the second author cited above for the definition of the associated GCM), and show that in the case of \(\tilde E_ 7\) and \(\tilde E_ 8\) singularities the associated GCM is not a topological invariant. They also show that in the case of \(\tilde E_ 6\)-singularities, the stable derivation algebra [in the sense of \textit{E. Schenkman}; see his paper in Am. J. Math. 73, 453-474 (1951; Zbl 0054.018)] of the solvable Lie algebra \(L(V)\) is not a topological invariant, even though the nilradical of \(L(V)\) is indeed a topological invariant in this case. Recall that \(L(V)\) is, by definition, the Lie algebra of all the derivations of the moduli algebra \(A(V)\) associated to the singularity \(V\). Lie algebra of derivations of the moduli algebra; generalized Cartan matrix; GCM; simple-elliptic surface singularities Craig Seeley and Stephen S.-T. Yau, Algebraic methods in the study of simple-elliptic singularities, Algebraic geometry (Chicago, IL, 1989) Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 216 -- 237. Singularities of surfaces or higher-dimensional varieties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Singularities in algebraic geometry Algebraic methods in the study of simple-elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of the paper is to obtain a complete list of weighted homogeneous Saito free divisors in the three-dimensional affine space over \(\mathbb C\). The approach is mainly based on properties of Lie algebras of vector fields tangent to reduced hypersurfaces at their non-singular points. A complete list of such Lie algebras consists of 17 weighted homogeneous polynomials. This set is naturally subdivided in three subsets containing 2, 7 and 8 elements related with discriminants of irreducible real reflection groups of type \(A_3,\) \(B_3\) and \(H_3,\) respectively. Then, some interesting relationships between 17 polynomials and root systems of types \(E_{6}, E_{7}\) and \(E_{8}\) as well as few examples in higher dimensional cases are briefly discussed.
In fact, the computational part of the work is a strong simplification of earlier results from [in: Algebraic analysis and number theory, RIMS Kôkyûroku 810, 85--94 (1992; Zbl 0966.17500)]. logarithmic vector fields; discriminants; Saito free singularities; Coxeter groups; Lie algebras; deformations J. Sekiguchi, \textit{A classification of weighted homogeneous Saito free divisors}, J. Math. Soc. Japan 61 (2009), 1071--1095. Complex surface and hypersurface singularities, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities of holomorphic vector fields and foliations A classification of weighted homogeneous Saito free divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Die einzige zwischen den Erzeugenden des graduierten Rings der Hilbertschen Modulformen zu \({\mathbb{Q}}(\sqrt{5})\) bestehende Relation, eine quadratische Relation für eine symmetrische Spitzenform vom Gewicht 15 über dem Ring der symmetrischen Hilbertschen Modulformen geraden Gewichts zu \({\mathbb{Q}}(\sqrt{5})\), wird, anders als bei \textit{H. L. Resnikoff} [Math. Ann. 208, 161-170 (1974; Zbl 0278.10029)] oder \textit{F. Hirzebruch} [Russ. Math. Surv. 31, No.5, 96-110 (1976); translation from Usp. Mat. Nauk 31, No.5(191), 153-166 (1976; Zbl 0335.14007); Lect. Notes Math. 627, 287-323 (1977; Zbl 0369.10017)], ohne Anleihen aus der Invariantentheorie oder algebraischen Geometrie auf völlig elementarem Wege bestimmt.
Weiter wird die Rationalität des Körpers der Hilbertschen Modulfunktionen zu \({\mathbb{Q}}(\sqrt{5})\), die bisher nur mit Mitteln der algebraischen Geometrie gezeigt werden konnte [\textit{E. Freitag}, J. Reine Angew. Math. 254, 1-16 (1972; Zbl 0241.10017), \textit{F. Hirzebruch}, Enseign. Math., II. Sér. 19, 183-281 (1973; Zbl 0285.14007)], hier elementar und konstruktiv, mit expliziter Angabe einer Transzendenzbasis, bewiesen. Die Erzeugenden sowohl des Rings der Hilbertschen Modulformen als auch des Körpers der Hilbertschen Modulfunktionen zu \({\mathbb{Q}}(\sqrt{5})\) werden dabei sämtlich explizit durch Thetanullwerte dargestellt. Vorgehensweise und Überlegung sind ähnlich wie im Fall \({\mathbb{Q}}(\sqrt{8})\) [Math. Ann. 266, 83-103 (1983; Zbl 0507.10022)]. elementary determination of generator relations; graded ring of Hilbert modular forms; rationality of field of Hilbert modular functions; explicit transcendence basis Müller, R, Hilbertsche modulformen und modulfunktionen zu \(\mathbb{Q}(\sqrt{5})\), Arch. Math., 45, 239-251, (1985) Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special surfaces Hilbertsche Modulformen und Modulfunktionen zu \({\mathbb{Q}}(\sqrt{5})\) (Hilbert modular forms and modular functions for \({\mathbb{Q}}(\sqrt{5})\)) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities H. Grundman führt ihre Untersuchungen über die Frage, welche Quotientenräume Hilbertscher Modulgruppen total-reeller Zahlkörper vom Grad \(n\), auf einem Produkt von \(n\) oberen Halbebenen operierend, das arithmetische Geschlecht 1 haben (nur dann kann der Funktionenkörper rational sein) zusammen mit der zweiten Autorin fort. Die Fälle \(n=1,2\) sind inzwischen klassisch. Den Fall \(n=3\) haben die beiden Autorinnen kürzlich endgültig gelöst [J. Number Theory 95, No. 1, 72--76 (2002; Zbl 1022.11022)], es gibt genau 33 Fälle mit dem arithmetischen Geschlecht 1. Für \(n=4\) hatte \textit{H. Grundman} [J. Number Theory 63, No. 1, 47--58 (1997; Zbl 0899.11021)] gezeigt, daß das arithmetische Geschlecht 1 für galoissche Zahlkörper vom Grad 4 mit einer Einheit der Norm \(-1\) nicht auftritt. In der vorliegenden Arbeit wird die Untersuchung auf alle total-reellen Zahlkörper vom Grad 4 ausgedehnt. Das arithmetische Geschlecht 1 kommt bei genau 3 Zahlkörpern vor. quotient spaces of Hilbert modular groups of totally real number fields H. G. Grundman and L. E. Lippincott, Hilbert modular fourfolds of arithmetic genus one, High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, Amer. Math. Soc., Providence, RI, 2004, pp. 217 -- 226. Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Rationality questions in algebraic geometry, Families, moduli, classification: algebraic theory, \(4\)-folds Hilbert modular fourfolds of arithmetic genus one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a complex, semisimple, simply connected algebraic group with Lie algebra \({\mathfrak g}\). We extend scalars to the power series field in one variable \({\mathbf C}((\pi))\), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of \({\mathfrak g}\otimes{\mathbf C}((\pi))\), i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig's topological construction of Springer's representations to the affine context. affine Weyl groups; Springer representations of affine groups; fixed point varieties on affine flag manifolds; complex semisimple simply connected algebraic groups; Lie algebras; nil-elliptic elements; representations D. S. Sage, A construction of representations of affine Weyl groups , Compositio Math. 108 (1997), 241--245. Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties A construction of representations of affine Weyl groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Le premier but de cet article est de construire des fonctions graphes modulaires sur les surfaces de Riemann compactes de genre arbitraire. Ces constructions sont données dans la section 2.
Le second but de cet article est de déterminer le comportement des fonctions graphes modulaires au voisinage du diviseur de la compactification de Deligne-Mumford paramètrant les courbes avec un nœud non séparant. En particulier, le théorème 1 donne le comportement de la fonction d'Arakelov-Green et le théorème 2 de l'invariant de Kawazumi-Zhang. Deux autres familles de fonctions graphes modulaires sont traitées dans les théorèmes 3 et 4. moduli space of curves; modular graph functions; Arakelov-Green function; Kawazumi-Zhang invariant Relationships between algebraic curves and physics, Planar graphs; geometric and topological aspects of graph theory, Teichmüller theory for Riemann surfaces, Arithmetic varieties and schemes; Arakelov theory; heights Higher genus modular graph functions, string invariants, and their exact asymptotics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field and \(\mathcal A\) be a \(k\)-category. For a \(k\)-linear endofunctor \(F\) of \(\mathcal A\) and a fixed object \(A\) of \(\mathcal A\), one considers a positively \(\mathbb{Z}\)-graded \(k\)-algebra \({\mathcal O}(F;A)=\bigoplus^\infty_{n=0}\text{Hom}_{\mathcal A}(A,F^nA)\), where the product is defined so that for \(u\colon A\to F^nA\) and \(v\colon A\to F^mA\), \(vu\) is the composite of \(u\colon A\to F^nA\) and \(F^nv\colon F^nA\to F^{n+m}A\). Call \({\mathcal O}(F;A)\) the orbit algebra of \(F\) at \(A\).
In this paper, the author mainly considers the following two cases separately. (1) Let \(\Sigma\) be a tame hereditary algebra over the field \(\mathbb{C}\) of complex numbers, and let \(\text{mod}(\Sigma)\) be the category of finite-dimensional right \(\Sigma\)-modules. Note that the Auslander-Reiten translation \(\text{Tr }D\) is an endofunctor of \(\text{mod}(\Sigma)\). It is then shown that for an indecomposable projective \(\Sigma\)-module \(P\) of defect \(-1\), the orbit algebra \({\mathcal O}(\text{Tr }D;P)\) of \(\text{Tr }D\) at \(P\), as a positively \(\mathbb{Z}\)-graded algebra, is isomorphic to the algebra of invariants of the action of a binary polyhedral group on the polynomial algebra \(\mathbb{C}[X,Y]\). (2) Let \(\Lambda\) be a wild canonical algebra over the field \(\mathbb{C}\) of complex numbers. Then the ordinary quiver of \(\Lambda\) has a unique source \(\vec o\) and a unique sink \(\vec c\). Let \(\text{mod}(\Lambda)\) be the category of finite-dimensional right \(\Lambda\)-modules. For a module \(M\in\text{mod}(\Lambda)\), let \((n_{\vec 0},\dots,n_{\vec c})\) be the dimension vector of \(M\). Then the rank of \(M\) is defined to be \(n_{\vec 0}-n_{\vec c}\). Consider the full subcategory \(\text{mod}_+(\Lambda)\) of \(\text{mod}(\Lambda)\) generated by the indecomposable modules of positive rank. Note that \(\text{mod}_+(\Lambda)\) is closed under extensions and the Auslander-Reiten translation \(D\text{Tr}\). Further, \(D\text{Tr}\) induces an endofunctor of \(\text{mod}_+(\Lambda)\). The author shows that for a nonpreprojective \(\Lambda\)-module \(M\) of rank one, the orbit algebra \({\mathcal O}(D\text{Tr};M)\) of \(D\text{Tr}\) at \(M\), as a positively \(\mathbb{Z}\)-graded algebra, is isomorphic to the algebra of entire automorphic forms with regard to an action of a Fuchsian group of the first kind on the upper complex half plane. By the above analysis, the author shows that the modern representation theory of finite-dimensional algebras has strong roots in classical mathematics. graded algebras; tame hereditary algebras; categories of finite-dimensional right modules; Auslander-Reiten translations; endofunctors; indecomposable projective modules; orbit algebras; algebras of invariants; wild canonical algebras; quivers; indecomposable modules; automorphic forms Lenzing, H.: \textit{Wild canonical algebras and rings of automorphic forms. In Finite-dimensional algebras and related topics (Ottawa, ON, 1992)}, volume 424 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 191-212. Kluwer Acad. Publ., Dordrecht, 1994 Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Representation-theoretic methods; automorphic representations over local and global fields, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Group actions on varieties or schemes (quotients), Automorphic forms, one variable Wild canonical algebras and rings of automorphic forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0562.00001.]
The theory of actions of formal groups on formal schemes is developed. Important applications are given. Since the theory of formal groups (over an arbitrary field k) is well established but the same techniques apply also to the theory of actions on formal schemes, the first paragraphs present a survey without proofs of the theory of formal groups. They are defined as functors G as well as by their contravariant (representating) algebras A and covariant Hopf algebras H. Also the necessary concepts of linearly compact spaces are discussed. Of prime interest is the relation between formal groups and their associated Lie algebras, which generates a category equivalence in characteristic zero, if one restricts oneself to infinitesimal formal groups. An additional restriction, the author calls it positively filtered Lie algebra, is important for the applications discussed below.
The Lie algebra of the formal scheme represented by the algebra of formal power series \(k[[x_ i| i\in I]]\), is generated by the partial derivatives \(\frac{\partial}{\partial x_ i}|_ 0\), so an intimate connection with analytical questions arises. The relation between variables and partial differential operators is studied under the concept of parameter systems. Effective tools for their calculation are developed. They are used to give a recursive algorithm to find the (unique) solution of the system of differential equations:
\[
\Phi_ j(0)=0;\quad \frac{d}{dt}\Phi_ j=\sum_{i}u_ i(t)(\frac{\partial}{\partial y_ i}|_ 0\circ y_ j)(\Phi)
\]
with \(\Phi_ j\in k[[t]]\), \(u_ i\in k[[t]]\) coordinate components of a curve in a positively filtered Lie algebra and \(y_ i\) the parameter system of the associated infinitesimal formal group A. - This is then applied to give a solution of the differential system
\[
\Psi (x,0)=x,\quad \frac{d}{dt}\Psi (x,t)=\sum u_ i(t)A_ i(\Psi (x,t))
\]
from control theory as discussed by \textit{M. Fliess} [Bull. Soc. Math. Fr. 109, 3-40 (1981; Zbl 0476.93021) and Invent. Math. 71, 521-537 (1983; Zbl 0513.93014)].
The concepts of controllability, observability and realization of control theory are also expressed and studied in the context of formal groups. In particular the existence of a minimal realization is shown by the fact that orbits exist in formal geometry. A different area of applications treated here is umbral calculus in combinatorics (of Rota and Roman), the interplay between polynomials and power series to obtain combinatorial identities. It is derived from and generalized to actions of the additive group \(G_ a\) and the multiplicative group \(G_ m\) on the affine line \({\mathbb{A}}\). So this technique opens a possibility to obtain also a generalization of the multivariate umbral calculus of Joni. actions of formal groups on formal schemes; Hopf algebras; parameter systems; controllability; observability; realization of control theory; multivariate umbral calculus U. Oberst,Actions of formal groups on formal schemes. Applications to control theory and combinatorics, inSeminaire d'Algebre (P. Dubreil and M.-P. Malliavin, eds.), Lecture Notes in Mathematics, No. 1146, Springer, Berlin, Heidelberg, New York, 1985. Formal groups, \(p\)-divisible groups, Group actions on varieties or schemes (quotients), Controllability, observability, and system structure, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Solvable, nilpotent (super)algebras Actions of formal groups on formal schemes. Applications to control theory and combinatorics. (With the cooperation of Arne Dür) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The note is devoted to the study of relations between algebraic and geometric descriptions of degeneration for finite dimensional modules over an associative algebra \(\Lambda\). At first the basic notions, definitions and examples of degeneration, partial orders, cancellation, etc. are discussed. Then the author explains main ideas of two recent results [\textit{K. Bongartz}, Adv. Math. 121, No. 2, 245-287 (1996; Zbl 0862.16007)] and [\textit{G. Zwara}, Proc. Am. Math. Soc. 127, No. 5, 1313-1322 (1999; see the following review Zbl 0927.16008)] where the partial orders \(\leq\) and \(\leq_{\text{deg}}\) are compared when \(\Lambda\) is representation directed or \(\Lambda\) is of finite representation type, respectively. finite dimensional modules; Artinian rings; associative algebras; degenerations of modules; partial orders; cancellation; preprojective modules; nilpotent matrices; Auslander-Reiten quivers; indecomposable modules; representation directed algebras Riedtmann, Ch.: Geometry of modules: degenerations. Contemp. math. 229, 281-291 (1998) Representations of associative Artinian rings, Finite rings and finite-dimensional associative algebras, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Geometry of modules: Degenerations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a finite-dimensional algebra over an algebraically closed field, a fundamental representation theory question is to determine whether the algebra has finite, tame or wild representation type. This work continues an extensive number of works by the author and others aimed at answering this question for distribution algebras of infinitesimal group schemes.
Let \(\mathcal G\) be a smooth algebraic group over an algebraically closed field of characteristic \(p\geq 3\), and let \({\mathcal G}_r\) denote the \(r\)-th Frobenius kernel of \(\mathcal G\). Of interest here is the representation theory of \({\mathcal G}_r\) or equivalently its distribution algebra \(\text{Dist}({\mathcal G}_r)\).
The main result is a determination up to Morita equivalence of the structure of those blocks of \(\text{Dist}({\mathcal G}_r)\) which have either finite or tame representation type. This determination was done previously by the author [Adv. Math. 155, No. 1, 49-83 (2000; Zbl 0970.16012)] in the case that \(\mathcal G\) is reductive. The proof is accomplished in part by investigating the nature of unipotent normal subgroups \(\mathcal U\) of \(\mathcal G\) and the quotient group \(\mathcal{G/U}\), and attempting to reduce the problem to known cases. A key tool in this is the use of the complexity of a module along with its connections to support varieties and properties of complexity under restriction of a module to a subgroup.
In addition, the author uses the main result to determine the possible components of the stable Auslander-Reiten quiver of \(\text{Dist}({\mathcal G}_r)\). smooth algebraic groups; blocks; representation types; Auslander-Reiten quivers; Frobenius kernels; distribution algebras; complexity; infinitesimal group schemes Farnsteiner, R.: Block representation type of Frobenius kernels of smooth groups. J. Reine Angew. Math. 586, 45--69 (2005) Representation theory for linear algebraic groups, Representation type (finite, tame, wild, etc.) of associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Group schemes Block representation type of Frobenius kernels of smooth groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Irreducible holomorphic symplectic manifolds are simply connected compact complex manifolds of Kähler type whose space of global holomorphic \(2\)-forms is generated by a closed non-degenerate \(2\)-form. In complex dimension two, the irreducible holomorphic symplectic manifolds are precisely the \(K3\)-surfaces. Higher dimensional examples are rare. To date, up to deformation equivalence, the only known examples are Hilbert schemes of zero-dimensional subschemes of length \(n\) on a \(K3\)-surface, generalised Kummer varieties and two examples of dimension \(6\) and \(10\) constructed by O'Grady.
In the paper under review the authors study moduli spaces of polarised irreducible holomorphic symplectic manifolds. For their main result they consider irreducible holomorphic symplectic manifolds which are deformation equivalent to Hilb\(^2(S)\) for a \(K3\) surface \(S\), equipped with a primitive polarisation of split type and of Beauville-degree \(2d\geq 24\). They prove that each irreducible component of the 20-dimensional moduli space of such polarised manifolds is of general type.
To prove this, they first show (for general polarised irreducible holomorphic symplectic manifolds) that each component of the moduli space is related by a finite dominant period map to a quotient of a homogeneous domain by an arithmetic group. The proof of the main result is similar to the proof the authors have given for the moduli space of polarised \(K3\) surfaces [Invent.\ Math.\ 169, No.\ 3, 519--567 (2007; Zbl 1128.14027)].
They use that the existence of a low-weight cusp form with suitable vanishing locus implies that the components of the arithmetic quotient are of general type. Such a cusp form is then constructed as a quasi-pull back of the Borcherds form (a modular form of weight 12 for \(O^+(2U\oplus3E_8(-1))\). This construction requires the existence of a vector of length \(2d\) in the lattice \(E_7\) which is orthogonal to at most 14 roots. The proof of the existence of such a vector is the technical heart of this article. In involves estimating the number of ways certain integers can be represented by various quadratic forms of odd rank. Beauville form; Fujiki invariant; irreducible hyperkähler manifold; locally symmetric variety of orthogonal type; period domain; Torelli theorem; modular form; cusp form; Weyl group; Zagier L-function; Cohen number; Siegel's formula V. Gritsenko, K. Hulek and G.\ K. Sankaran, Moduli spaces of irreducible symplectic manifolds, Compos. Math. 146 (2010), no. 2, 404-434. Moduli, classification: analytic theory; relations with modular forms, Sums of squares and representations by other particular quadratic forms, Other groups and their modular and automorphic forms (several variables), \(4\)-folds, Compact Kähler manifolds: generalizations, classification, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Moduli spaces of irreducible symplectic manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose that \(\Gamma\) is a finitely generated group. Denote by \(r_n(\Gamma)\) the number of irreducible \(n\)-dimensional complex representations of \(\Gamma\), up to equivalence. If the sequence \(r_n(\Gamma)\) is bounded by a polynomial in \(n\), then the representation zeta function \(\zeta_\Gamma(s)=\sum_n r_n(\Gamma)n^{-s}\) converges in a suitable half complex plane. \textit{A. Lubotzky} and \textit{B. Martin} [Isr. J. Math. 144, 293-316 (2004; Zbl 1134.20056)] proved that an arithmetic lattice in characteristic zero has the congruence subgroup property if and only if the sequence \(r_n(\Gamma)\) grows polynomially, or equivalently, if and only if the abscissa of convergence of \(\zeta_\Gamma(s)\) is finite.
The main result in this paper is the following: if an arithmetic lattice \(\Gamma\) in characteristic zero satisfies the congruence subgroup property, then the abscissa of convergence of \(\zeta_\Gamma(s)\) is a rational number. The proof follows a general strategy of Igusa and Denef. \textit{M. Larsen} and \textit{A. Lubotzky} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 351-390 (2008; Zbl 1142.22006)] proved that \(\Gamma\) contains a finite index subgroup \(\Delta\) such that the representation zeta function of \(\Delta\) has an Euler-like factorization. The author proves that the abscissa of convergence of \(\zeta_\Gamma(s)\) is unchanged when passing to a finite-index subgroup and previous results ensure that the abscissa of convergence is rational for every local factor of \(\zeta_\Delta(s)\). Therefore, in order to prove that the abscissa of convergence of the global zeta function is rational, it remains to understand the relation between the local zeta functions for different primes. Indeed, the paper is mainly an attempt to give an approximate formula to the local zeta functions, which is uniform in the prime \(p\). motivic integration; orbit method; arithmetic lattices; irreducible complex representations; algebraic groups; congruence subgroup property; polynomial representation growth; irreducible complex characters; representation zeta functions; subgroups of finite index; local zeta functions N., Avni., Arithmetic groups have rational representation growth, Annals of Mathematics, 174, 1009-1056, (2011) Representation theory for linear algebraic groups, Other Dirichlet series and zeta functions, Discrete subgroups of Lie groups, Arcs and motivic integration, Analysis on \(p\)-adic Lie groups, Subgroup theorems; subgroup growth Arithmetic groups have rational representation growth. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the note under review the authors show that all systems discussed in their previous note [Adv. Math. 38, 267--317 (1980; Zbl 0455.58017)] can be linearized according to a general scheme common to all of them, reminiscent of Mumford and van Moerbeke's treatment of the Toda lattice, and prove the independence of the linearization on the representation. The paper also contains a beautiful appendix in which the authors sketch the main concepts and theorems of the theory of correspondence in a style reminiscent of the classic Italian geometers. Toda type flow; flows of spinning top type; Lie algebra representation; theory of correspondences; linearization of Hamiltonian system; Jacobi varieties; representation theory; algebraic curve; Kac-Moody algebras M. Adler and P. van Moerbeke, ''Linearization of Hamiltonian systems, Jacobi varieties, and representation theory,'' Adv. Math.,38, No. 3, 318--379 (1980). Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Geometric quantization, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Rational and birational maps, Jacobians, Prym varieties Linearization of Hamiltonian systems, Jacobi varieties and representation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies variations of moduli spaces of representations of preprojective \(K\)-algebras by applying tilting theory, to deal with minimal resolutions of Kleinian singularities.
To a finite subgroup \(G\subset SL(2,K)\) a McKay quiver \(Q\) is associated, and a \(K\)-algebra \(\Lambda\) associated to \(Q\), called preprojective algebra. Different resolutions of quotient singularities \(\mathbb{A}^{n}/G\) are encoded by different moduli spaces \(\mathcal{M} _{\theta,d}({\Lambda})\) of \(\theta\)-semistable \(\Lambda\)-modules of dimension vector \(d\), in the sense of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)]. Hence, by studying variations of moduli spaces for different stability parameters \(\theta\) we can study quotient singularities.
In section 2, tilting modules \(I_{w}\) (generalization in homological algebra of being torsion or torsion free for a module) are constructed over preprojective algebras. If we denote by \(\mathcal{S}_{\theta}(\Lambda)\subset Mod \Lambda\) the full subcategory of \(\theta\)-semistable \(\Lambda\)-modules, in section 3 relations between the moduli spaces are given, by showing that the functors \(Hom_{\Lambda}(I_{w},-)\) and \(-\otimes_{\Lambda}I_{w}\) induce equivalences between the categories \(\mathcal{S}_{\theta}(\Lambda)\) and \(\mathcal{S}_{w\theta}(\Lambda)\) (c.f. Theorem 3.13) which preserve \(S\)-equivalence classes, where \(w\) are elements of the Coxeter group. This induces a bijection between the sets of closed points in the moduli spaces, and in section 4 the equivalence can be extended to the respective derived categories to show that the bijection can be extended to an isomorphism of \(K\)-schemes (c.f. Theorem 4.20), by using the functors of points.
Section 5 is devoted to use the previous results in the framework of Kleinian singularities to generalize some results of \textit{W. Crawley-Boevey} (see, e.g., [Am. J. Math. 122, No. 5, 1027--1037 (2000; Zbl 1001.14001)]). In section 6 a full example is provided.
Note that the proofs work even in the case where \(Q\) is a non-Dynkin quiver with no loops. Combined with the homological nature of the proofs, this is why the authors expect to use the results in higher dimensions. moduli spaces; preprojective algebras; tilting theory; McKay correspondence; Kleinian singularities Sekiya, Y.; Yamaura, K., \textit{tilting theoretical approach to moduli spaces over preprojective algebras}, Algebr. Represent. Theory, 16, 1733-1786, (2013) Fine and coarse moduli spaces, Representations of quivers and partially ordered sets, McKay correspondence, Homological functors on modules (Tor, Ext, etc.) in associative algebras, Families, moduli, classification: algebraic theory Tilting theoretical approach to moduli spaces over preprojective algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper consists of an interesting discussion of two classical strategies (due to Campedelli and Godeaux) to construct surfaces of general type with \(p_g=0\) and \(K^2= 1\) or \(K^2=2\). Roughly speaking, Campedelli's construction starts with a curve in \(\mathbb{P}^2\) or in \(\mathbb{P}^1\times\mathbb{P}^1\) with interesting configurations of singularities (which however is not so easy to produce), while Godeaux's method uses quotients by group actions. The first method, inspite of its beautiful geometry behind, has the drawback that it requires rather ingenious and delicate constructions involving many interesting singularities. The typical examples are plane curves of degree 10 with 6 triple tacnodes, or curves in \(\mathbb{P}^1 \times \mathbb{P}^2\) of bidegree (6,7) with 5 pairs of triple singularities on the fibres of the second projection \(\mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1\). The main conclusion of this discussion is that, as soon as one has a good knowledge of a class of surfaces (e.g. the class of surfaces arising from the fairly easy construction in Godeaux style), one may be able to use it to produce interesting configurations of singularities (as required by Campedelli's method). This discussion is divided by the author in four rounds. The first two present the background to the search for Campedelli configurations and the Godeaux constructions that lead to solutions. The idea is to take a known Godeaux or Campedelli surface \(X\) having an involution \(i\) with a fixed curve. Then the quotient \(S=X/i\) marked with the ramification curve \(C\) is a birational pair \(S\& C\) in the sense of Iitaka, after which the Campedelli configuration is obtained by passing to its minimal model. Round 3 puts together what is needed from Iitaka's theory. Round 4 attacks the question of the existence of \((-1)\)-elliptic curves \(\mathbb{Z}/5\) Godeaux surfaces directly in terms of the \(\mathbb{Z}/5\)-invariant quintic elliptic curves. Finally, questions related to the possible orders of the fundamental groups of the surfaces obtained are also considered. Godeaux surface; constructing surfaces of general type; configurations of singularities; Campedelli surface; fundamental groups Miles Reid, Campedelli versus Godeaux, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 309 -- 365. Surfaces of general type, Singularities of surfaces or higher-dimensional varieties, Families, moduli, classification: algebraic theory Campedelli versus Godeaux | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities These two volumes represent the second edition of the author's well-known and beautiful introductory book on algebraic geometry (1972; Zbl 0258.14001) which has been translated into several other languages (e.g. into English 1974, second edition 1977). This new edition differs from the first one because it was reworked and completed with great care. Here we only want to emphasize the differences between the two editions.
Chapter I begins with some new sections such as: an elementary discussion about plane curves, their singularities and the projective plane. Then some considerations about the zeta function, the theorem of Abhyankar- Moh, the Jacobian conjecture, the Grassmann variety and its Plücker embedding, associative algebras, determinantal varieties and the Tsen theorem and its application to the rationality of surfaces, are added.
Chapter II contains the following new things: more examples of smooth varieties, the varieties associated to associative algebras, Puiseux expansions, singularities of maps, the generic irreducibility or smoothness of morphisms, etc.
In chapter III the author added considerations about pencils of conics, a more detailed discussion about the cubic curves with emphasis to some arithmetical questions, and in chapter IV, the inequality of Riemann-Roch for surfaces, the geometry of the smooth cubic surface in \({\mathbb{P}}^ 3\), the singularities of a curve on a surface and their resolutions, and Du Val singularities of surfaces. The first volume ends with a (new) appendix of algebraic prerequisites.
The second volume contains the following new paragraphs: (1) The classification of the geometric objects, universal schemes, and the Hilbert scheme (in chapter VI; for this reason the paragraph in chapter I about Chow coordinates has been removed); (2) Connectivity of the fibers of a morphism of algebraic varieties; (3) The topology of the singularities of curves (both in chapter VII), and (4) Kähler manifolds and the Hodge theorem (in chapter VIII). A few exercises from the old edition disappeared, but many others have been included.
All in all these changes and completions made this remarkable textbook more updated and even more interesting. This second edition is highly recommended to every mathematician. plane curves; zeta function; Jacobian conjecture; Puiseux expansions; singularities; pencils of conics; universal schemes; Hilbert scheme I. R. Shafarevich, \textit{Basic Algebraic Geometry} (Nauka, Moscow, 1988; Springer, Berlin, 2013), Vol. 1. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Foundations of algebraic geometry, Local theory in algebraic geometry, Cycles and subschemes Basic algebraic geometry. (Osnovy algebraicheskoj geometrii.) Vol. 1: Algebraic manifolds in projective space. (Tom 1: Algebraicheskie mnogoobraziya v proektivnom prostranstve). Vol. 2: Schemes. Complex manifolds. (Tom 2: Skhemy. Kompleksnye mnogoobraziya). 2nd ed., rev. and compl | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The first edition of this meanwhile standard text on elliptic curves was published in 1987 (Zbl 0605.14032). Intended as an all-round introduction to the geometry and arithmetic of elliptic curves, the text divided naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. The first edition came with seventeen chapters that led the reader from the most elementary aspects of cubic equations, with full proofs given, up to the, by then, most advanced topics in the theory of elliptic curves, where the presentation became rather survey-like and sketchy. Nevertheless, Husemöller's text was and is a great first introduction to the world of elliptic curves, in their various aspects and trends, and a good guide to the current research literature as well.
The book under review is the second, up-dated and considerably enlarged edition of this standard text. Increased by about forty percent in volume, mainly by the addition of three new chapters and a few new sections to the old ones, this second edition builds on the original in several ways. Moreover, there are now three appendices which are partly written by co-authors, and which point out further recent developments and applications within the theory of elliptic curves.
The three new chapters are chapters 18, 19, and 20. The author provides here surveys of recent directions and extensions of the theory, thereby focusing on concepts, statements of new results, explanations of their strategy of proof, and relations to the more classic parts of the theory of elliptic curves.
In this vein, chapter 18 is devoted to the recent achievements towards Fermat's last theorem. As the author points out, this chapter is primarily designed to point out which material in the earlier chapters is relevant as background for studying \textit{A. Wiles}'s original work on the subject [Modular elliptic curves and Fermat's last theorem, Ann. Math. (2) 141, 443--551 (1995; Zbl 0823.11029)], together with the further developments due to the work of F. Diamond and R. Taylor in the late 1990s.
Basically, chapter 18 provides an introduction to the various research papers on the so-called modular curve conjecture (or Shimura-Taniyama-Weil conjecture) in survey form, with many comments and hints, but without proofs.
Chapter 19 deals with Calabi-Yau varieties as higher-dimensional analogues of elliptic curves. After a quick compilation of the relevant topics from Hermitian and Kählerian differential geometry, the various characterizations of Calabi-Yau varieties are explained and illustrated by examples. At the end of this survey, the Enriques classification for surfaces and a brief introduction to \(K3\)-surfaces is sketched. As the author points out, this chapter has been added for the sake of utility of the book for people interested in examples of fibrations of three-dimensional Calabi-Yau varieties. In view of the very fact that arithmetical Calabi-Yau varieties have recently become important in physics, especially in conformal field theory, the addition of this chapter may be seen as a service to a wider public, too.
Chapter 20 returns to the earlier material on families of elliptic curves, this time in the context of modern algebraic geometry (i.e., the theory of schemes) and analytic geometry. The author's main goal is here to point out some of the many areas of mathematics (and physics) in which families of elliptic curves play an important role. This includes, apart from a summary of the relevant concepts in algebraic and analytic theory, another survey on surfaces over curves, elliptic \(K3\)-surfaces, fibrations of 3-dimensional Calabi-Yau varieties, and examples of 3-dimensional Calabi-Yau hypersurfaces in weighted projective 4-space and their fiberings. In this regard, the new chapter 20 enhances the earlier material on elliptic fibrations by additional, more topical examples.
As to the new three appendices, the first one has been contributed by \textit{S. Theisen}. Entitled ``Calabi-Yau Manifolds and String Theory'', this short essay gives a physicist's view to Calabi-Yau varieties from a rather philosophical standpoint, thereby enfilading the mathematical discussion from this lookout.
Appendix II was written by \textit{O. Forster} and depicts ``Elliptic Curves in Algorithmic Number Theory and Cryptography''. This survey adds a discussion of the use of elliptic curves in arithmetical computing theory and coding theory to the overall panorama that the author of the book was striving for.
Appendix III, written by \textit{D. Husemöller} himself, provides another panoramic outlook, namely an elementary introduction to the relation between elliptic curves (given by Weierstrass cubics) and the homological (or homotopical) theory of modular forms. The central concept is here the so-called Weierstrass Hopf algebroid, in the categorical sense, and it is illustrated how this framework can be used to compute the homotopy of the so-called spectrum topological modular forms. The development of this topic is still in progress, and its discussion here leads the reader directly to the forefront of current research in the field.
Each appendix comes with its own list of special bibliographical references and according hints to them.
Apart from these enlargements of the book under review, the author has also taken the opportunity to make numerous minor additions to the original text, which has otherwise been left totally intact. Now as before, there are numerous exercises in each of the more elementary chapters, and there is again that (now fourth) appendix by \textit{R. Lawrence} providing solutions for them.
Of course, the bibliography has been up-dated and enlarged, too, and that by about 75 new references.
All together, this second edition of D. Husemöller's standard text on elliptic curves offers a much broader panoramic view to the subject than the first one, without having lost its introductory character with respect to the elementary parts, and it has certainly gained a good deal of topicality, appeal, power of inspiration, and educational value for a wider public. No doubt, this text will maintain its role as both a useful primer and a passionate invitation to the evergreen theory of elliptic curves and their applications. elliptic curves; algebraic curves; group schemes; modular functions; \(L\)-functions; theta functions; Fermat's last theorem; conjecture of Birch and Swinnerton-Dyer; Shimura-Taniyama-Weil conjecture; Calabi-Yau varieties; string theory; cryptography; Hopf algebroids; elliptic cohomology Husemöller, D.: Elliptic Curves. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (2004) Elliptic curves over global fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Elliptic curves, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Rational points, Special algebraic curves and curves of low genus, Global ground fields in algebraic geometry, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Research exposition (monographs, survey articles) pertaining to number theory, Calabi-Yau manifolds (algebro-geometric aspects), Elliptic genera, Research exposition (monographs, survey articles) pertaining to algebraic geometry Elliptic curves. With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a semisimple group over an algebraically closed field of `very good' characteristic \(p\). Let \(\varphi\colon\text{SL}_2\to G\) be a homomorphism and put \(X=d\varphi\left(\begin{smallmatrix} 0&1\\ 0&0\end{smallmatrix}\right)\). Then \(X\) is an unstable vector in the sense of geometric invariant theory. Now \(\varphi\) is called optimal for \(X\) if the cocharacter \({\mathbf G}_m\to G\) which one gets by identifying \({\mathbf G}_m\) with the diagonal subgroup of \(\text{SL}_2\) is optimal in the sense of Kempf and Rousseau.
It is shown that two optimal \(\text{SL}_2\)-homomorphisms are conjugate under the connected centralizer of \(X\). Applying this to the case that \(X\) is a regular nilpotent one deduces that there is a unique conjugacy class of principal homomorphisms \(\text{SL}_2\to G\). It is shown that the image of an optimal homomorphism is a completely reducible subgroup in the sense of Serre. It is also shown that optimal homomorphisms are good in the sense of Seitz, meaning that all weights of \({\mathbf G}_m\) in \(\text{Lie}(G)\) are \(\leq 2p-2\). Moreover the \(\text{SL}_2\)-module \(\text{Lie}(G)\) is tilting. Finally rationality questions are treated.
In an appendix Serre discusses Springer isomorphisms from the variety of unipotent elements in \(G\) to the variety of nilpotent elements in \(\text{Lie}(G)\). reductive groups; nilpotent orbits; completely reducible subgroups; principal homomorphisms; Springer isomorphisms; geometric invariant theory; optimal cocharacters; linear representations; weights; Lie algebras McNinch, G., Optimal \(S L(2)\)-homomorphisms, Comment. Math. Helv., 80, 2, 391-426, (2005) Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Representation theory for linear algebraic groups, Automorphisms of infinite groups, Geometric invariant theory, Group actions on varieties or schemes (quotients), Conjugacy classes for groups Optimal \(\text{SL}(2)\)-homomorphisms. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the third in a series of three papers about cancellation problems. In this part we use the techniques and results developed in the preceding papers (see review No. 57002 above and the preceding one) to give some further applications to the topology of four-manifolds. The main results concern smooth structures on elliptic surfaces with finite fundamental group, and the topological classification of four-manifolds with special fundamental groups.
Theorem A: Let \(X\) be an elliptic surface with finite fundamental group. If the geometric genus \(p_ g(X)>0\) then \(X\) has at least two smooth structures which remain distinct under blow-ups. If \(p_ g=0\) then \(X\#\overline{CP}^ 2\) has at least two smooth structures which remain distinct under further blow-ups.
We also show that topological 4-manifolds with odd order fundamental group, and large Euler characteristic are classified up to homeomorphism by explicit invariants. The precise statement includes a lower bound for the Euler characteristic in terms of an integer \(d(\pi)\) depending on the group. Let \(\text{Out}(\pi)\) denote the outer automorphism group of \(\pi\), and \(b_ 2(M)\) the second Betti number of a manifold \(M\). The type is the type (even or odd) of the intersection form on \(M\).
Theorem B: Let \(M\) be a closed oriented manifold of dimension four, and let \(\pi_ 1(M)=\pi\) be a finite group of odd order. When \(w_ 2(\widetilde M)=0\) (resp. \(w_ 2(\widetilde M)\neq 0)\), assume that \(b_ 2(M)-|\sigma(M)|>2d(\pi)\), (resp. \(>2d(\pi)+2)\). Then \(M\) is classified up to homeomorphism by the signature, Euler characteristic, type, Kirby-Siebenmann invariant, and fundamental class in \(H_ 4(\pi,Z)/\text{Out}(\pi)\).
Finally we give a classification theorem for manifolds with finite cyclic fundamental groups, and a further application to smooth structures:
Theorem C: Let \(M\) be a closed, oriented 4-manifold with finite cyclic fundamental group. Then \(M\) is classified up to homeomorphism by the fundamental group, the intersection form on \(H_ 2(M,Z)/\text{Tor} s\), the \(w_ 2\)-type, and the Kirby-Siebenmann invariant. Moreover, any isometry of the intersection form can be realized by a homeomorphism.
Corollary D: An algebraic surface with nontrivial finite cyclic fundamental group has at least two distinct smooth structures which remain distinct under blow-ups. smooth structures on elliptic surfaces with finite fundamental group; topological classification of four-manifolds with special fundamental groups; geometric genus; blow-ups; intersection form; signature; Euler characteristic; Kirby-Siebenmann invariant; fundamental class; manifolds with finite cyclic fundamental groups; algebraic surface with nontrivial finite cyclic fundamental group; distinct smooth structures Hambleton I., Kreck M.: Cancellation, Elliptic Surfaces and the Topology of Certain Four-Manifolds. J. Reine Angew. Math. 444, 79--100 (1993) Topology of Euclidean 4-space, 4-manifolds, Differentiable structures in differential topology, Algebraic topology on manifolds and differential topology, Special surfaces Cancellation, elliptic surfaces and the topology of certain four- manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Mordell's conjecture states that a curve \(C\) of genus \(g\geq 2\) over a global field \(K\) has only finitely many rational points. Here the \(K/k\)-Chow trace must be factored out if \(K\) is a function field with field of constants \(k\). The conjecture was proven independently by Manin and Grauert [see \textit{P. Samuel}, ``Lectures on old and new results on algebraic curves'', Tata Inst. Fund. Res. (Bombay 1966; Zbl 0165.24102), where the case of prime characteristic \(p>0\) is included ] in case \(K\) is a function field and -- recently -- by Faltings in case \(K\) is a number field.
Prior to Faltings' proof Dem'yanenko had established Fermat's last theorem for almost all twisted Fermat curves \(C_ b: X^ 4+Y^ 4=bZ^ 4\) \((b\in K^*\setminus K^{*4})\) and \(C_ b: X^ 6+Y^ 6=bZ^ 6\) \((b\in K^*\setminus K^{*6})\) by showing that \(C_ b\) has no rational points over \(K={\mathbb Q}\), i.e. \(C_ b({\mathbb Q})=\emptyset\), provided that an associated elliptic curve \(E_ b\) has rank \(\leq 1\) over \({\mathbb Q}\). In a later paper Dem'yanenko [see \textit{J. W. S. Cassels}, ``On a theorem of Dem'yanenko'', J. Lond. Math. Soc. 43, 61--66 (1968; Zbl 0165.24104)] proved Mordell's conjecture for those curves \(C\) over a number field \(K\) that have a rational map to an elliptic curve over \(K\). His method was generalized by \textit{Yu. I. Manin} [Math. USSR, Izv. 3 (1969), 433--438 (1970); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 33, 459--465 (1969; Zbl 0191.19601)].
Combining the Dem'yanenko-Manin idea with certain height estimates of his own [see Duke Math. J. 51, 395--403 (1984; Zbl 0579.14035)], the present author proves a general theorem in which all rational points on a twisted curve are obtained as fixed points of an automorphism. From this theorem he derives Demjanenko's result about the twisted Fermat curves \(C_b\) over an arbitrary number field \(K\) in place of \(\mathbb{Q}\) by showing that \(C_b(K)=\emptyset\). This is done in the first part of the paper.
In the second part, an estimate is obtained for the number of rational points of a twisted curve over a function field \(K\) of genus \(\gamma\) and transcendence degree one over an algebraically closed constant field \(k\) of characteristic zero. The rather strong bound depends on the twist, on the Jacobian variety and the genus of the twisted curve and on the genus \(\gamma\) of the function field \(K/k\). In the special case of the twisted Catalan curve \(C_{a,b}:aX^m+bY^n=1\) \((a\in K^*\setminus K^{*m}.b\in K^*\setminus K^{*n})\), if either \(\gamma\leq 1\) or the extension \(K(a^{1/m},b^{1/n})/K\) is ``sufficiently ramified'', the bound assumes the simple form \(|C_{a,b}(K)|\leq 4\cdot 5^{rk(J_{a,b} (K)^*)}\), where \(rk(J_{a,b}(K)^*)\) denotes the rank of the quotient \(J_{a,b}(K)^*\) of the Jacobian variety \(J_{a,b}(K)\) of \(C_{a,b}\) by the subgroup of constant rational points on \(J_{a,b}\).
The results are of particular interest in view of Coleman's question [see \textit{R. Coleman}, Duke Math. J. 52, 765--770 (1985; Zbl 0588.14015)] as to whether there exists a bound for \(C(K)\) depending only on the genus \(g\) of \(C\) and the rank of the Jacobian variety \(J(K)\) of \(C\) over \(K=\mathbb{Q}\). height function; Mordell's conjecture; twisted Fermat curves; dual pairs of type II; symplectic form; unitary groups; irreducible dual reductive pairs; parabolic subgroups; non-ramified type I dual reductive pairs; irreducible admissible representations; Hecke algebras DOI: 10.1112/plms/s3-55.3.465 Arithmetic ground fields for curves, Rational points, Jacobians, Prym varieties, Global ground fields in algebraic geometry, Special algebraic curves and curves of low genus Rational points on certain families of curves of genus at least 2 | 0 |
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