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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 The purpose of this paper is to find character formulae for every finite-dimensional simple module over the basic classical complex Lie superalgebra \(\mathfrak{g} = \mathfrak{osp}(m,2n)\). The results, without proofs and with a mistake in one statement, were enunciated in a previous paper by the second author [\textit{V. Serganova}, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 583--593 (1998; Zbl 0898.17002)]. Here, a different Borel subgroup and the language of weight diagrams are used in order to provide the needed proofs, which exploit geometrical methods adapted from Borel-Weil-Bott theory along the following lines:
1) Let \(G\) be the orthosymplectic Lie supergroup \(\text{OSP}(m,2n)\) and let \(P\) be a parabolic subgroup of \(G\). The multiplicities of simple \(\mathfrak{g}\)-submodules of the sheaf cohomology groups of a generalized supergrassmannian \(G/P\) are determined for certain \(P\).
2) These multiplicities are employed to express the characters of finite-dimensional simple \(\mathfrak{g}\)-modules as linear combinations of characters of the Euler characteristic of some invertible sheaves, characters which are determined by means of Borel-Weil-Bott theory. character formula; basic classical Lie superalgebra; cohomology of a generalized supergrassmannian; Borel-Weil-Bott theory; Euler characteristic; weight diagram C. Gruson, V. Serganova, \textit{Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras}, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852-892. Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Sheaf cohomology in algebraic topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Superalgebras Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras | 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 unitary representation \(T\) of a finite group \(G\) is called `frequent' if the dimension of the variety of all representations which are unitarily equivalent to \(T\) is maximal (among all unitary representations of \(G\) in the same finite-dimensional complex space). For example, the fundamental representation of \(G\) in the space of functions on \(G\) is frequent.
Let \(p_k\) and \(a_k\) be the multiplicities and the dimensions of the irreducible components of \(T\), so \(\sum p_ka_k=n\), the dimension of \(T\). For frequent \(T\) it is shown that: if \(n\) is divisible by the order of \(G\), then \(p_k\) are proportional to \(a_k\); as \(n\to\infty\), \(p_k\) are asymptotically proportional to \(a_k\).
Motivation came from the symmetries of the eigenvalues in magneto hydrodynamics. frequent representations of finite groups; unitary representations; varieties of representations V. I. Arnold, ''Frequent Representations,'' Moscow Math. J. 3(4), 1209--1221 (2003). Ordinary representations and characters, Asymptotic results on counting functions for algebraic and topological structures, Representations of finite symmetric groups, Group actions on varieties or schemes (quotients) Frequent representations. | 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 During the past fifteen years, there have been several attempts to solve a Schottky-type problem for Prym varieties, that is to characterize the locus of Prym varieties inside the corresponding moduli space of polarized abelian varieties either geometrically or by certain equations in theta constants. Some partial results concerning this problem have been obtained, in the meantime, by Shiota, Taimanov, Li, Mulase, Plaza-Martín, and others using different approaches. The paper under review also points in this direction and has two main objectives. First, the authors generalize some previous results of Shiota and Plaza-Martín to the more general case of Prym varieties associated with curves admitting an automorphism of prime order. Then they give an explicit description of the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of Sato's infinite Grassmannian. Using the formal approach developed by two of the authors [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, Equations of Hurwitz schemes in the infinite Grassmannian, Preprint http://arxiv.org/abs/math/0207091] in order to characterize Hurwitz schemes in the framework of inifinite Grassmannians, and extending it to their new concept of formal Prym varieties, the authors establish an analogue of the classical Krichever map as well as an explicit characterization of formal Prym varieties as subvarieties of the the Sato Grassmannian. Finally, in the last section of the present paper, explicit equations of the moduli spaces of curves with automorphisms of prime order are derived within the same framework. The latter formal approach is based on the results and methods of another foregoing work of two of the authors, and being concerned with the equations defining the moduli spaces of pointed curves in the infinite Grassmannian [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, J. Differ. Geom. 51, No. 3, 431--469 (1999; Zbl 1065.14512)]. Jacobians; automorphisms of curves; infinite Grassmannians; moduli spaces; Krichever correspondence; formal schemes E. Gómez González, J. M. Muñoz Porras, and F. J. Plaza Martín, Prym varieties, curves with automorphisms and the Sato Grassmannian, Math. Ann. 327 (2003), no. 4, 609 -- 639. Jacobians, Prym varieties, Automorphisms of curves, Families, moduli of curves (algebraic), Infinite-dimensional manifolds, Theta functions and curves; Schottky problem, Generalizations (algebraic spaces, stacks) Prym varieties, curves with automorphisms and the Sato Grassmannian | 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 Using the concept of characteristic polyhedron [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)], the authors continue to develop a constructive approach to the resolution of singularities in three-dimensional space in a more general context similar to [\textit{V. Cossart} and \textit{O. Piltant}, Math. Ann. 361, No. 1--2, 157--167 (2015; Zbl 1308.14008)]. The key idea is to determine the Hironaka polyhedron without passing to the completion of the local ring of a singularity. Thus, they prove that this is possible in a number of special situations, including the cases of local Henselian \(G\)-rings and singularities whose defining ideals satisfy certain numerical conditions on their standard bases. singularities; embedded resolutions; polyhedra; Hironaka's characteristic polyhedron; excellent rings; strong normalization; Hilbert-Samuel function; Hironaka schemes; standard bases Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Characteristic polyhedra of singularities without completion. II | 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 describe a method in deformation theory that David Mumford and the present author developed in 1966 [\textit{F. Oort} and \textit{D. Mumford}, Invent. Math. 5, 317--334 (1968; Zbl 0179.49901)]. deformation theory; finite group schemes; abelian varieties; Newton polygons; automorphisms of algebraic curves History of algebraic geometry, History of mathematics in the 20th century, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Local deformation theory, Artin approximation, etc., Automorphisms of curves, Algebraic moduli of abelian varieties, classification, Group schemes A method in deformation 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 purpose of this survey article is quite many-sided. On the one hand, it deals with the cohomology of Lie algebras of the form \({\mathfrak g}\otimes A\), where \({\mathfrak g}\) is a reductive Lie algebra and \(A\) is a finitely generated commutative \(\mathbb{C}\)-algebra. In this context, it briefly reports on some recent- (and forth-coming) joint work of the author's with I. Grojnowski and S. Fishel, where this kind of cohomology is explicitely computed in a special case, with remarkable combinatorial applications as well as with close relations to some earlier works of \textit{B. L. Feigin} and \textit{B. L. Tsygan} in the late 1980s [in: \(K\)-theory, arithmetic and geometry, Semin. Moscow Univ., Lect. Notes Math. 1289, 67--209, 210--239 (1987; Zbl 0635.18008, Zbl 0635.18009)]. On the other hand, the author exhibits the connection between the cohomology of Lie algebras and Hodge theory, notably with a view towards the framework set by the work of \textit{J.-L. Loday} and \textit{D. Quillen} [Comment. Math. Helv. 59, 565--591 (1984; Zbl 0565.17006)] and \textit{B. L. Tsygan} [Russ. Math. Surv. 38, No. 2, 198--199 (1983; Zbl 0526.17006)]. As the author points out, his and his collaborators computations reveal that the so-called Loday conjecture in cyclic homology fails to be true, in general, but can be confirmed in a few special cases. In another direction, the author's (and his collaborators') approach to Lie algebra cohomology allows to compute the Hodge-de Rham spectral sequence for the standard flag variety of a loop group \(LG\) of a reductive group \(G\). As for the detailed proofs, the reader will have to wait for the announced forthcoming paper by Fishel-Grojnowski-Teleman mentioned above. Hodge theory; cyclic homology; cohomology of Lie algebras; Hochschild homology; de Rham cohomology; Loday conjecture Transcendental methods, Hodge theory (algebro-geometric aspects), Cohomology of Lie (super)algebras, \(K\)-theory and homology; cyclic homology and cohomology, de Rham cohomology and algebraic geometry, Loop groups and related constructions, group-theoretic treatment, Homological methods in Lie (super)algebras Some Hodge theory from Lie 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 In previous work the author has developed a theory of operator vessels associated with operators \(A_1,A_2,\dots, A_n,B_1,B_2,\dots,B_n\) on a Banach space such that the differences \(A_j-B_j\) are small (e.g., of finite rank). A special kind of vessel can be constructed when the operators \(A_j,B_j\) are obtained as rational functions of operators \(A, B\) (i.e., \(A_j=r_j(A)\) and \(B_j=r_j(B))\) with rank one difference \(A-B\). The author makes a general conjecture about sufficient conditions under which a given vessel can be identified with one of the special kind. He proves this conjecture for \(n=2\). operator colligations; operator vessels; Livsic characteristic function; invariant subspaces; rational functions of operators Kravitsky, N, No article title, Integral Equations Operator Theory, 26, 60, (1996) Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc., Rational and birational maps, Equations and inequalities involving linear operators, with vector unknowns, Transformers, preservers (linear operators on spaces of linear operators), Functional calculus for linear operators Rational operator functions and Bezoutian operator vessels | 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 review in Zbl 0735.14010. Bibliography; geometric invariant theory; rationality of the field of invariants; constructive invariant theory; Hilbert's 14th problem; Poincaré series; categorical quotients; Russian conjecture Popov, V. L.; Vinberg, È. B., Invariant theory, (Algebraic Geometry. IV, Encyclopaedia of Mathematical Sciences, vol. 55, (1994), Springer-Verlag Berlin), (1989), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow, edited by A.N. Parshin and I.R. Shafarevich, vi+284 pp Geometric invariant theory, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, History of algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Representation theory for linear algebraic groups 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 actions of Hecke operators on the cohomology groups of groups; congruence relations; eigenvalues of Hecke operators on quaternion modular groups M. Kuga, W. Parry, and C. H. Sah, Group cohomology and Hecke operators , Manifolds and Lie groups (Notre Dame, Ind., 1980), Progr. Math., vol. 14, Birkhäuser Boston, Mass., 1981, pp. 223-266. Hecke-Petersson operators, differential operators (one variable), Congruences for modular and \(p\)-adic modular forms, Complex multiplication and abelian varieties, Homological methods in group theory Group cohomology and Hecke operators | 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 0752.00024.]
Let \(\text{Sh} (G,X)\) be the Shimura variety associated to a reductive group \(G\) over \(\mathbb{Q}\), and let \(\text{Sh}_ p (G,X) = \text{Sh} (G,X)/K_ p\), where \(K_ p\) is a compact open subgroup of \(G(\mathbb{Q}_ p)\). Let \(E\) be a reflex field of \(\text{Sh} (G,X)\), and let \(v\) be a prime of \(E\) lying over \(p\). If \(\mathbb{Q}^{\text{al}}_ p\) is the algebraic closure of \(E_ v\) and \(\mathbb{Q}^{\text{un}}_ p\) is the maximal unramified extension of \(\mathbb{Q}_ p\) contained in \(\mathbb{Q}^{\text{al}}_ p\), then \(\mathbb{F}\) denotes the residue field of \(\mathbb{Q}^{\text{un}}_ p \subset \mathbb{Q}^{\text{al}}_ p\). Let \({\mathfrak P}\) be the pseudomotivic groupoid associated with the Tannakian category of motives over \(\mathbb{F}\), and let \({\mathfrak G}_ G\) be the neutral groupoid defined by \(G\). Then a homomorphism \(\varphi : {\mathfrak G}_ G \to {\mathfrak P}\) defines a triple \((S (\varphi), \Phi (\varphi), \times (\varphi))\), where \(S(\varphi)\) is a set of the form \(I_ \varphi (\mathbb{Q})^ - \backslash X^ p (\varphi) \times X_ p (\varphi)\), \(\Phi (\varphi)\) is a Frobenius operator, and \(X(\varphi)\) is an action of \(G(\mathbb{A}^ p_ f)\) on \(S(\varphi)\) commuting with the action of \(\Phi (\varphi)\). Then the conjecture of \textit{R. P. Langlands} and \textit{M. Rapoport} [J. Reine Angew. Math. 378, 113-220 (1987; Zbl 0615.14014)] can be stated as
\[
\bigl( \text{Sh}_ p (\mathbb{F}), \Phi, \times \bigr) \approx \coprod_ \varphi \bigl( S (\varphi), \Phi (\varphi), \times (\varphi) \bigr),
\]
where the disjoint union is over a certain set of isomorphism classes of \(\varphi\). Let \({\mathcal V} (\xi)\) be the local system on \(\text{Sh} (X)\) defined by a representation \(\xi : G \to GL (V)\) of \(G\). Let \({\mathcal T} (g)\) be the Hecke operator defined by \(g \in G (\mathbb{A}_ f^ p)\), and let \({\mathcal T} (g)^{(r)}\) be the composite of \({\mathcal T} (g)\) with the \(r\)-th power of the Frobenius correspondence. In this paper the author derives from the conjecture of Langlands and Rapoport the formula for the sum
\[
\sum_{t'} \text{Tr} \bigl( {\mathcal T} (g)^{(r)} | {\mathcal V}_ t (g) \bigr),
\]
where \(t'\) runs over the elements of \(\text{Sh}_{K \cap gKg^{-1}} (G,X) (\mathbb{F})\) such that \({\mathcal T} (g)(t') = t\), as a sum of products of certain orbital integrals. He also introduces the notion of an integral canonical model for a Shimura variety, extends the conjecture of Langlands and Rapoport to Shimura varieties defined by groups whose derived group is not simply connected, and reviews results of R. Kottwitz concerning the stabilization. zeta functions; Picard modular surfaces; \(L\)-functions; automorphic representations; Shimura variety; Tannakian category of motives; Frobenius operator; conjecture of Langlands and Rapoport J. S. Milne, The points on a Shimura variety modulo a prime of good reduction, The Zeta Functions of Picard Modular Surfaces, University Montréal, Montreal (1992), 151-253. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Modular and Shimura varieties, Representation-theoretic methods; automorphic representations over local and global fields, Finite ground fields in algebraic geometry The points on a Shimura variety modulo a prime of good reduction | 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 \(\mathcal A\) be an Azumaya algebra over a locally Noetherian scheme \(X\). For \(\mathcal I\) a quasi-coherent \(\mathcal O_X\)-module, it is well-known that \(\mathcal A\otimes_{\mathcal O_X}\mathcal I\) is injective in the category of quasi-coherent \(\mathcal A\)-bimodules if and only if \(\mathcal I\) is injective in the category of quasi-coherent \(\mathcal O_X\)-modules. The purpose of this paper is to prove a similar result over the category of all \(\mathcal O_X\)-modules. Specifically, the author proves that \(\mathcal A\otimes_{\mathcal O_X}\mathcal I\) is injective in the category of all \(\mathcal A\)-bimodules if and only if \(\mathcal I\) is injective in the category of \(\mathcal O_X\)-modules; furthermore, these are equivalent to \(\mathcal A\otimes_{\mathcal O_X}\mathcal I\) being injective in the category of all left \(\mathcal A\)-modules.
The motivation for this result is to provide a dualizing complex for \(\mathcal A\), that is, a bounded complex of quasi-coherent \(\mathcal A\)-bimodules which are injective in the category of all left \(\mathcal A\)-modules as well as right \(\mathcal A\)-modules satisfying an additional duality condition. Such a complex enables the author to study coherent Hermitian Witt groups of Azumaya algebras with involution over \(X\).
In order to prove the theorem above, the strategy is to reduce to the affine case. What is shown is that for \(\mathcal A\) an Azumaya algebra over \(X=\text{Spec\,}R\) and \(\mathcal J\) a quasi-coherent \(\mathcal A\)-bimodule, then \(\mathcal J\) is injective in the category of quasi-coherent left \(\mathcal A\)-modules if and only if it is injective in the category of all left \(\mathcal A\)-modules.
As a consequence of the above, \(\mathcal Hom_{\mathcal O_X}(\mathcal{A,I})\) is injective as a left \(\mathcal A\)-module if and only if it is injective as an \(\mathcal A\)-bimodule, which is true if and only if the quasi-coherent \(\mathcal O_X\)-module \(\mathcal I\) is injective as an \(\mathcal O_X\)-module. A portion of this result is generalized to the case where \(\mathcal A\) is a coherent \(\mathcal O_X\)-algebra; namely that if \(\mathcal I\) is a quasi-coherent \(\mathcal O_X\)-module which is injective as an \(\mathcal O_X\)-module, then \(\mathcal Hom_{\mathcal O_X}(\mathcal{A,I})\) is injective as both a left \(\mathcal A\)-module as well as a right \(\mathcal A^{op}\)-module. Hence, a coherent \(\mathcal O_X\)-algebra has a dualizing complex if and only if \(X\) does. Azumaya algebras; locally Noetherian schemes; injective modules; quasi-coherent bimodules; dualizing complexes; coherent Hermitian Witt groups DOI: 10.1007/s00229-006-0046-2 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Schemes and morphisms, Projectives and injectives (category-theoretic aspects), Local cohomology and commutative rings, Syzygies, resolutions, complexes in associative algebras, Witt groups of rings On injective modules over Azumaya algebras over locally Noetherian 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 the entire collection see Zbl 0527.00017.]
This paper is concerned with the Milnor lattices of hypersurface singularities and Dynkin diagrams with respect to geometric (weakly distinguished and distinguished) bases of these lattices. It gives a summary of the author's results on the calculation of these invariants for some special classes of singularities, namely the singularities of Arnold's lists and the elliptic hypersurface singularities. These results are partly contained in the author's thesis [Math. Ann. 255, 463-498 (1981; Zbl 0438.32004)] and partly new. The paper starts with a general discussion of the notions of weakly distinguished and distinguished bases and the action of the braid group and symmetric group on the sets of these bases. Conjecture 2.3, stated in this context, has now been proven [cf. \textit{S. P. Humphries}, ''On weakly distinguished bases and free generating sets of free groups'', Q. J. Math., Oxf. II. Ser. (to appear)]. Then some general results on the discriminant quadratic forms of the Milnor lattices and on weakly distinguished bases are given. More details and further results can be found in the author's joint paper with \textit{C. T. C. Wall} [''Kodaira singularities and an extension of Arnold's strange duality'', Compos. Math. (to appear)] and for the general elliptic hypersurface case in the author's paper: ''The Milnor lattices of the elliptic hypersurface singularities'' (in preparation). Finally the author discusses a normal form for Dynkin diagrams with respect to distinguished bases of Arnold's bimodular singularities. These diagrams were used in a study of the deformation theory of these singularities [\textit{D. Balkenborg, R. Bauer} and \textit{F.-J. Bilitewski} (Diplomarbeit, Bonn 1984)]. discriminant quadratic form; weakly distinguished basis; Milnor lattices of hypersurface singularities; Dynkin diagrams; elliptic hypersurface singularities; action of the braid group; bimodular singularities W. Ebeling, ''Milnor Lattices and Geometric Bases of Some Special Singularities,'' in Noeuds, tresses et singularit és, Ed. by C. Weber (Enseign. Math. Univ. Genève, Genève, 1983), Monogr. Enseign. Math. 31, pp. 129--146; Enseign. Math. 29, 263--280 (1983). Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Braid groups; Artin groups, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex singularities, Quadratic forms over global rings and fields Milnor lattices and geometric bases of some special 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 0719.00018.]
This paper, as a continuation of [\textit{M. Kashiwara}, The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 407-433 (1990; Zbl 0727.17013)], completes the proof of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras. The proof consists of two parts: (1) the algebraic part --- the correspondence between \({\mathcal D}\)-modules on the flag variety and representations of the Kac-Moody Lie algebra, (2) the topological part --- the description of geometry of Schubert varieties in terms of the Kazhdan-Lusztig polynomials. The algebraic part is already established in the paper cited above and the paper under review is devoted to the topological part. There are two points in the proof except which the proof is similar to the finite dimensional case. The first one is the usage of the theory of mixed Hodge modules and the second one is the interpretation of the inverse Kazhdan-Lusztig polynomials as the coefficients of certain elements in the dual of the Hecke-Iwahori algebra.
Let \({\mathfrak h}\) be the Cartan subalgebra of a symmetrizable Kac-Moody Lie algebra and \(W\) the Weyl group. For \(w\in W\) define the action on \({\mathfrak h}^*\) by \(w\cdot\lambda=w(\lambda+\rho)-\rho\). Let \(P_{z,w}(q)\) be the Kazhdan-Lusztig polynomial and \(Q_{z,w}(q)\) the inverse Kazhdan- Lusztig polynomial. They are related by
\[
\sum_ w(-1)^{\ell(w)- \ell(y)}Q_{y,w}(q)P_{w,z}(q)=\delta_{y,z}.
\]
The main result of the paper is the following. For a dominant integral weight \(\lambda\in{\mathfrak h}^*\), one has
\[
ch L(w\cdot\lambda)=\sum_ z(-1)^{\ell(z)- \ell(w)}Q_{w,z}(1)ch M(z\cdot\lambda),
\]
or equivalently \(ch M(w\cdot\lambda)=\sum_ zP_{w,z}(1)ch L(z\cdot\lambda)\). Kazhdan-Lusztig conjecture; symmetrizable Kac-Moody Lie algebras; \({\mathcal D}\)-modules; flag variety; representations; geometry of Schubert varieties; Kazhdan-Lusztig polynomials; mixed Hodge modules O.J. Ganor, \textit{Supersymmetric interactions of a six-dimensional self-dual tensor and fixed-shape second quantized strings}, \textit{Phys. Rev.}\textbf{D 97} (2018) 041901 [arXiv:1710.06880] [INSPIRE]. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra. II: Intersection cohomologies of Schubert 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 We consider the Dynkin diagram \(A_ n\) with an arbitrary orientation \(\Omega\) and, for a given dimension \(d=(d_ 1,d_ 2,...,d_ n)\) we consider the corresponding variety \(L_ d\) of all representations of \((A_ n,\Omega)\) of dimension d. The group G, \(G=\prod^{n}_{i=1}GL(d_ i),\) acts naturally on the variety \(L_ d\). Let \({\mathcal R}(L_ d)\) be the ring of semi-invariants, i.e., the ring generated by the semi-invariant polynomials. In this paper we give explicitly a set of algebraic semi-invariant polynomials which generate \({\mathcal R}(L_ d).\)
Using a result due to Sato and Kimura we produce them as the reduced equations of the closure of the codimension 1 orbits of the given action. - To do this we have to classify the codimension 1 orbits which are explicitly constructed and given by their decomposition in terms of the irreducible representations of \(A_ n\). A similar result has been now obtained also for the representations of the Dynkin diagram \(D_ n\), and will appear in Trans. Am. Math. Soc. generators of ring of semi-invariants; Dynkin diagram; \(A_ n\); codimension 1 orbits S. Abeasis, Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type \(\mathbb{A}_n \) , Trans. Amer. Math. Soc.282 (1984), 463--485. Geometric invariant theory, Group actions on varieties or schemes (quotients) Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type \({\mathcal A}_ n\) | 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 The object of this paper is to study Hilbert functions of a set of points in the projective space \(\mathbb{P}^n_k\). In an earlier work [\textit{A. V. Geramita}, \textit{T. Harima} and \textit{Y. S. Shin} in: The curves seminar at Queen's, Vol. XII, Proc. Sem. Queen's Univ., Kingston 1998, Queen's Pap. Pure Appl. Math. 114, 67-96, Exposé II C (1998; Zbl 0943.13012)] the authors proved that there is a one to one correspondence between Hilbert functions of a set of points in the projective space \(\mathbb{P}^n_k\) and \(n\)-type vectors. Also given a \(n\)-type vector \({\mathcal F}\) they define a \(k\)-configuration of points of type \({\mathcal F}\). The main points of this paper are the followings:
1. Given \(X\) a \(k\)-configuration of points of type \({\mathcal F}\), the Betti numbers of the minimal resolution of the ideal \(I_X\) depends only on the Hilbert function \(H_{\mathcal F}\).
2. For any \(n\)-type vector \({\mathcal F}\) (and then a Hilbert function \(H_{\mathcal F})\), the authors can find a \(k\)-configuration of points of type \({\mathcal F}\) with Hilbert function \(H_{\mathcal F}\). They prove that \(k\)-configuration of points all have an extremal resolution.
3. The authors find all possible Hilbert functions for codimension 3 Gorenstein artinian \(k\)-algebras. configuration of points; Hilbert functions of a set of points Geramita, A.V., Harima, T., Shin, L.Y.S.: Extremal Point Sets and Gorenstein Ideals. Adv. Math. 152, 78--119 (2000) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Syzygies, resolutions, complexes and commutative rings Extremal point sets and Gorenstein ideals | 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 prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algebra is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not have a generic stabiliser. field of invariants; generic stabiliser; simple Lie algebra; seaweed subalgebra Panyushev, D., An extension of raïs' theorem and seaweed subalgebras of simple Lie algebras, Ann. Inst. Fourier (Grenoble), 55, 3, 693-715, (2005) Simple, semisimple, reductive (super)algebras, Graded Lie (super)algebras, Group actions on varieties or schemes (quotients) An extension of Raïs' theorem and seaweed subalgebras of simple Lie 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 relates the geometry of the maximum dimension locus of a real irreducible analytic germ \(X_ 0\subset {\mathbb{R}}^ n_ 0\) and the space of orders of the field of germs of meromorphic functions over \(X_ 0\), \({\mathcal O}(X_ 0)\). Results of the same nature are already known in the real algebraic case.
A formal half branch is defined to be a germ of non-constant \(C^{\infty}\)-mapping from \(]0,\epsilon[\) to \(X_ 0,c\), such that, if \(\hat c\) is the jet of c, an analytic function f is such that \(f(\hat c)=0\) iff for \(\epsilon\) sufficiently small for all \(t\in]0,\epsilon [\): \(f(c(t))=0\) \((t\to(t,e^{-1/t}))\) is not a formal half branch of \({\mathbb{R}}^ 2_ 0\), for example). The dimension of a formal half-branch is the dimension of the smallest analytic germ containing the image of c. - An order \(\alpha\) of \({\mathcal O}(X_ 0)\) is centered at a formal half- branch of \(X_ 0\) if every germ of analytic function over \(X_ 0\) positive on the image of c is positive for \(\alpha\). This is a geometric illustration of the theory of specialization in the real spectrum of the ring of germs of analytic functions on \(X_ 0\). Let \(\Omega\) be the set of orders on \({\mathcal O}(X_ 0)\), \(\Omega^*\) (resp. \(\Omega_ e\), \(e=1,...,d=\dim X_ 0)\) be the set of orders centered at a formal half- branch (resp. of dimension e). The author shows that \(\Omega_ 1,...,\Omega_ d\), \(\Omega-\Omega^*\) are disjoint and dense in \(\Omega\). Then he establishes a bijection between the clopens of \(\Omega\) and the regularly closed semi-analytic germs of the maximum dimension locus \(X^*_ 0\) of \(X_ 0\), which gives, as in the real algebraic case, a solution to Hilbert 17th problem. analytic germ; semi-analytic real spectrum; space of orders of the field of germs of meromorphic functions; formal half branch; maximum dimension locus; Hilbert 17th problem Ruiz, J.: Central orderings in fields of real meromorphic function germs. Preprint (1984) Real algebraic and real-analytic geometry, Real-analytic manifolds, real-analytic spaces, Germs of analytic sets, local parametrization, Real-analytic functions, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) Central orderings in fields of real meromorphic function germs | 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 concerns two types of objects and the deep relations between them: Twisted modular forms and Teichmüller curves. Two equivalent definitions of Teichmüller curves are as 1-dimensional subvarieties of the moduli space \(\mathcal{M}_{g}\) of genus \(g\) curves that are totally geodesic with respect to the Teichmüller metric, or as images in \(\mathcal{M}_{g}\) of orbits of the action of \(\mathrm{GL}_{2}(\mathbb{R})\) on the non-zero part of the cotangent bundle of \(\mathcal{M}_{g}\) (coming from flat surfaces). On the other hand, let a map \(\varphi\) from the upper half-plane \(\mathcal{H}\) to itself and a Fuchsian group \(\Gamma \subseteq \mathrm{SL}_{2}(\mathbb{R})\) be given, such that \(\Gamma\) is contained in \(\mathrm{SL}_{2}(\mathbb{K})\) for a real quadratic field \(\mathbb{K}\) with Galois automorphism \(\sigma\) over \(\mathbb{Q}\) and we have the equality \(\varphi\circ\gamma=\gamma^{\sigma}\circ\varphi\) for every \(\gamma\in\Gamma\). Then in addition to the usual factor of automorphy \(J\) of \(\mathrm{SL}_{2}(\mathbb{R})\) on \(\mathcal{H}\), the maps \(\varphi\) and \(\sigma\) produce a twisted factor of automorphy \(\widetilde{J}\), and a \textit{twisted} modular form, of some bi-weight \((k,l)\), involves \(J^{k}\widetilde{J}^{l}\) in its functional equations with respect to \(\Gamma\). These modular forms are intimately linked to embeddings of Teichmüller curves into Hilbert modular surfaces and Siegel modular threefolds.
As the paper is long and involves many notions and techniques, we shall describe the essence of each of its sections briefly. It consists of three parts. The first one (Sections 1--4) defines modular embeddings and describes the theory of twisted modular forms. A second part (Sections 5--8) presents the basic definitions of Teichmüller curves and gathers many results (some from a new view point) on the Teichmüller curve of discriminant 17. The last part (Sections 9--13) investigates theta functions of degree 2, and uses them to establish many results about the Teichmüller curves appearing in this paper.
We now indicate the content of each section separately. Section 1 presents embeddings of \(\mathcal{H}\) into \(\mathcal{H}^{2}\), and therefore of the analytic model \(\Gamma\backslash\mathcal{H}\) of a Teichmüller curve (where \(\Gamma\) is a Veech group) into a Hilbert modular surface, in a form that produces maps \(\varphi\) with the required commutation relations, and investigates the properties of \(\varphi\) (including the form of its expansion at a cusp). Section 2 describes the resulting theory of twisted modular forms, including growth of Fourier coefficients and examples arising from derivatives of \(\varphi\), in parallel to the corresponding theory of elliptic modular forms. In Section 3 some formulae are proved for the dimensions of spaces of twisted modular forms, the main tool being the Riemann-Roch Theorem (as usual), and introduces the important invariant \(\lambda_{2}\) of the Teichmüller curve in question. Recalling that modular forms of positive integral weight satisfy linear differential equations, a similar statement is established in Section 4 for twisted modular forms, with explicit descriptions of these equations is some cases.
Next, Section 5 introduces the basic theory of the embeddings of Hilbert modular surfaces inside the Siegel modular threefold \(\mathcal{A}_{2}\) (including the complement \(P\), called the \textit{reducible locus} of the moduli space \(\mathcal{M}_{2}\) of genus 2 curves), as well as quotes some results about Teichmüller curves in degree 2 and the twisted modular forms associated with their respective Veech groups. Relations between these objects are stated, including a characterization of unions of Teichmüller curves that is related to the natural foliation of the associated Hilbert modular surface and a construction using the differential operators mentioned above. Section 6--8 give the details of the case of discriminant 17: The explicit differential equations and their solutions (as power series) appear in Section 6, and some explicit as well as numerical expressions are determined in Section 7. Then Section 8 establishes the embedding of this Teichmüller curve into the Hilbert modular surface (as well as the equation on the latter that defines the former) and the structure of the ring of Hilbert modular forms with that discriminant, and concludes with proving the integrality of the Fourier expansions involved away from the primes lying over 2.
Section 9 then skims though the theory of theta functions and their derivatives on Siegel modular threefolds, as well as their restrictions to Hilbert modular surfaces. Section 10 proves the basic properties of certain binary quadratic forms called \textit{multiminimizers}, and relates them to cusps of Hilbert modular surfaces and their embeddded Teichmüller curves. Section 11 compares two compactifications of Hilbert modular surfaces, the classical one of Hirzebruch and a more modern one by Bainbridge, and considers the intersection of their boundary components with the Hirzebruch-Zagier divisors. Both compactifications are toroidal and involve continued fractions, but with different normalizations, thus giving sometimes different varieties. The fact that a certain Hilbert modular form of non-parallel weight \((3,9)\) (coming from products of derivatives of odd theta functions) cuts the Teichmüller curve inside the Hilbert modular surface is then shown in Section 12 to imply alone many of the important properties of these curves which were seen above. Finally, Section 13 deduces some applications of this theory, in particular the algebraicity of the Fourier coefficients of twisted modular forms, once the variable \(q\) is replaced by its multiple by a transcendental number of Gelfond-Schneider type. Teichmüller curves; Hilbert modular surfaces; theta functions; modular forms M. Möller and D. Zagier, Theta derivatives and Teichmüller curves , in preparation. Special algebraic curves and curves of low genus, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Theta series; Weil representation; theta correspondences, Arithmetic aspects of modular and Shimura varieties Modular embeddings of Teichmüller 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 We explicitly bound the Faltings height of a curve over \(\overline{\mathbb Q}\) polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualising sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of \(\mathrm{SL}_2(\mathbb Z)\) runs in polynomial time under the Riemann hypothesis for \(\zeta \)-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of \(\mathbb P_{\mathbb Z}^1\) with fixed branch locus. Arakelov theory; Arakelov-Green functions; Wronskian differential; Belyi degree; arithmetic surfaces; Riemann surfaces; curves; Arakelov invariants; Faltings height; discriminant; faltings' delta invariant; self-intersection of the dualising sheaf; branched covers Javanpeykar, A.: Polynomial bounds for Arakelov invariants of Belyi curves. With an appendix by Peter Bruin. Algebra Number Theory \textbf{8}(1), 89-140 (2014) Arithmetic varieties and schemes; Arakelov theory; heights, Dessins d'enfants theory, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic aspects of dessins d'enfants, Belyĭ theory, Heights, Riemann surfaces; Weierstrass points; gap sequences, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems Polynomial bounds for Arakelov invariants of Belyi curves. With an appendix by Peter Bruin. | 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 einer hoch interessanten Arbeit hat \textit{G. Shimura} [Ann. Math., II. Ser. 91, 144-222 (1970; Zbl 0237.14009)] allgemeine Reziprozitätsgesetze für spezielle Werte von Modulfunktionen mehrerer Variablen mit Hilfe der Theorie der Abelschen Mannigfaltigkeiten bewiesen.
Der Verf. gibt hier einen interessanten ``elementaren'' Beweis des Shimura Reziprozitätsgesetzes für spezielle Werte Siegelscher Modulfunktionen. Ähnliche Resultate für spezielle Werte Hilbertscher Modulfunktionen sind vom Autor in [J. Algebra 90, 567-605 (1984; Zbl 0549.10021), Prog. Math. 46, 49-92 (1984; Zbl 0549.10022) und Rev. Mat. Iberoam. 1, No.1, 85-119 (1985; Zbl 0599.10018)] angegeben worden. Shimura's reciprocity law; special values of arithmetic Siegel modular functions; Siegel space W.L. Baily, Jr., On the Proof of the Reciprocity Law for Arithmetic Siegel Modular Functions, Proc. Indian Acad. Sci. Math. Sci.97 (1987), no. 1-3 (1988), 21--30. Theta series; Weil representation; theta correspondences, Class field theory, Global ground fields in algebraic geometry On the proof of the reciprocity law for arithmetic Siegel modular 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 The first half of this paper explains how the D-module theory is used in the proof of the Kazhdan-Lusztig conjecture (character formula of the highest weight modules of semisimple Lie algebras, Beilinson-Bernstein and Brylinski-Kashiwara). It is a starting point as well as a foundation of the various applications of the D-module theory to the representation theory of semisimple Lie groups. The second half is devoted to a survey of the author's result [Publ. Res. Inst. Math. Sci. 23, 841-879 (1987; Zbl 0655.14004)] concerning a realization of the Hecke algebra of the Weyl group using the Hodge modules. D-module; Kazhdan-Lusztig conjecture; representation theory of semisimple Lie groups; Hecke algebra; Weyl group; Hodge modules Tanisaki, T.: Representations of semisimple Lie groups and D-modules. Sugaku Exposi- tions 4, 43-61 (1991) Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Semisimple Lie groups and their representations, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Representations of semisimple Lie groups and D-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 The cohomology of modular varieties defined by congruence subgroups of \(\mathrm{Sp}_ 4(\mathbb Z)\) whose levels lie between 2 and 4 is studied. Using a counting argument and the techniques of zeta functions, the authors completely determine the cohomology of a particular variety of this type.
See the review of the announcement in Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 66, 35 p. (1986; Zbl 0601.14021). cohomology of modular varieties; congruence subgroups of Sp(4,Z); levels; zeta functions Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Compactification of analytic spaces, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Cohomology of arithmetic groups, Arithmetic aspects of modular and Shimura varieties Moduli spaces of Riemann surfaces of genus two with level structures. I | 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's classic notes ``Introduction to Moduli Problems and Orbit spaces'' were based on a course of lectures he gave at the Tata Institute of Fundamental Research, Mumbai between January and March 1975. The published version of these lecture notes appeared in 1978 as Volume 51 of the series ``Tata Institute of Fundamental Research Lectures on Mathematics and Physics'' [Berlin-Heidelberg-New York Springer Verlag (1978; Zbl 0411.14003)], and were reviewed back then by P. Cherenack. The main goal of the author's course was to provide an introduction to the framework of geometric invariant theory (GIT) and its applications to the construction of various moduli spaces in algebraic geometry. This pioneering approach toward the classification theory of algebro-geometric objects had been developed by D. Mumford in the 1960s, that is, just about fifteen years prior to the author's lectures on the subject, and the first research monograph on GIT was \textit{D. Mumford's} book from 1965 [Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 34. Berlin-Heidelberg-New York: Springer-Verlag. VI, 145 p. (1965; Zbl 0147.39304)]. As Mumford's book is written in a highly concise, advanced and abstract style, it was (and still is) barely accessible to non-specialists in modern algebraic geometry. Due to this very fact, and in view of the crucial significance of moduli theory in contemporary mathematics, the author's notes were an attempt to explain Mumford's ideas in a simplified context, thereby working with algebraic varieties instead od schemes, on the one hand, and concentrating on carefully selected topics on the other. Actuaily, apart from the brilliant survey article ``Introduction to the Theory of Moduli'' by \textit{D. Mumford} and \textit{K. Suominen} [Algebraic Geom., Oslo 1970, Proc. 5th Nordic Summer-School Math., 171--222 (1972; Zbl 0242.14004)], the author's booklet from 1978 has been the only down-to-earth introduction to geometric invariant theory and moduli problems in algebraic geometry for several decades, and as such it has become one of the timeless classics on the subject. Indeed, generations of researchers in this area acquired their basic knowledge through these lecture notes, which unfortunately have been out of print for many years, and further generations can still profit a great deal from the study of this masterpiece of expository writing in the field.
The book under review is the long-desired reprint of the author's classic ``Introduction to Moduli Problems and Orbit Spaces''. Having left the well-tried original text totally unaltered, the Tata Institute of Fundamental Research has re-issued this classic in a new, modern print which replaces the nostalgic typescript from thirty-five years ago. As for the precise contents, we may refer to the review of the first edition (Zbl 0411.14003, loc. cit.), since no changes have been made. However, it seems appropriate, after so many years, to recall the topics treated in the five chapters of the book:
1. The concept of moduli (families of algebro-geometric objects, fine and coarse moduli spaces, universal families).
2. Moduli of endomorphisms of vector spaces (families of endomorphisms, semisimple and cyclic endomorphisms, moduli and quotients).
3. Quotients (actions of algebraic groups on varieties, reductive groups and Nagata's theorem, affine quotients, projective quotients, linearizations of group actions, (semi-)stable points).
4. Examples of (semi-)stable points (Mumford's criterion for stability, binary forms, plane cubics, \(n\)-ordered points on a line, sequences of linear subspaces).
5. Vector bundles over a curve (coherent sheaves over a curve, locally universal families for semi-stable bundle, existence of a fine moduli space, bundles over a singular curve).
As one can see from this table of contents, the material covered in the author's classic is still as topical as it was thirty-five years ago. Although a number of texts on the subject have appeared in the last few years, the book under review will maintain its unique role in the relevant literature for further decades to come. Therefore it is more than gratifying that the overdue reprint of it finally has become available for further generations of students and researchers. geometric invariant theory; reductive groups; moduli problems; moduli spaces; geometric quotients; moduli of vector bundles Newstead P. E., Introduction to Moduli Problems and Orbit Spaces (2012) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Collected or selected works; reprintings or translations of classics, Geometric invariant theory, Group actions on varieties or schemes (quotients), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles on curves and their moduli, Other algebraic groups (geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic) Introduction to moduli problems and orbit 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 We develop, for finite groups, a certain kind of resolutions of quotient spaces \(\mathbb{C}^n/G\) called pluri-toric resolutions. The pluri-toric resolutions extend the use of toric geometry, from the study of quotients by abelian finite groups to the study of quotients by arbitrary finite groups. This is done in such a way that the combinatorical and stratifying nature of toric resolutions is inherited by the pluri-toric resolutions.
We show that some of the properties of a pluri-toric resolution of \(\mathbb{C}^n/G\) can be deduced directly from the data of the group \(G\). One of these results states that a pluri-toric resolution, of a quotient by a finite subgroup of \(SL(n,\mathbb{C})\) where \(n\) is 2 or 3, is always crepant. The pluri-toric resolutions also give a generalised degree one McKay correspondence, i.e. a correspondence between certain conjugacy classes in \(G\) and the exceptional prime divisors in a pluri-toric resolution of \(\mathbb{C}^n/G\).
The pluri-toric resolutions are constructed in two steps. In the first step we use a toric resolution of the quotient space \(\mathbb{C}^n/A\), where \(A\) is a maximal abelian subgroup of \(G\), to construct a partial resolution, called a mono-toric partial resolution, of \(\mathbb{C}^n/G\). In the second step we take a mono-toric partial resolution for each maximal abelian subgroup of \(G\) and patch these together to a pluri-toric resolution of \(\mathbb{C}^n/G\). Even if the construction addresses the general case we mainly focus on the cases when \(\mathbb{C}^n/G\) is a surface or a threefold.
Our treatment leaves a number of open questions. resolutions of quotient spaces; pluri-toric resolutions; McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Equisingularity (topological and analytic), Homogeneous spaces and generalizations Pluri-toric resolutions of quotient 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 author extends standard monomial theory to the wonderful compactification \(X\) of a semisimple group \(G\) of adjoint type. Recall that \textit{R. Chirivì} and \textit{A. Maffei} have already constructed standard monomials for the more general situation of a wonderful compactification of a symmetric space [J. Algebra 261, No. 2, 310-326 (2003; Zbl 1055.14052)]. We fix a dominant weight \(\lambda\) and the corresponding line bundle \(\mathcal L_\lambda\) on \(X\). Then Chirivì and Maffei provide a basis of \(H^0(X,\mathcal L_\lambda)\) consisting of `standard monomials' with certain properties.
The author shows that in the present situation one can do more. First of all, the standard monomials are shown to behave well with respect to restriction to \(B\times B\)-orbit closures, not just \(G\times G\)-orbit closures. Recall that there are finitely many \(B\times B\)-orbits and that they have been classified by \textit{T. A. Springer} [J. Algebra 258, No. 1, 71-111 (2002; Zbl 1110.14047)]. There are degrees of freedom in the construction of Chirivì, Maffei and these the author exploits to arrange more properties familiar from the classical standard monomial theory for flag varieties.
The basis of \(H^0(X,\mathcal L_\lambda)\) is indexed by LS-paths again. And if \(Z\) is a \(G\times G\)-orbit closure, or more generally a \(B\times B\)-orbit closure, then the standard monomials that do not vanish on \(Z\) form a basis of \(H^0(Z,\mathcal L_\lambda)\). They can be characterized combinatorially. The many ingredients that are needed in the proof are explained clearly. standard monomials; group compactifications; wonderful compactifications; semisimple groups of adjoint type; orbit closures Appel, K, Standard monomials for wonderful group compactifications, J. Algebra, 310, 70-87, (2007) Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Standard monomials for wonderful group compactifications. | 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 0607.00004.]
If X is a non-smooth complex analytic space germ of a hypersurface \(f=0\) in some smooth germ P let \(\sin g X\) (resp. \(\sin g^*X)\) denote the subspace of X associated with the ideal \((f)+j(f)\) (resp. \((f)+m_ pj(f))\) in \(0_ p\), where \(m_ p\) is the maximal ideal of local ring \(0_ p\) of germs of analytic functions on P and j(f) is the jacobian ideal of f.
In an earlier paper [Invent. Math. 81, 427-447 (1985; Zbl 0627.14004)] joint with \textit{T. Gaffney} the first author has shown that the analytic type of X is completely determined by \(\sin g^*X\) and the dimension. By definition X is called harmonic if the analytic and the singular stratum of X, i.e. the maximal subspaces A and \(\Sigma\) of X along which X and resp. sing X are trivial, coincide. Otherwise X is called dissonant. In the reviewed paper the authors show that the harmonic singularities are exactly those which are determined by \(\sin g X\) and that the dissonance is actually an infinitesimal phenomenon. They also proves a Bertini type theorem on the singular stratum of a hypersurface and that ``most'' quasi-homogeneous hypersurfaces are harmonic. analytic type of a complex hyperspace germ; dissonant singularities; harmonic singularities; infinitesimal; quasi-homogeneous hypersurfaces Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties Harmonic and dissonant 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 authors define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. This category, which is a \(q\)-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type \(A\) and finite general linear groups. In this way, they obtain a categorification of the bosonic Fock space. They also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting. Heisenberg algebras; Hecke algebras; planar diagrammatics; finite general linear groups; Grothendieck groups; categorification A. Licata & A. Savage, ``Hecke algebras, finite general linear groups, and Heisenberg categorification'', Quantum Topol.4 (2013) no. 2, p. 125-185 Hecke algebras and their representations, Infinite-dimensional Lie (super)algebras, Module categories in associative algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Deformations of associative rings Hecke algebras, finite general linear groups, and Heisenberg categorification. | 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 fundamental domain \(\mathscr{R}_n\) for a cusp stabilizer of a Hilbert modular group \(\Gamma\) over a real quadratic field \(K=\mathbb{Q}(\sqrt{n})\). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of \(\mathcal{H}^2 \times \mathcal{H}^2\). The region \(\mathscr{R}_n\) is the product of \(\mathbb{R}^+\) with a 3-dimensional tower \(\mathcal{T}_n\) formed by deformations of lattices in the ring of integers \(\mathbb{Z}_K\), and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples. Hilbert modular surfaces; topological manifolds; geometric structures on manifolds; algebraic numbers; rings of algebraic integers; real and complex geometry; geometric constructions Modular and Shimura varieties Cusp shapes of Hilbert-Blumenthal 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 [For the entire collection see Zbl 0742.00065.]
Let \(G\) be a reductive group acting algebraically on an affine variety \(X\) over a field \(k\). This paper deals with the problem of finding an algorithm to compute a set of generators for the ring of invariant functions, \(k[X]^ G\), in the remaining unsolved case where the characteristic of \(k\) is prime. The author proves that there exists a computable subring \(A\) of \(k[X]^ G\) such that \(A\) is finitely generated over \(k\) and \(k[X]^ G\) is a finitely generated \(A\)-module. He also shows that if \(A\subset B\) are two integral domains which are finitely generated \(k\)-algebras, then the integral closure of \(A\) in \(B\) is a computable \(A\)-module of finite type. From this he concludes that \(k[X]^ G\) can be computed. prime characteristic; compute a set of generators for the ring of invariant functions Kempf, G. R.: More on computing invariants. Lecture notes in mathematics 1471, 87-89 (1991) Group actions on varieties or schemes (quotients), Geometric invariant theory, Computational aspects in algebraic geometry, Actions of groups on commutative rings; invariant theory More on computing 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 present book grew out of a one-semester course given by the author at Concordia University in 1986. The aim of these lectures was to introduce graduate students and researchers in other fields to the basic theory of abelian varieties, starting from the classical theory of compact Riemann surfaces and their Jacobians and ending up with a glance at some of the most famous recent topics in the arithmetic realm. The notes of this course, which have been distributed informally over the past few years, have gained a remarkable popularity among students and teachers, mainly for their efficient arrangement, enlightening style, self-containedness and -- nevertheless -- manageable conciseness. The book at issue preserves the style of these lectures and, in this way, makes them available to a wider class of readers who wish an independent, brief introduction to the subject of abelian varieties.
The first six chapters provide the basics on compact Riemann surfaces and their Jacobians. This includes the Riemann-Hurwitz formula for finite coverings, the existence of Weierstraß points, Schwarz's theorem on the finiteness of the automorphism group of a compact Riemann surface and other consequences from the Riemann-Roch theorem, the Abel-Jacobi theorem and the Riemann period relations, the construction of the Jacobian of a complex curve, divisors on complex tori and theta functions, spaces of theta functions and, finally, projective embeddings of complex tori with positive theta divisors (Lefschetz's theorem). -- This first part follows the elegant approach of \textit{E. Artin} and \textit{A. Weil}, as it was elaborated and presented in \textit{S. Lang}'s classic book ``Introduction to algebraic and abelian functions'' (1972; Zbl 0255.14001); 2nd edition 1982].
Chapter 7 gives a nice illustration of Lefschetz's embedding theorem by explicitly describing the embedding of an elliptic curve into \(\mathbb{P}^ 3\) as an intersection of two quadrics. This is a more down-to-earth version of D. Mumford's general approach [cf. \textit{D. Mumford}, ``Tata lectures on theta. I'', Prog. Math. 28 (1983; Zbl 0509.14049)], particularly suited for beginners. -- Chapter 8 discusses the Jacobian of the Fermat curve \(x^ n+y^ n+z^ n=0\), while the next three chapters deal with the modular curves associated to congruence subgroups of \(SL_ 2(R)\) and their Jacobians. This already arithmetically flavoured topic is treated by following parts of [\textit{G. Shimura}'s comprehensive book ``Introduction to the arithmetic theory of automorphic functions'' (1971; Zbl 0221.10029)] and adapting them to the more elementary purpose of the present introductory notes. -- Chapter 12 provides a very brief introduction to arbitrary abelian varieties and their basic properties, giving proofs in the special case of the complex groundfield and compact connected complex Lie groups as an illustration.
The final part of the book, chapters 13 to 15, gives an outlook to the theory of abelian varieties over number fields. This includes the concept of Tate modules of an abelian variety and the statement of the Tate conjecture, the deduction of Mordell's conjecture from Tate's conjecture via Arakelov theory and the Kodaira-Parshin construction, and the relation to Shafarevich's finiteness conjecture on the number of isomorphism classes of algebraic curves of genus \(g\) with good reduction over an algebraic number field. -- Of course, this final part is more sketchy than the others, but all the same very enlightening and motivating for deeper studies guided by more advanced textbooks and the very recent research literature.
Altogether, this book really leads the non-specialist from the origins of the theory of abelian varieties to the frontiers of todays research in the field. Lefschetz theorem; Riemann surfaces; Jacobians; Riemann-Roch theorem; Abel-Jacobi theorem; Riemann period relations; theta functions; embeddings of complex tori; modular curves; Tate conjecture; Arakelov theory V. Kumar Murty, Introduction to abelian varieties, CRM Monograph Series, vol. 3, American Mathematical Society, Providence, RI, 1993. Analytic theory of abelian varieties; abelian integrals and differentials, Theta functions and abelian varieties, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Arithmetic varieties and schemes; Arakelov theory; heights Introduction to abelian 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 Let \(\Gamma\) be a finite subgroup of \(SL(2, {\mathbb C})\), and \(X\) be the corresponding Kleinian singularity. Gonzalez-Springberg and Verdier showed that there is natural bijection between the irreducible components of the exceptional fibre in the minimal resolution of \(X\), and the non-trivial irreducible representations of \(\Gamma\). \textit{Y. Ito} and \textit{I. Nakamura} [see e.g. Proc. Japan Acad., Ser. A 72, No. 7, 135-138 (1996; Zbl 0881.14002)] found a beautiful new interpretation of this bijection, by using an interpretation of the minimal resolution of \(X\) as a subset of the Hilbert scheme of codimension \(|\Gamma|\) ideals in \({\mathbb C}[x,y]\).
In this paper the author gives a new proof of the result of Ito and Nakamura. This proof avoids a case by case analysis, contrary to the proof of Ito and Nakamura. Kleinian singularity; McKay correspondence; minimal resolution; Hilbert scheme Dlab, V.: Representations of valued graphs. In: Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 73. Presses de l'Université de Montréal, Montreal, Que (1980) Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) On the exceptional fibres of 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 Let \(p\) be a prime number and set \(q=p^ m\). Let \(X_{a,i}\) be the nonsingular algebraic curve over \(\mathbb F_ q\) with affine equation \(y^ p-y=ax+1/x+b_ i\), where \(a\in\mathbb F^*_ q\) and where \(b_ i\) is a fixed element of \(\mathbb F_ q\) with trace equal to \(i\in\mathbb F_ p\). It has long been known that the number of points of a curve of the above type is given by a Kloosterman sum and this has been used by Weil to deduce the classical estimate \((| K_ a| \leq 2\sqrt{q})\) on the absolute value of such Kloosterman sums.
We study the isogeny decomposition of the Jacobians of these curves and the \(p\)-torsion of their class groups. We find that the decomposition is determined to a large extent by the strict inequality \(| K_ a| <2\sqrt{q}\). It turns out -- surprisingly -- that the \(p\)-torsion is related to the characteristic polynomial of \(a\in\mathbb F^*_ q\) with respect to \(\mathbb F_ p\). The question of the variation of the \(p\)-order of \({\#}\text{Jac}(X_{a,i}(\mathbb F_ q))\) and that of the number of isogeny factors arose in the context of coding theory [cf. \textit{R. Schoof} and \textit{M. van der Vlugt}, J. Comb. Theory, Ser. A 57, No. 2, 163--186 (1991; Zbl 0729.11065) and the authors, J. Algebra 139, No. 1, 256--272 (1991; Zbl 0729.11066)]. The methods may also be used for other exponential sums such as multiple Kloosterman sums. p-torsion of class groups; number of points of Jacobian; number of points of algebraic curve over finite field; Kloosterman sum; isogeny decomposition of the Jacobians van der Geer, Gerard; van der Vlugt, Marcel, Kloosterman sums and the \textit{p}-torsion of certain Jacobians, Math. Ann., 290, 3, 549-563, (1991) Finite ground fields in algebraic geometry, Gauss and Kloosterman sums; generalizations, Exponential sums, Jacobians, Prym varieties, Enumerative problems (combinatorial problems) in algebraic geometry Kloosterman sums and the \(p\)-torsion of certain Jacobians | 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 is a continuation of the previous work by the author [Compos. Math. 111, 51--88 (1998; Zbl 0959.22012)] and is concerned with the cohomology groups of certain coherent sheaves on a Griffiths-Schmid variety associated to an anisotropic \(\mathbb Q\)-form of the unitary group \(\text{SU}(2,1)\). The author defines transforms relating this cohomology to the coherent cohomology of some sheaves on certain threefolds, which are fibered in projective lines over Picard modular surfaces. He explicitly describes the holomorphic and anti-holomorphic parts of the 1-cohomology of the Griffiths-Schmid variety in terms of classical Picard modular forms. This description provides an explicit generating system for the part of the 2-cohomology which corresponds to those automorphic representations whose archimedean components are degenerate limits of discrete series. automorphic forms; Dolbeault cohomology; unitary groups; Picard modular surfaces; cohomology groups; coherent sheaves; Griffiths-Schmid variety; anisotropic form of the unitary group SU(2,1); coherent cohomology; automorphic representations Carayol, H., Quelques relations entre LES cohomologies des variétés de Shimura et celles de Griffiths-schmid (cas du groupe \(S U(2, 1)\)), Compos. Math., 121, 305-335, (2000) Cohomology of arithmetic groups, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Representations of Lie and linear algebraic groups over global fields and adèle rings Some relations between the cohomologies of the Shimura varieties and the Griffiths-Schmid varieties (case of the group \(\text{SU}(2,1)\)) | 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 structure of the ring \({\mathcal D}(C)\) of differential operators on a singular projective curve \(C\) (over an algebraically closed field of characteristic zero) is described by work of \textit{I. M. Musson} in case \(C\) has positive genus [Arch. Math. 56, No. 1, 86-95 (1991; Zbl 0688.16002)], and by work of \textit{M. P. Holland} and \textit{J. T. Stafford} when \(C\) has genus zero and its normalization map is injective [J. Algebra 147, No. 1, 176-244 (1992; Zbl 0753.14014)]. Here the author extends a number of results from the latter paper to the case of an arbitrary singular, rational, projective curve \(C\). The strongest results are proved for ``sufficiently twisted'' rings of differential operators \({\mathcal D}_{\mathcal L}(C)\), where \({\mathcal L}\) is an invertible sheaf of sufficiently high degree. (When the normalization map is not injective, the untwisted algebra \({\mathcal D}(C)\) is considerably more complicated, and the results stated below can fail.) For example, the maximal finite dimensional factor algebra of \({\mathcal D}_{\mathcal L}(C)\) is a triangular matrix algebra of the form \({{M\;N} \choose {0\;F}}\) where \(M\) and \(N\) consist of matrices over the ground field while \(F\) is the maximal finite dimensional factor algebra of the ring of differential operators on an affine open subset of \(C\). It is also proved that the category of finitely generated left \({\mathcal D}_{\mathcal L}(C)\)-modules is equivalent to the category of quasi-coherent sheaves of \({\mathcal D}_C\)-modules, and that \({\mathcal D}_{\mathcal L} (C)\) is Morita equivalent to \({\mathcal D}_{{\mathcal L}'} (C)\) when both \({\mathcal L}\) and \({\mathcal L}'\) have sufficiently high degree. singular projective curves; normalization maps; rings of differential operators; invertible sheaf; maximal finite dimensional factor algebras; category of quasi-coherent sheaves Rings of differential operators (associative algebraic aspects), Special algebraic curves and curves of low genus, Sheaves of differential operators and their modules, \(D\)-modules, Commutative rings of differential operators and their modules, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Module categories in associative algebras Twisted rings of differential operators over projective rational 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 main aim of this paper is to characterize ideals \(I\) in the power series ring \(R = K [[x_1, \ldots, x_s]]\) that are finitely determined up to contact equivalence by proving that this is the case if and only if \(I\) is an isolated complete intersection singularity, provided \(\dim(R / I) > 0\) and \(K\) is an infinite field (of arbitrary characteristic). Here two ideals \(I\) and \(J\) are contact equivalent if the local \(K\)-algebras \(R / I\) and \(R / J\) are isomorphic. If \(I\) is minimally generated by \(a_1, \ldots, a_m\), we call \textit{I finitely contact determined} if it is contact equivalent to any ideal \(J\) that can be generated by \(b_1, \ldots, b_m\) with \(a_i - b_i \in \langle x_1, \ldots, x_s \rangle^k\) for some integer \(k\). We give also computable and semicontinuous determinacy bounds.
The above result is proved by considering left-right equivalence on the ring \(M_{m, n}\) of \(m \times n\) matrices \(A\) with entries in \(R\) and we show that the Fitting ideals of a finitely determined matrix in \(M_{m, n}\) have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in \(R\). Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture. singularities; finite determinacy; positive characteristic; algebraic group action; inseparable orbit action; specialization of power series; complete intersections Formal power series rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Local complex singularities Finite determinacy of matrices and ideals | 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 article is made up of the material of the Soviet-American symposium ``Algebraic groups and number theory'' which took place from 22-29 May 1991 in Minsk. The scientific programme covered current problems in the arithmetic theory of algebraic groups, the theory of discrete subgroups of Lie groups, algebraic geometry, and number theory. In the courses of lectures, new unsolved scientific problems were formulated, which will undoubtedly serve as the basis for further research in these areas. Symposium; Soviet-American symposium; Algebraic groups; Number theory; Minsk/Byelorussia; arithmetic theory of algebraic groups; discrete subgroups of Lie groups; algebraic geometry; number theory \textsc{Vladimir} P\textsc{latonov} and \textsc{Andrei} R\textsc{apinchuk}: \textit{Algebraic groups and number theory}, volume 139 of Pure and Applied Mathematics, Academic Press Inc., Boston, MA, 1994; translated from the 1991 Russian original by Rachel Rowen. Proceedings, conferences, collections, etc. pertaining to number theory, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to group theory, Proceedings, conferences, collections, etc. pertaining to topological groups Algebraic groups and number 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 compute zeros of Mellin transforms of modular cusp forms for \(SL_ 2({\mathbb{Z}})\). Such Mellin transforms are eigenforms of Hecke operators. We recall that, for all weights k and all dimensions of cusp forms, the Mellin transforms of cusp forms have infinitely many zeros of the form \(k/2+i t\), i.e., infinitely many zeros on the critical line.
A new basis theorem for the space of cusp forms is given which, together with the Selberg trace formula, renders practicable the explicit computations of the algebraic Fourier coefficients of cusp eigenforms required for the computations of the zeros.
The first forty of these Mellin transforms corresponding to cusp eigenforms of weight \(k\leq 50\) and dimension \(\leq 4\) are computed for the sections of the critical strips, \(\sigma +i t\), \(k-1<2\sigma<k+1\), - 40\(\leq t\leq 40\). The first few zeros lie on the respective critical lines \(k/2+i t\) and are simple. A measure argument, depending upon the Riemann hypothesis for finite fields, is given which shows that Hasse-Weil L- functions (including the above) lie among Dirichlet series which do satisfy Riemann hypotheses (but which need not have functional equations nor analytic continuations). zeros of Mellin transforms; modular cusp forms; eigenforms of Hecke operators; infinitely many zeros; critical line; explicit computations; algebraic Fourier coefficients of cusp eigenforms; Hasse-Weil L-functions H. R. P. Ferguson, R. D. Major, K. E. Powell, and H. G. Throolin, On zeros of Mellin transforms of \?\?\(_{2}\)(\?) cusp forms, Math. Comp. 42 (1984), no. 165, 241 -- 255. Holomorphic modular forms of integral weight, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Galois cohomology, Tables in numerical analysis, General theory of numerical methods in complex analysis (potential theory, etc.) On zeros of Mellin transforms of \(SL_ 2({\mathbb{Z}})\) cusp 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 The action of a connected reductive algebraic group \(G\) on \(G/P_-\), where \(P_-\) is a parabolic subgroup, differentiates to a representation of the Lie algebra \(\mathfrak g\) of \(G\) by vector fields on \(U_+\), the unipotent radical of a parabolic opposite to \(P_-\). The classical instances of this setting that we study in detail are the actions of \(\text{GL}_n\) on the Grassmannian of \(k\)-planes (\(1\leq k\leq n\)), of \(\text{SO}_n\) on the quadric of isotropic lines, and of \(\text{SO}_{2n}\) or \(\text{SP}_{2n}\) on their respective Grassmannians of maximal isotropic spaces; in each instance, \(U_+\) is one of the usual affine charts.
We show that both the polynomials on \(U_+\) and the polynomial vector fields on \(U_+\) form \(\mathfrak g\)-modules dual to parabolically induced modules, construct an explicit composition chain of the former module in the case where \(G\) is classical simple and \(U_+\) is Abelian -- these are exactly the cases above -- and indicate how this chain can be used to analyse the module of vector fields, as well.
We present two proofs of our main theorems: one uses the results of Enright and Shelton on classical Hermitian pairs, and the other is independent of their work. The latter proof mixes classical (and briefly reviewed) facts of representation theory with combinatorial and computational arguments, and is accessible to readers unfamiliar with the vast modern literature on category \(\mathcal O\). representation theory; Lie algebras; vector fields; category \(\mathcal O\); connected reductive algebraic groups; Grassmannians; parabolically induced modules; Hermitian pairs Draisma, J., Representation theory on the open Bruhat cell, J. Symb. Comput., 39, 279-303, (2005) Representation theory for linear algebraic groups, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Lie algebras of vector fields and related (super) algebras, Combinatorial aspects of representation theory Representation theory on the open Bruhat cell. | 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 determine all possible torsion groups of elliptic curves E with integral j-invariant over pure cubic number fields K. Except for the groups \({\mathbb{Z}}/2{\mathbb{Z}}\), \({\mathbb{Z}}/3{\mathbb{Z}}\) and \({\mathbb{Z}}/2{\mathbb{Z}}\oplus {\mathbb{Z}}/2{\mathbb{Z}}\), there exist only finitely many curves E and pure cubic fields K such that E over K has a given torsion group \(E_{TOR}(K)\), and they are all calculated here. The curves E over K with torsion group \(E_{TOR}(K)\cong {\mathbb{Z}}/2{\mathbb{Z}}\oplus {\mathbb{Z}}/2{\mathbb{Z}}\) have j- invariants belonging to a finite set. They are also calculated. A preliminary report on the results obtained was given by \textit{H. H. Müller}, \textit{H. Ströher}, and \textit{H. G. Zimmer} in Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 671-698 (1989; Zbl 0719.14021). torsion groups of elliptic curves with integral j-invariant; pure cubic number fields Fung, G.; Ströher, H.; Williams, H.; Zimmer, H.: Torsion groups of elliptic curves with integral j-invariant over pure cubic fields. J. number theory 36, 12-45 (1990) Elliptic curves, Computational aspects of algebraic curves, Cubic and quartic extensions, Global ground fields in algebraic geometry Torsion groups of elliptic curves with integral j-invariant over pure cubic 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 Over the past 25 years, the study of cohomological support varieties and representation-theoretic rank varieties has led to numerous results in the modular representation theory of finite groups, restricted Lie algebras, and other related structures. This work follows previous work of the authors [Am. J. Math. 127, No. 2, 379-420 (2005; Zbl 1072.20009), Erratum ibid. 128, No. 4, 1067-1068 (2006; Zbl 1098.20500)] in attempting to formulate a unifying theory for arbitrary finite group schemes (equivalently finite-dimensional cocommutative Hopf algebras) over arbitrary fields of prime characteristic. The paper contains several foundational results as well as a number of illuminating examples.
Let \(G\) be a finite group scheme over a field \(k\) of prime characteristic \(p\). Extending their previous notion of a \(p\)-point (defined over algebraically closed fields), the authors introduce the notion of a \(\pi\)-point of \(G\) which is a flat map of \(K\)-algebras \(K[t]/t^p\to KG\) which factors through the group algebra of a unipotent Abelian subgroup scheme for a field extension \(K/k\). The \(\Pi\)-points of \(G\), denoted \(\Pi(G)\), is the set of equivalence classes of such \(\pi\)-points under a certain specialization relation.
The first of several important results is that \(\Pi(G)\) is homeomorphic to the projectivized prime ideal spectrum of the (even-dimensional) cohomology ring of \(G\) over \(k\). For a finite-dimensional \(G\)-module \(M\), the \(\Pi\)-support of \(M\) is defined as a certain subset of \(\Pi(G)\) and can be identified cohomologically. Moreover, the authors extend this definition to arbitrary (i.e., even infinite-dimensional) \(G\)-modules. For an arbitrary module, the \(\Pi\)-support does not have a direct cohomological interpretation. The authors show that it does satisfy a number of nice properties and that every subset of \(\Pi(G)\) can be identified with the \(\Pi\)-support of some module. Another fundamental result is that the projectivity of a module can be detected by restriction to \(\pi\)-points which extends several known results in special cases. Further, the \(\Pi\)-support is used to determine the tensor-ideal thick subcategories of the stable module category of finite-dimensional \(G\)-modules, thus verifying a conjecture of \textit{M. Hovey, J. H. Palmieri}, and \textit{N. P. Strickland} [Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. Using this stable module category information, the authors give a scheme structure to \(\Pi(G)\) and show that the aforementioned homeomorphism of varieties can be extended to an isomorphism of schemes. group schemes; support varieties; rank varieties; \(p\)-points; thick subcategories; stable module categories; cohomology rings; finite-dimensional cocommutative Hopf algebras Friedlander, E. M.; Pevtsova, J., \({\Pi}\)-supports for modules for finite group schemes, Duke Math. J., 139, 2, 317-368, (2007) Representation theory for linear algebraic groups, Group schemes, Modular representations and characters, Cohomology theory for linear algebraic groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act \(\Pi\)-supports for modules for finite 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 Robin Hartshorne's popular textbook ``Algebraic geometry'' [Graduate Texts in Mathematics, 52. New York - Heidelberg - Berlin: Springer-Verlag (1977; Zbl 0367.14001)], is well known to everyone who ever tried to get acquainted with the basic modern aspects of the subject during the last three decades. Due to its comprehensiveness, versatility, and expository mastery, Hartshorne's ``Algebraic Geometry'' is by far the most widely used introductory text (and reference book) in this discipline of contemporary mathematics. Generations of algebraic geometers have acquired a profound fundamental knowledge of the subject from R. Hartshorne's book ever since its first appearance, and the book itself has never been out of print so far.
Now, more than thirty years after the publishing of his standard text on modern algebraic geometry, R. Hartshorne obliges with another book in the field. Being more specialized, the present volume is to provide an introduction to the main ideas of deformation theory in algebraic geometry and to illustrate their use in a number of typical concrete situations. Actually, as the author points out in the preface, this book has an amazingly long history that began exactly thirty years ago. Namely, in the fall semester of 1979, R. Hartshorne taught a course on algebraic deformation theory at Berkeley, mainly with the goal to understand completely Grothendieck's approach to the local study of the Hilbert scheme using cohomological methods. The handwritten notes of this course circulated quietly for many years until D. Eisenbud urged the author to complete and publish them. Then, five years ago, R. Hartshorne expanded the old notes into a rough draft, which he used to teach a course in the spring of 2005. Finally, he rewrote those notes once more and, with the addition of numerous exercises, turned them into the book under review.
As the present text is intended to be of introductory nature, no effort has been made to develop the theory of deformations in full generality. Instead, the author has preferred to elaborate the basic ideas underlying the theory, without letting them get buried in too many technical details. Also, the approach has been kept as elementary as possible, thereby assuming only a basic familiarity with the concepts and methods of algebraic geometry as developed in the author's above-mentioned standard text.
In this vein, and very much to the benefit of the rather unexperienced reader, the author has not striven for stating results in their most general form, nor has he attempted to use the more recent state-of-the-art framework of Grothendieck topologies and algebraic stacks to its full extent. Overall, the purpose of this book is to explain the basic concepts and methods of deformation theory, to bring forth clearly their fundamental essence, to show how they work in various standard situations, and to provide some instructive examples and applications from the literature.
As for the contents, the book is divided into four chapters which, altogether comprise twenty-nine sections. The author's guiding principle is to focus on four standard situations:
(A) Deformations of subschemes of a fixed ambient scheme \(X\).
(B) Deformations of line bundles on a fixed scheme \(X\).
(C) Deformations of coherent sheaves on a fixed scheme \(X\).
(D) Deformations of abstract schemes,including the local study of deformations of singularities as well as the global study of deformations of non-singular varieties (global moduli).
For each of these particular situations, a number of typical problems is discussed, with the ultimate goal to establish a global parameter space classifying the isomorphism classes of the objects in question and, moreover, to describe its geometric properties.
However, in this introductory text, the technically involved proofs of the existence of these global classifying spaces are not provided, as the author's primary goal is rather to lay the foundations of the respective deformation theory that allow to describe the local structure of the (assumed) global parameter space.
Chapter 1 is titled ``First-Order Deformations'' and deals with algebraic deformations over the ring \(D:=k[t]/(t^2)\) of dual numbers associated to an algebraically closed field \(k\). Starting with the concept of Hilbert scheme as a deformation space of a closed subscheme of the projective space \(\mathbb{P}^n_k\), which serves as a model in the sequel, deformations over \(D\) are discussed for the particular situations (A), (B) and (C). Then, after an introduction to the cotangent complex and the \(T^i\) functors of Lichtenbaum and Schlessinger, deformations of abstract schemes (as in situation (D)) are explained, thereby using the infinitesimal lifting property with regard to non-singular varieties.
Chapter 2 turns to the more general case of higher-order deformations, that is, to deformations over arbitrary Artin rings, together with the according obstruction theories for the respective situations (A), (B), (C), and (D). Along the way, three special cases are treated in greater detail, namely Cohen-Macaulay subschemes of codimension 2, locally complete intersection schemes, and Gorenstein subschemes in codimension 3. Additional illustrating material concerns the obstruction theory for local rings, the classical bound on the dimension of the Hilbert scheme of projective space curves, and Mumford's ``pathological'' example of a family of non-singular projective space curves whose Hilbert scheme is generically non-reduced.
Chapter 3 is devoted to the study of formal moduli spaces. Starting from the explicit case of plane curve singularities, the author discusses the general problem in terms of functors of Artin rings, including Schlessinger's criterion for pro-representability as the crucial technical tool in this context. In the sequel, this formal apparatus is applied to each of the standard situations (A), (B), (C) and (D), along with numerous concrete examples and applications. Further material concerns a comparison of embedded and abstract deformations, with a special view toward general surfaces in \(\mathbb{P}^4_k\) of degree greater than 3 the problem of algebraization of formal moduli (à la M. Artin), and -- as a further application -- the question of lifting varieties from characteristic \(p> 0\) to characteristic \(0\).
Chapter 4 comes with the headline ``Global Questions''. Here the methods of infinitesimal and formal deformations from the previous chapters are applied to study global moduli problems. After introducing the notions of fine moduli space and coarse moduli space, the Hilbert functor and the Picard functor are described as examples of representable moduli functors.
The rest of this concluding chapter is devoted to the discussion of a number of concrete classical moduli spaces and their geometric properties. More precisely, the author illuminates in detail the global moduli spaces of rational and elliptic curves, Mumford's concept of modular families of curves of higher genus (together with the idea of stacks), the moduli space of stable vector bundles over a curve, and the notions of formally smoothable schemes and smoothable singularities. As an application of the general theory, the reader encounters here Mori's theorem on the existence of rational curves in non-singular varieties in characteristic \(p>0\) whose canonical divisor is not numerically effective, on the one hand, and instructive examples of non-smoothable singularities on the other.
Much more material is covered by the huge number of further-leading exercises complementing each section of the book. These exercises provide much more of the respective theories as well as a wealth of additional examples. Most of the exercises are quite challenging, but they are also well-structured and equipped with guiding remarks or hints.
The author's approach to deformation theory shows lucidly how far the subject can be treated with relatively elementary methods, without providing the technically complicated proofs of the existence of the main classifying schemes, and without using more advanced toolkits like geometric invariant theory, Artin's approximation theorems, simplicial complexes, differential graded algebras, fibered categories, stacks, or derived categories.
No doubt, this masterly written book gives an excellent first introduction to algebraic deformation theory, and a perfect motivation for further, more advanced reading likewise. It is the author's masterful style of expository writing that makes this text particularly valuable for seasoned graduate students and for future researchers in the field. The list of 177 references at the end of the book, which the author frequently refers to throughout the text, is another special feature of the volume under review.
As for complementary and parallel reading, the recent monograph ``Deformations of Algebraic Schemes'' by \textit{E. Sernesi} [Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften 334. Berlin: Springer (2006; Zbl 1102.14001)] might be quite instructive and useful,especially in view of the technical prerequisites and subtleties as well as for working some of the exercises. deformation theory; infinitesimal methods; formal neighborhoods; algebraic moduli problems; formal moduli; Hilbert schemes; moduli of curves; moduli of vector bundles R. Hartshorne, \textit{Deformation Theory} (Springer, Berlin, 2010), Grad. Texts Math. 257. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, fibrations in algebraic geometry, Infinitesimal methods in algebraic geometry, Formal methods and deformations in algebraic geometry, Local deformation theory, Artin approximation, etc., Deformations of singularities, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) Deformation 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 part I see \textit{A. Szenes}, ibid. 587-597 (1993; see the preceding review).]
Let \({\mathcal M}^ C_ 0\) be the moduli space of semi-stable rank 2 vector bundles with fixed even degree determinant on a curve \(C\), and denote by \(L_ 0\) the generator of \(\text{Pic} {\mathcal M}^ C_ 0\). The authors prove the following formula
\[
\dim H^ 0 ({\mathcal M}^ C_ 0, L_ 0^{k-2}) = \sum^{k-1}_{J = +} (k/1 - \cos 2j \pi/k)^{g-1}.
\]
To prove this formula the authors use the Hecke correspondence between the two moduli spaces \({\mathcal M}^ C_ 0\) and \({\mathcal M}^ C_ 1\) to transfer the calculation from \({\mathcal M}^ C_ 0\) to \({\mathcal M}^ C_ 1\). Hilbert polynomials; moduli space of semi-stable rank 2 vector; Hecke correspondence A. Bertram and A. Szenes, ''Hilbert polynomials of moduli spaces of rank-\(2\). Vector bundles. II,'' Topology, vol. 32, iss. 3, pp. 599-609, 1993. Algebraic moduli problems, moduli of vector bundles, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert polynomials of moduli spaces of rank 2 vector bundles. II | 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 study certain \(\Delta\)-filtered modules for the Auslander algebra of \(k[T]/T^n\rtimes C_2\) where \(C_2\) is the cyclic group of order two. The motivation of this lies in the problem of describing the \(P\)-orbit structure for the action of a parabolic subgroup \(P\) of an orthogonal group. For any parabolic subgroup of an orthogonal group we construct a map from parabolic orbits to \(\Delta\)-filtered modules and show that in the case of the Richardson orbit, the resulting module has no self-extensions.
Let \(k\) be an algebraically closed of characteristic different from 2, and let \(G\) be a reductive algebraic group over \(k\), \(P\subset G\) a parabolic subgroup. Now \(P\) acts on its unipotent radical \(U\) by conjugation and on the nilradical \(\mathfrak n=\text{Lie\,}U\) by the adjoint action. By a fundamental result of \textit{R. W. Richardson} [Bull. Lond. Math. Soc. 6, 21-24 (1974; Zbl 0287.20036)], this action has an open dense orbit, the so-called `Richardson orbit' of \(P\). But in general, the number of orbits is not finite and it is a very hard problem to understand the orbit structure. The question of deciding whether \(P\) has a finite number of orbits in \(\mathfrak n\) has been asked by \textit{V. L. Popov} and \textit{G. Röhrle} [in Aust. Math. Soc. Lect. Ser. 9, 297-320 (1997; Zbl 0887.14020)].
If \(P\) is a parabolic subgroup of \(\text{SL}_N\), then there is an explicit description of the \(P\)-orbits in work of \textit{L. Hille} and \textit{G. Röhrle} [Transform. Groups 4, No. 1, 35-52 (1999; Zbl 0924.20035)], and \textit{T. Brüstle, L. Hille, C. M. Ringel}, and \textit{G. Röhrle} [Algebr. Represent. Theory 2, No. 3, 295-312 (1999; Zbl 0971.16007)], via a connection with a quasi-hereditary algebra, namely the Auslander algebra \(A_n\) of the truncated polynomial ring \(R_n:=k[T]/T^n\). They have shown that the \(P\)-orbits are in bijection with the isomorphism classes of certain \(\Delta\)-filtered modules of \(A_n\) with no self-extensions. This list has finitely many indecomposable modules, parametrized as \(\Delta(I)\) where \(I\) runs through the subsets of \(\{1,2,\dots,n\}\).
Our main goal in this paper is to establish an analogous correspondence between \(P\)-orbits for parabolic subgroups of the special orthogonal groups \(\text{SO}_N\) and certain \(\Delta\)-filtered modules for the Auslander algebra of \(k[T]/T^n\rtimes C_2\), the skew group ring of one considered by \textit{T. Brüstle} et al. [loc. cit.], where \(C_2\) is a cyclic group of order two. This article establishes the Auslander algebra of \(k[T]/T^n\rtimes C_2\) as the correct candidate for such a correspondence. There is one major difference as compared with \textit{T. Brüstle} et al. [loc. cit.]. In our situation, a list of all \(\Delta\)-filtered modules with no self extensions is difficult to obtain and may even be infinite. This must be expected however, as the construction of the Richardson elements for \(\text{SO}_N\) involves symmetric diagrams and hence gives rise to symmetric \(\Delta\)-dimension vectors. Here, we use signed sets \(I\), that is certain subsets of \(\{\pm 1,\pm 2,\dots,\pm n\}\). We associate to each signed set \(I\) another set \(J\) (a symmetric complement) and an extension \(E(I,J)\) that is \(\Delta\)-filtered with no self extensions and has the required symmetric \(\Delta\)-dimension vector. These extensions may then be used to construct a \(\Delta\)-filtered module that corresponds to the Richardson element, using the work of \textit{K. Baur} [J. Algebra 297, No. 1, 168-185 (2006; Zbl 1144.17004)] and \textit{K. Baur} and \textit{S. M. Goodwin} [Algebr. Represent. Theory 11, No. 3, 275-297 (2008; Zbl 1147.17011)] on orthogonal Lie algebras. connected reductive algebraic groups; orthogonal groups; parabolic subgroups; dense orbits; unipotent radical; Auslander algebras; truncated polynomial algebras; Richardson orbit; filtered modules; numbers of orbits Baur, K; Erdmann, K; Parker, A, {\(\Delta\)}-filtered modules and nilpotent orbits of a parabolic subgroup in \(O\)\_{}\{\(N\)\}, J. Pure Appl. Algebra, 215, 885-901, (2011) Representations of quivers and partially ordered sets, Lie algebras of linear algebraic groups, Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Coadjoint orbits; nilpotent varieties \(\Delta\)-filtered modules and nilpotent orbits of a parabolic subgroup in \(\mathrm O_N\). | 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 The present work is the author's habilitation, concerned with various aspects of the structure of zero-dimensional schemes \({\mathbf X}\) in the projective space \({\mathbb{P}}^d_K\) and the homogeneous coordinate rings of them, \(R = K[X_0, \ldots, X_d]/I_{\mathbf X}\). It is a summary of the author's research over the last eight years published in several mathematical journals, respectively available as preprints. The main technical tools of his investigations are Castelnuovo theory, Hilbert functions, the canonical module of the coordinate ring of \({\mathbf X}\), the Kähler module of differentials, and the canonical ideal.
In the second part the author discusses several conditions for uniformity, among them generic position, uniform position, cohomological uniformity, Cayley-Bacharach schemes and their hierarchy. The knowledge of the Hilbert function of \({\mathbf X}\), its growth behaviour, the relation to the Hilbert function of the canonical module provides important results of \({\mathbf X}\), in particular with respect to the above mentioned uniformity. The author's intention is an interplay between geometrical properties of \({\mathbf X}\) and the algebraic behaviour of \(R\). In particular there is an explicit description of the canonical ideal of \(R\). Furthermore there is also an description of the Kähler module of differentials in the case of a non-reduced \({\mathbf X}\).
Besides of these results the author's applications to several branches of mathematics described in the third part of this habilitation are interesting and original. The subjects he is interested in are the following: level schemes, hyperplane sections of curves, free resolutions, syzygy module of the canonical module, maximal Cayley Bacharach schemes, liaison, zero-dimensional subschemes of the projective plane, determinantal zero-dimensional schemes, coding theory, computer algebra. One of the main themes is an algebraic characterization of the Cayley-Bacharach property [see e.g. \textit{D. Eisenbud, M. Green} and \textit{J. Harris}, Bull. Am. Math. Soc., New Ser. 33, No. 3, 295-324 (1996; Zbl 0871.14024) for a survey]. The author characterizes Gorenstein schemes by the symmetry of the Hilbert function and the Cayley-Bacharach property. Moreover he discusses the question when a subscheme of a Caley-Bacharach scheme is again a Caley-Bacherach scheme. There is a connection of the Cayley-Bacharach property to the property of \(i\)-uniform position. This again is closely related to the Castelnuovo function introduced as the difference function of the Hilbert function of \({\mathbf X}\). One of the main results is a certain estimate shown for the case of the author's cohomological uniformity. A large part of the third section is devoted to the Castelnuovo theory, in particular its behaviour in prime characteristic and the relation to the Cayley-Bacherach property.
As applications the author is interested in the construction of \({\mathbf X}\) with given properties. He describes several methods based on liaison, determinantal schemes, etc. -- Another highlight is the author's study of reduced schemes \({\mathbf X}\) over a field of \(q = p^e\), \(p= \text{prime}\) characteristic of \(K,\) consisting only of \({\mathbb{F}}_q\)-rational points. In this situation there are strong restrictions on the Hilbert function. Some of these \({\mathbf X}\) define linear codes. It is shown that the invariants and geometric properties of \({\mathbf X}\) are closely related to length, dimension and minimal distance of the corresponding codes. In a final section the author summarizes the computer algebra methods for the construction of explicit samples of zero-dimensional schemes. Hilbert function; module of differentials; uniform position; generic position; liaison; coding theory; canonical module; Cayley Bacharach schemes; zero-dimensional subschemes; Castelnuovo function Kreuzer, M.: Beiträge zur theorie nulldimensionalen unterschemata projektiver räume. Regensburger math. Schr. 26 (1998) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, Linkage, complete intersections and determinantal ideals Contributions to the theory of zero-dimensional subschemes of 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 We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in \(\mathbb C^n\) by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants. complements of arrangements; vanishing cycles; second Lefschetz theorem; isolated singularities of functions on stratified spaces; monodromy Relations with arrangements of hyperplanes, Configurations and arrangements of linear subspaces Complements of hypersurfaces, variation maps, and minimal models of arrangements | 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 \(T\) be a tree with \(n\) vertices, and let \(\mathbb{K}\) be a field. Let \({\alpha}=(\alpha_{t})_{t\in T}\) be an assignment to each vertex of \(T\) a value in \(\mathbb{K}.\) One can then consider the affine scheme \(X_{T}({\alpha})\) given by the equations \(x_{t}x_{t}^{\prime}=1+\alpha_{t}\prod x_{s}\), where the product ranges over all \(s\) adjacent to \(t\). This paper is concerned with computing the number of points \(N_{T}({\alpha})\) on this variety in the case where \(\mathbb{K}\) has \(q\) elements, focusing on three types of trees. Specifically, the trees considered are Dynkin diagrams of type \(\mathbb{A} ,\mathbb{D},\) and \(\mathbb{E}\) in the case where each \(\alpha_{i}\) is invertible.
Suppose that \(T=\mathbb{A}_{n}\). If \(n\) is even then \(X_{\mathbb{A}_{n} }(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{A}_{n}}(1,\dots,1)\): this is proved by studying full domino tilings of the tree. From this it follows that \(N_{\mathbb{A}_{n}}=(q^{n+2}-1)/(q^{2}-1).\) On the other hand, if \(n\) is odd we have \(X_{\mathbb{A}_{n}}(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{A}_{n}}(\alpha,1,\dots,1)\) for some \(\alpha\) depending on the \(\alpha_{i}\) with \(i\) odd: this follows from the partial domino tiling of \(\mathbb{A}_{n}\) leaving the first vertex uncovered. If \(\alpha\neq(-1)^{(n+1)/2}\) then \(N_{\mathbb{A} _{n}}=(q^{(n+1)/2}-1)(q^{(n+3)/2-1})/(q^{2}-1).\) Write \(X_{\mathbb{A} _{n}}(\alpha)=X_{\mathbb{A}_{n}}(\alpha,1,\dots ,1)\) (here \(n\) may be even or odd). If \(n\) is odd, and \(\alpha=(-1)^{(n+1)/2}\) then \(X_{n}(\alpha)\) has a unique singular point, namely \(x_{2k+1}=x_{2k+1}^{\prime}=0,\) \(x_{2k} =x_{2k}^{\prime}=-(-1)^{(n/2+k)}\). In all other cases, \(X_{n}(\alpha)\) is smooth.
While the focus is on counting the number of points on the varieties, further results are obtained when \(T=\mathbb{A}_{n}.\) Let \(Z_{\mathbb{A}_{n}}\) (resp. \(Y_{\mathbb{A}_{n}}\)) be the union of all \(X_{\mathbb{A}_{n}}(\alpha)\) not necessarily invertible (resp. necessarily invertible). Then \(Z_{\mathbb{A}_{n}}\) has \(q^{n+1}\) points and \(Y_{\mathbb{A}_{n}}\) is a smooth open subset of \(Z_{\mathbb{A}_{n}}\;\)having \((q^{n+2}+(-1)^{n+1})/(q+1)\) points.\ For \(0\leq i\leq n+1\) the non-zero cohomology groups with compact support for \(Y_{\mathbb{A} _{n}}\) are \(H_{c}^{i+n+1}(Y_{\mathbb{A}_{n}})\cong \mathbb{Q}(i)\), the Tate Hodge structure of weight \(i\). Also, for \(n\) even \(H_{c}^{i+n}(X_{\mathbb{A}_{n}}(1))\cong\mathbb{Q}(i)\) for all even \(0\leq i\leq n.\)
Suppose now that \(T=\mathbb{D}_{n}.\) By partial domino tilings if \(n\) is even \(X_{\mathbb{D}_{n}}(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{D}_{n}}(\alpha,\beta,1,\dots,1)\) for some \(\alpha \),\(\beta\) which depend on the \(\alpha_{i};\) if \(n\) is odd then \(X_{\mathbb{D} _{n}}(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{D}_{n} }(\alpha,1,\dots,1)\). Formulas are given for \(N_{\mathbb{D} _{n}}\) -- there are six different cases to consider.
Finally, for the diagrams of type \(\mathbb{E}\) different formulas are obtained for the different exceptional cases. Using domino tilings and counting, we have \(X_{\mathbb{E}_{6}}({\alpha})\cong X_{\mathbb{E} _{6}}(1,\dots,1)\) has \(q^{6}+q^{4}+q^{3}+q^{2}+1\) points and \(X_{\mathbb{E}_{8}}({\alpha})\cong X_{\mathbb{E}_{8} }(1,\dots,1)\) has \(q^{8}+q^{6}+q^{5}+q^{4}+q^{3}+q^{2}+1\) points. As \(\mathbb{E}_{7}\) affords only a partial domino tiling we get \(X_{\mathbb{E}_{7}}({\alpha})\cong X_{\mathbb{E}_{7} }(1,\dots,1,\alpha)\) where \(\alpha\) is the value on the last vertex on the long branch. Provided \(\alpha\neq-1\) we have \(q^{7}+q^{5} -q^{2}-1,\) \(N_{\mathbb{E}_{7}}(1,\dots,-1)\) has \(q^{7} +2q^{5}+q^{3}-q^{2}-1\) points. cluster algebras; affine varieties; Dynkin diagrams; finite fields Rational points, Representations of quivers and partially ordered sets, Cluster algebras, Finite ground fields in algebraic geometry On the number of points over finite fields on varieties related to 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 In this work, classical modular forms are generalized to the so-called Picard modular forms. The moduli spaces of principally polarized abelian varieties, having an elliptic element \(M\) of the modular group as an automorphism, are studied. These kinds of elements are called varieties of Picard type. It is obtained that the ring of Picard modular forms of even weight is a direct sum. The ring of Hermitian forms and the Hermitian modular group are discussed, and the genus 1 case is considered as an example. The result is the direct sum of varieties of weight a multiple of 4.
Finally Picard modular forms are discussed, and it is shown that they form a ring which only depends on the isomorphism class of \(M\). Also this ring is generated by 4 polynomials which can be expressed in terms of 3 polynomials \(P_6\), \(P_{12}\) and \(P_{18}\). varieties of Picard type; Picard modular forms; moduli spaces of principally polarized abelian varieties; Hermitian forms; Hermitian modular group --, On Picard modular forms.Math. Nachr. 184 (1997), 259--273. Other groups and their modular and automorphic forms (several variables), Modular and Shimura varieties, Complex multiplication and moduli of abelian varieties On Picard modular 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 Let K/F be a field extension of degree 2 and let A be a central simple K- algebra. A classical result of Albert says that A admits an involution of the second kind if and only if \(Cor_{K/F}([A])=1\). One of the main purposes of the paper is to extend this result to Galois extensions K/F of higher degree. Let \(\sigma_ 1=1,\sigma_ 2,...,\sigma_ n\) be the elements of G and let \(A_ i\) be the algebra A with the K-action twisted through \(\sigma_ i^{-1}\). A generalized involution of the second kind is an anti-imbedding \(A\to A_ n\otimes_ K...\otimes A_ 2\) with further properties. A generalized involution of the second kind exists for A if and only if \(Index\quad (Cor_{K/F}([A]))| (Index([A])^{n-2}).\) Nontrivial examples are split algebras and certain cyclic algebras. In the split case, (classical) involutions of the second kind arise from nondegenerate Hermitian forms. This construction is also generalized to arbitrary Galois extensions. The paper ends with some explicit examples. In particular, the author shows how his approach can be used to compute the corestriction of cyclic algebras of degree 3. This paper is a sequel to [ibid. 57, 449-465 (1979; Zbl 0408.16016) and 66, 205-219 (1980; Zbl 0451.16013)], where involutions of the first kind are generalized. central simple K-algebra; involution of the second kind; Galois extensions; generalized involution; split algebras; cyclic algebras; Hermitian forms; corestriction Finite rings and finite-dimensional associative algebras, Division rings and semisimple Artin rings, Rings with involution; Lie, Jordan and other nonassociative structures, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Brauer groups of schemes, Algebraic numbers; rings of algebraic integers On the corestriction homomorphism and generalized involutions of the second kind | 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 verify the monodromy conjecture [\textit{J. Denef} and \textit{F. Loeser}, J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)] about topological zeta functions for a large class of singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions depending on four variables. They explain the novelty of their approach as follows. Thus, it is known that in the case of three variables all singularities close to a non-degenerate one are non-degenerate as well [\textit{A. Lemahieu} and \textit{L. Van Proeyen}, Trans. Am. Math. Soc. 363, No. 9, 4801--4829 (2011; Zbl 1248.14012)]. Next, in the setting of \textit{E. Artal Bartolo} et al. [Quasi-ordinary power series and their zeta functions. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1095.14005)], all singularities close to a quasi-ordinary one are also quasi-ordinary. However, in contrast to these papers, a new phenomenon arises in the four-dimensional case: there are degenerate singularities arbitrarily close to a nonisolated non-degenerate singularity; this is one of the main difficulties of the proof pointed out by the authors.
The paper is divided into several parts. In the introduction, the authors give a detailed review of a number of previously obtained results and the corresponding extensive bibliography, explain the essence of their approach, and formulate the main statements. Then the monodromy conjecture for the topological zeta function and the main properties of Newton polyhedra are discussed, and configurations of faces of the Newton polyhedron that do not ensure the existence of the corresponding pole of the topological zeta function are studied. After that, the authors study face configurations, which, on the contrary, always nontrivially contribute to the multiplicity of the expected monodromy eigenvalue and prove the conjecture for a certain class of non-degenerate singularities in arbitrary dimension. The last section contains a proof of the monodromy conjecture for non-degenerate singularities of functions depending on four variables. In the appendix, it is briefly discussed some basic concepts related to the geometry of the lattice, and the necessary results used in the article. nonisolated singularities; non-degenerate singularities; topological zeta functions; monodromy conjecture; toric varieties; Newton polytopes; non-convenient Newton polyhedra; eigenvalues of monodromy; corners; hypermodular function; lattice geometry Toric varieties, Newton polyhedra, Okounkov bodies, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) On the monodromy conjecture for non-degenerate hypersurfaces | 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 generalizes some known results of \textit{G. Harder} on cohomology of semi-simple split algebraic groups [Invent. Math. 4, 165- 191 (1967; Zbl 0158.395)] to a larger class of global base fields. cohomology of semi-simple split algebraic groups J.-C. Douai , Cohomologie des schémas en groupes sur les courbes définies sur les corps quasi-finis ...., Journal of Algebra 103 , n^\circ 1 ( 1986 ), 273 - 284 . MR 860706 | Zbl 0604.14034 Group varieties, Cohomology theory for linear algebraic groups Cohomologie des schémas en groupes sur les courbes définies sur les corps quasi-finis et loi de réciprocité. (Cohomology of group schemes over curves defined over quasi-finite fields and reciprocity law) | 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 Although being one of the comparatively recent creations in mathematics, abstract commutative algebra has a long and fascinating genesis. Its development into a beautiful, deep and widely applied mathematical discipline, in its own right, must be understood as a function of that of algebraic number theory and algebraic geometry, both of which essentially gave birth to it.
In the second half of the 19th century, two concrete classes of commutative rings (and their ideal theory) marked the beginning of what is now called commutative algebra: rings of integers of algebraic number fields, on the one hand, and polynomial rings occurring in classical algebraic geometry and invariant theory, on the other hand. In the first half of the 20th century, after the basics of abstract algebra had been established, commutative algebra grew into an independent subject, mainly under the influence of E. Noether, E. Artin, W. Krull, B. L. van der Waerden, and others. In the 1940s this abstract and general framework was applied, in turn, to give classical algebraic geometry both a completely new footing and a new toolkit for further investigations. In this context, the epoch-making innovations by Chevalley, Zariski, and Weil not only created a revolution in algebraic geometry, but also had a very strong impact on the quickened growing of commutative algebra itself in the following decades.
The 1950s and 1960s saw the development of the structural theory of local rings, the foundations of algebraic multiplicity theory, Nagata's counter-examples to Hilbert's 14th problem, the introduction of homological methods into commutative algebra, and other pioneering achievements. However, the indisputably most characteristic mark of this period was A. Grothendieck's creation of the theory of schemes, the (till now) ultimate revolution of algebraic geometry. His foundational work culminated in a far-reaching alliance of commutative algebra and algebraic geometry, which also made is possible, in turn, to apply geometric methods as a tool in commutative algebra.
In the following decades, geometric and homological methods have substantially engraved the vigorous research activities in commutative algebra. At present, commutative algebra is an independent, abstract, deep, and smoothly polished subject for its own sake, on the one hand, and an indispensible conceptual, methodical, and technical resource for modern algebraic and complex analytic geometry, on the other hand. Studying or actively pursuing research in geometry or number theory requires today a profound knowledge of commutative algebra; however, most textbooks on algebraic or complex analytic geometry usually assume such a knowledge from the beginning, often refer to the (undoubtedly excellent) great standard texts on abstract commutative algebra, or survey a minimal account of the basic results, mostly in a series of appendices. The reason for that is quite clear, for the fusion of the two subjects is close to such an extent that the necessary prerequisites from commutative algebra would occupy a considerable part of the text, thereby possibly discouraging the reader who is mainly interested in the geometric aspects of the subject. Conversely, most textbooks on commutative algebra present the material in its purely algebraic and perfectly polished abstract form, with at most a few elementary hints and applications to the related geometry. This situation really creates a dilemma for both students and teachers of algebraic geometry or commutative algebra.
The present book under review aims at toning down this traditional, nevertheless somewhat artificial discrepancy. The author, in person one of the leading experts in both fields, has tried to write on commutative algebra in a way that makes the heritage, the geometric character, and the geometric applications of the subject as apparent as possible. A first attempt in this direction, namely to offer a textbook that mixes algebra and geometry in an organic manner, has been successfully carried out by \textit{E. Kunz} in 1980. His textbook ``Einführung in die kommutative Algebra und algebraische Geometrie.'' Braunschweig etc.: Friedr. Vieweg (1980; Zbl 0432.13001) and the English translation ``Introduction to commutative algebra and algebraic geometry.'' Boston etc.: Birkhäuser (1985; Zbl 0563.13001), (reprint 2013; Zbl 1263.13001) provided an exposition of the basic definitions and results in both commutative algebra and algebraic geometry, centered around the (at this time) recent solution of Serre's problem on projective modules over polynomial rings or, respectively, Kronecker's longstanding problem on complete intersections in projective spaces. Along this road, E. Kunz gave a very natural introduction to commutative algebra and algebraic geometry, especially emphasizing the concrete elementary nature of the objects which were at the beginning of both subjects.
The book under review aims at the same goal; however, it does so under much wider aspects. In fact, the author strives for a rather complete and up-to-date exposition of the present state of commutative algebra, with all its old and new links to modern algebraic geometry. As he says in the preface, his precise goal has been, from the beginning, to cover at least all the material that graduate students in algebraic geometry should have at their disposal, in particular those studying \textit{R. Hartshorne}'s matchless modern textbook ``Algebraic geometry'' [New York etc.: Springer-Verlag (1977; Zbl 0367.14001)] and, perhaps subsequently, \textit{A. Grothendieck}'s and \textit{J. Dieudonné}'s ``Éléments de géométrie algébrique'' [Publ. Math., Inst. Haut. Étud. Sci. I--IV (1960- 1967; Zbl 0118.36206; Zbl 0122.16102; Zbl 0136.15901; Zbl 0135.39701; Zbl 0144.19904; Zbl 0153.02202)].
According to this strategy, the text is subdivided into three major parts and six appendices.
After a beautiful introduction, providing readers of different background knowledge or expertise with instructions for both self-teaching and teaching, and after a brief synopsis of the elementary definitions concerning rings, ideals and modules, part I of the book under review discusses the ``Basic constructions'' in commutative algebra. This first part consists of seven separate chapters: Chapter 1 is still introductory and surveys some of the history of commutative algebra in number theory, algebraic curve theory, one-dimensional complex analysis, and invariant theory. It also explains the dictionary ``Commutative algebra -- projective algebraic geometry'', including Hilbert's basis theorem, Hilbert's syzygy theorem, Hilbert's Nullstellensatz, graded rings, and the Hilbert polynomials.
Chapter 2 deals with the localization principle in commutative algebra, with an analysis of zero-dimensional rings, and chapter 3 turns to associated prime ideals and the primary decomposition in noetherian rings. Chapter 4 discusses Bourbaki's proof of Hilbert's Nullstellensatz, Nakayama's lemma, integral dependence, and the normalization process, whereas chapter 5 goes back to graded rings and modules, including the construction of the blow-up algebra, Krull's intersection theorem, and their geometric interpretation. Chapter 6 introduces the concept of flatness, the Tor-functor, and derives the most important flatness criteria. Completions of rings and Hensel's lemma are presented in chapter 7, together with their geometric significance, and Cohen's structure theorems are also found here. -- Each chapter comes with a large number of well-prepared exercises, most of which concern additional theoretical material and results. The same holds for the following chapters, whereby hints and solutions for selected exercises are given at the end of the book. By this method, the author manages to cover even more interesting and recent material, at least in outlines.
Part II of the book is entitled ``Dimension theory''. It consists of nine more chapters. Chapter 8 illustrates the history of dimension theory in its topological, geometric, and algebraic aspects. In this complexity, it is much more advanced than the following ones, and (as the author says) it is actually meant to be read for motivation and ``for culture only''. Chapter 9 gives then the fundamental definitions of algebraic dimension theory (Krull dimension of a ring), with special emphasis on the case of dimension zero. Chapter 10 covers Krull's principal ideal theorem, regular sequences, parameter systems, regular local rings, and their algebro-geometric meanings. Chapter 11 is entitled ``Dimension and codimension one'' and treats normal rings, discrete valuation rings, Serre's criterion, Dedekind domains, and the ideal class group. Hilbert-Samuel functions and polynomials, together with their natural appearance in multiplicity theory are discussed in chapter 12. The geometric aspects of dimension theory, above all Noether normalization and the finiteness of the integral closure of an affine ring, are the subject of chapter 13, and the following chapter 14 is devoted to both classic and modern elimination theory. -- Chapter 15 gives, for the first time in a textbook on commutative algebra, an account of the fast growing computational part of the subject. Gröbner bases, initial ideals, and their applications to constructive module theory and projective algebraic geometry are the principal items here. The rather mathematical approach, relative to the usual more computational ones, is particularly convenient for algebraic geometers, and a set of seven computer algebra projects, at the end of the chapter, shows how the computational possibilities of this approach lead to new (at least conjectural) insights. -- Chapter 16 is concerned with the differential calculus in commutative algebra, that is with modules of differentials, tangent and cotangent bundles, smoothness and generic smoothness, the Jacobi criterion, infinitesimal automorphisms, and some deformation theory.
Part III of the book is devoted to the homological methods in commutative algebra. Chapter 17 deals with regular sequences by means of the Koszul complex, and with applications of the Koszul complex to the study of the cotangent bundle of \(\mathbb{P}^n\). Chapter 18 discusses the notion of depth and the Cohen-Macaulay property. The significance of the Cohen- Macaulay property is illustrated from various viewpoints, ranging from Hartshorne's theorem on connectedness in codimension one to the theorem on flatness over a regular base to primeness criteria using Serre's characterization of normality. -- The homological theory of regular local rings occupies chapter 19. This chapter contains, apart from the standard material on projective dimension, minimal resolutions, global dimension, and the Auslander-Buchsbaum formula, also an application to the factoriality of local rings via stably free modules. Chapter 20 concerns free resolutions and their role in algebra and algebraic geometry. The author, who has contributed to this topic by a good deal of his own research in the past, presents here various criteria of exactness, mainly based on the approach via Fitting ideals and Fitting invariants, and he gives some very instructive applications, e.g., the Hilbert-Burch theorem characterizing ideals of projective dimension one, and, at the end, an algebraic treatment of Castelnuovo-Mumford regularity. The concluding chapter 21 gives an account of duality theory for local Cohen-Macaulay rings, and some parts of the theory of Gorenstein rings. This includes the discussion of the canonical module and its properties, maximal Cohen- Macaulay modules and their duality theory, and the theory of linkage à la Peskine and Szpiro from the algebraic point of view. An interesting feature in this chapter, among many others in the text, is the treatment of the canonical module via reduction to the case of an Artinian ring. This makes the whole topic pleasantly concrete and lucid, at least for the beginner.
The main text is followed by seven appendices, in which the author provides both some more technical material from algebra, as it is needed in the course of the text, and some furthergoing topics related to it. In brevity, these appendices are the following:
1. Field theory (transcendency degree, separability, \(p\)-bases);
2. Multilinear algebra (including divided powers and Schur functors);
3. Homological algebra (projective modules, injective modules, complexes, homology, derived functors, Ext and Tor, double complexes, spectral sequences, and derived categories);
4. Local cohomology (local and global cohomology, local duality, depth and dimension via local cohomology);
5. Category theory (categories, functors, natural transformations, adjoint functors, limits, representable functors, and Yoneda's lemma);
6. Limits and colimits (flat modules as limits of free modules);
7. Where next? (hints for further reading).
The book ends with a section containing hints and solutions for more than one hundred selected exercises spread over the entire text. The bibliography is extremely rich and carefully selected, being a true help for both the reader and the interested expert.
Altogether, the book under review has filled a longstanding need for a text on commutative algebra which thoroughly reflects the naturally grown relations to algebraic geometry. Containing numerous novel results and presentations, the book is still fairly self-contained, accessible for beginners, and a treasure for teachers and researchers in both fields. The consequent mixing of algebra and geometry, from the beginning to the end, has made it impossible to present commutative algebra in its most systematic and abstract perfection, as it has been done, for example, in the great standard textbooks of \textit{O. Zariski} and \textit{P. Samuel} [cf. ``Commutative algebra'', Vol. I. Princeton etc.: D. van Nostrand Company (1958; Zbl 0081.26501) and II (1960; Zbl 0121.27801); reprints, respectively, New York: Springer-Verlag (1975; Zbl 0313.13001) and 1976 (Zbl 0322.13001)], \textit{N. Bourbaki} [``Commutative algebra'', Chapters 1--7. Paris: Hermann (1972; Zbl 0279.13001), 2nd printing Berlin etc.: Springer-Verlag (1989; Zbl 0666.13001)], and \textit{H. Matsumura} [``Commutative algebra.'' New York: W. A. Benjamin (1970; Zbl 0211.06501); 2nd edition (1980; Zbl 0441.13001)]. However, the book under review, apart from having the compensating advantage of combining algebra and geometry in a natural manner, at least touches upon numerous recent development and results not yet contained in any other textbook. In this sense, it should be regarded as an unique and excellent enrichment of the existing literature in commutative algebra and algebraic geometry, just as a new standard text among the celebrated others, and as a highly welcome supplement to them. bibliography; Hilbert's basis theorem; dictionary: commutative algebra-projective algebraic geometry; Hilbert's syzygy theorem; Hilbert's Nullstellensatz; Hilbert polynomials; dimension theory; Dedekind domains; Hilbert-Samuel functions; elimination theory; computer algebra; modules of differentials; homological methods; Koszul complex; Cohen-Macaulay property; duality theory; linkage Eisenbud D, \textit{Commutative Algebra: With a View Toward Algebraic Geometry}, 150, Springer New York, 1995. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, History of commutative algebra, General commutative ring theory, Theory of modules and ideals in commutative rings, Actions of groups on commutative rings; invariant theory, Dimension theory, depth, related commutative rings (catenary, etc.) Commutative algebra. With a view toward 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 This article can be regarded as a report on the progress in Hodge theory since 1975.
In the first section, we review Deligne's construction of mixed Hodge structures on the cohomology of all algebraic varieties, and simplicial varieties, over \(\mathbb{C}\), for it sets the tone of much of the work that followed [cf. \textit{P. Deligne}, Théorie de Hodge. I--III, Actes Congr. internat. Math. 1970, Part I, 425-430 (1971; Zbl 0219.14006), Publ. Math., Inst. Hautes Étud. Sci. 40(1971), 5-57 (1972; Zbl 0219.14007) and ibid. 44(1974), 5-77 (1975; Zbl 0237.14003)]. -- Much of section 2 also contains review material, namely the theorem of \textit{W. Schmid} [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] on degeneration of Hodge structures in the abstract, i.e., in the absence of any hypothesis that the variation of Hodge structure arises from a family of smooth projective varieties. These are the nilpotent orbit theorem and the SL(2)-orbit theorem. -- Section 3 is about \(L_ 2\)-cohomology. It is an integral part of the subject. There is a general relation between \(L_ 2\)-cohomology and harmonic forms, which gives a key motivation for its introduction: it was the most familiar way to establish the existence of useful Hodge decompositions.
Ironically, it was not by \(L_ 2\)-cohomology that Hodge structures for the intersection homology groups of singular projective varieties were finally produced. Instead, it came out of work in algebraic analysis (i.e., \(\mathbb{D}\)-modules), a subject that also has blossomed during the past 15 years, as we report in section 4. The starting point is the so- called Riemann-Hilbert correspondence, which asserts that taking the de Rham complex sets up an equivalence of categories between holonomic \(\mathbb{D}\)-modules with regular singularities and perverse sheaves (and likewise for their derived categories). The idea is to equip such \(\mathbb{D}\)-modules with Hodge (and weight) filtrations, so that they induce (mixed) Hodge structures on hypercohomology. -- Section 5 is devoted to Deligne cohomology and to its generalization, by Beilinson, to noncompact varieties. -- Several approaches have been used to put mixed Hodge structures on homotopy groups of algebraic varieties, discussed in section 6. The method of \textit{J. W. Morgan} [Publ. Math., Inst. Hautes Étud. Sci. 48, 137-204 (1978; Zbl 0401.14003) and ibid. 64, 185 (1986; Zbl 0617.14013)] is based on Sullivan's theory of minimal models for differential graded algebras. The method of \textit{R. M. Hain} [\(K\)-Theory 1, 271-324 (1987; Zbl 0637.55006) and ibid. 1, No. 5, 481-497 (1987; Zbl 0657.14004)] is based on Chen's method of iterated integrals. Alternative approaches to mixed Hodge theory on homotopy groups, due Deligne and to Navarro Aznar, are only briefly mentioned.
The notion of a variation of mixed Hodge structure is a very natural generalization of that of a variation of Hodge structure. The idea, naturally, is that a variation of mixed Hodge structure on \(X\) must at least yield a filtered local system \((\mathbb{V},W)\) on \(X\), and that the graded local systems \(\text{Gr}^ W_ k \mathbb{V}\) should be honest polarized variations of Hodge structure. In section 7, we present such a notion, that of admissible variations of mixed Hodge structure. The definition is due to Steenbrink and Zucker in the case of curves, and to Kashiwara in the general case.
Section 8 is devoted to the question of determining which local systems on a given quasi-projective manifold underlie a variation of Hodge structure. We primarily discuss very recent work of \textit{C. Simpson}, which, in our opinion, promises to have profound repercussions in Hodge theory. He uses the nonlinear P.D.E. methods of differential geometry to obtain a correspondence between irreducible vector bundles on a compact Kähler manifold and stable Higgs bundles with vanishing total Chern class (see theorem 8.9). A Higgs bundle is a vector bundle \({\mathcal E}\), together with an operator-valued one-form \(\theta\) on \({\mathcal E}\) of square 0. \(\mathbb{C}^*\) acts on Higgs bundles in the obvious way, by dilating \(\theta\). The remarkable fact is that the fixed points of this \(\mathbb{C}^*\)-action correspond exactly to complex variations of Hodge structure (corollary 8.10), and real variations correspond to fixed points which give self-dual Higgs bundles (see proposition 8.12). Some striking applications are presented. Bibliography; \(L_ 2\)-cohomology; \(\mathbb{D}\)-modules; Beilinson cohomology; regulator maps; Grothendieck motives; degeneration of Hodge structures; intersection homology; Riemann-Hilbert correspondence; Deligne cohomology; variation of mixed Hodge structure; Higgs bundles Brylinski, J.-L., Zucker, S.: An overview of recent advances in Hodge theory. In: Several Complex Variables VI. Encyclopedia Math. Sci., vol. 69, pp. 39--142. Springer, Berlin (1990) Transcendental methods, Hodge theory (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), History of algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), History of mathematics in the 20th century An overview of recent advances in Hodge 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 Let \(Q=(Q_0,Q_1,t,h)\) be a quiver with set of vertices \(Q_0\), set of arrows \(Q_1\) and functions \(t,h\colon Q_1\to Q_0\) attaching to an arrow \(\alpha\) its tail \(t(\alpha)\) and head \(h(\alpha)\). A representation of \(Q\) of dimension vector \(\underline n=(n_i\mid i\in Q_0)\) is a collection of linear maps \(L=(L_\alpha\colon V_{t(\alpha)}\to V_{h(\alpha)}\mid\alpha\in Q_1)\), where \(V_i\) is an \(n_i\)-dimensional vector space associated with \(i\in Q_0\). Let the base field \(K\) be infinite and of any characteristic. Identifying \(V_i\) with the space \(K^{n_i}\) of column vectors, and \(L_\alpha\) with an \(n_{h(\alpha)}\times n_{t(\alpha)}\) matrix operating by left multiplication, the representation \(L\) corresponds to a point in the affine space \(R=R(Q,\underline n)=\bigoplus_{\alpha\in Q_1}K^{n_{h(\alpha)}\times n_{t(\alpha)}}\). The group \(\text{GL}(\underline n)=\prod_{i\in Q_0}\text{GL}_{n_i}(K)\) acts on \(R\) such that \(g\cdot L=(g_{h(\alpha)}L_\alpha g_{t(\alpha)}^{-1}\mid\alpha\in Q_1)\) for any \(g\in\text{GL}(\underline n)\).
The purpose of the paper under review is to describe the algebra of \(\text{GL}(\underline n)\)-semi-invariants (i.e., the algebra of \(\text{SL}(\underline n)\)-invariants) of the quiver \(Q\). There is a canonical way to blow up an arbitrary quiver to obtain a bipartite quiver, i.e., a quiver for which any vertex is either a source or a sink. The crucial observation of the authors is that this construction corresponds to forming an associated fibre bundle. Algebraically this gives induction of modules over algebraic groups. Applying Frobenius reciprocity, the authors conclude that the semi-invariants of the original quiver can be obtained from those of the enlarged bipartite quiver.
The authors define generating semi-invariants of bipartite quivers as partial polarizations of determinants of block matrices. The proof is very natural and is based on a translation of the first fundamental theorem of classical invariant theory. As an application the authors obtain the generators of the algebra of invariants of \(d\)-tuples of \(n\times n\) matrices under simultaneous conjugation in the case of a field of any characteristic [see \textit{S. Donkin}, Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)].
As the authors mention in the paper, similar results in the case of characteristic 0 have been independently obtained by \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)]. matrix invariants; representations of quivers; semi-invariants of quivers; algebras of semi-invariants; fibre bundles; bipartite quivers Mátyás Domokos and, A. N. zubkov. semi-invariants of quivers as determinants, Transformation Groups, 6, 9-24, (2001) Trace rings and invariant theory (associative rings and algebras), Representations of quivers and partially ordered sets, Geometric invariant theory, Vector and tensor algebra, theory of invariants, Actions of groups and semigroups; invariant theory (associative rings and algebras), Representation theory for linear algebraic groups Semi-invariants of quivers as determinants | 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 concerns the global dimension of the rings of global sections \(D^ \lambda = \Gamma (\mathbb{P}^ n, {\mathcal D}^ \lambda)\) of the sheaves of twisted differential operators \({\mathcal D}^ \lambda\) on \(\mathbb{P}^ n\), which are indexed by \(\mathbb{C}\). It turns out that the answer for \(\text{gldim }D^ \lambda\) can be \(n\), \(2n\), or \(\infty\). These cases arise when \(\lambda \in \mathbb{C} \setminus \mathbb{Z}\), \(n \in \mathbb{Z} \setminus\{ -1, \dots, -n-1\}\), \(n \in \{-1, \dots, -n-1\}\).
For the finite cases the main technique used is the Beilinson-Bernstein localisation theorem in combination with the spectral sequence
\[
H^ q(\mathbb{P}^ n, \text{Ext}^ p({\mathcal F}, {\mathcal G})) \implies \text{Ext}^ n({\mathcal F}, {\mathcal G}).
\]
In the infinite case an ad hoc method is used to produce a module of infinite projective dimension. A complete answer for the global dimension of rings of global sections of twisted differential operators on flag varieties (equivalently, minimal primitive factors of enveloping algebras of semisimple Lie algebras) is known. In detail, suppose that \(\mathfrak g\) is a semisimple Lie algebra with Cartan subalgebra \(\mathfrak h\) and let \(\lambda \in {\mathfrak h}^*\). Associated to \(\lambda\), there is a sheaf of twisted differential operators on the flag variety, \(X\), denoted by \({\mathcal D}^ \lambda\). One writes \(D^{\lambda} := U({\mathfrak g}) \text{ann }M (\lambda) \cong \Gamma (X, {\mathcal D}^ \lambda)\). There is no loss of generality in assuming that \(\lambda\) is antidominant. One says that \(\lambda\) is regular if its stabiliser in the Weyl group is trivial. \textit{T. J. Hodges} and \textit{S. P. Smith} [J. Lond. Math. Soc., II. Ser. 32, 411-418 (1985; Zbl 0588.17009)] showed that if \(\lambda\) is regular then \(\text{gldim }D^ \lambda\) is finite (in fact they found an upper bound and, appealing to an earlier result of Levasseur, pointed out that it is attained). \textit{A. Joseph} and \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 49, 361-384 (1984; Zbl 0543.17004)] showed that \(\text{gldim }D^ \lambda\) is infinite, if \(\lambda\) is singular. \textit{H. Hecht} and \textit{D. Miličić} [Proc. Am. Math. Soc. 108, 249-254 (1990; Zbl 0714.22011)] have also obtained this latter result by showing that the localisation functor \({\mathcal D}^ \lambda \otimes \underline{\phantom m}\) has infinite cohomological dimension.
There do not appear to be analogous general results, for the global dimension of the rings of global sections of twisted differential operators on complete homogeneous spaces, in the literature. global dimension; rings of global sections; sheaves of twisted differential operators; spectral sequence; flag varieties; enveloping algebras of semisimple Lie algebras; Weyl group DOI: 10.1112/blms/24.2.148 Rings of differential operators (associative algebraic aspects), Universal enveloping (super)algebras, Homological dimension in associative algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Arithmetic ground fields for curves, Commutative rings of differential operators and their modules, Sheaves of differential operators and their modules, \(D\)-modules The global dimension of rings of differential operators on 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 This paper gives a characterization of a class of algebras called concealed-canonical algebras.
The original definition is the following: An algebra \(\Sigma\) is called concealed-canonical if it is obtained as the endomorphism ring of a tilted module, \(T\), over a canonical algebra, where all the indecomposable summands of \(T\) have strictly positive rank. This rank is defined via the Euler quadratic form.
There is a characterization of these algebras via the shape of their Auslander-Reiten quivers. One of the theorems says that they can be characterized via the fact that their A-R quivers have a sincere separating tubular family of stable tubes. Another equivalence is the existence of a small hereditary Noetherian \(k\)-category with an Artinian center \(k\) and without projectives and a torsion free tilting object \(T\), with endomorphism ring \(\Sigma\).
One of the main tools is the fact that the paper shows that in this case the indecomposables can be divided into three big classes, \(\text{mod}_+(\Sigma)\), \(\text{mod}_0(\Sigma)\) and \(\text{mod}_-(\Sigma)\), the modules in the first one have strictly positive rank, the ones in the middle one have zero rank, and the ones in the last one have negative rank. Also there are no nonzero maps between modules from the right to the left. Moreover, each one of them is a union of connected components of the Auslander-Reiten quiver. Also \(\text{mod}_0(\Sigma)\) is uniserial and decomposes into a coproduct of uniserial categories.
The paper uses various techniques from algebraic geometry. concealed-canonical algebras; separating exact subcategories; components of Auslander-Reiten quivers; quadratic forms; Artin algebras; tame hereditary algebras Lenzing, H.; de la Peña, J. A., Concealed-canonical algebras and separating tubular families, Proc. Lond. Math. Soc., 78, 513-540, (1999) Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Vector bundles on curves and their moduli, Representations of quivers and partially ordered sets Concealed-canonical algebras and separating tubular families. | 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 reduction algebraic group, \(\rho:G\to\text{GL}(V)\) a finite dimensional complex representation and \(\pi:V\to V//G\) the categorical quotient of the action of \(G\) on \(V\) induced by \(\rho\). There is the conjecture in invariant theory that if \(G\) is a connected semisimple group and \(\pi\) an equidimensional morphism, then \(V//G\) is smooth (and hence isomorphic to an affine space). This conjecture was formulated twenty years ago and since then has been verified for simple groups, for semisimple groups and their irreducible representations \(\rho\) (the verification followed a posteriori from the corresponding classification results) and for tori.
In the present paper this conjecture is proved for a large class of the semisimple groups having exactly two simple factors (also in an a posteriori way, via classification of the equidimensional representations of these groups). complex reduction algebraic groups; complex representations; actions; invariant theory; connected semisimple groups; equidimensional morphisms; simple groups; irreducible representations; equidimensional representations DOI: 10.1006/jabr.1993.1024 Representation theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Group actions on varieties or schemes (quotients), Geometric invariant theory, Classical groups (algebro-geometric aspects) Equidimensional representations of \(2\)-simple 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 \(X\) be a non-singular complex projective variety of dimension \(d\) and let \(Y\) be a divisor with normal crossing, but whose irreducible components may be singular. By considering the singular stratification of \(X\) induced by \(Y\), we can define \(N^*(Y)\), the union of conormal bundles of the strata. It is a closed Lagrangian subset of \(T^*X\). We consider regular holonomic \({\mathcal D}\)-modules on \(X\) whose characteristic varieties are contained in \(N^*(Y)\). Such regular holonomic \({\mathcal D}\)-modules form an abelian category, in which each object is of finite length. These are called regular holonomic \({\mathcal D}\)-modules on \((X,Y)\). The present paper solves the moduli problem for regular holonomic \({\mathcal D}\)-modules. This is done by extending the notion of pre-\({\mathcal D}\)-modules to our case, defining semi-stability, and constructing a moduli for these, using geometric-invariant-theoretic methods and Simpson's conjecture of moduli for semi-stable \(\Lambda\)-modules. Also, a moduli is constructed for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism with various good properties. moduli; holonomic \(D\)-module; Simpson's conjecture of moduli for semi-stable \(\Lambda\)-modules; conormal bundles; Lagrangian subset; Riemann-Hilbert correspondence Nitsure, N.: Moduli of regular holonomic D-modules with normal crossing singularities. Duke math. J. 99, 1-39 (1999) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, Complex-analytic moduli problems Moduli of regular holonomic \(\mathcal D\)-modules with normal crossing 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 \(F\) be a base field, and let \(\mathcal F\colon Fields/F\to Sets\) be a functor from the category of field extensions of \(F\) to the category of sets. An element \(\alpha\in\mathcal F(E)\) is said to be defined over a subfield \(K\) of \(E\) if \(\alpha\) is in the image of the morphism \(\mathcal F(K)\to\mathcal F(E)\). The essential dimension \(\text{ed}^{\mathcal F}(\alpha)\) of \(\alpha\) is the minimum of the transcendence degrees of \(K\) over \(F\) where \(K\) runs over all fields of definition of \(\alpha\). If \(p\) is a prime, the essential \(p\)-dimension \(\text{ed}^{\mathcal F}_p(\alpha)\) of \(\alpha\) is the minimum of \(\text{ed}^{\mathcal F}(\alpha_{E'})\) where \(E'\) runs over all prime-to-\(p\) extensions of \(E\). The essential \(p\)-dimension \(\text{ed}_p(\mathcal F)\) of \(\mathcal F\) is the supremum of \(\text{ed}^{\mathcal F}_p(\alpha)\) for \(\alpha\in\mathcal F(E)\) and \(E\) an extension of \(F\).
Let \(m,n\) be positive integers with \(m\) dividing \(n\). The functor \(Alg_{n,m}\) is defined as the functor that takes a field extension \(E\) of \(F\) to the set of isomorphism classes of central simple \(E\)-algebras of degree \(n\) and period dividing \(m\). In the paper under review, the authors give lower and upper bounds for the essential \(p\)-dimension of this functor when \(p\) is different from \(\text{char}(F)\). Recall that if \(p^r\) and \(p^s\) are the largest powers of \(p\) dividing \(n\) and \(m\) respectively, then we have \(\text{ed}_p(Alg_{n,m})=\text{ed}_p(Alg_{p^r,p^s})\), so it is sufficient to consider \(\text{ed}_p(Alg_{p^r,p^s})\). One then has the inequality \((r-1)2^{r-1}\leq\text{ed}_p(Alg_{p^r,p^s})\leq p^{2r-2}+p^{r-s}\) if \(p=2\) and \(s=1\) and \((r-1)p^r+p^{r-s}\leq\text{ed}_p(Alg_{p^r,p^s})\leq p^{2r-2}+p^{r-s}\) otherwise.
The key points in the proof are as follows. First, the lower bound for the essential \(p\)-dimension of \(Alg_{p^r,p^s}\) is expressed in terms of the essential \(p\)-dimension of a certain algebraic torus. Second, a result of \textit{R. Lötscher, M. MacDonald, A. Meyer} and \textit{Z. Reichstein} [``Essential dimension of algebraic tori'', J. Reine Angew. Math. (to appear)] is used to calculate the essential \(p\)-dimension of this torus. Finally, to prove the upper bound, the authors prove an inequality that bounds \(\text{ed}_p(Alg_{p^r,p^s})\) by \(\text{ed}_p(Alg_{p^r})=\text{ed}_p(Alg_{p^r,p^r})\) and use a result of \textit{A. Ruozzi} [J. Algebra 328, No. 1, 488-494 (2011; Zbl 1252.16016)] to obtain the final inequality.
A corollary of this result is that if \(\text{char}(F)\neq 2\), then \(\text{ed}_2(Alg_{8,2})=\text{ed}(Alg_{8,2})=8\). This proves the existence of a central simple algebra of degree 8 and period 2 over a field \(F\) which cannot be written as the tensor product of three quaternion algebras. essential dimension; central simple algebras; Brauer groups; algebraic tori Baek, S.; Merkurjev, A., Essential dimension of central simple algebras, Acta Math., 209, 1-27, (2012) Finite-dimensional division rings, Brauer groups (algebraic aspects), Group actions on varieties or schemes (quotients), Linear algebraic groups over arbitrary fields, Galois cohomology of linear algebraic groups Essential dimension of central simple 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 K be an algebraic function-field in one variable over an algebraically closed constant field k of characteristic \(p\geq 0\), denote by \(g=g_ k\) the genus of K and let \(K_ 0\) be a rational subfield of K. It is a classical result of Hurwitz that if \(k={\mathbb{C}}\) and \(g\geq 2\), then the group Aut(K/\({\mathbb{C}})\) of automorphisms of K over \({\mathbb{C}}\) is finite and the order \(ord(Aut(K/{\mathbb{C}}))\leq 84(g-1).\) The method of proof works over any k of characteristic 0. \textit{H. L. Schmid} [J. Reine Angew. Math. 179, 5-15 (1938; Zbl 0019.00301)] considered the cases in which \(p>0\) and, using work of \textit{F. K. Schmidt}, proved that the group Aut(K/k) is finite. \textit{P. Roquette} [Math. Z. 117, 157-163 (1970; Zbl 0194.353)] showed that if \(p>g+1\geq 3\), then the Hurwitz bound \(ord(Aut K/k)\leq 84(g-1)\) holds also in this case, except in the case \(p\geq 5\), and \(K=k(x,y)\), \(y^ 2=x^ p-x\). Moreover he showed that in that case \(G=Aut(K/k)\) is a central extension of \(C_ 2\) (the cyclic group of order 2) by PGL(2,p).
It is natural to ask which finite groups arise as automorphism groups of function fields. Evidently one requires \(g\geq 2\) and in the case \(k={\mathbb{C}}\) \textit{L. Greenberg} [Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 207-226 (1974; Zbl 0295.20053)] showed that every finite group is realizable as the automorphism group of some K/\({\mathbb{C}}\) and that is true in the case \(p>0\) also (for references and a history of the problem, see Chapter 1 of this thesis).
Consider in particular the case when K is hyperelliptic. The author addresses the question as to which finite groups are realizable as some Aut(K/k) in that case and in certain natural generalizations of it (see below). A hyperelliptic function-field K possesses exactly one rational subfield \(K_ 0\) such that \([K:K_ 0]=2\) and there exists a subgroup Z of \(G=Aut(K/k)\) such that Z is in the centre Z(G) and \(Z=Gal(K/K_ 0)\). But then Aut(K/k) is a central extension of Z by a finite subgroup of \(Aut(K_ 0/k)\) and all those groups actually arise as automorphism groups of hyperelliptic function fields [\textit{K. Iwasawa}, Ann. Math., II. Ser. 58, 548-572 (1953; Zbl 0051.266)].
The author gives a complete solution of the problem of determining all those extensions, including their defining equations, and also of the following more general one, which is suggested by the foregoing. A function-field of type \(F[G_ 0| q,p]\) is a function-field (in one variable) over an algebraically closed field of constants k of characteristic \(p\geq 0\) such that: 1) there exist finite subgroups Z and G of Aut(K/k), with Z a subgroup of the centre Z(G) of G, \(Z\neq G\); \(Z\cong C_ q\) where \(q\neq p\) is a prime and \(C_ q\) denotes the cyclic group of order q; \(G/Z\cong G_ 0\); and 2) the fixed field of Z is rational. For fields of that type, G is a central extension of \(Z\cong C_ q\) by a finite subgroup \(G_ 0\) of \(Aut(K_ 0/k)\) and so the first task is to determine all such rational function fields. The group \(G_ 0\) acts on the places of \(K_ 0/k\) and one finds the orbits of \(G_ 0\). The author determines the central extensions of \(C_ q\) by finite subgroups of \(Aut(K_ 0/k)\) in the cases when \(G_ 0\) is an elementary Abelian q-group or a semi-direct product of an Abelian q-group with a cyclic group. He goes on to give a detailed study of function-fields \(F[G_ 0| q,p]\) and in particular of extensions of automorphisms \(\sigma \in Aut(K_ 0/k)\) to K and of the ramification type G(K,G) of K, which is defined as follows. With the foregoing notation, let \(K_ 1\) be the fixed field of G and \(K_ 0\) the fixed field of Z and let \(Q_ 1,...,Q_ r\) be the places of \(K_ 1\) ramified in \(K_ 0/K_ 1\), \(e_ 1,...,e_ r\) their ramification indices in \(K/K_ 1\), then \(T(K,G)=(G;e_ 1,...,e_ r)\). If \(q\neq p\), then the isomorphism type of G is determined completely by T(K,G), but if \(q=p\) then that is no longer the case and he obtains the appropriate analogues.
The thesis is largely self contained, the necessary results concerning group extensions, for example, being included and the greater part of it is concerned with a very detailed discussion of the function-field of type \(F[G_ 0| q,p]\) and their defining equations, for groups G. finite groups; automorphism groups of function fields; hyperelliptic function-field R. Brandt, Über die Automorphismengruppen von algebraischen Funktionenkörpern, PhD thesis, Universität Essen, 1988. Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On the groups of automorphisms of algebraic function 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 [For the entire collection see Zbl 0597.00007.]
Let G be a connected and simply connected \({\mathbb{Q}}\)-split semisimple algebraic group over \({\mathbb{Q}}\) with a \({\mathbb{Q}}\)-involution \(\sigma\) and a \({\mathbb{Q}}\)-split maximal torus T such that \(\sigma (t)=t^{-1}\) for all t in T. If \(X:=G/H\) with \(H:=\{g\in G|\sigma (g)=g\}\), every \(G_{{\mathbb{R}}}\)-orbit in \(X_{{\mathbb{R}}}\) is a semisimple non-Riemannian symmetric space of \(\epsilon\)-involution type in the sense of Oshima- Sekiguchi. This paper deals with the definition and detailed investigation of Eisenstein series attached to such symmetric spaces. These Eisenstein series turn out to possess analyticity properties like their counterparts in the case of Riemannian symmetric spaces; they admit a continuation as meromorphic functions on \(X(T)\otimes_{{\mathbb{Z}}} {\mathbb{C}}\) where X(T) is the group of rational characters of T, with explicit functional equations.
Included as a special case of these results, viz. for \(G=SL(n+1)\), is the author's main theorem in an earlier paper [Ann. Math., II. Ser. 116, 177- 212 (1982; Zbl 0497.10012)] where Eisenstein series are related to zeta functions for prehomogeneous vector spaces. However, the method used by the author in his earlier papers studying such zeta functions seems to work as well in the present generality, without major modification. The precise relationship between the Eisenstein series defined by the author and those constructed by Oshima-Sekiguchi for symmetric spaces as above is, according to the author, still to be clarified. semisimple symmetric spaces of Chevalley groups; semisimple algebraic group; non-Riemannian symmetric space; Eisenstein series; functional equations; zeta functions for prehomogeneous vector spaces F. SATO, Eisenstein series on semisimple symmetric spaces of Chevalley groups, Advanced Studie in Pure Math. 7, Automorphic Forms and Number Theory, (I. Satake, ed.), Kinokuniya, Tokyo, 1985, 295-332. Harmonic analysis on homogeneous spaces, Holomorphic modular forms of integral weight, Zeta functions and \(L\)-functions of number fields, Homogeneous spaces and generalizations, Linear algebraic groups over global fields and their integers Eisenstein series on semisimple symmetric spaces of Chevalley 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 Roughly speaking, cluster algebras, introduced recently by \textit{S.~Fomin} and \textit{A.~Zele\-vinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)], are defined by \(n\)-regular trees whose vertices correspond to \(n\)-tuples of cluster variables and edges describing birational transformations between two \(n\)-tuples of variables. Model examples of cluster algebras are coordinate rings of double Bruhat cells. Also every \(m\times n\), \(m\leq n\), integer matrix \(Z\), such that the matrix \([(DZ)_{ij}]_{i,j\leq m}\) is skew-symmetric for a certain positive integer diagonal \(m\times m\) matrix \(D\), defines a natural cluster algebra \({\mathcal A}(Z)\) of geometric type.
The authors introduce in the paper the cluster manifold \(\mathcal X\) as a ``handy'' nonsingular part of \(\text{ Spec}({\mathcal A}(Z))\) and a Poisson structure on \(\mathcal X\) which is compatible with the cluster algebra structure in the sense that the Poisson bracket is homogeneously quadratic in any set of cluster variables. Then edge transformations describe simply transvections with respect to the Poisson structure. Poisson and topological properties of the union of generic orbits of a toric action on this Poisson variety are studied.
The second goal of the paper is to extend calculations of the number of connected components in double Bruhat cells to a more general setting of geometric cluster algebras and compatible Poisson structures. Namely, given a cluster algebra \(\mathcal A\) over the reals, the number of connected components in the union of generic symplectic leaves of any compatible Poisson structure on \(\mathcal X\) is computed. Finally, the general formula is applied to a special case of Grassmannian coordinate ring. cluster algebras; Poisson brackets; toric action; Grassmannians; Poisson-Lie groups; Sklyanin bracket M. Gekhtman, M. Shapiro, A. Vainshtein, \textit{Cluster algebras and Poisson geometry}, Mosc. Math. J. \textbf{3} (2003), no. 3, 899-934, 1199. Poisson manifolds; Poisson groupoids and algebroids, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Associative rings and algebras arising under various constructions, Simple, semisimple, reductive (super)algebras Cluster algebras and Poisson 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 Sei M eine reell-analytische Mannigfaltigkeit. Der Verf. beweist folgende Sätze: (1) Sei X eine kompakte irreduzible analytische Menge in M und \(f: X\to {\mathbb{R}}\) eine nirgends negative analytische Funktion. Dann ist f eine Summe von Quadraten meromorpher Funktionen. - (2) Sei \(I\subset {\mathcal O}(M)\) ein endlich erzeugtes Ideal und sei die Nullstellenmenge Z(I) von I kompakt. Dann gilt für das von Z(I) erzeugte Ideal IZ(I): (a) IZ(I)\(=^ R\sqrt{I}\), wobei \({}^ R\sqrt{I}\) das reelle Radikal von I bezeichnet; (b) IZ(I)\(=I \Leftrightarrow I\) ist reell.
Zentrales Beweismittel für die Sätze ist die Theorie der totalen Ordnungen von Ringen, vor allem der folgende Satz: Sei \(I\subset {\mathcal O}(M)\) ein Primideal and \(X=Z(I)\). Es existiere ein \(\psi\in I\), derart, daß \(\{\psi =0\}\) kompakt ist. Sind dann \(f_ 1,...,f_ m\in {\mathcal O}(M)\), derart, daß ihre Klassen modulo I in einer totalen Ordnung auf \({\mathcal O}(M)/I\) positiv sind, so ist \(\{f_ 1>0,...,f_ m>0\}\cap X\neq \emptyset.\) real nullstellensatz; Hilbert's 17th problem; sum of squares of; meromorphic functions; real radical Ruiz, Jesús M., On Hilbert's 17th problem and real Nullstellensatz for global analytic functions, Math. Z., 0025-5874, 190, 3, 447-454, (1985) Real algebraic and real-analytic geometry, Real-analytic manifolds, real-analytic spaces On Hilbert's 17th problem and real Nullstellensatz for global analytic 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 This paper contains a complete list of all finite Auslander-Reiten quivers of local Gorenstein orders over a complete Dedekind domain of finite lattice type. For each translation quiver \(\Gamma\) in this list, a Gorenstein order \(\Lambda\) is explicitly indicated, with \(\Gamma\) as its Auslander-Reiten quiver. In each case, indecomposable \(\Lambda\)-lattices are described. finite Auslander-Reiten quivers; local Gorenstein orders; finite lattice type; indecomposable \(\Lambda \) -lattices Wiedemann, A.: Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities. J. pure appl. Algebra 41, No. 2-3, 305-329 (1986) Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Representation theory of associative rings and algebras, Singularities in algebraic geometry Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve 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 \(p\) be a prime number and \(N\) an integer prime to \(p\). Let \(X_1(pN)\) denote the modular curve of level \(pN\), and let \(I=I(N)\) be the Igusa curve of level \(N\) modulo \(p\). Let \(K\) be a finite extension of \(\mathbb{Q}_p (\zeta) \), where \(\zeta\) is a \(p\)th root of unity, let \(M\) denote the space of cusp forms of weight 2 on \(X_1(Np)\) over \(K\), and let \(M^0\) denote the subspace of \(M\) consisting of forms whose associated differentials have residue zero at all supersingular points. The space \(M^0\) is spanned by the \(p\)-fold forms and by the forms \(f\) such that \(\sum_{d\in (\mathbb{Z}/p \mathbb{Z})^\times}\langle d\rangle f=0\). The author constructs a pairing \(M^0\times M^0\to\mathbb{Q}_p\) which relates well to the Hecke action. When the forms are ordinary, the pairing can be described in terms of the Serre-Tate invariant. A similar pairing can be defined on forms of higher weight, and the author sketches out this definition.
To study the integrality properties of the pairing, the author first describes integral structures that can be put on the de Rham cohomology of a wide open space (proofs are given in an appendix). This leads to several results on integrality and reduction properties. For example, let \(f\) be an eigenform of weight \(k\), \(2<k\leq p\), on \(X_1(N)\) modulo \(p\), and let \(F\) be an eigenform of weight 2 on \(X_1(Np)\) lifting \(f\). Let \(F'\) be the normalization of the form obtained from \(F\) by an Atkin-Lehner automorphism associated to a \(p\)th root of unity. In earlier work [Invent. Math. 110, No. 2, 263-281 (1992; Zbl 0770.11024)], \textit{R. Coleman} and \textit{J. F. Voloch} constructed a cohomology class \([\bar F']\) in the first de Rham cohomology group of the Igusa curve \(I\) associated to the reduction of \(F'\). In this paper, the author obtains a natural cohomology class \([F']\) in the first crystalline cohomology group of \(I\) associated to \(F'\) itself (the reduction of \([F']\) is a multiple of \([\bar F'])\). Let \(\rho_f\) be the Galois representation attached to \(f\). By work of \textit{B. H. Gross} [Duke Math. J. 61, No. 2, 445-517 (1990; Zbl 0743.11030)], the triviality of the class \([\bar F']\) is equivalent to \(\rho_f\) being tamely ramified at \(p\). The author shows that the reduction of \([F']\) is zero precisely when knowledge of the ramification of \(\rho_f\) does not determine the splitting behavior of the restriction of \(\rho_f\) to an inertia group at \(p\). elliptic modular forms; pairing; Hecke action; Serre-Tate invariant; forms of higher weight; integrality properties; de Rham cohomology; reduction properties; Igusa curve; crystalline cohomology; Galois representation Robert F. Coleman, A \?-adic inner product on elliptic modular forms, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 125 -- 151. \(p\)-adic theory, local fields, Galois representations, Arithmetic aspects of modular and Shimura varieties, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Local ground fields in algebraic geometry A \(p\)-adic inner product on elliptic modular forms. -- Appendix: Forms on an annulus | 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 Singularities of plane curves occupy a central position in singularity theory: numerous definitions of equisingularity coincide and the classification up to this is well known, and has been the starting point of many investigations. This manuscript, which originates from 1973, gives an account of the theory from the viewpoint of maximal contact; in particular, equisingularity class is characterised via `simpler' curves having maximal contact with the given one. This is given for reducible, as well as for irreducible curves. Moreover an algorithm to determine the class (via Newton polygons and blowing-up) is described, but is not fully implemented. The theory is developed in detail, but not independently of other versions of classification. This viewpoint, unlike some of the alternatives, works well in finite characteristic. Singularities of plane curves; equisingularity; maximal contact; Newton polygons; blowing-up; finite characteristic M. Lejeune , Sur l'équivalence des singularités des courbes algébroïdes planes . Coefficients de Newton. Thèse. Singularities in algebraic geometry, Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Finite ground fields in algebraic geometry, Analytic subsets of affine space On the equivalence of singularities of plane algebroid curves. Newton coefficients | 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 Cohomology rings of finite groups provide an interesting and important class of commutative rings. Examples can be explored by computer calculations, which in turn lead to conjectures about their properties. The author explains what these properties are, what might be conjecturally true of them, and what has been proven so far.
To give a feeling of what the article deals with we cite some of the titles of the sections: 1. Properties of Cohomology Rings (Graded Commutativity and Elementary Abelian Groups, Functorial Properties, Varieties and Quillen's Theorem, Dimension). 2. Depth and Detection (Depth and Regular Sequences, Detecting Cohomology). 3. Computing Group Cohomology (Projective Resolutions, Products and Relations, Gröbner Bases). 4. Regularity and Tests for Completion (Homogeneous Parameters, A Sample Test for Completion, Quasiregular Sequences and Regularity). cohomology rings of finite groups Jon F. Carlson, Cohomology, computations, and commutative algebra, Notices Amer. Math. Soc. 52 (2005), no. 4, 426 -- 434. Cohomology of groups, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Semialgebraic sets and related spaces, Software, source code, etc. for problems pertaining to group theory Cohomology, computations, and commutative 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 The aim of this book written by the well known Soviet mathematicians Yu. I. Manin and A. A. Panchishkin is to inform about classical methods and results in number theory and to present recent achievements in number theory associated with applications of the theory of automorphic forms and automorphic representations in number theory.
Section I is entitled ``Problems and methods''. In chapter 1 (``Elementary number theory'') the authors deal with well-known facts from elementary number theory such as decomposition of any natural number into prime factors, the Euclidean algorithm, Chinese remainder theorem, Gauss' quadratic reciprocity law and such problems of analytic number theory as the distribution of prime numbers in natural sequences. Also the authors discuss problems arising in the theory of representation of integers by quadratic forms, Diophantine approximations of irrational numbers and connections with the theory of continued fractions.
In chapter 2 (``Selected modern problems in elementary number theory'') the authors discuss such computational problems in number theory as the search for a prime divisor of a given large natural number and their applications to asymmetric coding, reliable tests (for primality). Also the authors give a sketch of a proof of the irrationality of \(\zeta\) (3) presented by Apéry in 1978.
Section II is called ``Ideas and Theories''. In chapter 1 (``Induction and Recursion'') the authors explain some methods of logic, namely induction and recursion in their connections with number theory. The authors show that the notions of Diophantine sets, partially recursive functions, recursively enumerable sets, algorithmic unsolvability play an important role in number theory. The authors emphasize the importance of Gödel's incompleteness theory.
In chapter 2 (``Arithmetic of algebraic numbers'') the authors communicate some facts about Galois extensions of number fields, actions of the Frobenius elements, decomposition of prime ideals in field extensions, Hilbert's norm residue symbol and its properties, Artin's reciprocity map, and Galois cohomology.
In chapter 3 (``Arithmetic of algebraic varieties'') the authors deal with problems of solvability of Diophantine equations. They consider Diophantine problems connected with problems of algebraic geometry, discuss the importance of such notions from algebraic geometry as schemes of finite type over rings, cycles and divisors on varieties and schemes, regular differential forms on varieties, linear space of the divisor, heights associated with the divisor in number theory. The authors give a sketch of Faltings' approach to the Tate conjecture about isogenies of abelian varieties.
In chapter 4 (``Zeta-functions and modular forms'') the authors introduce the reader into an extremely wide area of problems arising from the connections between the theory of zeta- and L-functions and the theory of automorphic forms and automorphic representations. The authors begin with a definition of zeta-functions of an arithmetic scheme and discuss the famous Deligne result about the ``Weil conjecture'' for the zeta-function attached to the scheme over a finite field, give an example of application of the Deligne estimate to problems of estimates of trigonometric sums. The authors briefly present the theory of L-functions associated with rational representations of Galois groups, Artin formalism, the theory of Hecke characters and Tate theory. One of the most important part of the modern theory of L-functions connects with the theory of automorphic forms. The authors give a sketch of the theory of modular forms and discuss the problems of representability of Dirichlet series in terms of their Euler product. Discussing connections between modular forms and Galois representations the authors present important conjectures of number theory such as Taniyama-Weil conjecture, Birch- Swinnerton-Dyer conjecture, Ramanujan-Petersson conjecture, Artin conjecture, Sato-Tate conjecture, Serre conjecture. In conclusion of this chapter the authors discuss Langlands' functoriality principle.
The list of references contains 457 titles.
The present book is written at a high mathematical level and can be considered as a good introduction to number theory. Diophantine sets; recursively enumerable sets; Arithmetic of algebraic varieties; Diophantine problems; Tate conjecture; isogenies of abelian varieties; Zeta-functions; L-functions; automorphic forms; automorphic representations; arithmetic scheme; Weil conjecture; modular forms; Galois representations Research exposition (monographs, survey articles) pertaining to number theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Discontinuous groups and automorphic forms, Arithmetic algebraic geometry (Diophantine geometry), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Abelian varieties and schemes, Arithmetic problems in algebraic geometry; Diophantine geometry, Connections of number theory and logic, Algebraic number theory: global fields, Elementary number theory Introduction to number 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 This paper is a continuation of [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, 129-160 (1995; Zbl 0827.11024)]. It is concerned with the study of the \(p\)-adic valuations of eigenvalues of the Hecke operator \(U_p\) acting on certain spaces of cusp forms of level divisible by \(p\) for the congruence subgroup \(\Gamma_1 (pN)\) of \(\text{SL}_2 (\mathbb{Z})\), \(p\nmid N\). For a positive integer \(M\), a non-negative integer \(k\), and a commutative \(\mathbb{Q}\)-algebra \(R\), let \(S_{k+2} (\Gamma_1 (M); R)= S_{k+2} (\Gamma_1 (M); \mathbb{Q}) \otimes_\mathbb{Q} R\), with \(S_{k+2} (\Gamma_1 (M); \mathbb{Q})\) the \(\mathbb{Q}\)-vector space of cusp forms (for \(\Gamma_1 (M)\)) of weight \(k+2\) all of whose Fourier coefficients at the standard cusp \(\infty\) are rational numbers. On the space \(S_{k+2} (\Gamma_1 (M); R)\) one has an action of Hecke operators \(T_\ell\) for all primes \(\ell \nmid M\), \(U_\ell\) for \(\ell \mid M\), and the diamond operators \(\langle d\rangle_M\) for \(d\in (\mathbb{Z}/ M\mathbb{Z})^\times\). In particular, for \(M= pN\), \(p\nmid N\), and \(R= \mathbb{Q}_p\), one has a decomposition
\[
S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) \cong \bigoplus^{p- 2}_{j= 0} S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^j),
\]
where \(\chi: (\mathbb{Z}/ p\mathbb{Z} )^\times\to \mathbb{Z}_p\) is the Teichmüller character, and where \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi)\) consists of those \(f\in S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p)\) such that \(\langle d\rangle_p f= \chi(d) f\) for all \(d\in (\mathbb{Z}/ p\mathbb{Z})^\times\).
In the sequel \(p\) will always be an odd prime, \(N\geq 5\), \(0\leq k< p\), and \(a\) will be an integer with \(0< a< p-1\). Then the action of \(U_p\) on \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^a)\) (which is a space of newforms at \(p\)) is semi-simple with eigenvalues algebraic integers of absolute value \(p^{(k+ 1)/2}\). \(U_p\) also acts on \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p )^{p-\text{old}}\), the space of forms which are old at \(p\). The eigenvalues of \(U_p\) are again algebraic integers of absolute value \(p^{(k+ 1)/2}\). Write
\[
S= S(k, b)= \begin{cases} S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^b) &\text{ if }b\not\equiv 0\pmod {p-1}\\ S_{k+2} (\Gamma_1 (pN); \mathbb{Q}_p )^{p-\text{old}} &\text{ if } b\equiv 0 \pmod {p-1}. \end{cases}
\]
For an interval \(I\subset \mathbb{R}\) let \(S_I= \bigoplus_{\lambda\in I} S_\lambda\), where \(S_\lambda \subset S\) is such that all of the eigenvalues of \(U_p\) on \(S_\lambda\) have slope \(\lambda\). In [loc. cit.] it was shown that, for \(N\geq 5\) and \(0< a< p-1\), the Newton polygon \(P(k, a)= \text{det} (1- U_p T|S(k,a))\) is bounded below by an explicit Hodge polygon. For any Newton polygon its associated contact polygon is defined as the highest Hodge polygon lying on or below it and having the same endpoints. In the context of [loc. cit.], one defines \(t^i= t^i (k, a)\), \(i= 1, \dots, k\), as the number of units that the slope \(i\) edge of the Hodge polygon should be raised so that it meets the slope \(i\) edge of the contact polygon of \(P(k, a)\) (or equivalently, so that it meets the Newton polygon). Let \(i\) be an integer \(1\leq i\leq k\). As a first result one has that \(t^i (k, a)= 0\) if one has \(i\leq a\) and \(k+ 1- i\leq p- 1- a\). For \(i= a+1\), \(t^i (k, a) =0\) if and only if \(S_2 (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^{k+ 1-i} )_{(0, 1)}= 0\). If \(k+ 1-i= p-a\) then \(t^i (k, a)= 0\) if and only if \(S_2 (\Gamma_1 (pN); \mathbb{Q}_p) (\chi^{-i} )_{(0, 1)} =0\). For \(i> a+ 1\) or \(k+ 1-i> p-a\) one has \(t^i (k, a)> 0\).
The main results of the paper give the dimensions of \(S_{k+2} (\Gamma_1 (pN); \mathbb{Q}) (\chi^a)_I\) for suitable values of \(k\), \(a\) and the interval \(I\subset \mathbb{R}\). The following cases are dealt with:
(i) \(i\leq a\), \(k+ 1- i\leq p- 1-a\), and \(I= [i]\);
(ii) \(i+1\leq a\), \(k+ 1- i\leq p- 1- a\), and \(I= (i, i+1)\);
(iii) \(i=0\), \(a\leq p-1- k\), and \(I= (i, i+1)\);
(iv) \(i=k\), \(a\geq k\), and \(I= (i, i+1)\);
(v) \(k>1\), \(a= (p-1)- (k-1)\), \(i=1\), and \(I= (0, 1)\);
(vi) \(k> 1\), \(a= k-1\), \(i=k\), and \(I= (0, 1)\).
Actually, (i), (ii), (iii) and (iv) also hold for \(N\leq 2\wedge k\equiv a\pmod 2\). These results follow from a comparison of modular forms and cohomology, much in the spirit of [loc. cit.], in a motivic setting. Various complicated expressions for suitable (crystalline) cohomology groups of the motive at hand are proved. These proofs take up the greater part of the text. Galois representation; Gauss-Manin connection; cohomology groups of motives; \(p\)-adic valuations of eigenvalues of the Hecke operator; spaces of cusp forms; Fourier coefficients; diamond operators; Newton polygon; Hodge polygon; contact polygon; modular forms 41. Ulmer, Douglas L. On the Fourier coefficients of modular forms. II \textit{Math. Ann.}304 (1996) 363--422 Math Reviews MR1371772 Fourier coefficients of automorphic forms, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Congruences for modular and \(p\)-adic modular forms On the Fourier coefficients of modular forms. II | 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 \(A_n=K\langle x_1,\dots,x_n\rangle\) be a free associative algebra over a field \(K\) of arbitrary characteristic. The main result of the paper under review gives an estimate of the minimal degree of the elements of the subalgebra of \(A_n\) generated by two algebraically independent elements \(f\) and \(g\) under the natural conditions that the leading monomials of \(f\) and \(g\) are algebraically dependent and the degrees of \(f\) and \(g\) do not divide each other. The result is stated in terms of the degree of the commutator \([f,g]\).
The paper is motivated by the estimates of the degree for \(K[x_1,\dots,x_k]\) obtained by \textit{I. P. Shestakov} and \textit{U. U. Umirbaev} [J. Am. Math. Soc. 17, No. 1, 181--196 (2004; Zbl 1044.17014); ibid. 17, No. 1, 197--227 (2004; Zbl 1056.14085)] over a field of characteristic 0 with techniques based on Poisson algebras. These estimates were essential in the description of the tame automorphisms of \(K[x,y,z]\) and the proof that the Nagata automorphism is wild. Later \textit{L. Makar-Limanov} and \textit{J.-T. Yu} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 533--541 (2008; Zbl 1137.16029)] used the lemma on radicals for the Malcev-Neumann algebras of formal power series [\textit{G. M. Bergman}, Bull. Malays. Math. Soc., II. Ser. 1, 29--41 (1978; Zbl 0395.16002); ibid. 2, 41--42 (1979; Zbl 0409.16001)] to obtain a lower bound for the degree of the nonconstant elements in the subalgebra generated by two elements \(f,g\) in \(K[x_1,\dots,x_k]\) and \(K\langle x_1,\dots,x_k\rangle\), again in characteristic 0.
Now the authors use the lemma of Bergman on the centralizers established in his papers cited above. The lemma states that if \(R\) is a commutative ring and \(S\) is an ordered semigroup, and \(a\) is an element of the Malcev-Neumann algebra \(R((S))\) with an invertible leading term \(a_uu\) (\(a_u\in R\), \(u\in S\)), then there exists an element \(f\) with leading term 1, such that the element \(c=f^{-1}af\) has support entirely in the centralizer of \(u\) in \(S\). The results of the present paper are essential for the important results on the automorphisms of \(K\langle x,y,z\rangle\) over an arbitrary field \(K\) obtained by \textit{A. Belov-Kanel} and \textit{J.-T. Yu} [Sel. Math., New Ser. 17, No. 4, 935--945 (2011; Zbl 1232.13005); ibid. 18, No. 4, 799--802 (2012; Zbl 1268.16025)]. degree estimates; subalgebras; free associative algebras; commutators; Mal'cev-Neumann algebras; centralizers; ordered groups Li, Y.-C.; Yu, J.-T., Degree estimate for subalgebras, J. Algebra, 362, 92-98, (2012) Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Polynomial rings and ideals; rings of integer-valued polynomials, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Automorphisms and endomorphisms, Computational aspects of associative rings (general theory) Degree estimate for subalgebras. | 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 volume is the third of a three volume series on the theory of theta functions. Like the first volume (1983; Zbl 0509.14049) and the second one (1984; Zbl 0549.14014), it is based on D. Mumford's lectures held at the Tata Institute of Fundamental Research (Bombay) during November 1978 -- March 1979. Whilst Volumes I and II are devoted to the classical, geometric and analytic theory of complex abelian varieties and their associated theta functions, together with their most recent applications to moduli spaces of curves and Jacobian varieties, non-linear partial differential equations and special integrable Hamiltonian systems, this final volume focusses on the purely algebraic theory of theta functions. The aim of the authors is to compare and clarify the deep interrelations between the different ways of viewing (and using) theta functions in analysis, algebraic geometry, and representation theory. To be more precise, theta functions occur
(i) as analytic functions depending on complex vectors and Riemann matrices (in the classical setting),
(ii) as matrix coefficients of representations of Heisenberg groups and/or metaplectic groups, and
(iii) as sections of line bundles on abelian varieties and/or their moduli spaces (in the general setting of algebraic geometry over arbitrary groundfields),
and these three conceptually and methodically completely different frameworks for theta functions are shown to be equivalent, on an ultimate and deep level, in the course of the book.
In this regard, the present third volume, consisting of the concluding fourth chapter of the whole series, reflects the recent progress that has been made, to a great extent by the authors themselves, in understanding the ubiquity and crucial role of theta functions in various branches of mathematics. - In particular, the algebraic aspect of theta functions, which is the relevant one when viewing them as sections of line bundles on families of polarized abelian varieties over arbitrary base schemes, has been introduced and developed about 25 years ago by \textit{D. Mumford} in his celebrated three part paper ``On the equations defining abelian varieties'' published in Invent. Math. 1, 287--354 (1966), 3, 75--135 and 215--244 (1967; Zbl 0219.14024). As this paper is highly advanced and barely accessible to non-specialists, and since the development of this general (algebraic) approach is still far from being completed, the authors also give a simplified explicit treatment of the algebraic theory of theta functions, including D. Mumford's basic original ideas and, moreover, related further results by \textit{G. Kempf}, \textit{I. Barsotti}, \textit{J.-I. Igusa}, \textit{L. Moret-Bailly} and the co-autor \textit{P. Norman}. The structure of the book is as follows:
Section 1 provides an introduction to Heisenberg groups. These are locally compact complex Lie groups which contain the unit circle \(\mathbb{C}^*_ 1\) as a normal subgroup, in the center, such that the factor group is abelian and locally compact. The central result, within this introduction, is the main theorem about representations of Heisenberg groups, which is due to Stone, Von Neumann, and Mackey. --- In section 2 the theory of Heisenberg representations is specialized to the real case, that is, to the case where the factor group of a Heisenberg group is a real vector space. This includes the construction of special realizations of such Heisenberg representations via elements in the Siegel upper-half space, particularly the so-called Fock representation, and the introduction of theta functions as matrix coefficients for these special realizations. --- Section 3 deals with the interrelation between the theta functions associated with Heisenberg representations, and the classical ones defined by complex tori. This link is explained by showing how the representation theory of finite Heisenberg groups occurs, in a natural way, in the study of sections of line bundles on complex abelian varieties.
In section 4 the authors turn to the purely algebraic theory of theta functions, and introduce the reader to the adelic methods developed by \textit{D. Mumford} in his original approach to that topic. In the course of this section, they discuss the \(p\)-adic (adelic) version of Heisenberg groups and their significance for the study of towers of abelian varieties, isogenies and (symmetric) line bundles on abelian varieties. This is then used, in section 5, to define theta functions algebraically, namely as certain functions on the tower of an abelian variety, which are associated with sections of ample symmetric line bundles on the corresponding abelian variety. The fact that the construction and the basic properties of these algebraic theta functions are quite parallel to the classical analytical theory, is very skillfully demonstrated. In an appendix I to section 5, a scheme-theoretical version (with a view towards relative abelian schemes) of the algebraic theta functions is briefly sketched, and the following appendix II provides a panorama relating all the important Heisenberg representations.
Section 6 is devoted to a further generalization of theta functions, namely to the construction of theta series associated with positive definite (rational) quadratic forms. This is used, in the sequel, to study the algebra of theta functions, in particular the polynomial identities satisfied by them. Among those theta relations which are of protruding importance, is the algebraic generalization of Riemann's theta relation. Its special reformulations and interpretations, mainly in terms of the Heisenberg action on towers of sections of ample degree one symmetric line bundles on abelian varieties, are fully explained in section 7, whereas in the following section 8 the classical functional equation (with respect to points in the Siegel upper-half space) of Riemann's theta function is generalized to the algebraic theta functions characterized as matrix coefficients of the representations of the real Heisenberg groups. Then, in section 9, a further very natural and important generalization of the theta functions is investigated. The authors introduce theta functions depending not only on quadratic forms, as in section 6, but also on a fixed spherical harmonic polynomial. This is done from three different points of view, namely
(i) by differentiating analytic theta functions with respect to polynomial partial differential operators,
(ii) by the representation-theoretic treatment of theta functions and their transformation laws with respect to pluri-harmonic functions and, finally,
(iii) by using the purely algebraic method of defining analytic modular forms (as theta functions) for pluri-harmonic polynomials and (positive definite rational) quadratic forms, which is essentially due to \textit{I. Barsotti} [cf. Sympos. Math., Roma 3, 247--277 (1970; Zbl 0194.52201)].
The very general theory of algebraic theta functions explained in this section leads directly to the frontiers of current research on moduli theory for general abelian schemes [cf., e.g. \textit{L. Moret-Bailly}, ``Pinceaux de variétés abéliennes'', Astérisque 129 (1985; Zbl 0595.14032)].
The concluding section 10 is devoted to one of the main applications of theta functions in algebraic geometry, namely to the study of the homogeneous coordinate ring of an abelian variety. The topic discussed here originated with \textit{D. Mumford's} earlier work on the explicit computation of bases for linear systems on abelian varieties [cf. Lectures at C.I.M.E. \(3^ 0\) Ciclo Varenna, 1969, Quest. algebraic Varieties, 29--100 (1970; Zbl 0198.25801) and the paper in Invent. Math. 1 and 3 cited above]. Subsequently, these questions have been investigated by \textit{S. Koizumi}, \textit{T. Sekiguchi}, \textit{G. R. Kempf}, and others. Two approaches have proved to be far-reaching: the direct study of linear systems by theta functions and the more abstract cohomological methods together with the use of the finite Heisenberg group (cf. section 3 of this book). Following closely the recent work by \textit{G. R. Kempf} [cf. Am. J. Math. 111, No. 1, 65--94 (1989; Zbl 0673.14023)], the authors demonstrate here how these two basic methods work. The book ends with some related comments on the projective embeddings of the moduli spaces for abelian varieties with level-\(n\) structures.
Altogether, this third volume represents the point of culmination of this extremely beautiful and important series of D. Mumford's lectures. It throws a bridge, unique in literature, between the classical theory of theta functions, as developed in the first two volumes, and the most recent ideas, generalizations and developments in the current research. In the last few sections many open problems are raised, and the entire fascinating, enlightening and masterly presentation of that highly advanced topic should be appealing to both experts in the field and ambitious (graduate) students, just as to physicists working in quantum field theory.
This book is an indispensable guide to the current literature! metaplectic groups; algebraic theory of theta functions; representations of Heisenberg groups; sections of line bundles on complex abelian varieties; isogenies; tower of an abelian variety; theta relations; homogeneous coordinate ring of an abelian variety D. Mumford, \textit{Tata lectures on theta} (1988). Theta functions and abelian varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Algebraic moduli of abelian varieties, classification, Theta series; Weil representation; theta correspondences Tata lectures on theta. III | 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 describe the derived category of a tubular algebra \(\Sigma\) over an arbitrary base field and show that its automorphism group acts with at most three orbits on the set of tubular families. We call this number of orbits the index of \(\Sigma\). It is known that the index of a tubular algebra is one for \(k=\overline k\) and may be two for \(k=\mathbb{R}\). We show that index three actually occurs for \(k=\mathbb{Q}\). derived categories; tubular algebras; automorphism groups; numbers of orbits; weighted projective lines Representations of associative Artinian rings, Derived categories, triangulated categories, Vector bundles on curves and their moduli A tubular algebra with three types of separating tubular families. | 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 0575.00008.]
The classification of singular points which can occur on normal quartic surfaces has been known over seventy years [\textit{C. Jessop}, ''Quartic surfaces with singular points'' (Cambridge 1916)]. Still unknown are all possible combinations of the singularities. The author gives a partial answer to this problem by applying the theory of \textit{E. Looijenga} of rational surfaces with effective anti-canonical divisor [Ann. Math., II. Ser. 114, 267-322 (1981; Zbl 0509.14035)].
Let \(\pi: \bar X\to X\) be a minimal resolution of a normal quartic surface X. Then X is either a K3-surface, or a rational surface, or a ruled irrational surface. In the first case all singularities of X are double rational points. The author considers the second case in which X has one minimal elliptic singularity and double rational points. The fundamental cycle D of the elliptic singularity represents the anticanonical class \(-K_{\bar X}\). By technical reasons the author restricts himself to the case D is irreducible and \(D^ 2=-1\), \(-2\), or \(-3\). This corresponds to either simple elliptic singularities of type \(\tilde E_ 8, \tilde E_ 7, \tilde E_ 6\), or cusp singularities \(T_{2,3,7}, T_{2,4,5}, T_{3,3,4}\), or unimodal exceptional singularities \(E_{12}, Z_{11}, Q_{10}\). Let L be the orthogonal complement of \(K_{\bar X}\) in Pic\((\bar X)\) and let \(\lambda =\pi^*(H)\) where H is a plane section of X. Then \(\lambda\in L\), \(\lambda^ 2=4\) and \(\lambda \cdot f>2\) for every isotropic vector f in L. The exceptional curves of double rational singularities of X correspond to irreducible components of the finite root system in the sublattice \(L'=Ker(res:\;L\to Pic(D)).\) Together with Looijenga's theorem of Torelli type for rational surfaces this allows the author to reduce the problem to the classification of the vectors \(\lambda_ 0\) as above in the abstract lattice \(L_ 0\) isomorphic to L and symmetric root systems in the sublattice \(L_ 0/{\mathbb{Z}}\lambda_ 0\) of \(({\mathbb{Z}}\lambda_ 0)^{\perp}\otimes {\mathbb{Q}}\). The latter are derived by elementary transformations known in the Lie theory.
The following is an example of the results obtained in the paper: Assume that X has a singularity \(\tilde E_ 8\). Then its other singularities are double rational points corresponding to the Dynkin diagrams obtained from the Dynkin diagram of either type \(B_ 9\) or type \(E_ 8\) by elementary operations repeated twice such that the resulting set of vertices has no vertex associated to the short simple root.
The same approach is applied to the classification of singularities of plane sextics by considering the double cover of the plane branched over the sextic.
Note that the classification of singular quartic surfaces with triple points was given by more direct computational method in the works of \textit{T. Takahashi, K. Watanabe} and \textit{T. Higuchi} [Sci. Rep. Yokohama Natl. Univ., Sect. I 29, 47-70; 71-94 (1982; Zbl 0586.14030)]. Dynkin graphs; minimal resolution of a normal quartic surface; classification of singularities of plane sextics; classification of singular quartic surfaces Urabe, T.: Singularities in a certain class of quartic surfaces and sextic curves and Dynkin graphs. Proc. 1984 Vancouver Conf. Alg. Geom., CMS Conf. Proc.6, 477-497 (1986) Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Singularities in algebraic geometry Singularities in a certain class of quartic surfaces and sextic curves and Dynkin graphs | 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 Cayley-Algebra Cay ist nichtassoziativ, nicht-kommutativ und hat als \({\mathbb{C}}\)-Vektorraum die Dimension 8. Der von den Kommutatoren der Elemente von Cay aufgespannte Untervektorraum V ist 7-dimensional. Die Gruppe G der Algebraautomorphismen von Cay ist die einfache komplexe algebraische Gruppe vom Typ \(G_ 2\). Sie operiert treu auf V. - Der Autor bestimmt Erzeugende und Relationen von \({\mathbb{C}}[V^ m]^ G\), der Algebra der G-invarianten Polynomfunktionen auf \(V^ m (m\in {\mathbb{N}})\). algebra of invariant polynomial functions; Cay Schwarz, G. W., Invariant theory of \(G_2\), Bull. Amer. Math. Soc. (N.S.), 9, 3, 335-338, (1983) Geometric invariant theory, Group actions on varieties or schemes (quotients), Vector and tensor algebra, theory of invariants, Automorphisms, derivations, other operators (nonassociative rings and algebras) Invariant theory of \(G_ 2\) | 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 Let \(G \subset \text{GL}(2,\mathbb{C})\) be a finite subgroup, and \(Y=G\text{-Hilb}(\mathbb{C}^2)\) be the Hilbert scheme of \(G\)-clusters \(\mathcal{Z}\subset \mathbb{C}^2 \). By definition \(\mathcal{Z}\) is a 0-dimensional subscheme of \(\mathbb{C}^2\) of length \(|G|\) such that \(H^0(\mathcal{O}_{\mathcal{Z}})\) is isomorphic to the regular representation of \(G\). It is known that \(Y\) is isomorphic to the minimal resolution of the singularity \( \mathbb{C}^2/G \).
The paper under review gives an explicit description of a natural affine open covering \(Y\) in the case that \(G\) is small binary dihedral group. The open covering is in bijection with the set of bases for \(H^0(\mathcal{O}_{\mathcal{Z}})\) which are called the \(G\)-graphs. All the possible \(G\)-graphs are explicitly calculated by interpreting the action of \(G\) as the cyclic action of its maximal normal abelian subgroup \(H\) of index 2 followed by a dihedral involution. As an application the classification of the \(G\)-graphs is used to list the special representations of any small binary dihedral group. \(G\)-graphs; special representations; binary dihedral groups; McKay correspondence Alvaro Nolla de Celis, \(G\)-graphs and special representations for binary dihedral groups in \(\mathrm{GL}(2,\mathbf{C})\). (to appear in Glasgow Mathematical Journal). McKay correspondence, Parametrization (Chow and Hilbert schemes) \(G\)-graphs and special representations for binary dihedral groups in \(\mathrm{GL}(2,\mathbb C)\) | 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 shown that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic element \(n_t\) is \(t^l\) where \(l\) is the rank of the associated finite type Lie algebra. semisimple groups; Borel subalgebras; Coxeter numbers; Euler characteristic; Iwahori subalgebras; affine flag varieties; Lie algebras Kenneth Fan, C, Euler characteristic of certain affine flag varieties, Transform. Groups, 1, 35-39, (1996) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Structure theory for Lie algebras and superalgebras Euler characteristic of certain affine 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 F. Klein studied simple singularities, classifying them as quotients of $\mathbb{C}^2$ by the action of a finite subgroup $\Gamma \subseteq \mathrm{SU}_2$. P. Du Val showed that the exceptional divisors of the minimal resolution of the isolated singularity of such a quotient form an arrangement of projective lines whose dual graph is a simply-laced Dynkin diagram $\Delta(\Gamma)$; thus the quotient $\mathbb{C}^2/\Gamma$ is called a simple singularity of type $\Delta(\Gamma)$. P. Slodowy then extended the definition of a simple singularity to the non simply-laced types by adding a second finite subgroup $\Gamma' \subseteq \mathrm{SU}_2$ such that $\Gamma'\supseteq \Gamma$ as a normal subgroup; $\Gamma'/\Gamma =\Omega$ acts on $\mathbb{C}^2/\Gamma$ and this action can be lifted to the minimal resolution of the singularity, inducing an action on the exceptional divisors, which corresponds to a group of automorphisms of the Dynkin diagram of $\mathbb{C}^2/\Gamma$. \par A deformation of a simple singularity $(X_0,\Omega)$ is an $\Omega$-equivariant deformation of the singularity $X_0$ with a trivial action of the automorphism group $\Omega$ on the base space. Setting $\pi : X \rightarrow Y$ as a deformation of $X_0$, a deformation $\psi : X' \rightarrow Y'$ of $X_0$ is induced from $\pi$ by a morphism $\varphi : Y' \rightarrow Y$ if there exist a morphism $\Phi :X' \rightarrow Y'$ such that $\pi\circ \Phi= \varphi\circ\psi$, and given $X_0 \stackrel{i}{\hookrightarrow} X$ and $X_0\stackrel{j}{\hookrightarrow} X'$, $\Phi\circ j=i$. \par A semiuniversal deformation $\pi_0 : X \rightarrow Y$ of a simple singularity $(X_0,\Omega)$ is a deformation of $(X,\Omega)$ such that any other deformation $\psi : X' \rightarrow Y'$ of $(X,\Omega)$ is induced from $\pi_0$ by an $\Omega$-equivariant morphism $\varphi : Y' \rightarrow Y$ with a uniquely determined differential $d_{y'} \varphi :T_{y'} Y' \rightarrow T_y Y$. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$ by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type $X=D_s$, $E_6$ or $E_7$. \par Letting $\alpha : X_\Gamma \rightarrow \mathfrak{h}/W$ to be the semiuniversal deformation of a simple singularity of type $\Delta(\Gamma)=A_{2r-1}$ ($r \geq 2$), $D_{r}$ ($r \geq 4$) or $E_6$ obtained by the construction of H. Cassens and P. Slodowy, and $\mathfrak{h}$ and $W$ being the Cartan subalgebra and the associated Weyl group of the simple Lie algebra $\mathfrak{g}$ of the same type, respectively, they showed that $\Omega$ of the Dynkin diagram of $\mathfrak{g}$ acts on $X_\Gamma$ and $\mathfrak{h}/W$ such that $\alpha$ is $\Omega$-equivariant. Slodowy then showed that taking the restriction $\alpha^\Omega$ of $\alpha$ over the $\Omega$-fixed points of $\mathfrak{h}/W$ leads to a semiuniversal deformation of a simple singularity, which is inhomogeneous. As $\alpha$ is $\Omega$-equivariant, there is an action of $\Omega$ on every fiber of $\alpha^\Omega$, and the quotient leads to a new morphism $\overline{\alpha^\Omega}$, which is a non-semiuniversal deformation of a simple singularity of homogeneous type $\Delta(\Gamma')$. \par Let $\Gamma$ be a finite subgroup of $\mathrm{SU}_2$, $R$ its regular representation, $N$ its natural representation as a subgroup of $\mathrm{SU}_2$, and $\Delta(\Gamma)$ the associated Dynkin diagram. If $\Omega$ acting on $M(\Gamma)=(\mathrm{End}(R) \otimes N)^\Gamma$ is symplectic, then $\widetilde{\alpha}:X_{\Gamma}\times_{\mathfrak{h}/W}\mathfrak{h}\rightarrow \mathfrak{h}$ and $\alpha:X_{\Gamma}\rightarrow \mathfrak{h}/W$ can be made into $\Omega$-equivariant maps (Theorem 1.4, page 388): letting $M(\Gamma)$ to be the representation space of a McKay quiver built on a Dynkin diagram of type $A_{2r-1}$, $D_{r}$ or $E_6$, there exists a symplectic action of $\Omega=\Gamma'/\Gamma$ on $M(\Gamma)$, inducing the natural action on the singularity $\mathbb{C}^2/\Gamma$; this action then turns $\alpha$ into an $\Omega$-equivariant morphism. \par After A. Caradot shows that the morphism $\alpha^{\Omega} : X_{\Gamma, \Omega} \rightarrow (\mathfrak{h}/W)^{\Omega}$ is $\Omega$-invariant, it follows that $\Omega$ acts on each fiber of $\alpha^{\Omega}$, and the fibers can be quotiented. Furthermore it is known that $(\alpha^{\Omega})^{-1}(\overline{0}) = X_{\Gamma,0}=\mathbb{C}^2/\Gamma$, and thus $(\alpha^{\Omega})^{-1}(\overline{0})/\Omega=X_{\Gamma,0}/\Omega = (\mathbb{C}^2/\Gamma)/(\Gamma'/\Gamma) \cong \mathbb{C}^2/\Gamma'$, which is a simple singularity since $\Gamma'\subseteq \mathrm{SU}_2$ is finite. Thus, $\overline{\alpha^{\Omega}} : X_{\Gamma, \Omega} /\Omega \rightarrow (\mathfrak{h}/W)^{\Omega}$ is a deformation of the simple singularity $\mathbb{C}^2/\Gamma'$ of type $\Delta(\Gamma')$, where the deformation $\overline{\alpha^{\Omega}}$ is obtained through $\Delta(\Gamma)-\Delta(\Gamma,\Gamma')-\Delta(\Gamma')$-procedure. \par Caradot also studies the regularity of the fibers of $\overline{\alpha^{ \Omega}}$ (Theorem 2.3, page 390): assuming $\alpha^\Omega$ is the semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$, every fiber of the quotient $\overline{\alpha^\Omega}$ is singular. \par Finally, after stating a conjecture (Conjecture 3.1, page 396) that there exists a subset $\Theta$ of simple roots of the root system of type $\Delta(\Gamma')$ such that the Dynkin diagram associated to the singular configuration of any fiber of $\overline{\alpha^\Omega}$ is a subdiagram of the Dynkin diagram of type $\Delta(\Gamma')$ containing the vertices associated to $\Theta$, the author proves the conjecture for the types $A_3-B_2-D_4$, $A_5-B_3-D_5$, $D_4-C_3-D_6$, $D_4-G_2-E_6$, $D_4-G_2-E_7$, and $E_6-F_4-E_7$. (Theorem 3.2, page 397). deformations of simple singularities; simple root systems; simple singularities of inhomogeneous types; singular configurations Deformations of singularities, Root systems, Representation theory for linear algebraic groups Root systems and quotients of deformations of simple 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 This paper considers the fundamental group \(\pi_Y\) of the complement of a branch curve of a generic projection of an algebraic surface \(Y\) to \(\mathbb{C}\text{P}^2\). There is no precise description of this group but there is a precise description of the kernels of homomorphisms of \(\pi_Y\) into \(\widetilde{\text{Sp}}(n,\mathbb{Z}/3)\). finite quotients of braid groups; symplectic groups; fundamental groups; complements of branch curves; algebraic surfaces B. Moishezon,Finite quotients of braid groups related to symplectic groups over \(\mathbb{Z}\)/3, preprint. Braid groups; Artin groups, Coverings of curves, fundamental group, Coverings in algebraic geometry Finite quotients of braid groups related to symplectic groups over \(\mathbb{Z}/3\) | 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 \(G\) denote a connected reductive group \(G\) acting on an affine variety \(X.\) Let \(U \subset G\) be a maximal unipotent subgroup. By the result of Hadziev and Grosshans the covariant algebra \(k[X]^U\) is finitely generated. The author studies invariants of nonreductive subgroups of reductive groups. He uses a classification of actions by certain notions. One of that is the complexity, i.e. the minimal codimension of the orbits. The second one is the rank defined in terms of the semi-invariant functions of the function field \(k(X).\)
One of the main results of the paper is a method that reduces the problem of finding rank and complexity to certain problems about actions of reductive groups. This method allows to find not only the rank of an action but the rank group and even the saturated rank semigroup for actions on affine varieties. Another theme of the paper is the investigation of the covariant algebra. One part is concerned with the reduction of the calculation of the covariant algebra to a certain `smaller' action. A second part is concerned with the `symmetry' of Poincaré series of the covariant algebra in case \(X\) is factorial and has rational singularities. As an application there is a classification of spherical nilpotent orbits (with respect to adjoint representation) in simple Lie algebras and a classification of affine homogeneous spaces of complexity 1 for simple algebraic groups. geometric invariant theory; connected reductive group; maximal unipotent subgroup; complexity; minimal codimension; rank; actions on affine varieties; Poincaré series; simple Lie algebra D. I. Panyushev, Complexity and rank of actions in invariant theory,'' J. Math. Sci. (New York), 95, No. 1, 1925--1985 (1999). Geometric invariant theory, Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients) Complexity and rank of actions in 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 authors give a partial solution to the following problem: when a parabolic subgroup \(P\) of \(\text{GL}_n(\Bbbk)\) acts with a finite number of orbits on the Lie algebra \(\mathfrak q_u\) of the unipotent radical \(Q_u\) of a parabolic group \(Q\supset P\) such that \([\mathfrak q_u,\mathfrak q_u]\) is Abelian? Here \(\Bbbk\) is any algebraically closed field and the action is the adjoint one.
More precisely, let \(Fl(\underline e)\) be any flag variety of flags in \(\Bbbk^n\) and let \(P(\underline e)\) be the stabilizer of the standard flag \(0\subset\Bbbk^{e_1}\subset\Bbbk^{e_2}\subset\cdots\subset \Bbbk^{e_{t-1}}\subset\Bbbk^{e_t}=\Bbbk^n\) of signature \(\underline e=\{e_i\}\). Let \(\underline a=(a_1,a_2,a_3)\in\mathbb{Z}^3_{\geq 1}\) be a triple such that \(a_1+a_2+a_3=t\) and let \(Q(\underline a,\underline e)\) be the parabolic subgroup \(P(e_{a_1},e_{a_1+a_2},e_{a_1+a_2+a_3})\) of \(\text{GL}_n(\Bbbk)\). Observe that \(Q(\underline a,\underline e)\) contains \(P(\underline e)\) for each \(\underline a\). The authors classify the triples \(\underline a\) such that \(P(\underline e)\) acts with a finite number of orbits on \(\mathfrak q_u(\underline a,\underline e):=\text{Lie}(R_u(Q(\underline a,\underline e))\) for any \(\underline e\). We remark that in this work it is defined \(\underline e\), but the authors prefer to denote \(P(\underline e)\) by \(P(\mathbf d)\), where \(\mathbf d\) is \(t\)-pla \((e_1,e_2-e_1,\dots,e_i-e_{i-1},\dots,e_t-e_{t-1})\).
Utilizing the lexicographic order on the \(\underline a\), one can reduce itself to study a finite number of cases: more precisely the ones in Tables 1 and 2. First, the authors exhibit, for any \(\underline a\) in Table 1, an \(\underline e\) such that \(P(\underline e)\) acts with an infinite number of orbits on \(\mathfrak q_u(\underline a,\underline e)\). The proof is an elementary dimension counting argument. Observe that, given \(\underline a\) as in Table 1, there is not a classification of the \(\underline e\) such that \(P(\underline e)\) acts with a finite number of orbits on \(\mathfrak q_u(\underline a,\underline e)\).
The most difficult part of the work is the proof that, for any \(\underline a\) in Table 2, \(P(\underline e)\) acts with a finite number of orbits on \(\mathfrak q_u(\underline a,\underline e)\) for all \(\underline e\). To each \(\underline a\) the authors associate a quiver \(\mathcal Q(\underline a)\) with vertices \(\{1,\dots,t\}\) and an algebra \(\mathcal A(\underline a)\) obtained as a quotient of the path algebra of \(\mathcal Q(\underline a)\). With the same arguments of \textit{Th. Brüstle} and \textit{L. Hille} [J. Algebra 226, No. 1, 347-360 (2000; Zbl 0968.20023)] the authors show that \(\mathcal Q(\underline a)\) is quasi-hereditary and that the isoclasses of \(\Delta\)-filtered \(\mathcal A(\underline a)\)-modules with \(\Delta\)-dimension vector \(\mathbf d\) are in bijective correspondence with the orbits of \(P\) on \(\mathfrak q_u\). They prove also that the category \(\mathcal F(\mathcal A(\underline a),\Delta)\) of \(\Delta\)-filtered \(\mathcal A(\underline a)\)-modules is the full subcategory of \(\mathcal A(\underline a)\)-modules defined by the injectivity of the linear maps associated to some fixed arrows of \(\mathcal Q(\underline a)\). Moreover, such a \(\mathcal A(\underline a)\)-module has \(\Delta\)-dimension vector \(\mathbf d\) if and only if the vector space associated to \(i\) has dimension \(e_i\) for each \(i\).
Next, the authors show that, given any \(\underline a\) in Table 2, \(\mathcal A(\underline a)\) has finite \(\Delta\)-representation type by calculating the Auslander-Reiten quiver of \(\mathcal F(\mathcal A(\underline a),\Delta)\). They use standard methods as explained by \textit{M. Auslander, I. Reiten} and \textit{S. O. Smalø} [Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics. 36. Cambridge: Cambridge University Press (1995; Zbl 0834.16001)] and \textit{C. M. Ringel} [Tame algebras and integral quadratic forms. Lect. Notes Math. 1099. Berlin: Springer-Verlag (1984; Zbl 0546.16013)]. They give the Auslander-Reiten quivers so obtained in Appendix A.
Finally, the authors prove that one can reduce itself to study the triples \(\underline a\) in Tables 1 and 2. Indeed, if \(\underline a'\leq\underline a\), then there is an embedding of \(\mathcal F(\mathcal A(\underline a'),\Delta)\) in \(\mathcal F(\mathcal A(\underline a),\Delta)\). Thus, if there are finitely many \(P(\underline e)\)-orbits in \(\mathfrak q_u(\underline a,\underline e)\) for all \(\underline e\), then there are finitely many \(P(\underline e')\)-orbits in \(\mathfrak q_u(\underline a',\underline e')\) for all \(\underline e'\). group actions; parabolic subgroups; numbers of orbits; Lie algebras S. M. Goodwin, L. Hille, and G. Röhrle, ''Orbits of Parabolic Subgroups on Metabelian Ideals,'' arXiv: 0711.3711. Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients) Orbits of parabolic subgroups on metabelian ideals. | 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 connected closed subgroup \(G \subset \mathrm{GL}(n,\mathbb{R})\) that is invariant under transposition is called a real reductive Lie group. The Lie algebra \(\mathfrak{g}\) of \(G\) can be considered as a subalgebra of the linear maps \(\mathrm{gl}(n,\mathbb{R})\) on \(\mathbb{R}^n\), with the exponential map \(\exp\) given by the regular exponential map of matrices. In that case, there exists a scalar product \(\langle \cdot, \cdot \rangle\) on \(V = \mathbb{R}^n\) such that \(G\) can be written as \[ G = K \exp(\mathfrak{p})\] with \(K = G \cap O(V)\) and \(\mathfrak{p} = \mathfrak{g} \cap \mathrm{Sym}(V)\). Here we denote by \(O(V)\) the orthogonal group with respect to the scalar product \(\langle \cdot, \cdot \rangle\) and by \(\mathrm{Sym}(V)\) the set of symmetric endomorphisms on \(V\). If \(\mathfrak{k}\) is the Lie algebra of the group \(K\), which is moreover the maximal compact subgroup of \(G\), then \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) is the Cartan decomposition of \(\mathfrak{g}\). We can define the map \(\psi: G \times V \to \mathbb{R}\) via \[\psi(g,x) = \frac{1}{2}\left( \langle g x, gx \rangle - \langle x, x \rangle \right)\] and this is a Kempf-Ness function as defined in the third section of the paper. The study of real reductive groups via the Kempf-Ness function and corresponding gradient map, in particular their convexity properties, is very active and has for example lead to a better understanding of metrics with negative Ricci curvature on solvmanifolds, see [\textit{J. Deré} and \textit{J. Lauret}, Math. Nachr. 292, No. 7, 1462--1481 (2019; Zbl 1425.53042)].
In the paper under review, the author gives numerous different results about the image of the gradient map. For example, the image of the gradient map for abelian subalgebras of \(\mathfrak{p}\) is computed, which generalizes a result of Kac and Peterson from the complex numbers to the reals [\textit{V. G. Kac} and \textit{D. H. Peterson}, Invent. Math. 76, 1--14 (1984; Zbl 0534.17008)]. On the other hand, a new proof of the Hilbert-Mumford criterion for real reductive groups is given in the last section. gradient maps; real reductive representations; real reductive Lie groups; geometric invariant theory Representations of Lie and linear algebraic groups over real fields: analytic methods, Momentum maps; symplectic reduction, Geometric invariant theory Convexity properties of gradient maps associated to real reductive representations | 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 this paper is to discuss some previous results due to the author, some conjectures due to him, to announce new conjectures, and to present some progress made in the calculation of local fundamental groups of algebraic varieties. The paper contains no proofs. Galois group; resolution of singularities; local fundamental groups of algebraic varieties Abhyankar S S, Local fundamental groups of algebraic varieties,Proc. Am. Math. Soc. 125 (1997) 1635--1641 Separable extensions, Galois theory, Coverings of curves, fundamental group, Simple groups: alternating groups and groups of Lie type, Extensions, wreath products, and other compositions of groups Local fundamental groups of 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 provides new upper bounds for the so called symmetric tensor rank of multiplication in finite extensions \(\mathbb F_{q^n}\) of a finite field \(\mathbb F_q,\,\, q=p^m\). Associated with the bilinear multiplication \(\mathbb F_{q^n}\times \mathbb F_{q^n}\rightarrow \mathbb F_{q^n}\) we get a tensor \(T=\sum_{i=1}^r x_i^*\otimes y_i^*\otimes c_i\in \mathbb F_{q^n}^*\otimes \mathbb F_{q^n}^*\otimes \mathbb F_{q^n}\). The minimum \(r\) in such an expression it is called the tensor rank of the multiplication in \(\mathbb F_{q^n}\) while the minimum \(r\) for a decomposition \(T=\sum_{i=1}^r x_i^*\otimes x_i^*\otimes c_i\) it is called the the symmetric tensor rank \(\mu_q^{\mathrm{sym}}\) of the multiplication.
Upper bounds for \(\mu_q^{\mathrm{sym}}\) can be deduced from results of \textit{D. Chudnovsky} and \textit{G. Chudnovsky} [J. Complex. 4, No. 4, 285--316 (1988; Zbl 0668.68040)]. In particular there exists a constant \(C_q\) such that \(\mu_q^{\mathrm{sym}}\leq C_qn\). Theorem 5 gathers the best previous estimations for \(C_q\). The aim of the present paper is to improve these results in the case \(q=p, p^2\) and \(p\geq 5\).
The used tools are the construction of families of modular curves \(\{X_i\}\) with increasing genus \(g_i\) attaining the Drinfeld-Vladut bound and such that \(\lim_{i\rightarrow \infty}g_{i+1}/g_i =1\) as well as bounds on gaps between two consecutive primes (Theorem 6).
Section 2.1 studies the case \(q=p^2\) (Proposition 7) and Section 2.2 the case of prime fields (Proposition 10). finite fields; symmetric tensor rank; algebraic function field; tower of function fields; modular curve; Shimura curve Computational aspects of algebraic curves, Effectivity, complexity and computational aspects of algebraic geometry, Number-theoretic algorithms; complexity, Computational aspects of field theory and polynomials Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain 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 In Invent. Math. 95, No. 1, 1-11 (1989; Zbl 0676.14009), \textit{G. Ellingsrud} and \textit{C. Peskine} proved that the Hilbert scheme of (smooth) non-general-type surfaces in \(\mathbb{P}^ 4\) consists only of a finite number of components. In particular, there is a maximum degree \(d_ 0\) for such surfaces. There are known examples of non-general-type surfaces of degree 15, so that \(d_ 0 \geq 15\). Although there is not an explicit upper bound for \(d_ 0\) in the quoted paper, the bound that could be obtained from the proof would be extremely high (about 10,000). In the paper under review, the authors prove that \(d_ 0 \leq 105\). They use essentially the same kind of inequalities as Ellingsrud and Peskine, but improve some of them by using the machinery of initial ideals. More precisely, for a codimension-two subvariety in a projective space, they get a set of (numerical) invariants for the generic initial ideal associated to its homogeneous ideal (in the case of curves, these invariants correspond to the numerical character introduced by Gruson and Peskine). This is what allows them to give the required bounds for the sectional genus and the Euler-Poincaré characteristic of (the structure sheaf of) a surface in \(\mathbb{P}^ 4\) in terms of the degree and the postulation, which is the main ingredient to get a bound for \(d_ 0\). With this method, the upper bound for the sectional genus coincides with the one obtained by Gruson and Peskine (and then used in the proof by Ellingsrud and Peskine), while the lower bound for the Euler-Poincaré characteristic is better than the one used by Ellingsrud and Peskine.
Finally, it could be worth to mention that, using also initial ideas, \textit{Shelly Cook} seems to have succeeded in improving the previous bound for \(d_ 0\). The last reference the reviewer had was that she succeeded in proving \(d_ 0 \leq 76\). number of components of Hilbert scheme; non-general-type surfaces; initial ideals; codimension-two subvariety Braun, R; Floystad, G, A bound for the degree of smooth surfaces in \({\mathbb{P}}^4\) not of general type, Compositio Math., 93, 211-229, (1994) Surfaces of general type, Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry A bound for the degree of smooth surfaces in \(\mathbb{P}^ 4\) not of general type | 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 Automorphisms of algebraic curves of the algebras \(\langle\{I_{n_i}^{[3]}\},+,\cdot\rangle\) and \(\langle\{P_{n_i}^{[3]}\},+,\cdot\rangle\) with group of symmetries \([3]\) are described. The automorphisms of the algebra \(\langle\{I_{n_i}^{[3]}\},+,\cdot\rangle\) are the group coinciding with the dihedral group. The automorphisms of the algebra \(\langle\{P_{n_i}^{[3]}\},+,\cdot\rangle\) form a partial algebra. automorphisms; partial algebras; groups of symmetries; dihedral groups; algebraic curves Automorphism groups of groups, Automorphisms of curves Algebras of automorphisms of curves with group of symmetries \([3]\). | 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 commutative ring \(R(P(t))=\mathbb C[t^{\pm 1},u\mid u^2=P(t)]\), where \(P(t)=\sum_{i=0}^na_it^i=\prod_{k=1}^n(t-\alpha_i)\) with \(\alpha_i\in \mathbb C\) pairwise distinct, is the coordinate ring of a hyperelliptic curve when \(n>4\). The Lie algebra \(\mathcal {R}(P(t))=\operatorname{Der}(R(P(t)))\) of derivations is called the hyperelliptic Lie algebra associated to \(P(t)\) and is a particular type of multipoint Krichever-Novikov algebra. In this paper, we describe the universal central extension of \(\operatorname{Der}(R(P(t)))\) in terms of certain families of polynomials which in a particular case are associated Legendre polynomials. Moreover we describe certain families of polynomials that arise in the study of the group of units for the ring \(R(P(t)\)), where \(P(t)=t^4-2bt^2+1\). In this study, pairs of Chebyshev polynomials \((U_n,T_n)\) arise as particular cases of a pairs \((r_n,s_n)\) with \(r_n+s_n\sqrt{P(t)}\) a unit in \(R(P(t)\)). We explicitly describe these polynomial pairs as coefficients of certain generating functions and show that certain of these polynomials satisfy particular second-order linear differential equations. Krichever-Novikov algebras; automorphism groups; Pell's equation; associated Legendre polynomials; universal central extensions; superelliptic Lie algebras; superelliptic curves; DJKM algebras; Fáa di Bruno's formula; Bell polynomials Infinite-dimensional Lie (super)algebras, Riemann surfaces; Weierstrass points; gap sequences, Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) Certain families of polynomials arising in the study of hyperelliptic Lie 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 simply connected semisimple complex Lie group with Lie algebra \(\mathfrak g\). Let \({\mathcal O} \subset {\mathfrak g}\) be a nilpotent adjoint \(G\)-orbit and \(M\) a \(G\)-homogeneous cover of \(\mathcal O\). This paper is concerned with the symplectic and algebraic geometry of \(M\). The authors show that \(M\) admits a \(G\)-equivalent normalization \(X\) into which it embeds as a dense open subset. The ring \(R\) of regular functions on \(X\) has a natural Poisson structure which is graded by a \(\mathbb{C}^*\)- action, thanks to the nilpotence of \(\mathcal O\). The authors show that this Poisson structure turns the 2-graded piece \(R[2]\) of \(R\) into a semisimple Lie algebra containing \(\mathfrak g\) and \(R[0] + R[1]\) into a Heisenberg Lie algebra on which \({\mathfrak g}':= R[2]\) acts by derivations. Pairs \(({\mathfrak g},{\mathfrak g}')\) arising in this way with \({\mathfrak g}' \neq {\mathfrak g}\) are completely classified; it is striking that each such pair arises from a space \(M\) that can be identified with the minimal nilpotent orbit in \({\mathfrak g}'\). It can happen that \(\mathfrak g\) is classical while \({\mathfrak g}'\) is not; thus classical groups can ``see'' some of the structure of exceptional ones. Finally, in the last section, it is shown that part of the above classification serves to classify partial flag varieties \(G/P\) admitting a group of holomorphic automorphisms larger than \(G\). homogeneous cover; ring of regular functions; simply connected semisimple complex Lie group; Lie algebra; nilpotent adjoint \(G\)-orbit; Poisson structure; semisimple Lie algebra; Heisenberg Lie algebra; minimal nilpotent orbit; flag varieties; group of holomorphic automorphisms R. Brylinski and B. Kostant, \textit{Nilpotent orbits, normality, and Hamiltonian group actions}, \textit{J. Am. Math. Soc.}\textbf{7} (1994) 269 [math/9204227]. Semisimple Lie groups and their representations, Geometric quantization, Lie algebras of Lie groups, Complex Lie groups, group actions on complex spaces, Group actions on varieties or schemes (quotients), Solvable, nilpotent (super)algebras, Simple, semisimple, reductive (super)algebras Nilpotent orbits, normality, and Hamiltonian group actions | 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 \(X\) be a variety over a number field \(k\). One has inclusions \(X(k) \subset \overline{X(k)} \subset X(\mathbb{A}_k)^{\mathrm{\'{e}t,Br}} \subset X(\mathbb{A}_k)^{\mathrm{Br}} \subset X(\mathbb{A}_k)\) (the closure taken inside \(X(\mathbb{A}_k)\)). The authors give an elementary construction of a smooth projective surface \(X\) over an arbitrary number field \(k\) with \(X(k) = \emptyset\) and \(X(\mathbb{A}_k)^{\mathrm{\'{e}t,Br}}\) infinite (so \(X\) is a counterexample to the Hasse principle which is not explained by the étale Brauer-Manin obstruction), and smooth projective geometrically integral surface \(X/k\) which is a counterexample to weak approximation and with \(|X(k)| = 1\) and \(X(\mathbb{A}_k)^{\mathrm{\'{e}t,Br}}\) infinite. rational points; Hasse principle; weak approximation; Brauer-Manin obstruction; Brauer groups of schemes Yonatan Harpaz & Alexei N. Skorobogatov, ``Singular curves and the étale Brauer-Manin obstruction for surfaces'', Ann. Sci. Éc. Norm. Supér.47 (2014) no. 4, p. 765-778 Rational points, Brauer groups of schemes Singular curves and the étale Brauer-Manin obstruction for 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 A new conjecture due to John McKay claims that there exists a link between (1) the conjugacy classes of the Monster sporadic group and its offspring, and (2) the Picard groups of bases in certain elliptically fibered Calabi-Yau threefolds. These Calabi-Yau spaces arise as F-theory duals of point-like instantons on ADE type quotient singularities. We believe that this conjecture, may it be true or false, connects the Monster with a fascinating area of mathematical physics which is yet to be fully explored and exploited by mathematicians. This article aims to clarify the statement of McKay's conjecture and to embed it into the mathematical context of heterotic/F-theory string-string dualities. conjugacy classes of the Monster sporadic group; Picard groups; elliptically fibered Calabi-Yau threefolds; F-theory; McKay's conjecture String and superstring theories; other extended objects (e.g., branes) in quantum field theory, \(3\)-folds, Simple groups: sporadic groups, Anomalies in quantum field theory Friendly giant meets pointlike instantons? On a new conjecture by John McKay | 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 Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) parametrizing closed subschemes of \(\mathbb P^n\) with Hilbert polynomial \(p(t)\) has been of great interest every since Grothendieck constructed it in the early 1960s. Early results include the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)] and smoothness of \(\mathrm{Hilb}^{p(t)} (\mathbb P^2)\) due to \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. \textit{A. Reeves} and \textit{M. Stillman} showed that every non-empty Hilbert scheme contains a smooth Borel-fixed point [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] and \textit{A. P. Staal} classified those with exactly one such fixed point, which are necessarily smooth and irreducible [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)].
The main result classifies Hilbert schemes with two Borel-fixed points over a field \(k\) of characteristic zero. To describe the result, express the Hilbert polynomial \(p(t)\) in the form used by \textit{Gotzmann}, namely
\[
p(t) = \sum_{i=1}^m \binom{t+\lambda_i-i}{\lambda_i-1}
\]
where \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_m \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. Writing \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\), the theorem lists for exactly which \(\mathbf{\lambda}\) the Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) has two Borel-fixed points and further determines when it is (a) smooth, (b) irreducible and singular or (c) a union of two components. In each case the irreducible components are normal and Cohen-Macaulay and the singularities of the Hilbert scheme appear as cones over certain Segre embeddings of \(\mathbb P^a \times \mathbb P^b\). Since the writing of his paper, (a) \textit{A. P. Staal} [``Hilbert schemes with two Borel-fixed points in arbitrary characteristic'', Preprint, \url{arXiv:2107.02204}] has shown that the theorem is valid in all characteristics with a small modification when char \(k=2\) and (b) \textit{R. Skjelnes} and \textit{G. G. Smith} [J. Reine Angew. Math. 794, 281--305 (2023; Zbl 07640144)] have classified the smooth Hilbert schemes are described their geometry.
Despite the difficulty of the content, the paper is readably written. Section 1 gives preliminaries on Borel-fixed (strongly stable) ideals and the resolution of \textit{S. Eliahou} and \textit{M. Kervaire} [J. Algebra 129, No. 1, 1--25 (1990; Zbl 0701.13006)], while Section 2 identifies the tuples \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\) corresponding to Hilbert schemes with two components. Section 3 uses the comparison theorem of \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] to compute the tangent space of the non-lexicographic Borel-fixed ideal \(I(\mathbf{\lambda})\) and give a partial basis for the second cohomology group of \(k[x_0,\dots,x_n]/I(\mathbf{\lambda})\). These are used in Section 4 where the main theorem is proved to describe the universal deformation space of \(I(\mathbf{\lambda})\) and hence the nature of singularities of the Hilbert schemes. Finally in Section 5 the author gives examples of Hilbert schemes with three Borel-fixed points. The last three examples relate to Hilbert schemes studied in the literature [\textit{S. Katz}, in: Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 231--242 (1994; Zbl 0839.14001); \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004); \textit{D. Chen} et al., Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)]. Hilbert scheme; singularities; Borel-fixed points; deformations of ideals Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Hilbert schemes with two Borel-fixed points | 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 gives a full proof account of the state of the art of the theory of polylogarithms from a geometric point of view as advocated by the author. The case of the dilogarithm being known since some time by work of Gabrielov et al., Bloch, Wigner, Zagier,\( \dots\), the underlying paper focusses on the trilogarithm.
Classically the \(p\)-th polylogarithm \(\text{Li}_p (z)\) is defined as the analytic continuation of the expression \(\text{Li}_p(z)=\sum_{n=1}^\infty {z^n \over n^p}\), \(|z|\leq 1\). With \(\text{Li}_1(z)= - \log(1-z)\), one has the inductive formula \(\text{Li}_p(z)=\int^z_0 \text{Li}_{p-1} (z) {dt \over t}\) with its (multivalued) continuation to \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\). It turns out to be advantageous to consider modified functions \({\mathcal L}_p (z)={\mathcal R}_p \left( \sum^p_{j=0} {2^j B_j \over j!} (\log |z |)^j \cdot \text{Li}_{p-j} (z) \right)\), where the \(B_j\) are the Bernoulli numbers, and \({\mathcal R}_m\) denotes the real part for odd \(m\) and the imaginary part for even \(m\), \(\text{Li}_0 (z):= -{1 \over 2}\). Thus one has
\[
{\mathcal L}_2 (z)={\mathfrak I} \bigl( \text{Li}_2 (z) \bigr) + \arg (1-z) \cdot \log |z |,
\]
the Bloch-Wigner function. For the present paper the most interesting function becomes
\[
{\mathcal L}_3 (z)={\mathfrak R} \Bigl( \text{Li}_3 (z)-\log |z |\cdot \text{Li}_2 (z)+ \textstyle {{1\over 3}} \log^2 |z|\cdot \text{Li}_1(z) \Bigr).
\]
The functions \({\mathcal L}_p(z)\) are single-valued, real analytic on \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\) and continuous at \(0,1,\infty\). In particular, \({\mathcal L}_3 (0)= {\mathcal L}_3 (\infty) =0\) and \({\mathcal L}_3 (1)= \zeta_\mathbb{Q} (3)\), the Riemann zeta-function at \(s=3\). The functions \({\mathcal L}_p (z)\) admit a Hodge theoretic interpretation.
As a first goal one tries to find the generic functional equation for the (modified) trilogarithm. Motivated by results on the Bloch-Wigner function where the functional equations of the dilogarithm are related to the cross-ratio of four points in \(\mathbb{P}^1\), in the case of the trilogarithm one considers configurations of six (or seven) points in \(\mathbb{P}^2\). By an explicit geometric reasoning the main result is obtained: The generic functional equation for the trilogarithm \({\mathcal M}_3\) (a specific alternating sum of \({\mathcal L}_3\)'s) is a seven term identity for \({\mathcal M}_3\) on a configuration \((l_0, \dots, l_6)\) of seven points in \(\mathbb{P}^2\) that can be given explicitly in terms of the coordinates of the points. In particular, the Spence-Kummer relation may be derived. Furthermore, it is shown that the trilogarithm is determined by its functional equation.
Polylogarithms show up in other contexts, e.g. in connection with algebraic \(K\)-theory, motivic cohomology, characteristic classes, continuous cohomology, the Dedekind zeta function of an arbitrary number field, \(\dots\), etc. In some cases one can prove interesting results, e.g. for a number field \(F\) one can express (up to a non-zero rational factor) the value of \(\zeta_F (2)\) in terms of the dilogarithm \({\mathcal L}_2\) at specific values of its argument depending on the (complex) embeddings of \(F\). A similar result holds for \(\zeta_F (3) \) in terms of \({\mathcal L}_3\). For general \(\zeta_F (n)\), \(n=4,\dots\), Zagier stated the conjecture that they can be expressed, analogously to \(\zeta_F (2)\) and \(\zeta_F (3)\), in terms of \({\mathcal L}_n\). This fits very well in Beilinson's world where values of \(L\)-functions at special values of their arguments are given (up to non-zero rational factors) by the volume of the regulator map which is itself a map from algebraic \(K\)-groups to Deligne-Beilinson cohomology. As a matter of fact, the theory of polylogarithms is closely related to \(K\)-theory. One of its main building blocks is a certain complex \(\Gamma_F (n)\) (the existence of which was originally conjectured by Beilinson and Lichtenbaum) of the form:
\[
\Gamma_F (n): {\mathcal B}_n(F) @> \delta>> {\mathcal B}_{n-1} (F) \otimes F^\times @>\delta>> \cdots @>\delta>> {\mathcal B}_2 (F) \otimes \wedge^{n-2} F^\times @>\delta>> \wedge^n F^\times,
\]
where \({\mathcal B}_m (F) = \mathbb{Z} [\mathbb{P}^1 _F]/{\mathcal R}_m (F)\), with \({\mathcal R}_m (F) \subset \mathbb{Z} [\mathbb{P}^1_F]\) reflecting the functional equations of the classical \(m\)-polylogarithm. Here \({\mathcal B}_n (F)\) is placed in degree one, and the \(\delta\)'s are explicitly defined. For the \(K\)-groups of \(F\) one has \(K_n (F )_\mathbb{Q} = \text{Prim} H_n (GL_n(F), \mathbb{Q})\) and by the canonical filtration on \(H_n (GL_n (F), \mathbb{Q})\) implied by \(\text{Im} (H_n (GL_{n-1} (F), \mathbb{Q}) \to H_n (GL_n(F), \mathbb{Q}))\), one obtains a filtration \(K_n(F)_\mathbb{Q} \supset K_n^{(1)} (F)_\mathbb{Q} \supset K_n^{(2)} (F)_\mathbb{Q} \supset \cdots \). Let \(K_n^{[i]} (F)_\mathbb{Q}: = K_n^{(i)} (F)_\mathbb{Q}/K_n^{(i+1)} (F)_\mathbb{Q}\). Then one has Conjecture A: \(K_{2n-i}^{[n-i]} (F)_\mathbb{Q} = H^i (\Gamma_F(n) \otimes \mathbb{Q})\). On the other hand, Beilinson conjectured the existence of a mixed Tate category \({\mathcal M}_T(F)\) which should be Tannakian. Thus the formalism of Tannakian categories implies that \({\mathcal M}_T (F)\) is equivalent to the category of finite-dimensional representations of some graded pro-Lie algebra \(L(F)_\bullet = \oplus^{-\infty}_{i= -1} L(F)_i\). One may state Conjecture B: (i) \(L(F)_{\leq-2}\) is a free graded pro-Lie algebra such that the dual of the space of its degree \(-n\) generators is isomorphic to \({\mathcal B}_n (F)_\mathbb{Q}\); (ii) The dual map to the action of the quotient \(L(F)_\bullet /L(F)_{\leq -2}\) on the space of degree \(-(n-1)\) generators of \(L(F)_{\leq -2}\) is just the differential \(\delta: {\mathcal B}_n (F)_\mathbb{Q} \to({\mathcal B}_{n-1} (F)\otimes F^\times)_\mathbb{Q}\). It is shown that in Beilinson's world Conjecture A is equivalent to Conjecture B. Conjecture B has some deep consequences, e.g. its truth implies the truth of a conjecture of Bogomolov, and also of a conjecture due to Shafarevich which says that the commutant of \(\text{Gal} (\overline \mathbb{Q}/ \mathbb{Q})\) is a free profinite group.
Let \(F\) be an arbitrary field and define \(B_p(F): = \mathbb{Z} [\mathbb{P}^1_F \backslash \{0,1, \infty\}]/R_p(F)\), \(p\leq 3\), where the \(R_p(F)\) again reflect the functional equations of the classical \(p\)-th polylogarithm. One defines the complex \(B_3(F) \otimes \mathbb{Q}\) as follows: \(B_3(F)_\mathbb{Q} @>\delta>> (B_2(F) \otimes F^\times)_\mathbb{Q} @>\delta>> (\wedge^3 F^\times)_\mathbb{Q}\), with \(B_3 (F)_\mathbb{Q}\) placed in degree 1, and \(\delta \{x\} = [x] \otimes x\) and \(\delta ([x] \otimes y) = (1-x) \wedge x \wedge y\) for a generator \(\{x\}\) of \(B_3(F)\) and a generator \([x]\) of \(B_2(F)\). Then there are canonical maps \(c_1: K_5^{[2]} (F)_\mathbb{Q} \to H^1 (B_3(F) \otimes \mathbb{Q})\) and \(c_2: K_4^{[1]} (F)_\mathbb{Q} \to H^2 (B_3(F) \otimes \mathbb{Q})\). It is conjectured that \(c_1\) and \(c_2\) are isomorphisms. This should be related to results of Suslin on Milnor \(K\)-groups.
Other subjects discussed are duality of configurations, projective duality, and an explicit formula for the Grassmannian trilogarithm.
Many unsolved questions and deep conjectures remain. algebraic \(K\)-theory; values of \(L\)-functions; Beilinson conjecture; finite dimensional representations of graded pro-Lie algebra; polylogarithms; Bernoulli numbers; Riemann zeta-function; generic functional equation; trilogarithm; Bloch-Wigner function; Spence-Kummer relation; motivic cohomology; characteristic classes; Dedekind zeta function; number field; dilogarithm; regulator map; Deligne-Beilinson cohomology; mixed Tate category; Tannakian categories; duality of configurations; Grassmannian trilogarithm M. Prausa, \textit{epsilon: A tool to find a canonical basis of master integrals}, arXiv:1701.00725 [INSPIRE]. Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Zeta functions and \(L\)-functions of number fields, \(K\)-theory of global fields, Other functions defined by series and integrals Geometry of configurations, polylogarithms, and motivic 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 We show that a von Neumann algebra is finite if and only if its Grassmannians are small in a certain sense related to the atlas of affine coordinates. von Neumann algebra; finite; Grassmannians; affine coordinates General theory of von Neumann algebras, Grassmannians, Schubert varieties, flag manifolds Affine coordinates and finiteness | 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 totally real number field of degree \(n\) over \(\mathbb Q\) with ring of integers \(\mathcal O_k\), and let \(\mathcal H^n\) be the product of \(n\) copies of the Poincaré upper half plane. Then the quasi-projective variety \(PSL_2 (\mathcal O_k) \backslash \mathcal H^n\) can be compactified in a natural way by adding \(h_k\) points, where \(h_k\) is the class number of \(k\). The Hilbert modular variety of \(k\) is obtained by desingularizing this compact variety. In this paper the author proves that Hilbert modular varieties of Galois quartic fields containing units of norm \(-1\) have genus greater than one and therefore are not rational. Hilbert modular groups; Hilbert modular varieties; quartic fields H. G. Grundman, Hilbert modular varieties of Galois quartic fields, J. Number Theory 63 (1997), no. 1, 47 -- 58. 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, Rational points, Modular and Shimura varieties Hilbert modular varieties of Galois quartic 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 [For the entire collection see Zbl 0619.00009.]
The author gives an exposition of some of the basic ideas in computing the invariants of a representation, V, of a reductive group, G, with special emphasis on finding an upper bound for the degree of the invariants in a generating set. The ring of invariant functions, I, is generated by a finite set, J, of homogeneous polynomials, called `primary invariants', which are algebraically independent over the field k. A result of \textit{M. Hochster} and \textit{J. L. Roberts} [Adv. Math. 13, 115- 175 (1974; Zbl 0289.14010)] says that there exists a finite set, S, of secondary invariants such that I is the free k[J]-module with basis \(\{\) \(1\}\cup S\). The author has shown that the maximum degree of any secondary invariant is not greater than the sum of the degrees of the primary invariants [see the author, Mich. Math. J. 26, 19-32 (1979; Zbl 0409.13004)]. From this the author easily deduces that the ring I is generated by invariants of degree \(\leq (\dim V)C(m)\) where C(m) is the least common multiple of the numbers \(\leq m\) and m is the maximum degree of the elements in J. The author show how one may compute m using estimates from \textit{V. Popov} [see Astérisque 87-88, 303-334 (1981; Zbl 0491.14004)] and by determining J explicitly for the special case of a torus. ring of invariant functions; primary invariants Kempf, G.R.: Computing invariants. In: Invariant Theory. Lect. Notes in Math., vol. 1278, pp. 81--94. Springer-Verlag, Berlin (1987) Geometric invariant theory, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups Computing 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 This paper gives reasonable and explicit bounds on the central fiber of the degeneration of surfaces of general type with given \(\chi({\mathcal O}_X)\) and \(K^2_X\).
Section 1 presents reasonable bounds for the number of singularities outside the double curve on the central fiber of a relatively minimal permissible degeneration of surfaces of general type (theorem 10), and the number of components in a relative canonical model (theorem 11).
In section 2, we figure out how to change discrepancies for the iteration of singularity of class \(T\) (theorem 15). This provides several inductive formulas used in section 3 for a singularity of class \(T\). Section 3 clarifies the reason behind the bound of the index of singularity of the central fiber under the simplifying assumption that the central fiber is irreducible, with resolution a surface of general type (theorem 23). Also we present explicit bounds for the index of singularity for this case in theorem 23. degenerations of surfaces of general type; number of singularities; number of components; index of singularity Lee, Y., Numerical bounds for degenerations of surfaces of general type, International Journal of Mathematics, 10, 79-92, (1999) Global theory and resolution of singularities (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Surfaces of general type, Minimal model program (Mori theory, extremal rays) Numerical bounds for degenerations of surfaces of general type | 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 total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper [Int. Math. Res. Not. 2015, No. 23, 12590--12619 (2015; Zbl 1387.11088)], the authors observed that in a particular example two total masses coming from two different weightings are dual to each other. We discuss the problem of how generally such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples. mass formulas; local Galois representations; quotient singularities; dualities; McKay correspondence; equisingularities; stringy invariants Galois theory, Ramification and extension theory, Varieties over finite and local fields, Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence Mass formulas for local Galois representations and quotient singularities. II: Dualities and resolution of 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 From the preface: ``Le volume que voici est un hommage à Marcel-Paul Schützenberger offert par ses amis et disciples. Bien que ceux qui ont contribué à ce volume ne forment qu'une faible partie de l'ensemble total de ses amis, élèves, disciples, admirateurs et épigones, la diversité de leurs contributions témoigne qu'ils
appartiennent à des espèces variées. Le résultat est ce livre aux facettes multiples qui comprend des raisonnements mathématiques tout aussi bien que des analyses historiques.
Table of contents: Préface (7--10); Portrait (11--13);
Publications de M.-P. Schützenberger (155 items until 1988) (14--24);
A. Lentin, Ode à Marco (25--32).
Réflexions sur le langage et la connaissance.
M. Eden, Man, God and Science (34--43); G. Gadoffre, Érasme et le merdier gaulois (44--54); M. Borillo, Cogniticiens, encore un effort (55--58); Gian-Carlo Rota, Intentionality today (59--69); Moshé Flato, Physique et vérité (70--80); Charles Galperin, Le physicien dépaysé (81--94); André Lichnerowicz, L'activité mathématique (95--105); Morris Halle, Apprentissage et savoir. Quelques réflexions à propos de certains développements récents en phonologie (106--120); Maurice Gross, Sur la détermination de quantités dans les langues naturelles (121--135); Zellig Harris, On the mathematics of language (136--141).
Mathématiques concrètes:
\textit{Roger C. Lyndon}, Words and infinite permutations (143--152);
\textit{C. Procesi}, The toric variety associated to Weyl chambers (153--161);
\textit{Gerard Lallement}, On \({\mathcal D}\)-representations of semigroups (162--170);
\textit{Christophe Reutenauer}, Dimensions and characters of the derived series of the free Lie algebra (171--184);
\textit{Aldo de Luca}, On the Burnside problem for semigroups (185--200);
\textit{Pierre Barrucand} and \textit{Alain Lascoux}, Une formule de Cayley et les caractères èquerres du groupe symétrique [A Cayley formula and ``èquerre'' characters of the symmetric group] (201--207);
\textit{Dominique Foata}, Permutations colorées et tableaux gauches [Colored permutations and skew tableaux] (208--220);
\textit{Alain Lascoux}, Diviser! [Divide!] (221--231);
\textit{Robert Cori} and \textit{Antonio Machi}, Cartes, hypercartes et leurs groups d'automorphismes [Maps, hypermaps and their groups of automorphisms] (232--245);
\textit{Adalbert Kerber}, La théorie combinatoire sous-tendant la théorie des représentations linéaires des groupes symétriques finis [The combinatorial theory underlying the theory of linear representations of finite symmetric groups] (246--253);
\textit{Shi He}, On the Lie Shan-Lan identity (254--264);
\textit{G. X. Viennot}, Trees (265--297);
\textit{Jean Berstel}, Trace de droites, fractions continues et morphismes itérés [Drawing lines, continued fractions and iterated morphisms] (298--309);
\textit{Christian Choffrut}, Fonctions rationnelles et distances discrètes [Rational functions and discrete distances] (310--327);
\textit{Michel Fliess}, Automatique, algèbre différentielle et causalité [Automatic control, differential algebra and causality] (328--334);
\textit{Georges Hansel} and \textit{Dominique Perrin}, Mesures de probabilités rationnelles [Rational probability measures] (335--357);
\textit{Antonio Restivo}, Codes with constraints (358--366);
\textit{P. Rosenstiehl}, Grammaires acycliques de zigzags du plan [Acyclic grammars of zigzags in the plane] (367--383);
\textit{Imre Simon}, The nondeterministic complexity of a finite automaton (384--400).
The articles of this volume will not be reviewed individually. Mots; words; trees; permutations; toric variety; Weyl chambers; semigroups; Lie algebra; Burnside problem for semigroups; symmetric group; skew tableaux; hypermaps; combinatorial theory; representations; continued fractions; differential algebra; probability measures; grammars of zigzags; complexity; finite automaton M. Lothaire , Mots . Hermès Paris 1990 . MR 1252659 | Zbl 0862.05001 Proceedings, conferences, collections, etc. pertaining to combinatorics, Proceedings, conferences, collections, etc. pertaining to computer science, Proceedings, conferences, collections, etc. pertaining to group theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Collections of articles of miscellaneous specific interest, Festschriften Words. Miscellany offered to M.-P. Schützenberger | 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 \({\mathfrak q}={\mathfrak q}(A)\) be the Kac-Moody-Lie algebra, associated to a symmetrizable generalized Cartan matrix \(A=(a_{ij})_{1\leq i,j\leq\ell }\) and let \(S\subset\{1,...,\ell\}\) be the subset of finite type. Denote by \({\mathfrak p}_ S\) the corresponding ''parabolic''; \({\mathfrak r}\) ''the maximal'' reductive subalgebra of \({\mathfrak p}_ S\) and \({\mathfrak u}^-\) the orthocomplement of \({\mathfrak p}_ S\) (so that \({\mathfrak u}^- \oplus {\mathfrak p}_ S={\mathfrak q})\). Let \(L(\lambda_ 0)\) be the quasi- simple \({\mathfrak q}\)-module, with highest weight \(\lambda_ 0\). Further, let \((\Lambda (\mathfrak u^{-},L(\lambda_ 0)^ t),\partial)\) be the standard chain complex associated to the Lie algebra \(\mathfrak u^{-}\), with coefficients in the right module \(L(\lambda_ 0)^ t\) and let \((C(\mathfrak q,\mathfrak r),d)\) denote the standard co-chain complex associated to the Lie algebra pair \((\mathfrak q,\mathfrak r)\) (with trivial coefficient \(\mathbb C\)). There is associated (in general infinite dimensional) a group \(G\) (resp. a ``parabolic'' subgroup \(p_ S)\) with \(\mathfrak q'\) (resp. \(\mathfrak p_ S\cap\mathfrak q'\)). The flag variety \(G/P_ S\) admits a Bruhat cell decomposition with cells \(\{V_ w\}\), parametrized by \(w\in W_ S\backslash W\cong W^ 1_ S\) (\(W\) is the Weyl group for \(\mathfrak q)\).
In this paper, we explicitly compute the action of the Laplacian \(\Delta =\partial\partial^*+\partial^*\partial\) on \(\Lambda (\mathfrak u^{-}, L(\lambda_ 0)^ t).\) Further, we use this to prove the ``disjointness'' of the operators \(d\) and \(\partial\), acting on \(C(\mathfrak q,\mathfrak r)\). This gives rise to a ``Hodge type'' decomposition, with respect to the pair \(d,\partial\) (\(d,\partial\) are not adjoints of each other), of the space \(C(\mathfrak q, \mathfrak r)\). In particular, every \(d\) cohomology class in \(C(\mathfrak q,\mathfrak r)\) has a unique \(d,\partial\) closed representative. The ``Hodge type'' decomposition also gives, by a slight refinement of the arguments, that \(H^*_ d(\mathfrak q,\mathfrak r)\) is bi-graded; \(H_ d^{p,q}(\mathfrak q,\mathfrak r)=0\) unless \(p=q\) and \(H_ d^{p,p}(\mathfrak q,\mathfrak r)\) is a vector space with a ``canonical'' \(\mathbb C\)-basis \(\{s^ w\}_{w\in W^ 1_ S}\) with \(w=p\).
Finally (and this was our main interest) we prove that, properly defined, \(\int_{V_{w'}}s^ w=0\) unless \(w=w'\) and \(\int_{V_ w}s^ w>0\).
When \(\mathfrak q\) is a finite-dimensional semisimple Lie algebra, all these results are due to Kostant. In the infinite dimensional situation, Garland has computed \(\Delta\) for a special case. Most of the other results are new (as far as is known to the author). action of Laplacian; Hodge type decomposition; Kac-Moody-Lie algebra; chain complex; flag variety Cohomology of Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Hodge theory in global analysis, Grassmannians, Schubert varieties, flag manifolds Geometry of Schubert cells and cohomology of Kac-Moody-Lie 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 Some technical results on the deformations of varieties of general type and on permanence of semi-log-canonical singularities are proved. These results are applied to show that the connected component of the moduli space of stable surfaces containing the moduli point of a product of stable curves is the product of the moduli spaces of the curves, assuming the curves have different genera. An application of this result shows that even after compactifying the moduli space and fixing numerical invariants, the moduli spaces are still very disconnected. semi-log-canonical singularities; components of moduli schemes Michael A. van Opstall, Moduli of products of curves, Arch. Math. (Basel) 84 (2005), no. 2, 148 -- 154. Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry Moduli of products of 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 Let \(F\) be a field, \(R\) a ring and let \({\mathcal C}(F,R)\) be the category of Chow motives with coefficients in \(R\). If \(X\) is a smooth projective variety over \(F\) its motive \(M(X)\in{\mathcal C}(F, R)\) is split if it is the finite sum of Tate motives, it is geometrically split if it splits over a field extension of \(F\). For a gemetrically split variety \(X\) let's denote by \(\overline X\) the scalar extension of \(X\) to a splitting field of its motive and by \(\overline{\mathrm{CH}}(X)\) the subring of \(F\)-rational cycles in \(\mathrm{CH}(\overline X)\). A variety \(X\) is said to satisfy the nilpotence principle if for every fled extension \(E/F\) the kernel of the base-change homomorphism \(\text{End}_F(M(X))\to\text{End}_E(M(X))\) consists of nilpotents. If the ring \(R\) is finite any shift \(N(k)\) of any summand of a geometrically split \(F\)-variety satisfying the nilpotence principle also satisfies the Krull-Schmidt principe, i.e. every direct summand decomposition can be refined to a unique complete decomposition.
Let \(R=\mathbb{F}_p\) and let \(D\) be a central division \(F\)-algebra of degree \(p^n\). Let \(X(p^n,D)\) be the generalized Severi-Brauer variety of right ideals in \(D\) of reduced dimension \(p^m\), with \(0\leq m\leq n\). In particular \(X(p^n,D)= \text{Spec}\,F\) and \(X(1,D)\) is the usual Severi-Brauer variety of \(D\).
It has been proved by Karpenko that the motive with coefficients in \(R=\mathbb{F}_p\) of the Severi -Brauer variety \(X= X (1,D)\) is indecomposable. Also the motive with coefficients in \(\mathbb{F}_2\) of the variety \(X(2,D)\) is indecomposable.
In this paper the author proves the following results, showing that these are the only cases where the motive of a generalized Severi-Brauer variety is indecomposable.
Theorem 1. Let \(D\) be a central division \(F\)-algebra of degree \(p^n\) with \(n\geq 1\). Let \(X\) be the Severi-Brauer variety \(X(1,D)\) and \(Y\) a variety satisfying the nilpotence principle, such that \(Y\) is split over the function field of \(X\). Then for any integer \(k\) the number of copies \(M(X)(k)\) in the complete motivic decomposition of \(Y\) is equal to \(\dim_{\mathbb{F}_p}\overline{\mathrm{CH}}_{\dim Y-k}(X\times Y)\), where \(f\) is the projection onto the second factor.
Corollary 1. Let \(D\) be a central division \(F\)-algebra of degree \(p^n\) with \(n\geq 1\). The motive with coefficients in \(\mathbb{F}_p\) of the variety \(X(p^m,D)\) is decomposable for \(p=2\), \(1< m< n\) and for \(p> 2\) and \(0< m< n\). In these cases the motive \(M(X(1, D))(k)\) is a summand of \(M(X(p^M,D)\) for \(2\leq k\leq p^n- p^m\).
Note that in the case \(p= 3\), \(m= 1\), \(n= 2\) and \(p= 2\), \(m= 2\), \(n= 3\), \(M(X(1, D))(k)\) for \(2\leq k\leq p^n- p^m\) gives a complete list of indecomposable motivic summands of the variety \(X(p^m,D)\). central simple algebras; generalized Severi-Brauer varieties; Chow groups and motives Zhykhovich, M., \textit{decompositions of motives of generalized Severi-Brauer varieties}, Doc. Math., 17, 151-165, (2012) Affine algebraic groups, hyperalgebra constructions, Algebraic cycles Decompositions of motives of generalized Severi-Brauer 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 \textit{K. Kato} and \textit{C. Nakayama} [Kodai Math. J. 22, No. 2, 161-186 (1999; Zbl 0957.14015)] constructed a ringed space \((X^{\log},{\mathcal O}_X^{\log})\) over a given fs log analytic space \((X,{\mathcal M}_X)\) and proved a log version of the Riemann-Hilbert correspondence on them. In the case where the fs log analytic space \((X,{\mathcal M}_X)\) is the one corresponding to a divisor \(D\) with normal crossings on a complex manifold \(X\), i.e., \({\mathcal M}_X: =\{f\in{\mathcal O}_X\mid f\) is invertible outside \(D_{\text{red}}\}\), the projection \(\tau_X:X^{\log}\to X\) is nothing but the real oriented blowing-up of \(X\) along \(D_{\text{red}}\).
Let us consider a relative case. Let \(f:X\to\Delta\) be a proper surjective flat morphism of a complex manifold onto an open disc such that \(f\) is smooth over the punctured disc \(\Delta^*: =\Delta-\{0\}\) and that the central fiber \(X_0: =f^{-1} (0)\) is a reduced divisor with simple normal crossings. Let \(Y\) be a divisor on \(X\), flat with respect to \(f\). We assume that \(X_0+Y\) is also a divisor with simple normal crossings. Then, by the paper cited above, we can construct a map \(f^{\log}: X^{\log}\to\Delta^{\log}\) and a subspace \(Y^{\log}\) of \(X^{\log}\) over the given ones and we have a commutative diagram:
\[
\begin{tikzcd} (X^{\log}, Y^{\log})\ar[r,"\tau_X"]\ar[d,"f^{\log}" '] & (X,Y) \ar[d,"f"]\\ \Delta^{\log} \ar[r,"\tau_\Delta" '] & \Delta \rlap{\, .}\end{tikzcd}\tag{1}
\]
The main result in the present paper is that the family
\[
\overset\circ f^{\log}: (X^{\log}-Y^{\log}{)} \Delta^{\log} \tag{2}
\]
of open spaces is locally piecewise \(C^\infty\) trivial over the base \(\Delta^{\log}\). As a consequence, we see that the family (2) is the one which recovers the vanishing cycles, in the most naive sense, of the degenerating family \({\overset\circ f}: (X-Y)\to \Delta\).
This implies, in particular, that \(L_Z:=R^q \left(\overset\circ f^{\log}\right)_*\mathbb{Z}\) is a locally constant sheaf of \(\mathbb{Z}\)-modules on \(\Delta^{\text{log}}\). On the other hand, \textit{J. Steenbrink} and \textit{S. Zucker} [Invent. Math. 80, 489-542 (1985; Zbl 0626.14007)] showed that \({\mathcal V}:=R^qf_* \Omega^\bullet_{X/ \Delta}(\log (X_0+Y))\) is a free \({\mathcal O}_\Delta\)-module with the Gauss-Manin connection \(\Delta\). We thus have
\[
{\mathcal V} \simeq (\tau_\Delta)_* ({\mathcal O}_\Delta^{\text{log}}\otimes_CL_C) \text{ on }\Delta
\]
under the log version of the Riemann-Hilbert correspondence established in the cited paper by Kato and Nakayama. As a corollary, we have two types of integral structure of the degenerate variation of mixed Hodge structure on \({\mathcal V}\). recovery of vanishing cycles; log geometry; Riemann-Hilbert correspondence; integral structure of the degenerate variation of mixed Hodge structure S. USUI, Recovery of vanishing cycles by log geometry, Tôhoku Math. J., 53 (1) (2001), pp. 1-36. Zbl1015.14005 MR1808639 Variation of Hodge structures (algebro-geometric aspects), Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations Recovery of vanishing cycles by log geometry | 0 |
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