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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 abstract: Let \(G\) be a finite subgroup of \(GL (3,\mathbb{C})\). Then \(G\) acts on \(\mathbb{C}^ 3\). It is well known that \(\mathbb{C}^ 3/G\) is Gorenstein if and only if \(G \subseteq SL (3,\mathbb{C})\). In chapter one, we sketch the classification of finite subgroups of \(SL(3,\mathbb{C})\). We include two more types \((J)\) and \((K)\) which were usually missed in the work of many mathematicians. In chapter 2, we give a general method to find invariant polynomials and their relations of finite subgroups of \(GL(3,\mathbb{C})\). The method is in practice substantially better than the classical method due to Noether. In chapter 3, we recall some properties of quotient varieties and prove that \(\mathbb{C}^ 3/G\) has isolated singularities if and only if \(G\) is abelian and 1 is not an eigenvalue of \(g\) in \(G\). We also apply the method in chapter 2 to find minimal generators of the ring of invariant polynomials as well as their relations. finite subgroups of \(SL (3,\mathbb{C})\); quotient varieties; isolated singularities; ring of invariant polynomials S. Yau and Y. Yu, \textit{Gorenstein quotient singularities in dimension three}, \textit{Mem. Amer. Math. Soc.}\textbf{505}, American Mathematical Society, Providence RI U.S.A., (1993). Singularities in algebraic geometry, Homogeneous spaces and generalizations, Local complex singularities, Linear algebraic groups over the reals, the complexes, the quaternions, Geometric invariant theory, Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects) Gorenstein quotient singularities in dimension three | 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 surveys the remarkable work of \textit{R.~Borcherds} [Invent. Math. 120, 161--213 (1995; Zbl 0932.11028)] about a multiplicative correspondence between classical modular forms with poles at cusps and meromorphic modular forms on complex varieties of the form
\[
SO(n) \times SO(2) \backslash SO(n,2) / \Gamma,
\]
where \(\Gamma\) is an arithmetic subgroup of the real Lie algebra \(SO(n,2)\). The article formulates Borcherds' results in various examples and discusses connections with other topics in mathematics such as generalized Kac-Moody algebras, \(K3\) surfaces, mirror symmetry, string theory, and Donaldson invariants. automorphic forms; Kac-Moody algebras; theta correspondence; theta functions; multiplicative correspondence; modular forms; meromorphic modular forms; K3 surfaces; mirror symmetry; string theory; Donaldson invariants Kontsevich, M., Product formulas for modular forms on O(2, n) (after R.borcherds), Astérisque, 245, 41, (1997) Other groups and their modular and automorphic forms (several variables), Forms of half-integer weight; nonholomorphic modular forms, \(K3\) surfaces and Enriques surfaces, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Relationship to Lie algebras and finite simple groups, Differentiable structures in differential topology Product formulas for modular forms on \(O(2,n)\) (after R. Borcherds) | 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 0716.00007.]
This is a survey paper in which the emphasis is made on the similarities and differences between the theory of invariants of commuting and noncommuting variables. Such aspects as: (a) formula for the Hilbert series of the algebra of invariants, (b) finite generatedness of this algebra, (c) freeness of this algebra, are considered in both of the theories. The cases of a finite group and an infinite group are considered resp. in the first and in the second sections. The classical case of the group \(SL(2,\mathbb{C})\) is considered in the next section, including a discussion of the asymptotics of the coefficients of the Hilbert series. The last section contains the formulation of some results of T. Tambour on the noncommutative invariant theory of \(SL(2,\mathbb{C})\). Some interesting and intriguing conjectures and research proposals are formulated. theory of invariants; Hilbert series; finite generatedness; freeness; asymptotics; noncommutative invariant theory of \(SL(2,\mathbb{C})\) Almkvist, Gert: Commutative and noncommutative invariant theory, Banach center publ. 26, 259-268 (1990) Representation theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Vector and tensor algebra, theory of invariants, Trace rings and invariant theory (associative rings and algebras), Group actions on varieties or schemes (quotients) Commutative and noncommutative 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 Let RG be the group ring of a finite group G over a commutative ring R with identity. If M, N are left RG-progenerators and \(f\in Hom_{RG}(M,N)\), then \(M_ f=Hom_{RG}(N,M)\) is a finitely generated R-module that is an R-algebra subject to the operation \(x\circ y=xfy\). The set \(M_{{\mathcal C}}(R)\) of classes of a certain equivalence on the set of all algebras \(M_ f\) is a commutative monoid with zero subject to the operation induced by \(\otimes_ R\). This modifies the idea used for a similar construction by \textit{F. R. DeMeyer} and \textit{T. J. Ford} [J. Algebra 113, 379-398 (1988; Zbl 0654.13016)]. If R is a field, then \(M_{{\mathcal C}}(R)\) has no nonzero nilpotents and the group of units of \(M_{{\mathcal C}}(R)\) coincides with the Schur group of R. Moreover, a computation of \(M_{{\mathcal C}}(R)\) is given in the case where R is a field of characteristic zero.
Reviewer's note: As observed in the paper, because of the sandwich multiplication in \(M_ f\), the monoids \(M_{{\mathcal C}}(R)\) can be used to classify via Munn algebras the semigroup rings of finite completely 0- simple semigroups. However, this does not apply to semigroup rings of arbitrary finite semigroups as the author seems to claim. group ring; finite group; progenerators; commutative monoid; group of units; Schur group; Munn algebras; semigroup rings; finite completely 0- simple semigroups Sundhir, N. R.: On classification of semigroup rings. Semigroup forum 40, 159-179 (1990) Group rings, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Semigroup rings, multiplicative semigroups of rings, Brauer groups of schemes On classification of semigroup rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0716.00007.]
The author reports on various aspects of his research during the eighties; a common theme is that of finite dimensional algebras of arbitrary fields. The first circle of ideas concerns unirationality of conic bundles. As a sample consider the following problem: Suppose that \(A\) is a finite dimensional central simple algebra over \(k(x)\) with a \(k\)-point. Does \(A\) have a \(k\)-rational splitting field? If \(k\) is a real field, the answer is yes. --- From Dokl. Akad. Nauk BSSR 29, 1061-1064 (1985; Zbl 0609.14030), the author cites the following of his results concerning the circumstances of the problem: Theorem. If \(k\) is a Henselian field then \(A\) has a \(k\)-rational splitting field.
\textit{I. I. Voronovich} has the following theorem which illustrates the connection between algebras and unirationality [Dokl. Akad Nauk BSSR 30, 773-775 (1986; Zbl 0609.12014)]: Suppose that \(k\) is perfect and any nonvoid absolutely irreducible algebraic variety of \(k\) has a \(k\)-point. Then \(A\) has a rational splitting field and any conic bundle over \(k\) is unirational. (Conic bundles are certain algebraic surfaces fibered over curves of genus zero with similar fibers: see the summary for the precise definitions.)
A second circle of ideas concerns algebraic groups. Let \(G\) be a quasisimple and isotropic algebraic group over \(k\) and \(G_ k\) the group of rational elements over \(k\); let \(G_ k^ + \) be the subgroup generated by all unipotent elements of \(G_ k\) contained in the unipotent radical of some parabolic subgroup of \(G\), defined over \(k\). The Whitehead group of \(G\) is \(W(G_ k) =G_ k/G_ k^ + \), and the Kneser-Tits conjecture reads: If \(G\) is simply connected, then \(W(G_ k)=\{1\}\). The author reports on the status of the situation for classical groups over a finite dimensional division algebra \(D\) over a \(p\)-adic or algebraic number field [joint work with \textit{V. P. Platonov} in Math. USSR, Izv. 25, 573-599 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 6, 1266-1294 (1984; Zbl 0579.16011)], and joint work with \textit{A. P. Monastyrnyj}, Preprint 8 (318), Inst. Mat. Akad. Nauk BSSR, Minsk (April 1988); see Sov. Math., Dokl. 42, No. 2, 351-335 (1991); translation from Dokl. Akac. Nauk SSSR 314, No. 1, 110-114 (1990)]. Roughly, \(W(G_ k)\) is trivial for \(B_ \ell\) and \(C_ \ell\), and nontrivial for \(A_ \ell\) and \(D_ \ell\). finite dimensional algebras; unirationality of conic bundles; central simple algebra; rational splitting field Rational and unirational varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Finite-dimensional division rings, Linear algebraic groups over arbitrary fields \(K\)-unirationality of conic bundles, the Kneser-Tits conjecture for spinor groups and 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 This report concerns very recent developments in the theory of arc spaces, motivic integration, and the McKay correspondence. To a large extent, these developments have been propelled by the work of J. Denef and F. Loeser, during the last decade, and it is the author's main goal to provide a profound survey on their results published in numerous papers and preprints. However, this report is by far more than just a survey, since the author presents some of their results somewhat differently, very much so for the benefit of a better understanding of this comparatively new, rapidly progressing and highly advanced field of research in algebraic geometry. Therefore this report contains much more detailed proofs than usual for a seminar talk, and that is why it may be regarded as a research paper, too.
An \(n\)-jet of an arc in an algebraic variety \(X\) defined over an algebraically closed field \(k\) is a \(k[[t]]/(t^{n+1})\)-valued point of \(X\). The set of such \(n\)-jets are the closed points of a variety \(L_n(X)\) over \(k\) and the arc space \(L(X)\) of the variety \(X\) is by definition the projective limit \(\displaystyle \varprojlim_n L_n(X)\).
The first systematic study of arc spaces goes back to the work of \textit{J. Nash}, written in 1968 and finally published in 1995 [Duke Math. J. 81, 31--38 (1995; Zbl 0880.14010)]. The renewed interest in arc spaces arose in the 1990's, namely in the context of mirror symmetry and \textit{V. Batyrev}'s study of Calabi-Yau manifolds [in: New trends in algebraic geometry. Set. Pap. Euro Conf., Warwick 1996, Lond. Math. Soc. Lect. Note Ser. 264, 1--11 (1999; Zbl 0955.14028)], on the one hand, and in the study of \(p\)-adic integration techniques in complex algebraic geometry by \textit{J. Denef} and \textit{F. Loeser} [Invent. Math. 135, 201--232 (1999; Zbl 0928.14004)], on the other hand.
The underlying techniques developed here, now going under the name ``motivic integration'', have led to an avalanche of far-reaching application in the meantime, including the so-called ``stringy invariants'' of singularities, a complex analogue of Igusa's local zeta function, a motivic version of the classical Thom-Sebastiani property, and the motivic McKay correspondence.
In the present article, the author masterly succeeds in covering, explaining, and partially revising these recent achievements, guided by the various original research papers that appeared during the past ten years. The paper is divided into eight main sections treating the following topics:
1. The arc space and its measure;
2. The transformation rule (of Denef-Loeser-Kontsevich);
3. The basic formula (in the Grothendieck ring);
4. The motivic nearby fiber;
5. The motivic zeta function (of Denef-Loeser);
6. The motivic convolution (including the abstract Thom-Sebastiani property);
7. The McKay correspondence (after Batyrev, Denef-Loeser, Reid);
8. A (new) proof of the Denef-Loeser-Kontsevich transformation rule.
All in all, this extended report provides a highly valuable introduction to and a systematic overview of the recent and extremely powerful method of motivic integration. Also, this article is a very useful guide through the vast original literature on the subject. motivic zeta function; motivic convolution; families of varieties; Grothendieck ring; \(n\)-jet spaces; motivic measures; stringy invariants of singularities; arc spaces; Thom-Sebastiani property; motivic McKay correspondence Looijenga, E., Motivic measures, Séminaire Bourbaki 874, Astérisque, 276, 267-297, (2002) Motivic cohomology; motivic homotopy theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Rational points, Families, moduli, classification: algebraic theory, Structure of families (Picard-Lefschetz, monodromy, etc.) Motivic measures | 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 Gabrielov numbers describe certain Coxeter-Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold's strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup \(G\) of \(\mathrm{SL}(3,\mathbb C)\) using the Gabrielov numbers of the cusp singularity and data of the group \(G\). Here we consider a crepant resolution \(Y \to \mathbb C^3/G\) and the preimage \(Z\) of the image of the Milnor fibre of the cusp singularity under the natural projection \(\mathbb C^3 \to\mathbb C^3/G\). Using the McKay correspondence, we compute the homology of the pair \((Y,Z)\). We construct a basis of the relative homology group \(H_3(Y,Z;\mathbb Q)\) with a Coxeter-Dynkin diagram where one can read off the Gabrielov numbers. cusp singularity; group action; crepant resolution; McKay correspondence; Coxeter-Dynkin diagram; Gabrielov numbers Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory, McKay correspondence, Group actions on varieties or schemes (quotients) A geometric definition of Gabrielov numbers | 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 recall the form factors \(f_{N,N}^{(j)}\) corresponding to the \(\lambda\)-extension \(C(N,N;\lambda)\) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a ``Russian-doll'' nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral \(E\)). The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure, the ``scaled'' linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the \(n\)-particle contributions \(\xi^{(n)}\) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for \(n=1,2,3,4\) and, only modulo a prime, for \(n=5\) and 6, thus providing a large set of (possible) new singularities of the \(\xi^{(n)}\). We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition \(1+3\omega+4\omega^2=0\), that occurs in the linear differential equation of \(\xi^{(3)}\), actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion. sigma form of Painlevé VI; two-point correlation functions of the lattice Ising model; Fuchsian linear differential equations; complete elliptic integrals; elliptic representation of Painlevé VI; scaling limit of the Ising model; susceptibility of the Ising model; singular behaviour; Fuchsian linear differential equations; apparent singularities; Landau singularities; pinch singularities; modular forms; Landen transformation; isogenies of elliptic curves; complex multiplication; Heegner numbers; moduli space of curves; pointed curves S. Boukraa, S. Hassani, J.-M. Maillard, B. M. McCoy, W. P. Orrick, and N. Zenine, ``Holonomy of the Ising model form factors,'' Journal of Physics A, vol. 40, no. 1, pp. 75-111, 2007. Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, General theory of ordinary differential operators, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Ordinary differential operators, Functional-differential equations (including equations with delayed, advanced or state-dependent argument), Elliptic curves, Continuum limits in quantum field theory, Other functions defined by series and integrals, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics From holonomy of the Ising model form factors to \(n\)-fold integrals and the theory of elliptic 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 For a review of the English original (1977) see Zbl 0346.20020. algebras of invariants; representation theory; finite reflection groups; graded polynomial algebras Linear algebraic groups and related topics, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Vector and tensor algebra, theory of invariants, Classical groups (algebro-geometric aspects) Invariant theory. Transl. from the English by V. L. Popov | 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 algebra on which a meaningful notion of positivity exists. A `positivstellensatz' for \(\mathcal{A}\) is a theorem saying that elements in \(\mathcal{A}\) that are positive have an algebraic description turning positivity `evident'. The classical example is Artin's solution to Hilbert's 17th problem according to which a rational function on \(\mathbb{R}^n\) that is nonnegative is a sum of squares of such functions. Positivstellensätze for commutative algebras have meanwhile a rich history that is reported in the books by Bochnak, Coste, Roy; Delzell and Prestel; and Marshall.
More recently, positivstellensätze for noncommutative algebras have been proved. At the beginning of this endeavour we have \textit{J. W. Helton} [Ann. Math. (2) 156, No. 2, 675--694 (2002; Zbl 1033.12001)]. In the present paper, positivstellensätze are proved for path algebras, matrices over certain noncommutative and commutative algebras, crossed product algebras, matrix polynomials in intervals, matrices over fields, and cyclic algebras.
Sections 1 and 2 give much of the necessary definitions and notations fundamental for what follows: Let \(\mathcal{A}\) be a \(*\)-algebra, i.e., an algebra over \({\mathsf L}={\mathsf R}\) or \({\mathsf C}\) with involution, where \({\mathsf R}\) is a real field and \({\mathsf C}={\mathsf R}(\sqrt{-1}).\) If \((\mathcal{V},\langle \cdot,\cdot \rangle)\) is a unitary space (over \(\mathbb{C}\)), let \(\mathcal{L(V)}\) be the algebra of linear mappings of \(\mathcal{V}\) and let \(\mathcal{L}^+(\mathcal{V})\) denote the largest \(O^*\)-subalgebra; i.e., the largest subalgebra of \(\mathcal{L (V)}\) with the property that for each of its elements \(A\) there exists an adjoint \(A^*\) in it -- i.e., \(\forall v,w \in \mathcal{V}\) we have \(\langle Av,w\rangle= \langle v,A^*w\rangle.\) Recall that the standard notion of positivity on \(\mathcal{L(V)}\) is given by \(A>0\) iff \(\langle Av,v\rangle>0\) for \(v\neq 0.\) A \(*\)-representation of a \(*\)-algebra \(\mathcal{A}\) over \(\mathbb{C}\) is a homomorphism \(\pi:\mathcal{A} \rightarrow \mathcal{L^+(V)}.\)
Let \(\mathcal{A}_h=\{a\in \mathcal{A}:a^*=a\}\) (with \(\mathcal{A}\) over \({\mathsf L}\)) be the self adjoint or Hermitian elements of \(\mathcal{A}\), and \(\mathcal{A}^\circ=\{x\in \mathcal{A}\setminus \{0\}: x \text{ is not zero divisor}\}.\) A set \(\mathcal{C}\subseteq \mathcal{A}_h\) is a quadratic module if it is closed under nonnegative linear combinations, contains 1, and satisfies \(x^*\mathcal{C}x\subseteq \mathcal{C}\) for all \(x\in \mathcal{A}.\) The smallest quadratic module in \(\mathcal{A},\) denoted \(\sum \mathcal{A}^2,\) is the set of sums of form \(x_1^*x_1+\cdots+x_m^*x_m\) with \(x_i\in \mathcal{A}.\) A quadratic module \(\mathcal{C}\) defines a noncommutative preordering if the product of any two commuting elements in \(\mathcal{C}\) belongs to \(\mathcal{C}.\) Denote the smallest noncommutative preordering by \(\sum_{ nc} \mathcal{A}^2.\) Given \(a\in \mathcal{A},\) a set \(\mathcal{S}_a \subseteq \mathcal{A}_h\) containing \(a,\) so that for all \(x\in \mathcal{A},\) \(x^*\mathcal{S}_a x \subseteq \mathcal{S}_a;\) and \(bc=cb,~ b\in \mathcal{S}_a,~ c\in \sum_{nc} \mathcal{A}^2\) together imply \(bc\in \mathcal{S}_a\) is a denominator set for \(a.\) Finally, given a \(*\)-algebra \(\mathcal{A}\) over \({\mathsf L}\) with \(*\)-subalgebra \(\mathcal{B},\) certain linear and involution-preserving surjections \(p:\mathcal{A} \rightarrow \mathcal{B}\) so that \(p(\sum \mathcal{A}^2) \subseteq \sum \mathcal{B}^2\) are called strong conditional expectations.
With these definitions and a notion of positivity on \(\mathcal{A},\) let \(\mathcal{A}_+\) be the positive elements in \(\mathcal{A}.\) A positivstellensatz of type I, II, III, IV, V is satisfied by \(x\in \mathcal{A}_+\) according to the following: type I: if \(x\in \sum \mathcal{A}^2;\) type II: if there exists a \(c\in \mathcal{A}^\circ\) so that \(c^*xc \in \sum \mathcal{A}^2;\) type III: if there is a \(c\in \mathcal{A}^\circ \cap \sum \mathcal{A}^2\) so that \(xc=cx\) and \(xc \in \sum \mathcal{A}^2;\) type IV: if \(x\in \sum_{nc} \mathcal{A}^2;\) type V: if there is \(c_x\in \mathcal{S}_x\) so that \(\sum_{nc} \mathcal{A}^2.\) Algebra \(\mathcal{A}\) is said to satisfy a positivstellensatz of type X if all its positive elements satisfy the respective positivstellensatz.
Section 3 recalls the notion of the path algebra \(\mathbb{C} \Gamma\), defined in a natural way from a directed multigraph \(\Gamma\) that contains with each arrow \(b\) also its reverse \(b^*.\) Let \(\mathcal{F}=\mathbb{C}\langle a_1,\dots,a_m,a_1^*,\dots,a_m^* \rangle\) be the free \(*\)-algebra over \(\mathbb{C}\) on \(m\) generators. In Proposition 3.1, the authors generalize the main theorem of Helton [loc. cit.]: a Hermitian element in the matrix algebra \(M_n(\mathcal{F})\) satisfies \(\rho(X)\geq 0\) for every finite dimensional representation \(\rho,\) if and only if \(X\in \sum M_n(\mathcal{F})^2.\) They then define a strong conditional expectation from \(M_n(\mathcal{F})\) onto \(\mathbb{C} \Gamma\) to prove a similar type I positivstellensatz for the path algebra in a manner different from \textit{S. Popovych} [J. Algebra 324, No. 9, 2418--2431 (2010; Zbl 1219.46068)]: a Hermitian \(X\) in \(\mathbb{C} \Gamma\) allows only nonnegative images under finite dimensional \(*\)-representations if and only if \(X=\sum_{j=0}^k X_j^* X_j\) for some \(X_j\in \mathbb{C}\Gamma.\)
Section 4 proves an Artin theorem for matrices over noncommutative zero divisor free unital \(*\)-algebras that satisfy the left Ore condition: given \(a\in \mathcal{A}, s\in \mathcal{A}^\circ=\mathcal{A}\setminus\{0\},\) there exist \(b\in \mathcal{A}, t\in \mathcal{A}^\circ\) so that \(ta=bs.\) First diagonalizability results for Hermitian matrices over \(\mathcal{A}\) are shown. Next suppose \(\mathcal{A}\) to be an \(O^*\)-algebra on a unitary space. Then natural definitions of positivity yield \(\mathcal{A}_+\) and \(M_n(\mathcal{A})_+.\) Theorem 4.8 shows that a type II positivstellensatz holds for the matrix algebra \(M_n(\mathcal{A})\) provided it holds for \(\mathcal{A}\) itself.
In Section 5 for a unital \(*\)-algebra \(\mathcal{A},\) and a finite group \(G\) of \(*\)-automorphisms of \(\mathcal{A},\) the crossed product \(\mathcal{A} \times_\alpha G\) is defined -- see \textit{R. S. Pierce} [Associative algebras. New York-Heidelberg-Berlin: Springer (1982; Zbl 0497.16001)]. One then has the regular covariant representation \(\pi_{creg}:\mathcal{A} \times_\alpha G \rightarrow \mathcal{L}(\oplus_{g\in G} \mathcal{V})\) which defines in analogy to earlier observations a notion of positivity via \((\mathcal{A} \times_\alpha G)_+=\{x\in \mathcal{A} \times_\alpha G: \pi_{creg}(x) \geq 0\}.\) Then if a positivstellensatz of type I or type III holds for \(M_n(\mathcal{A})\) then - again via a strong conditional expectation -- the respective theorem will hold for \(\mathcal{A} \times_\alpha G.\) Also if \(\mathcal{A}\) is an \(O^*\)-algebra without zero divisors and \(\mathcal{A}^\circ\) is left Ore, we have: if a type II satz holds for \(\mathcal{A}\) then it holds for \(\mathcal{A} \times_\alpha G.\)
Section 6 assumes \(\mathcal{A}\) to be a finitely generated unital and commutative \(*\)-algebra without zero divisors over \(\mathbb{R}\) or \(\mathbb{C}.\) By \(\hat{\mathcal{A}}\) the set of characters is understood and it is assumed that \(\sum_j a_j^*a_j=0\) implies \(a_1=a_2= \cdots=a_n=0.\) This can be guaranteed if \(\hat{\mathcal{A}}\) separates points. Positivity notions are given by \(\mathcal{A}_+ =\{a\in \mathcal{A}: \chi(a)\geq 0 \text{ for } \chi \in \hat{\mathcal{A}}\}\) and \(M_n(\mathcal{A})_+=\{A: (\chi(a_{ij}))\geq 0 \text{ for }\chi \in \hat{\mathcal{A}}\}.\) Then a type II positivstellensatz holds for \(M_n(\mathcal{A}).\) The proof uses induction on \(n\) and diagonalization.
Section 7 recalls a matrix version of the classical Fejér-Riesz theorem due to \textit{M. Rosenblum} [J. Math. Anal. Appl. 23, 139--147 (1968; Zbl 0159.43102)] and defines a conditional expectation to show the following interesting result. Theorem 7.3: Let \(F(x) \in M_n(\mathbb{C}[x])\) be a self adjoint matrix polynomial. Then \(F(x)\) is positive semidefinite in every point \(x\in [a,b]\) if and only if \(F=G_1^*G_1+(b-x)(x-a)G_2^*G_2\) for some \(G_1,G_2 \in M_n(\mathbb{C}[x]).\) Similar results are shown for half open intervals.
Section 8 is concerned with positivstellensätze for \(M_n({\mathsf L})\) with \({\mathsf L}\) as in section 2. If \(B=\text{diag}(1,\lambda_1,\dots, \lambda_{n-1})\) with \(\lambda_i\in \mathbb{R}^\circ\), then for \(X\in M_n({\mathsf L})\) one defines \(X^\tau = B^{-1} X^* B.\) This yields an involution which is associated to the inner product \(\langle x,y \rangle_1 :=\langle Bx,y \rangle,\) where the right hand side uses the standard inner product. Write \(P_B({\mathsf L})\) for the preordering generated by the \(\lambda_i.\) An ordering \(p\) of \({\mathsf R}\) that contains \(\lambda_1, \dots, \lambda_{n-1}\) is a \(*\)-ordering. The positivity is defined by \((M_n({\mathsf L}), \tau)_+\) being the set of matrices \(X\) for which \(X=X^\tau\) and all the principal minors of \(X\) belonging to \(P_B({\mathsf L}).\) It is shown that a type IV positivstellensatz holds for \((M_n({\mathsf L}),\tau).\) \textit{I. Klep} and \textit{T. Unger} have shown that a type I satz cannot hold for \(n\geq 3;\) see [J. Algebra 324, No. 2, 256--268 (2010; Zbl 1262.16039)].
Section 9 uses results and notation of the previous section to show positivstellensätze for cyclic algebras. Let \({\mathsf L}\) be as before. Let \( {\mathsf L/K} \) be a Galois extension with group \(\mathbb{Z}/(n),\) and \(\mathfrak{A}\) be the cyclic algebra associated to \({\mathsf L/K} \) (see Pierce [loc. cit., Ch. XV]). Positivity is defined by \(\mathfrak{A}_+\) being the set of elements \(y=y^*\) for which \(\mathfrak{p}(x^*yx)\in P_B({\mathsf L} )\) for all \(x\in \mathfrak{A},\) where \(\mathfrak{p}:\mathfrak{A} \rightarrow {\mathsf L} \) is a certain canonical projection. A type IV positivstellensatz holds for \(\mathfrak{A}.\)
Section 10 illustrates the results of the previous sections having in common that \(\mathcal{A}_+=\{x\in \mathcal{A}_h: \pi(x) \geq 0 \text{ for all finite dimensional }*\)-representations matrices over ring; conditional expectation; Positivstellensätze; sums of squares; noncommutative associative algebras; algebras with involutions; Ore condition; diagonalization; quivers; path algebra; cyclic algebra; enveloping algebra of Lie algebra; path algebras; crossed product algebras; matrix polynomials; preordering; Lie algebras; Weyl algebras Savchuk, Y; Schmüdgen, K, Positivstellensätze for algebras of matrices, Linear Algebra Appl., 43, 758-788, (2012) Positive matrices and their generalizations; cones of matrices, Matrices over special rings (quaternions, finite fields, etc.), Real algebraic and real-analytic geometry, Representation theory of associative rings and algebras, Universal enveloping (super)algebras, Hermitian, skew-Hermitian, and related matrices Positivstellensätze for algebras of matrices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\iota\colon\mathbb C^*\to G\) be a homomorphism from the multiplicative group to a connected reductive algebraic group \(G\) over \(\mathbb C\). The action of \(\mathbb C^*\) provides a grading on the Lie algebra \(LG\) of \(G\). Let \(G^\iota\) denote the centralizer of the image of \(\iota\). For some nonzero \(n\) let \(L_nG\) be the degree \(n\) part of \(LG\).
The paper presents an algorithm to determine multiplicities \(m_{i;\mathcal L,\mathcal L'}\) of local systems in intersection cohomology sheaves associated to \(G^\iota\)-equivariant local systems on \(G^\iota\)-orbits in \(L_nG\). This algorithm is described in section 2 in terms of combinatorial data associated to the situation. The algorithm produces a square matrix \((c_{\xi,\xi'})\) with entries in \(\mathbb Q(v)\), where \(v\) is an indeterminate. Section 3 discusses perverse sheaves. The connection between the square matrix and the multiplicities \(m_{i;\mathcal L,\mathcal L'}\) is treated in section 4. Termination of the algorithm is not clear from the description in section 2, but this apparently follows from geometry. perverse sheaves; graded Lie algebras; canonical bases; connected reductive algebraic groups; algorithms; multiplicities of local systems; intersection cohomology sheaves Lusztig, G, Graded Lie algebras and intersection cohomology, representation theory of algebraic groups and quantum groups, Progr. Math., 284, 191-244, (2010) Representation theory for linear algebraic groups, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Cohomology theory for linear algebraic groups, Lie algebras of linear algebraic groups, Graded Lie (super)algebras Graded Lie algebras and intersection 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 Moduli spaces of stable coherent sheaves on a surface are of much interest for both mathematics and physics. Yoshioka computed generating functions of Poincaré polynomials of such moduli spaces if the surface is \(\mathbb P^2\) and the rank of the sheaves is 2. Motivated by physical arguments, this paper investigates the modular properties of these generating functions. It is shown that these functions can be written in terms of the Lerch sum and theta function. Based on this, we prove a conjecture by Vafa and Witten, which expresses the generating functions of Euler numbers as a mixed mock modular form. Moreover, we derive an exact formula of Rademacher-type for the Fourier coefficients of this function. This formula requires a generalization of the classical Circle Method. This is the first example of an exact formula for the Fourier coefficients of mixed mock modular forms, which is of independent mathematical interest. generating functions of Poincaré polynomials of moduli spaces; Lerch sum; theta function; conjecture by Vafa and Witten; mixed mock modular form; exact formula of Rademacher-type; Fourier coefficients Bringmann, K.; Manschot, J., From sheaves on \(\mathbb{P}^2\) to a generalization of the Rademacher expansion, A. J. Math., 135, 1039-1065, (2013) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fourier coefficients of automorphic forms, Discontinuous groups and automorphic forms, Applications of the Hardy-Littlewood method, Families, fibrations in algebraic geometry, Analytic sheaves and cohomology groups From sheaves on \(\mathbb{P}^2\) to a generalization of the Rademacher expansion | 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 determine the possible finite groups \(G\) of symplectic automorphisms of hyperkähler manifolds which are deformation equivalent to the second Hilbert scheme of a \(K3\) surface. The authors prove that \(G\) has such an action if, and only if, it is isomorphic to a subgroup of either the Mathieu group \(M_{23}\) having at least four orbits in its natural permutation representation on \(24\) elements, or one of two groups \(3^{1+4}\):\(2.2^{2}\) and \(3^{4}\):\(A_{6}\) associated to \(\mathcal{S}\)-lattices in the Leech lattice. The authors describe in detail those \(G\) which are maximal with respect to these properties, and in most cases they determine all deformation equivalence classes of such group actions. The authors also compare their results with the predictions of Mathieu Moonshine. hyper-Kähler manifolds; Hilbert schemes; finite group actions; sporadic groups; Moonshine Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Finite groups of transformations in algebraic topology (including Smith theory), Jacobi forms, Birational automorphisms, Cremona group and generalizations Finite groups of symplectic automorphisms of Hyperkähler manifolds of type \(K3\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Lambda=KQ/I\) be a basic finite-dimensional algebra over an algebraically closed field \(K\) and let \(J\) be the Jacobson radical of \(\Lambda\). For a finite dimensional left \(\Lambda\)-module \(M\) denote by \(s\) the number of indecomposable summands of \(M\) and by \(m=\dim_K\Hom_\Lambda(P,JM)-\dim_K\Hom_\Lambda(M,JM)\), where \(P\) is a projective cover of \(M\). A degeneration \(M'\) of \(M\) is said to be `top-stable', if \(M/JM\cong M'/JM'\).
The following theorem is one of the main results of the paper.
Theorem A. Suppose \(T=M/JM\) is a direct sum of \(t\) pairwise non-isomorphic simple \(\Lambda\)-modules and \(P\) a projective cover of \(T\). Write \(M\) in the form \(M\cong P/C\) with \(C\subseteq JP\).
(1) The lengths of chains of proper top-stable degenerations of \(M\) are bounded above by \(m+t-s\).
(2) Existence. The following conditions are equivalent: (a) \(M\) has a proper top-stable degeneration; (b) \(m+t-s>0\); (c) either \(M\) is not a direct sum of local modules, or else \(C\) fails to be invariant under all homomorphisms \(P\to JP\); (d) \(C\) fails to be invariant under all endomorphisms of \(P\).
(3) Unique existence. The representation \(M\) has a unique proper top-stable degeneration if and only if \(M\) is a direct sum of local modules and \(m=1\). If \(m=0\) and \(t-s=1\), \(M\) has precisely two distinct proper top-stable degenerations. (For all values \(m+t-s\geq 2\), infinitely many top-stable degenerations can be realized.)
(4) Bases. If \(M'\) is a top-stable degeneration of \(M\), then \(M\) and \(M'\) share a basis consisting of paths in the quiver \(Q\).
(5) The maximal top-stable degenerations of \(M\) always possess a fine moduli space, \(\max\text{-topdeg}(M)\), classifying them up to isomorphism. This moduli space is a projective variety of dimension at most \(\max\{0,m+(t-s)-1\}\).
(6) The case \(m=0\). Let \(M=\bigoplus_{1\leq k\leq s}M_k\) be a decomposition of \(M\) into indecomposable summands. Then \(M\) has only finitely many top-stable degenerations, and each of them is a direct sum of top-stable degenerations of the \(M_k\). Moreover, \(M\) degenerates top-stably to \(M'\) precisely when \(M\leq M'\) in the Ext-order.
A degeneration \(M'\) of \(M\) is said to be `layer-stable', if \(J^lM/J^{l+1}M\cong J^lM'/J^{l+1}M'\). The following theorem is proved.
Theorem B (Layer stable degenerations). Let \(M\), \(s\) and \(t\) be as in Theorem A.
(1) If \(M\) is a direct sum of local modules, that is, if \(t-s=0\), then \(M\) has no proper layer-stable degenerations.
(2) If \(m=0\), then every minimal layer-stable degeneration of \(M\) is of the form \(M=U\oplus M/U\), where \(U\subseteq M\) is a layer-stably embedded submodule (meaning that \(J^l=J^lM\cap U\) for all \(l\)).
(3) On the other hand, for any positive integer \(r\), there exists an indecomposable finite-dimensional module \(M\) with squarefree top, over a suitable finite-dimensional algebra \(\Lambda\), such that \(M\) has a \(\mathbb{P}^r\)-family of pairwise non-isomorphic indecomposable layer-stable degenerations. degenerations of modules; representations of quivers; top-stable degenerations; finite-dimensional representations; finite-dimensional algebras; radical layerings; layer-stable degenerations -, Top-stable degenerations of finite dimensional representations I, posted at www.math.ucsb.edu/\( \sim \)birge/papers.html. Representations of associative Artinian rings, Representations of quivers and partially ordered sets, Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras), Fibrations, degenerations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Top-stable degenerations of finite-dimensional representations. 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 For stable regular singular parabolic connections with given spectral type on smooth projective curves, the authors define a moduli space. The latter is smooth and has a relative symplectic structure. The authors define an isomonodromic deformation on the moduli space and prove that the deformation has the geometric Painlevé property (which in turn implies the usual Painlevé property a differential equation to have only ordinary poles as movable singularities). regular singular connection of spectral type; moduli space of parabolic connections; symplectic structure; Riemann-Hilbert correspondence; geometric Painlevé property; isomonodromic deformation of linear connection; higher dimensional Painlevé equations Algebraic moduli problems, moduli of vector bundles, Isomonodromic deformations for ordinary differential equations in the complex domain, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies Moduli of regular singular parabolic connections with given spectral type on smooth projective 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 In this article, the authors consider some Hilbert modular threefolds for the totally real cubic field \(K\) of discriminant \(49\) explicitly. They also investigate certain geometric properties. In particular, as a subvariety of a Hilbert modular threefold, they provide a surface of degree \(8\) in \(\mathbb{P}^3(\mathbb{C})\) with \(84\) singularities of type \(A_2\).
More precisely, the main objects of the article are, for the maximal totally real subfield \(K= \mathbb{Q}(\zeta_7+\zeta_7^{-1})\) of \(\mathbb{Q}(\zeta_7)\), and the prime ideal \(p\) in \(K\) above \(7\), a Hilbert modular threefold \(X^\circ = \Gamma(p) \! \setminus \! \mathfrak{H}^3\), the minimal compactification \(X=\Gamma(p) \! \setminus \! (\mathfrak{H}^3\cup \mathbb{P}^1(K))\) of \(X^\circ\), a singular toroidal compactification \(X_{\mathrm{ch}}\) which corresponds to taking a convex hull, a smooth toroidal compactification \(X_{\mathrm{sm}}\) corresponding to a certain fan data giving by subdividing fan data for \(X_{\mathrm{ch}}\), and the Galois orbits \(X_{\mathrm{Gal}} = X/\mathrm{Gal}(K/\mathbb{Q})\) of \(X\). Here, the action of the Galois group \(\mathrm{Gal}(K/\mathbb{Q})\) on \(X\) is induced by one on the real places of \(K/\mathbb{Q}\). The authors call \(X_{\mathrm{Gal}}\) the symmetric Hilbert modular threefold.
The authors obtain the following results: {\parindent=6mm \begin{itemize} \item[{\(\bullet\)}] They determine the exceptional divisors of natural projections \(X_{\mathrm{sm}} \to X_{\mathrm{ch}}\) and \(X_{\mathrm{ch}} \to X\) (\S3). \item [{\(\bullet\)}] They calculate the intersection numbers of these exceptional divisors on \(X_{\mathrm{ch}}\) and \(X_{\mathrm{sm}}\) explicitly and give the dimension of the vector spaces of Hilbert modular forms of parallel weight \(1\) and \(2\) (\S4). \item [{\(\bullet\)}] They determine the all Eisenstein series of parallel weight \(1\) and \(2\) and give an equation \(P(x_0, x_1, x_2, x_3, x_4)\) ``of degree \(8\)'' which the Eisenstein series satisfy (\S5 and \S6). \item [{\(\bullet\)}] They show that \(X_{\mathrm{Gal}}\) is the hypersurface of the weighted projective space \(\mathbb{P}(1,1,1,1,2)\) defined by \(P=0\) (\S7). \item [{\(\bullet\)}] They prove that \(X_{\mathrm{ch}}\) is the canonical model of \(X\) (\S8). \item [{\(\bullet\)}] They give an octic \(W\) in \(\mathbb{P}^3\) with \(84\) singularities of type \(A_2\) as the intersection of the above hypersurface and the hypersurface \(x_4=0\) (\S9).
\end{itemize}} Finally, they remark that the octic \(W\) has the best lower bound of the number of singularities of type \(A_2\) for octics to date. Indeed, this example is better than the known bound \(98\) for the octic studied in the paper [\textit{Y. Miyaoka}, Math. Ann. 268, 159--171 (1984; Zbl 0521.14013)]. Hilbert modular threefolds; toroidal compactification; Eisenstein series; symmetric Hilbert modular varieties; canonical model; resolution of singularities; Hilbert modular forms; octic Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Modular and Shimura varieties On Hilbert modular threefolds of discriminant 49 | 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 paper we study subalgebras \(\mathcal A\) of the algebra \(\mathcal D(X)\) of differential operators on a smooth variety \(X\) that are big in the following sense: using the order of a differential operator, the ring \(\mathcal D(X)\) is equipped with a filtration. Its associated graded algebra \(\overline{\mathcal D}(X)\) is commutative and can be regarded as the set of regular functions on the cotangent bundle of \(X\). The subalgebra \(\mathcal A\) inherits a filtration from \(\mathcal D(X)\), and its associated graded algebra \(\overline{\mathcal A}\) is a subalgebra of \(\overline{\mathcal D}(X)\). We call \(\mathcal A\) graded cofinite in \(\mathcal D(X)\) if \(\overline{\mathcal D}(X)\) is a finitely generated \(\overline{\mathcal A}\)-module.
Our guiding example of a graded cofinite subalgebra is the algebra of invariants \(\mathcal D(X)^W\), where \(W\) is a finite group acting on \(X\). Other examples can be constructed as follows. Let \(\varphi\colon X\to Y\) be a finite dominant morphism onto a normal variety \(Y\). Then we put
\[
\mathcal D(X,Y)=\{D\in\mathcal D(X)\mid D(\mathcal O(Y))\subseteq\mathcal O(Y)\}.
\]
We show (Corollary 3.6) that this subalgebra is graded cofinite if and only if the ramification of \(\varphi\) is uniform -- that is, if the ramification degree of \(\varphi\) along a divisor \(D\subset X\) depends only on the image \(\varphi(D)\).
It should be noted that these two constructions are in fact more or less equivalent. In Theorem 3.1 we show that \(\mathcal D(X)^W=\mathcal D(X,X/W)\). Conversely, we show in Proposition 3.3 that \(\mathcal D(X,Y)=\mathcal D(\widetilde X)^W\), where \(\widetilde X\to X\) is a suitable finite cover of \(X\) and \(W\) is a finite group acting on \(\widetilde X\).
Our main result is that, up to automorphisms, every graded cofinite subalgebra is of the form just described.
1.1. Theorem. Let \(X\) be a smooth variety and \(A\) a graded cofinite subalgebra of \(\mathcal D(X)\). Then there is an automorphism \(\Phi\) of \(\mathcal D(X)\), inducing the identity on \(\overline{\mathcal D}(X)\), such that \(\mathcal A=\Phi\mathcal D(X,Y)\) for some uniformly ramified morphism \(\varphi\colon X\to Y\). subalgebras of algebras of differential operators; smooth varieties; associated graded algebras; regular functions; cotangent bundle; filtrations; graded cofinite subalgebras; algebras of invariants; finite group actions doi:10.1307/mmj/1144437435 Rings of differential operators (associative algebraic aspects), Actions of groups and semigroups; invariant theory (associative rings and algebras), Actions of groups on commutative rings; invariant theory, Graded rings and modules (associative rings and algebras), Group actions on varieties or schemes (quotients) Graded cofinite rings of differential 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 We describe all connected components of the space of hyperbolic Gorenstein quasi-homogeneous surface singularities. We prove that any connected component is homeomorphic to a quotient of \(\mathbb R^d\) by a discrete group. Gorenstein singularities; \(\mathbb Q\)-Gorenstein singularities; quasi-homogeneous surface singularities; higher spin structures; moduli spaces; Arf functions; lifts of Fuchsian groups Complex surface and hypersurface singularities, Vector bundles on curves and their moduli, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Teichmüller theory for Riemann surfaces Topological invariants and moduli of Gorenstein 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 \(I \subseteq R = k[x_ 0, \dots, x_ n]\) be the ideal of \(s\) points in \(\mathbb{P}^ n (k)\), \(k\) a field. For a sufficiently generic position of these points conjectures have been made about the minimum values of the minimal number of generators of \(I\) and of the Cohen-Macaulay type of the homogeneous coordinate ring \(R/I\) of the points -- and also about the existence of points in generic position reaching those conjectured values.
The author states here a minimal resolution conjecture with explicitly stated conjectured values of the Betti numbers. This conjecture implies both the ideal generation conjecture and the Cohen-Macaulay type conjecture (see \S2, p.14). Some positive answers for the minimal resolution conjecture are given in section 3. Moreover the author proves the Cohen-Macaulay type conjecture for \(s\) points if \(n + 1 \leq s \leq {n + 2 \choose n}\) (see proposition 3.5 and theorem 3.6). Hilbert functions; points in generic position; ideal of points; number of generators; Cohen-Macaulay type; minimal resolution conjecture; Betti numbers Lorenzini, A.: The Minimal Resolution Conjecture. J. Algebra156, 5--35 (1993). Commutative rings and modules of finite generation or presentation; number of generators, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings The minimal resolution conjecture | 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, \(\text{Set}\) the category of sets, \(\text{Fields}/F\) the category of field extensions of \(F\) and field \(F\)-homomorphisms, and let \(\mathcal T\colon\text{Fields}/F\to\text{Sets}\) be a functor (called an ``algebraic structure''). For instance, \(\mathcal T(E)\) with \(E\in\text{Fields}/F\) can be the sets \(\text{Alg}_{n,m}\) of isomorphism classes of central simple \(E\)-algebras of degree \(n\) and exponent \(m\) (for an arbitrary fixed divisor \(m\) of \(n\)), torsors (principal homogeneous spaces) over \(E\) under a given algebraic group, etc. For fields \(E,E'\in\text{Fields}/F\), a field homomorphism \(f\colon E\to E'\) over \(F\) and \(\alpha\in\mathcal T(E)\), we write \(\alpha_{E'}\) for the image of \(\alpha\) under the morphism \(\mathcal T(f)\colon\mathcal T(E)\to\mathcal{T}(E')\). An element \(\alpha\in\mathcal T(E)\) is said to be defined over an intermediate field \(K\) of \(E/F\) (and \(K\) is called a field of definition of \(\alpha\)), if there exists \(\beta\in\mathcal T(K)\), such that \(\beta_E=\alpha\), where \(f\) is the natural embedding of \(K\) into \(E\). The essential dimension \(\text{ed}(\alpha)\) of an algebraic structure is defined to be the minimum of transcendence degrees \(\text{trd}(K/F)\), taken over the fields of definition \(K\) of \(\alpha\). For each prime number \(p\), the essential \(p\)-dimension \(\text{ed}_p(\alpha)\) is defined as \(\min\{\text{ed}(\alpha_L)\}\), where \(L\) ranges over all field extensions of \(E\) of degree prime to \(p\).
The paper under review computes upper bounds for \(\text{ed}(\text{Alg}_{n,2})\) and \(\text{ed}_2(\text{Alg}_{n,2})\). It continues the research in this direction carried out by \textit{A. S. Merkurjev} and the author [in Acta Math. 209, No. 1, 1-27 (2012; Zbl 1258.16023)]. Its starting point are the following results of the quoted joint paper: (i) \(\text{ed}(\text{Alg}_{n,2})=\text{ed}(\text{Alg}_{2^r,2})\) and \(\text{ed}_2(\text{Alg}_{n,2})=\text{ed}_2(\text{Alg}_{2^r,2})\), where \(2^r\) is the largest power of \(2\) dividing \(n\); (ii) \(\text{ed}(\text{Alg}_{4,2})=\text{ed}_2(\text{Alg}_{4,2})=4\) and \(\text{ed}(\text{Alg}_{8,2})=\text{ed}_2(\text{Alg}_{8,2})=8\), provided that \(\text{char}(F)\neq 2\) (for the case of \(\text{char}(F)=2\), see the author's paper [C. R., Math., Acad. Sci. Paris 349, No. 7-8, 375-378 (2011; Zbl 1237.16020)].
When \(\text{char}(F)\neq 2\) and \(n=2^r\geq 8\), the paper under review shows that \(\text{ed}(\text{Alg}_{n,2})\leq (n-1)(n-2)/2\). Concerning \(\text{ed}_2(\text{Alg}_{n,2})\), it proves that if \(n=2^r\geq 8\), then \(\text{ed}_2(\text{Alg}_{n,2})\leq n^2/4\) in case \(\text{char}(F)=2\), and \(\text{ed}_2(\text{Alg}_{n,2})\leq n^2/16+n/2\), provided that \(\text{char}(F)\neq 2\). In addition, the author obtains that \(\text{ed}_2(\text{Alg}_{16,2})=24\) whenever \(\text{char}(F)\neq 2\). The proof relies on a natural interpretation of \(\text{Alg}_{n,2}\) in terms of nonabelian Galois cohomology. essential dimension; essential \(2\)-dimension; central simple algebras; fields of definition; involutions; categories of field extensions; transcendence degrees; Brauer groups; cyclic algebras A. Vishik, \textit{Direct summands in the motives of quadrics}, preprint, 1999, available at http://www.maths.nott.ac.uk/personal/av/papers.html. Finite-dimensional division rings, Brauer groups (algebraic aspects), Group actions on varieties or schemes (quotients), Integral representations of finite groups, Linear algebraic groups over arbitrary fields, Galois cohomology of linear algebraic groups Essential dimension of simple algebras with involutions. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a graduate text with an integrated treatment of several complex variables and complex algebraic geometry, with applications to the structure and representation theory of complex semisimple Lie groups.
The book begins with an exposition of the Cauchy-Riemann equations, Mittag-Leffler and Weierstrass theorems, partitions of unity, Cauchy's formula, power series expansions, Hartog's theorem and domain of holomorphy (Chs 1-2, Selected Problems in One Complex Variable, Holomorphic Functions of Several Variables).
The behavior of holomorphic functions in small neighborhoods of a point is discussed, leading to the notion of the germ of a function at a point. The algebraic properties of the local rings of regular and holomorphic functions, first on \(\mathbb C^n\) and then on varieties, are studied. The basic tools for this study are the Weierstrass preparation and division theorems. These allow to reduce problems involving germs of holomorphic functions in \(n \) variables to problems involving polynomials with coefficients which are germs of holomorphic functions in \(n-1\) variables. These results lead to the fact that the local ring of germs of holomorphic functions is Noetherian, and to the implicit and inverse mapping theorems, which do not have analogues for the local rings of rational function (Ch. 3, Local Rings and Varieties). The exact connection between germs of holomorphic varieties and ideals in the ring of germ of holomorphic functions is stated (Ch. 4, The Nullstellensatz). Three notions of dimension of the local ring -- topological, geometric, and tangential -- are discussed. The attention is focused on holomorphic varieties, turning to the study of dimension for algebraic varieties (Ch. 5, Dimension).
Abstract sheaf theory and sheaf cohomology provide the formal machinery for passing from local to global solutions for wide variety of problems, as well for classifying the obstructions to doing so when local solutions do not give rise to global solutions. Sheaves are defined and sheaf cohomology is developed as an application of homological algebra to the category of sheaves on a topological space. These results are used to give brief developments of two classical cohomology theories -- de Rham and Čech (Chs 6-7, Homological Algebra, Sheaves and Sheaf Cohomology). Abstract algebraic and holomorphic varieties are defined and classes of quasi-coherent and coherent algebraic and analytic sheaves on such varieties are studied. It is shown that the category of coherent analytic sheaves on holomorphic variety \(X\) is a full abelian subcategory of the category of sheaves of \(_X\mathcal H\)-modules (Chs 8-9, Coherent Algebraic Sheaves, Coherent Analytic Sheaves).
The category of Stein spaces is considered and the ground work for proving Cartan's theorems is laid. The key result here is a vanishing theorem which states that a coherent analytic sheaf defined in a neighborhood of a compact polydisc has vanishing higher cohomology on the polydisc (Ch. 10, Stein Spaces).
The approximation argument used to finally prove Cartan's theorems requires knowing that a coherent analytic sheaf has the structure of a Fréchet sheaf. It is shown that there always is such a structure and that it is unique subject to certain condition. It is also proved that morphisms between coherent sheaves are automatically continuous for this structure. Then Cartan's theorems, that on a Stein space, a coherent analytic sheaf has a rich supply of global sections and every coherent analytic sheaf is acyclic, are proved. The Cartan-Serre theorem, that the cohomology modules of a coherent algebraic sheaf on a compact holomorphic variety are finite dimensional, is also proved. It plays a key role, along with Cartan's theorems, in the proof of Serre's theorem (Ch. 11, Fréchet Sheaves -- Cartan's Theorems).
Several complex variables and complex algebraic geometry are not just similar; they are equivalent when done in the context of projective varieties. This is the content of the famous Serre's GAGA theorems. Complete proofs of these results are given after first studying the cohomology of coherent sheaves on projective spaces (Chs 12-13, Projective Varieties, Algebraic vs. Analytic -- Serre's Theorems).
The final three Chs 14-16, Lie Groups and Their Representations, Algebraic Groups, The Borel-Weil-Bott Theorem, are devoted to the study of complex semisimple Lie groups and their finite dimensional representations. The subject does provide significant insight into how the preceding results are used in practice. A survey of basic results from harmonic analysis -- specifically, topological groups, Lie groups, Lie algebras and group representations, with an emphasis on compact groups and the Peter-Weyl theorem, the structure of complex semisimple Lie groups and Lie algebras, and the finite dimensional representations of semisimple Lie algebras -- is included in order to provide the formulation and Miličić's proof of the Borel-Weil-Bott theorem which pinpoints the relationship between finite dimensional holomorphic representations of a complex semisimple Lie group \(G\) and the cohomologies of \(G\)-equivariant holomorphic line bundles on a projective variety constructed from \(G\). A brief introduction to the theory of complex algebraic groups is developed just enough to prove the key structure theorems for complex semisimple Lie groups using algebraic group methods. In particular, it provides nice applications of the algebraic geometry, the sheaf theory, the Cartan-Serre theorem, the material on projective varieties, Serre's theorems, and of course, the background material on algebraic groups and general Lie theory. For example, it is proved that a connected and semisimple complex Lie group is actually an algebraic group.
Each chapter ends with an exercise set. Many exercises involve filling in details of proofs in the text or proving results that are needed elsewhere in the text, while others supplement the text by exploring examples or additional material.
The book can serve as an excellent text for a graduate course on modern methods of complex analysis, as well as a useful reference for those working in analysis. holomorphic functions of several variables; algebraic geometry; complex semisimple Lie groups; abstract harmonic analysis; local rings and varieties; nullstellensatz; dimension; homological algebra; sheaf cohomology; coherent algebraic sheaves; coherent analytic sheaves; Stein spaces; Fréchet sheaves; Cartan's theorem; projective varieties; Serre's theorems; Dolbeault cohomology; chains of syzygies; Cartan's factorization; amalgamation of syzygies; algebraic groups; Borel-Weil-Bott theorem; equivariant line bundles; flag variety; Casimir operator J. L. Taylor, \textit{Several Complex Variables with Connections to Algebraic Geometry and Lie groups}, Graduate Studies in Mathematics, \textbf{46}, AMS, Providence, RI, 2002. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, General properties and structure of complex Lie groups, Simple, semisimple, reductive (super)algebras, Semisimple Lie groups and their representations, Analysis on other specific Lie groups Several complex variables with connections to algebraic geometry and Lie 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 Im Zuge einer Reihe von Arbeiten von H. Grundman zur Frage, welche Quotientenräume Hilbertscher Modulgruppen total-reeller Zahlkörper vom Grad \(n\), auf einem Produkt von \(n\) oberen Halbebenen operierend, das arithmetische Geschlecht 1 haben (nur dann kann der Funktionenkörper rational sein), hatten die Verfasserinnen kürzlich [Fields Inst. Commun. 41, 217--226 (2004; Zbl 1098.11030)] den Fall \(n=4\) vollständig gelöst. Die Untersuchungen beruhen sämtlich auf der Auswertung der einzelnen Terme in der Formel für das arithmetische Geschlecht.
In der vorliegenden Arbeit zeigen die Verfasserinnen nun, wie man im Fall \(n=4\) alle Terme mit Hilfe der gegenwärtig zur Verfügung stehenden Computer-Algebra-Systeme explizit berechnen kann und damit das arithmetische Geschlecht, unabhängig von der Frage, ab welcher Diskriminante es sicher größer als eins ist, bestimmen kann. Als Beispiel fügen sie eine Liste des arithmetischen Geschlechts für die 210 total-reellen Zahlkörper vom Grad 4 mit kleinster Diskriminante an. quotient spaces of Hilbert modular groups of totally real number fields Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Rationality questions in algebraic geometry, Families, moduli, classification: algebraic theory, \(4\)-folds, Values of arithmetic functions; tables Computing the arithmetic genus of Hilbert modular fourfolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hilbert schemes \(\text{Hilb}^n(X)\) of \(n\)-tuples on a complex projective manifold \(X\) are natural compactifications of their configuration spaces of unordered distinct \(n\)-tuples of points on \(X\). Their geometry is determined by the geometry of \(X\) itself and the geometry of ``punctual'' Hilbert schemes of all zero-dimensional subschemes on the affine plane that are supported at the origin. This geometry is most attractive when \(X\) is a surface, since the Hilbert schemes are themselves irreducible projective manifolds. This leads to the problem of the explicit determination of the geometry or the topological invariants of Hilbert schemes \(\text{Hilb}^n(X)\) from the corresponding data of the manifold \(X\) itself.
These lecture notes are one of the essential references for the understanding of the classical results and the recent developments of the topic, in particular the author's work [Invent. Math. 136, 157--207 (1999; Zbl 0919.14001)]. The present paper includes complete proofs and precise references to the literature for further readings.
The lectures treat the following points:
Configurations of unordered \(n\)-tuples. Set-theoretical discussion of the Hilbert scheme \(\text{Hilb}^n(X)\), the symmetric quotient \(\text{S}^n(X)\), the Hilbert-Chow morphism and the punctual Hilbert scheme.
The geometry of Hilbert schemes. Representability of the moduli functor \(\underline{\text{Hilb}}(X)\) (Theorem of Grothendieck), study of the tangent space, smoothness of \(\text{Hilb}^n(X)\) for \(X\) a smooth quasiprojective surface (Theorem of Fogarty), construction of the Hilbert-Chow morphism, induction schemes, irreducibility of the punctual Hilbert scheme (Theorem of Briançon).
The cohomology of \(\text{Hilb}^n(X)\). Computation of the Betti numbers (Theorem of Göttsche), structure of irreducible representation of a Heisenberg Lie algebra of the whole cohomology of Hilbert schemes of points on \(X\) for all values of \(n\) together (Theorem of Nakajima).
Vertex algebras. Exposition of the vertex algebra point of view for the understanding of the whole cohomology space of Hilbert schemes of points.
The ring structure. Study of the interplay between the vertex algebra structure induced by the Nakajima's operators and the ring structure of the cohomology investigated by construction of multiplication operators. This results in an explicit algorithm for the computation of the ring (Theorems of Lehn). A precise treatment is devoted to the Hilbert scheme on the affine plane with an explicit description of the cohomology ring (Theorem of Lehn-Sorger), and finally to the Hilbert scheme on a \(K3\) surface (Theorem of Lehn-Sorger) in relation with orbifold cohomology (Conjecture of Ruan). Hilbert schemes of points; vertex algebras Lehn, M.: Lectures on hilbert schemes. CRM Proc Lect Notes \textbf{38}, 1-30 Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Lectures on Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that the orbit closure of a directing module is regular in codimension one. In particular, this result gives information about a distinguished irreducible component of a module variety.
Given a finite-dimensional \(k\)-algebra \(\Lambda\) and an element \(\mathbf d\) of the Grothendieck group \(K_0(\Lambda)\) of the category of \(\Lambda\)-modules, one defines the variety \(\text{mod}^{\mathbf d}_\Lambda(k)\) of \(\Lambda\)-modules of dimension vector \(\mathbf d\). A product \(\text{GL}_{\mathbf d}(k)\) of general linear groups acts on \(\text{mod}^{\mathbf d}_\Lambda(k)\) in such a way that the \(\text{GL}_{\mathbf d}(k)\)-orbits correspond to the isomorphism classes of \(\Lambda\)-modules of dimension vector \(\mathbf d\). In particular, the author has shown [J. Math. Soc. Japan 54, No. 3, 609-620 (2002; Zbl 1048.16004)] that, if \(\mathbf d\) is the dimension vector of a directing module and \(\Lambda\) is tame, then \(\text{mod}^{\mathbf d}_\Lambda(k)\) is normal if and only if it is irreducible. Here a \(\Lambda\)-module \(M\) is called directing if there is no sequence \((X_0,\dots,X_n)\) of indecomposable \(\Lambda\)-modules such that \(\Hom_\Lambda(X_{i-1},X_i)\neq 0\) for each \(i\in [1,n]\), there exists an \(i\in [1,n-1]\) such that the Auslander-Reiten translate of \(X_{i+1}\) equals \(X_{i-1}\), and \(X_0\) and \(X_n\) are direct summands of \(M\). In general, if \(M\) is a directing module, then the closure \(\overline{{\mathcal O}(M)}\) of the \(\text{GL}_{\mathbf d}(k)\)-orbit \(\mathcal O(M)\) of \(M\) is an irreducible component of \(\text{mod}^{\mathbf d}_\Lambda(k)\). Thus the above result naturally raises the question about properties of \(\overline{{\mathcal O}(M)}\).
The question about the properties of \(\overline{{\mathcal O}(M)}\) for a directing module \(M\) is a special case of another geometric problem investigated in the representation theory of finite-dimensional algebras, namely, the study of the properties of the orbit closures in the module varieties. In particular, \textit{G. Zwara} and the author proved [J. Algebra 298, No. 1, 120-133 (2006; Zbl 1131.16010)] (using, among other things, the results of the author and \textit{A. Skowroński} [J. Algebra 215, No. 2, 603-643 (1999; Zbl 0965.16009)] that, if \(M\) is an indecomposable directing module, then \(\overline{{\mathcal O}(M)}\) is a normal variety. Recall that the normal varieties are regular in codimension one; that is, the set of singular points is of codimension at least two. Thus, the following main result of the paper is the first step in order to generalize the above result about the closures of indecomposable directing modules over tame algebras to arbitrary directing modules over arbitrary algebras.
Main Theorem. If \(M\) is a directing module, then \(\overline{{\mathcal O}(M)}\) is regular in codimension one.
The paper is organized as follows. In Section 1 we recall the definitions of quivers and their representations. We also describe the properties of directing modules that are needed in the proof of our main result. In Section 2 we discuss interpretations of extension groups that are useful in geometric investigations. Next, in Section 3 we define the module schemes and some schemes connected with them. Finally, in Section 4 we prove the main result of the paper.
The main idea of the proof is the following. Let \(M\) be a directing module over an algebra \(\Lambda\). We first observe that each minimal degeneration \(N\) of \(M\) (that is, a module \(N\) whose orbit is maximal in \(\overline{{\mathcal O}(M)}\setminus\mathcal O(M)\)) is of the form \(N=U\oplus V\) for a short exact sequence
\[
\xi:0\to U\to M\to V\to 0.
\]
Now we use a connection between the tangent space \(T_N\text{mod}^{\mathbf d}_\Lambda(k)\) to the module variety at \(N\) and the first extension group. As a consequence, it follows that \(\text{Ext}^2_\Lambda(V,U)\) measures the difference between \(\dim\overline{{\mathcal O}(M)}\) and \(\dim_kT_N\text{mod}^{\mathbf d}_\Lambda(k)\). On the other hand, we show for a general minimal degeneration \(N\) of \(M\) that, if
\[
\xi_1:0\to U\to W_1\to U\to 0\quad\text{ and }\quad\xi_2:0\to V\to W_2\to V\to 0
\]
are short exact sequences, then \((\xi_1,\xi_2)\) corresponds to an element of \(T_N\overline{{\mathcal O}(M)}\) if and only if the sequences \(\xi_1\circ\xi\) and \(\xi\circ\xi_2\) determine the elements in \(\text{Ext}^2_\Lambda(V,U)\) that differ by the sign. We prove that the space of such pairs of sequences is of codimension \(\dim_k\text{Ext}^2_\Lambda(V,U)\) in \(\text{Ext}^1_\Lambda(U,U)\times\text{Ext}^1_\Lambda(V, V)\), and this will complete the proof. finite-dimensional algebras; categories of modules; actions of general linear groups; Auslander-Reiten translations; irreducible components; indecomposable directing modules; normal varieties; degenerations; short exact sequences; module varieties Bobiński, G., Orbit closures of directing modules are regular in codimension one, J. lond. math. soc. (2), 79, 211-224, (2009) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Group actions on affine varieties, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Orbit closures of directing modules are regular in codimension one. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities By deforming an algebra of functions of some geometrical object we get a quantum group, which is a new algebra of ``functions'' on some noncommutative space. The author shows that a good candidate for this space is the category of graded modules of a noncommutative algebra, i.e., the ``quantum space'' as introduced by Artin, Tate and Van den Bergh. deforming algebras of functions; quantum groups; noncommutative spaces; categories of graded modules Vancliff M., Algebras and representation theory Graded rings and modules (associative rings and algebras), , Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Clifford algebras, spinors, Rings arising from noncommutative algebraic geometry, Quantum groups and related algebraic methods applied to problems in quantum theory Non-commutative spaces for graded quantum groups and graded Clifford 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 \(\Sigma\) be a finite dimensional algebra over an algebraically closed field. By [\textit{D. Happel, I. Reiten} and \textit{S. Smalø}, ``Tilting in abelian categories and quasitilted algebras'' (Mem. Am. Math. Soc. 575, 1996; Zbl 0849.16011)], \(\Sigma\) is called quasitilted if it is the endomorphism ring of a tilting object in a hereditary abelian category. The paper under review deals with quasitilted algebras of canonical type, i.e. those where \(\Sigma\) can be realized as the endomorphism ring of a tilting object in a hereditary abelian category which is derived-equivalent to a category of coherent sheaves \(\text{coh }\mathbb{X}\) for a weighted projective line \(\mathbb{X}\) in the sense of \textit{W. Geigle} and \textit{H. Lenzing} [in: Singularities, representations of algebras and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)].
The authors first determine all hereditary categories which can occur in this context. Then the structure of tilting objects in such categories is investigated in detail. The corresponding endomorphism rings are characterized as semiregular branch enlargements of concealed canonical algebras, or equivalently by the property that the module category admits a sincere separating family of semiregular standard tubes. The results generalize the characterizations obtained for tilting sheaves in \(\text{coh }\mathbb{X}\) given by \textit{H. Lenzing} and the reviewer [in Representations of algebras, CMS Conf. Proc. 18, 455-473 (1996; Zbl 0863.16013)] and by \textit{H. Lenzing} and \textit{J. A. de la Peña} [``Concealed-canonical algebras and algebras with a separating tubular family'', Proc. Lond. Math. Soc. (to appear)]. Further the module category for a quasitilted algebra of canonical type is studied in detail, in particular the structure of the Auslander-Reiten components is described completely. \(K\)-theoretical results concerning those algebras are also given.
A conjecture says that a hereditary category with a tilting object is derived-equivalent either to a category \(\text{coh }\mathbb{X}\) or else to a category \(\text{mod }\Delta\) where \(\Delta\) is a hereditary algebra. As a consequence of their results the authors are able to confirm this conjecture in case \(\Sigma\) is of tame representation type. finite dimensional algebras; hereditary Abelian categories; quasitilted algebras of canonical type; categories of coherent sheaves; tilting objects; endomorphism rings; sincere separating families of semiregular standard tubes; Auslander-Reiten components; tame representation type Lenzing, H.; Skowroński, A., Quasi-tilted algebras of canonical type, Colloq. Math., 71, 161-181, (1996) Representations of quivers and partially ordered sets, Vector bundles on curves and their moduli, Representation type (finite, tame, wild, etc.) of associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Derived categories, triangulated categories, Module categories in associative algebras Quasi-tilted algebras of canonical 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 \textit{B. H. Gross} and \textit{D. B. Zagier} proved a formula which relates the central derivatives of certain Rankin \(L\)-series and the heights of certain Heegner points on elliptic curves [Invent. Math. 84, 225--320 (1986; Zbl 0608.14019)]. Combined with work of Goldfeld this formula gives a solution to Gauss' problem on class numbers and combined with work of Kolyvagin gives evidence for the rank statement in the conjecture of Birch and Swinnerton-Dyer. \textit{B. H. Gross} has proposed a program to generalize this formula to totally real fields with anticyclotomic characters [Modular forms, Symp. Durham/Engl. 1983, 87--105 (1984; Zbl 0559.14011)]. In the present paper the author works out the weight 2 case of the program.
Let \(F\) be a totally real field with ring of adeles \(\mathbb A\). Let \(\phi\) be a Hilbert modular form of weight \((2,\ldots,2,0,\ldots,0)\) over \(F\), which is a cuspidal newform of level \(N\) and has trivial central character. Let \(K\) be a totally imaginary quadratic extension of \(F\). Let \(\chi\) be a character of finite order of \({\mathbb A}_K^\times/K^\times{\mathbb A}^\times\). The author studies the Rankin-Selberg convolution function \(L(s,\chi,\phi)\). His main formula expresses the central derivative \(L'(1,\chi,\phi)\) in terms of the heights of CM-points on a Shimura curve, when \(\phi\) is holomorphic and the sign of the functional equation of \(L(s,\chi,\phi)\) is \(-1\). One immediate application is to generalize the work of Kolyvagin-Logachev and Bertolini-Darmon to obtain some evidence toward the Birch and Swinnerton-Dyer conjecture in the rank 1 case. The details should be given in other papers. If \(\phi\) has possible nonholomorphic components and the sign of the functional equation of \(L(s,\chi,\phi)\) is \(+1\) the author proves an explicit formula for \(L(1,\chi,\phi)\), which has an application to the distribution of CM-points on locally symmetric varieties covered by \(({\mathcal H}^+)^n\) where \({\mathcal H}^+\) is the upper half plane and \(n\) is the number of real places of \(F\) where \(\phi\) has weight \(0\). Other application should be on BSD-conjecture in the rank 0 case. Gross-Zagier formula; heights of Heegner points on modular curves; derivatives of \(L\)-series of cusp forms; Hilbert modular form; Automorphic forms on GL(2); Rankin-Selberg \(L\)-function; kernel functions; Geometric pairing of CM-cycles; Shimura curves and CM-points Zhang, SW, Gross-Zagier formula for \(GL_2\), Asian J. Math., 5, 183-290, (2001) Arithmetic aspects of modular and Shimura varieties, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Heights, Modular and Shimura varieties, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Gross-Zagier formula for \(\text{GL}_ 2\). | 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 ``Inverse Galois Theory is concerned with the question which finite groups occur as Galois groups over a given field \(K\)''. The first sentence of the preface sums up the problem this book is dealing with. By extending the work of Scholz and Reichardt, Shafarevich was able to show that over the rationals (and therefore over any number field) each finite solvable group occurs as a Galois group (the relevant techniques are presented in Chapter IV on embedding problems, where the theorem of Scholz and Reichardt is proved). Thus the present book addresses mainly the case of nonsolvable groups.
The first edition was reviewed in [Zbl 0940.12001]; in the second edition a number of additions and corrections were made, and the tables with polynomials whose Galois groups are transitive of degree up to \(14\) have been updated. In addition, the algebraic version of the Katz algorithm going back to Dettweiler and Reiter has been included in Section III.10
The main change, however, is the addition of a new Chapter V on additive polynomials. These results are based on [\textit{B. H. Matzat}, Dev. Math. 11, 233--268 (2004; Zbl 1111.12002)] and [\textit{G. Malle}, Pac. J. Math. 212, No. 1, 157--167 (2003; Zbl 1048.12003)] and the results of various theses by Garcia Lopez, \textit{M. Albert} and \textit{A. Maier} [Isr. J. Math. 186, 125--195 (2011; Zbl 1295.12007)] and \textit{D. Stichel} [J. Commut. Algebra 6, No. 4, 587--603 (2014; Zbl 1312.12001)], and provide an constructive approach to an existence theorem due to \textit{M. V. Nori} [in: Algebraic geometry and its applications. Collections of papers from Shreeram S. Abhyankar's 60th birthday conference, 1990. New York: Springer-Verlag, 209--212 (1994; Zbl 0811.14031)].
This second edition will continue to be the standard book for everyone interested in Inverse Galois Theory. Galois group; inverse Galois theory; Mathieu groups; finite simple groups; embedding problems; rigidity method; Hilbertian fields; function fields; absolute Galois group; generating polynomials of Galois groups Research exposition (monographs, survey articles) pertaining to field theory, Inverse Galois theory, Separable extensions, Galois theory, Galois cohomology, Rigid analytic geometry Inverse Galois 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 Special constructions of moduli spaces in the category of complex spaces, but ''fitting into the philosophy of constructing moduli spaces as quotients with respect to group operations'', are presented. The paper is an apt starting point for those interested in studying such constructions. As a consequence the moduli space of simple holomorphic bundles E (for which \(End(E)={\mathbb{C}})\) on a fixed compact complex space X is shown to exist. Lie groups; moduli space of simple holomorphic bundles S. Kosarew and C. Okonek, Global moduli spaces and simple holomorphic bundles, Mathematica Gottingensis, Heft 10, 1987. Complex-analytic moduli problems, Complex Lie groups, group actions on complex spaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles Global moduli spaces and simple holomorphic bundles | 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 Hilbert modular forms; Eisenstein series; rationality conjecture; special values of Hecke L-functions of CM-fields Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Complex multiplication and abelian varieties, Complex multiplication and moduli of abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Propriétés de rationalité de valeur spéciales de fonctions \(L\) attachées aux corps \(CM\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the final instalment in a series of four papers on fixed point ratios for actions of classical groups [see \textit{T. C. Burness}, part I, J. Algebra 309, No. 1, 69-79 (2007; Zbl 1128.20003), for more details on the set-up].
The main theorem of these papers is the following: if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then either \(\text{fpr}(x)\lesssim|x^G|^{-\frac 12}\) for all \(x\in G\) of prime order, or the pair \((G,\Omega)\) is one of a small number of known exceptions.
The proof of this result first reduces to primitive actions and then considers the different possibilities for the stabiliser \(G_\omega\) of a point \(\omega\in\Omega\); in the paper under review the proof of the result is completed by considering the cases that \(G_\omega\) is a subgroup from Aschbacher's family \(\mathcal S\) [see \textit{M. Aschbacher}, Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)], or is a subgroup in a small additional collection \(\mathcal N\). The groups in \(\mathcal N\) are introduced in the second paper of the series [\textit{T. C. Burness}, part II, J. Algebra 309, No. 1, 80-138 (2007; Zbl 1128.20004)] and arise because of the existence of certain outer automorphisms, including the triality automorphisms for orthogonal groups in dimension 8. finite classical groups; fixed point ratios; primitive permutation groups; monodromy groups; permutation representations; finite almost simple groups; maximal subgroups of classical groups Timothy C. Burness, Fixed point ratios in actions of finite classical groups. III, J. Algebra 314 (2007), no. 2, 693 -- 748. , https://doi.org/10.1016/j.jalgebra.2007.01.011 Timothy C. Burness, Fixed point ratios in actions of finite classical groups. IV, J. Algebra 314 (2007), no. 2, 749 -- 788. Primitive groups, Linear algebraic groups over finite fields, Representation theory for linear algebraic groups, Riemann surfaces; Weierstrass points; gap sequences, Group actions on varieties or schemes (quotients) Fixed point ratios in actions of finite classical groups. IV. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that the dimension of a reductive algebraic group \(G\) acting regularly and effectively on a connected reduced algebraic variety of dimension \(n\) over an algebraically closed field of characteristic 0 is at most \(n^ 2+n\) [ \textit{H. Carayol}, Bull. Sci. Math., II. Ser. 99, 135-143 (1975; Zbl 0333.20031)]. The author reproves this result using some standard facts from the theory of semi-simple Lie algebras. In fact, he improves the estimate by showing that \(\dim(G)\leq(n-c)^ 2+2(n-c)+c\), where \(c\) is the dimension of the center of \(G\). dimension of a reductive algebraic group; semi-simple Lie algebras Group actions on varieties or schemes (quotients), Lie algebras of Lie groups Dimensions of reductive automorphism 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 These are the exposés the author made in the seminars of Bourbaki, Cartan, Chevalley and Delange-Pisot-Poitou, mainly in the period 1950-1960, with four more up to 1999.
The list of topics is as follows: Extensions of locally compact groups (after Iwasawa and Gleason, 1949-1950), Algebraic applications of the cohomology of groups I, Theory of simple algebras II (1950-1951), Automorphic functions of one variable: application of the Riemann-Roch theorem (1953-1954)), Two theorems on the completely continuous functions (1953-1954), Analytic sheaves on a projective space (1953-1954), Automorphic functions (1953-1954), Linear representations (homomorphisms) and homogeneous Kählerian spaces of the compact Lie groups (after A. Borel, and A. Weil, 1953-1954), The spaces \(K(\Pi,n)\) (1954-1955), The homotopy groups of bouquets of spheres (1954-1955), Algebraic fibred spaces (1958), Universal morphisms and Albanese varieties (1958-1959), Universal morphisms and differentials of the third kind (1958-1959), Rationality of the functions \(\zeta\) of the algebraic varieties (after B. Dwork, 1959-1960), Ramified coverings of the projective plane (after S. Abhyankar, 1959-1960), Finite groups with periodic cohomologies (after R. Swan, 1960-1961), \(p\)-adic dependence of the exponentials (1965-1966), \(p\)-divisible groups (1966-1967), Rational points of the modular curves \(X_0(N)\) (after B. Mazur, 1977-1978), Finite subgroups of the Lie groups (1998-1999).
The value of this book is in development of concepts and theories in their historical context. Everyone working in the areas covered will find something useful including unsolved problems. Notes throughout the text are useful and interesting, but there is no index. Exposés de séminaires; Bourbaki; locally compact groups; cohomology of groups; Riemann-Roch theorem; automorphic functions; compact Lie groups; homotopy groups; fibred algebraic spaces; Albanese varieties Collected or selected works; reprintings or translations of classics, History of number theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, History of topological groups Exposés de séminaires. 1950--1999. (Exposes of seminars. 1950--1999). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose that \(R\) is an affine PI-algebra over an algebraically closed field \(k\). Let \(G\) be an affine algebraic \(k\)-group and \(k[G]\) its Hopf algebra of regular functions. An action of \(G\) on \(R\) is rational if \(A\) has \(k[G]\)-comodule algebra structure. A prime ideal \(P\) in \(R\) is \(G\)-rational if the algebra of \(G\)-invariants in the center of \(R/P\) coincides with \(k\). If the group \(G\) acts rationally by \(k\)-automorphisms then the following are equivalent: (i) the set of \(G\)-rational ideals is finite; (ii) the set of all \(G\)-prime ideals is finite; (iii) \(G\) has finitely many orbits in the set of 1-rational prime ideals. If \(G\) is an algebraic \(k\)-torus acting rationally then the \(G\)-spectrum of \(R\) is finite if and only if the action of \(G\) on the center of \(R\) is multiplicity-free. It means that for any character of \(G\) in \(k\) the weight center has dimension at most one. affine PI-algebras; affine algebraic groups; prime ideals; multiplicity free actions; group actions; Hopf algebras of regular functions; prime spectra; algebras of invariants Trace rings and invariant theory (associative rings and algebras), Actions of groups and semigroups; invariant theory (associative rings and algebras), Ideals in associative algebras, Group actions on varieties or schemes (quotients), Hopf algebras and their applications Rational group actions on affine PI-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 [For the entire collection see Zbl 0614.00007.]
In characteristic 0, a rational double point is a \(quotient\quad X\) of \(k^ 2\) (k is the field) by a finite \(subgroup\quad G\) of SL(2,k). The McKay correspondence establishes a one-to-one correspondence between the irreducible representations of G and the vertices of the extended Dynkin diagram associated to the minimal desingularization of X. Moreover, it says how to read on the Dynkin diagram the tensor product of the standard representation \(G\subset SL(2,k)\) with any other one. We refer to it as the multiplicative structure. One may rephrase it in erms of irreducible reflexive modules on X, knowing that they are in one-to-one correspondence with the irreducible representations of G, and that \(G\subset SL(2,k)\) corresponds to the Kähler one \(forms\quad \Omega^ 1_ X\) (as a reflexive module).
In characteristic p\(>0\), where the group G no longer exists in general, the one-to-one correspondence between irreducible reflexive modules and vertices of the extended Dynkin diagram is still true (Gonzalez- Sprinberg, Verdier and Artin, Verdier). In the paper under review the authors complete the picture in characteristic \(p.\) Replacing \(\Omega^ 1_ X\) by \(\Omega\), the unique non trivial extension of the maximal ideal by \({\mathcal O}\) (which was also considered by other people, among them M. Auslander), they show that the multiplicative behavior remains true, except in few cases which are studied precisely. rational double point; McKay correspondence; Dynkin diagram; characteristic p Gonzalez-Sprinberg, G.; Verdier, J. -L.: Structure multiplicative de modules réflexifs sur LES points doubles rationnels. Travaux en cours 22, 79-100 (1987) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Structure multiplicative des modules réflexifs sur les points doubles rationnels. (Multiplicative structure of the reflexive modules on the rational double 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 Concerns the article ibid. 63, No.1, 108-113 (1988). The mistakes pointed out in this erratum are already mentioned in the review Zbl 0658.20024 by G. Mislin. cohomology of finite Chevalley groups; cohomology stability; connected split reductive group scheme; change of fields; algebra retract; elementary abelian \(\ell \)-subgroups; cohomology algebras; integral cohomology; cohomological restriction map Cohomology theory for linear algebraic groups, Linear algebraic groups over finite fields, Homology of classifying spaces and characteristic classes in algebraic topology, Group schemes, Cohomology of groups Erratum: Multiplicative stability for the cohomology of finite 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 Operads were introduced by the second author in a purely topological framework. Here they are proposed to serve the study of various kinds of algebras: associative, associative and commutative, Lie, Poisson, DGA's, etc., occurring in algebraic topology, algebraic geometry, differential geometry and string theory. Much motivation comes from the theory of mixed Tate motives in algebraic geometry.
Fix a commutative ground ring \(k\) and write \(\Sigma_j\) for the symmetric group on \(j\) elements. Tensor products are always over \(k\). An operad (of \(k\)-modules) consists of \(k\)-modules (i.e., differential \(\mathbb{Z}\)-graded chain complexes over \(k)\) \({\mathcal C} (j)\), \(j \geq 0\), together with a unit map \(\eta : k \to {\mathcal C} (1)\), a right action of \(\Sigma_j\) on each \({\mathcal C} (j)\) for each \(j\) and maps \(\gamma : {\mathcal C} (m) \otimes {\mathcal C} (j_1) \otimes \cdots \otimes {\mathcal C} (j_m) \to {\mathcal C} (j)\), for \(m \geq 1\) and \(j_s \geq 0\), where \(\sum j_s = j\). The \(\gamma\) are supposed to satisfy suitable associativity and \(\Sigma\)-equivariance requirements and to be unital. Neglecting the \(\Sigma\)-action one gets a non-\(\Sigma\) operad. Working with unital \(k\)-algebras \(A\) one may think of \(1 \in A\) as a map \(k \to A\). It is sensible to insist that \({\mathcal C} (0) = k\), and one says that \({\mathcal C}\) is a unital operad. If \({\mathcal C}\) is unital, one has augmentations \(\varepsilon = \gamma : {\mathcal C} (j) \cong {\mathcal C} (j) \otimes {\mathcal C} (0)^j \to {\mathcal C} (0) = k\). \({\mathcal C}\) is called acyclic if the augmentations are quasi-isomorphisms, and \({\mathcal C}\) is said to be \(\Sigma\)-free (or \(\Sigma\)-projective) if \({\mathcal C} (j)\) is \(k [\Sigma_j]\)-free (or \(k [\Sigma_j]\)-projective) for each \(j\). \({\mathcal C}\) is called an \(E_\infty\) operad if it is both acyclic and \(\Sigma\)-free and if each \({\mathcal C} (j)\) is concentrated in nonnegative degrees. A non-\(\Sigma\) operad is called an \(A_\infty\) operad if it is acyclic and each \({\mathcal C} (j)\) is concentrated in nonnegative degrees.
Let \({\mathcal C}\) be an operad. A \({\mathcal C}\)-algebra is a \(k\)-module \(A\) with maps \(\theta : {\mathcal C} (j) \otimes A^j \to A\), \(j \geq 0\), that are associative, unital and equivariant in a suitable way. For a \({\mathcal C}\)-algebra \(A\) an \(A\)-module is a \(k\)-module \(M\) together with maps \(\lambda : {\mathcal C} (j) \otimes A^{j - 1} \otimes M \to M\) satisfying, as always, suitable associativity, unitality and equivariance requirements. To an operad \({\mathcal C}\) one may associate a monad \(C\), and, as a matter of fact, \({\mathcal C}\)-algebras correspond to \(C\)-algebras. Define the unital operad \({\mathcal M}\) by \({\mathcal M} (j) = k [\Sigma_j]\) as a right \(k [\Sigma_j]\)-module (concentrated in degree 0). An \({\mathcal M}\)-algebra \(A\) is now a DGA. An \(A\)-module \(M\) is then just an \(A\)-bimodule in the classical sense. Similarly, for the unital operad \({\mathcal N}\), \({\mathcal N} (j) = k\) for all \(j\), an \({\mathcal N}\)-algebra is just a commutative DGA, and an \(A\)-module is just an \(A\)-module in the classical sense. Another example is provided by the nonunital operad \({\mathcal L}\) whose algebras are just the Lie-algebras over \(k\), where one assumes that \(k\) is a field of characteristic other than 2 or 3. Here, for an \({\mathcal L}\)-algebra \(L\), an \(L\)-module is just a Lie-algebra module in the classical sense.
Actually, operads can be defined in any symmetric monoidal category, e.g. the category of topological spaces under Cartesian product. To the operad of spaces \({\mathcal E}\) one can associate the homology operad of \(k\)-modules \(H_* ({\mathcal E})\). For a space \(X\) set \(EX = \coprod {\mathcal E} (j) \times_{\Sigma_j} X^j\). Then one has the result: Let \( E\) be the monad in the category of spaces associated to \({\mathcal E}\) and let \(E_H\) be the monad in the category of \(k\)-modules associated to \(H_* ({\mathcal E})\). If \(k\) is a field of characteristic zero, then \(H_* (EX) \cong E_H H_* (X)\). Several other themes of interest are discussed, e.g. operads coming from iterated loop spaces, leading to the notion of \(n\)-Lie algebras and \(n \)-braid algebras, the little \(n\)-cubes operad \({\mathcal C}_n\) (whose homology operad is shown to be isomorphic to \({\mathcal L})\), algebras over the operad \(H_* ({\mathcal C}_{n + 1}) \) (which are shown to be just the \(n\)-braid algebras), and homology operations in characteristic \(p\).
Part II deals with partial algebras, modules, etc. \(k\) is assumed to be a Dedekind domain. For a \(\Sigma\)-projective operad \({\mathcal C}\) of simplicial \(k\)-modules and a partial \({\mathcal C}\)-algebra \(A\) there is a functor \(V\) that assigns a quasi-isomorphic \({\mathcal C}\)-algebra \(VA\) to \(A\). Similarly for a partial \(A\)-module \(M\) there exists a quasi-isomorphic \(A\)-module \(VM\). In case \(k\) is a field of characteristic zero, every operad is \(\Sigma\)-projective, so one may drop the condition in the statement. Similarly, with \(k\) a field of characteristic zero, for an acyclic operad \({\mathcal C}\) of \(k\)-modules, there is a functor \(W\) that assigns a quasi-isomorphic DGA \(WA\) to a \({\mathcal C}\)-algebra \(A\). Several results of this kind are treated. The motivating example comes from Bloch's higher Chow groups. Let \(X\) be a smooth, quasi-projective variety over a field \(F\). Bloch defined an Adams graded simplicial abelian group \({\mathcal Z} (X)\) with free groups \({\mathcal Z}^r (X,q)\) of \(q\)-simplices in Adams grading \(r\). There is a partially defined intersection product on this graded simplicial abelian group. In Adams degree \(r\) and simplicial degree \(q\), the domain \({\mathcal Z} (X)_j\) of the \(j\)-fold product is the sum over all partitions \(\{r_1, \dots, r_j\}\) of \(r\) of the subgroups of \(\otimes^j_{i = 1} {\mathcal Z}^{r_i} (X,q)\) spanned by those \(j\)-tuples of simplices all intersections of subsets of which meet all faces (of \(X \times \Delta^q)\) properly. An essential result is the moving lemma which gives a quasi-isomorphism \({\mathcal Z} (X)_j \to {\mathcal Z} (X)^j\). Bloch's (integral) higher Chow groups are defined by \(CH^r (X,q) = H_q ({\mathcal Z}^r (X,*); \mathbb{Z})\), and Bloch proved that \(CH^r (X,q) \otimes \mathbb{Q} \simeq (K_q (X) \otimes \mathbb{Q})^{(r)}\), where the superscript \((r)\) means the \(n^r\)-eigenspace for the Adams operations \(\psi^n\) for any \(n > 1\). One chooses an \(E_\infty\) operad \({\mathcal C}\) of simplicial abelian groups and regards the partial commutative simple rings as partial \({\mathcal C}\)-algebras. Apply the functor \(V\) to obtain genuine \({\mathcal C}\)-algebras. There is another functor \(C_\#\) which converts \({\mathcal C}\)-algebras to algebras over the associated \(E_\infty\) operad \(C_\# {\mathcal C}\) of chain complexes. Let \({\mathcal A} (X)\) be the \(E_\infty\) algebra \(C_\# V ({\mathcal Z} (X)_*)\) and let \({\mathcal A}_\mathbb{Q} (X) = C_\# W ({\mathcal Z} (X)_* \otimes \mathbb{Q})\), then one shows: \({\mathcal A} (X) \otimes \mathbb{Q}\) is an \(E_\infty\) algebra, and there is a quasi-isomorphism \({\mathcal A} (X) \otimes \mathbb{Q} \to {\mathcal A}_\mathbb{Q} (X)\), of \(E_\infty\) algebras. Writing \({\mathcal N}^{2r - p} (X) (r) = {\mathcal A}_p (X)\), a candidate for motivic cohomology becomes \(H^i_{\text{Mot}} (X, \mathbb{Q} (r)) : = H^i ({\mathcal N}_\mathbb{Q} (X)) (r)\). Let \({\mathcal N} = {\mathcal N} (\text{Spec} (F))\). The \(E_\infty\) algebras \({\mathcal N} (X)\) can be regarded as \({\mathcal N}\)-modules and thus as objects of the derived category \({\mathcal D}_{\mathcal N}\). Deligne proposed this derived category as a derived category of integral mixed Tate modules.
Before constructing the derived category of \(E_\infty\) modules, it turns out to be fruitful to treat derived categories of modules over a DGA from a topological point of view. This is done in Part III. The basic notion, to be compared with a CW-spectrum in stable homotopy theory, is that of a cell \(A\)-module, where \(A\) is an associative, unital, but not necessarily commutative DGA over \(k\). A cell \(A\)-module \(M\) is the union of an expanding sequence of sub \(A\)-modules \(M_n\) such that \(M_0 = 0\) and \(M_{n + 1}\) is the cofiber of a map \(\pi_n : F_n \to M_n\), with \(F_n\) a direct sum of `sphere modules'. Cofiber sequences are just exact triangles in the formalism of triangulated categories. Let \({\mathcal M}_A\) denote the category of \(A\)-modules, and write \(h {\mathcal M}_A\) for its homotopy category. The derived category \({\mathcal D}_A\) is obtained from \(h {\mathcal M}_A\) by formally inverting quasi-isomorphisms of \(A\)-modules. One has several results on cell \(A\)-modules: (i) Whidehead's theorem which can be reformulated by saying that a quasi-isomorphism between cell \(A\)-modules is a homotopy equivalence; (ii) the cellular approximation theorem which says that for any \(A\)-module \(M\) there exists a quasi-isomorphic cell \(A\)-module \(\Gamma M\), and (iii) Brown's representability theorem. One has a formalism for Tor and Ext as derived tensor product and Hom. For \(A\)-modules \(M\) and \(N\), one defines \(M \otimes^L_A N = M \otimes_A \Gamma N\) and \(R \Hom_A (M,N) = \Hom_A (\Gamma M,N)\). One gets the well-known formula: \({\mathcal D}_A (K \otimes^L M,N) \simeq {\mathcal D}_k (K,R \Hom_A (M,N))\) for \(k\)-modules \(K\). There is a corresponding formalism for relative cell \(A\)-modules. For commutative \(A\), \({\mathcal D}_A\) can be given the structure of a symmetric monoidal (= tensor) category with internal Hom's.
Part IV is concerned with rational derived categories and mixed Tate motives. Let \(k\) be a field of characteristic zero, and let \(A\) be a DGA, here in the sense of a commutative, differential graded, and Adams graded \(k\)-algebra. Thus \(A\) is bigraded via \(k\)-submodules \(A^q (r)\), \(q \in \mathbb{Z}\) and \(r \geq 0\). One assumes \(A^q (r) = 0\) unless \(2r \geq q\), and \(A\) cohomologically connected. \({\mathcal D}_A\) will stand for the derived category of cohomologically bounded below \(A\)-modules. Let \({\mathcal H}_A\) be the full subcategory of \({\mathcal D}_A\) consisting of the cell \(A\)-modules \(M\) with \(H^q (k \otimes_A M) = 0\) for \(q \neq 0\), and let \({\mathcal F} {\mathcal H}_A\) be the full subcategory of \({\mathcal H}_A\) consisting of the modules \(M\) with finite-dimensional \(H^0 (k \otimes_A M)\). Define \(\omega (M) = H^0 (k \otimes_AM)\). The following results are proved: (i) the triangulated category \({\mathcal D}_A\) has a \(t\)-structure whose heart is \({\mathcal H}_A\). In particular, \({\mathcal H}_A\) is abelian. Also, \({\mathcal F} {\mathcal H}_A\) is a (graded) neutral Tannakian category over \(k\) with fiber functor \(\omega\); (ii) assume \(A\) is a connected DGA, then the following categories are equivalent: (a) the heart \({\mathcal H}_A\) of \({\mathcal D}_A\), (b) the category of comodules over the Hopf algebra \(\chi_A = H^0 \overline B (A)\) (the bar construction), (c) the category of generalized nilpotent representations of the co-Lie algebra \(\gamma_A\) (the \(k\)-module of indecomposable elements of \(\chi_A)\); (iii) the derived category of the bounded below complexes in \({\mathcal H}_A\) is equivalent to the derived category of modules over the DGA \(\wedge \gamma_A [-1]\) \((\wedge \gamma_A [-1]\) is the 1-minimal model of \(A)\). Now, with the notation of Part II, one may reformulate a well-known conjecture of Beilinson-Soulé: \({\mathcal N}_\mathbb{Q} (X)\) is cohomologically connected. Specializing to the case \(X = \text{Spec} (F)\), write \({\mathcal N}\) for the \(E_\infty\) algebra \({\mathcal N} (\text{Spec} (F))\) and \({\mathcal N}_\mathbb{Q}\) for the commutative DGA \({\mathcal N}_\mathbb{Q} (\text{Spec} (F))\). Write \(\chi_{\text{Mot}}\) for the Hopf algebra \(\chi_{{\mathcal N}_\mathbb{Q}} = H^0 \overline B ({\mathcal N}_\mathbb{Q})\), and define the category of (rational) mixed Tate motives over the field \(F\), \({\mathcal M} {\mathcal T} {\mathcal M} (F)\), as the category of finite-dimensional comodules over \(\chi_{\text{Mot}}\). The above results may be applied to obtain: (i) assume the Beilinson-Soulé conjecture holds for \(\text{Spec} (F)\), then \({\mathcal M} {\mathcal T} {\mathcal M}(F)\) is equivalent to \({\mathcal F} {\mathcal H}_{{\mathcal N}_\mathbb{Q}}\) (this implies the equivalence of the definitions of mixed Tate motives by Bloch and the first author on the one hand, and by Deligne on the other hand); (ii) if \({\mathcal N}_\mathbb{Q}\) is a \(K (\pi, 1)\), then \(\text{Ext}^p_{{\mathcal M} {\mathcal T} {\mathcal M} (F)} (\mathbb{Q}, \mathbb{Q} (r)) \simeq gr^r_\gamma (K_{2r - p} (F) \otimes \mathbb{Q})\).
In Part V it is shown that, for a commutative ring \(k\) and an \(E_\infty\) operad \({\mathcal C}\), the derived category of modules over a \({\mathcal C}\)-algebra looks very much like the derived category of modules over a commutative DGA as discussed in Part III. Let \({\mathcal C}\) be an operad, and write \(\mathbb{C}\) for \({\mathcal C} (1)\). Then \(\mathbb{C}\) is a DGA over \(k\), usually not commutative, but it is homotopy commutative if \({\mathcal C}\) is an \(E_\infty\) operad. For \(\mathbb{C}\)-modules \(M\) and \(N\) one defines the operadic tensor product \(M \boxtimes N\) to be the \(\mathbb{C}\)-module \(M \boxtimes N = {\mathcal C} (2) \otimes_{\mathbb{C} \otimes \mathbb{C}} M \otimes N\). Also, for two (left) \(\mathbb{C}\)-modules \(M\) and \(N\) one defines \(\Hom^\boxtimes (M,N) = \Hom_\mathbb{C} ({\mathcal C} (2) \otimes_\mathbb{C} M,N)\). Of fundamental importance is the existence of a special \(E_\infty\) operad \({\mathcal C}\), the linear isometries operad, for which the tensor product \(\boxtimes\) is commutative and associative, but not unital at the module level. It becomes unital in the derived category. It is shown that, up to quasi-isomorphism, it is no loss of generality to take the preferred \(E_\infty\) linear isometries operad \({\mathcal C}\). Using the cellular approximation theorem for \(\mathbb{C}\)-modules, \(\boxtimes\) and \(\Hom^\boxtimes\) induce derived versions \(\boxtimes^L\) and \(R \Hom^\boxtimes\) in \({\mathcal D}_\mathbb{C}\). The forgetful functor from \(\mathbb{C}\)-modules to \(k\)-modules induces an equivalence \({\mathcal D}_\mathbb{C} \to {\mathcal D}_k\), under which \(\boxtimes^L\) is carried to \(\otimes^L\). Similarly for \(R \Hom^\boxtimes\) and \(R \Hom\). A formula such as \({\mathcal D}_\mathbb{C} (M,N) \simeq {\mathcal D}_\mathbb{C} (M \boxtimes^L k,N) \simeq {\mathcal D}_\mathbb{C} (M,R \Hom^\boxtimes (k,N))\) should come as no surprise, it implies that there is a natural isomorphism \(N \to R \Hom^\boxtimes (k,N)\) in \({\mathcal D}_\mathbb{C}\). Three more products are introduced to deal with questions of unitality. They are used to define a tensor product \(\boxtimes_A\) and a Hom-functor \(\Hom^\boxtimes_A (M,N)\) for modules over \(A_\infty\) algebras. \(\boxtimes_A\) has the property that, for augmented \(A\) and a cell \(A\)-module \(N\), the unit map \(\lambda : A \boxtimes_A N\to N\) is a quasi-isomorphism. One also defines \(R \Hom^\boxtimes_A\). It satisfies \({\mathcal D}_A (L \boxtimes^L M,N) \simeq {\mathcal D}_\mathbb{C} (L,R \Hom^\boxtimes_A (M,N))\). As a matter of fact, one can carry out homological algebra in this generalized context, e.g. one may construct Hochschild homology of \(A_\infty\) algebras. One has Eilenberg-Moore spectral sequences to compute the homology of the derived tensor product \(M \boxtimes^L_A N\) and the derived Hom functor \(R \Hom^\boxtimes_A (M,N)\) in terms of the classical Tor and Ext groups \(\text{Tor}^*_{H^* (A)} (H^* (M)\), \(H^* (N))\) and \(\text{Ext}^*_{H^* (A)} (H^* (M)\), \(H^* (N))\). For an \(E_\infty\) algebra \(A\) and an \(A\)-module \(M\) one defines \(M^\vee = \Hom^\boxtimes_A (M,A)\), and one says that a cell \(A\)-module \(M\) is strongly dualizable if it has a coevaluation map \(\eta : A \to M \boxtimes_A M^\vee\) with the usual properties. For such \(M\) the natural maps \(\rho : M \to M^{\vee \vee}\) and \(\nu : M^\vee \boxtimes_A N \to \Hom^\boxtimes_A (M,N)\) induce isomorphisms in \({\mathcal D}_A\). It follows that, for a strongly dualizable \(M\) and any \(A\)-module \(N\), one has \(\text{Tor}^n_A (M^\vee, N) \simeq \text{Ext}^n_A (M,N)\).
After reading this highly interesting booklet of only 140 pages, the reader will be impressed by the overwhelming quantity of material hidden in it and he or she will generously acknowledge the SMF for publishing it as a double (!) volume in the Astérisque series. monads; higher Chow complex; operads; unital \(k\)-algebras; little \(n\)-cubes operad; tensor category; braid algebras; mixed Tate motives; symmetric monoidal category; operad of spaces; iterated loop spaces; higher Chow groups; Adams operations; derived category; integral mixed Tate modules; derived categories of modules; DGA; triangulated category; Tannakian category; Hopf algebra; co-Lie algebra; Beilinson-Soulé conjecture; operadic tensor product; cellular approximation theorem Kriz, I.; May, J. P., Operads, algebras, modules and motives, Astérisque, 233, (1995), iv+145 pp Research exposition (monographs, survey articles) pertaining to category theory, Applied homological algebra and category theory in algebraic topology, Research exposition (monographs, survey articles) pertaining to \(K\)-theory, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Homological algebra in category theory, derived categories and functors, Higher algebraic \(K\)-theory, \(K\)-theory in geometry, Generalizations (algebraic spaces, stacks), Derived categories, triangulated categories Operads, algebras, modules and motives | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group acting on a finite set \(\Omega\), and let \(C_\Omega(x)\) denote the set of points in \(\Omega\) fixed by \(x\in G\). Define the `fixed point ratio of \(x\)' to be the proportion of points in \(\Omega\) fixed by \(x\); i.e. \(\text{fpr}(x)=|C_\Omega(x)|/|\Omega|\). This paper is the first in a series of four papers which study this ratio for finite classical groups [see also \textit{T. C. Burness}, part II, ibid. 309, No. 1, 80-138 (2007; see the following review Zbl 1128.20004), part III, ibid. 314, No. 2, 693-748 (2007; Zbl 1133.20003), part IV, ibid. 314, No. 2, 749-788 (2007; Zbl 1133.20004)].
The strategy is to reduce the problem to a study of conjugacy classes in \(G\), using the simple observation that if the action of \(G\) is transitive, then \(\text{fpr}(x)=|x^G \cap H|/|x^G|\), where \(H\) is the stabiliser of some point in \(\Omega\). This naturally leads to a consideration of the maximal subgroups in a classical group, famously classified by \textit{M. Aschbacher} [Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)]. Certain of the subgroups in Aschbacher's classification can be naturally called `subspace subgroups'; roughly speaking, these arise as stabilisers of subspaces in the natural module for \(G\) (e.g., parabolic subgroups). By extension, a transitive \(G\)-action is called a `subspace action' if the stabiliser of a point of \(\Omega\) is a subspace subgroup; such actions give rise to relatively large fixed point ratios.
The aim of this series of papers is to provide concrete bounds on the fixed point ratio for `non-subspace' actions. The main result says that, with a small number of exceptions (which are also dealt with), if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then \(\text{fpr}(x)\lesssim |x^G|^{-\frac 12}\) for all elements \(x\in G\) of prime order. The proof of the theorem is contained in the later papers, which run through each of the relevant Aschbacher classes in turn.
In the current paper the author introduces the main result and describes how it can be applied to the study of minimal bases for primitive permutation groups. He also outlines how the main result may be useful in classifying those primitive permutation groups which arise as the monodromy groups of a branched covering \(\varphi\colon X\to\mathbb{P}^1\mathbb{C}\), where \(X\) is a compact connected Riemann surface. finite classical groups; fixed point ratios; primitive permutation groups; monodromy groups; permutation representations; finite almost simple groups; maximal subgroups of classical groups DOI: 10.1016/j.jalgebra.2006.05.024 Primitive groups, Linear algebraic groups over finite fields, Representation theory for linear algebraic groups, Riemann surfaces; Weierstrass points; gap sequences, Group actions on varieties or schemes (quotients) Fixed point ratios in actions of finite classical groups. 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 We show that the types of singularities of Schubert varieties in the flag varieties \(\text{Flag}_n\), \(n \in {\mathbb N}\), are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type~\({\mathbb A}\). Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians \(\text{Grass}(n,a) \times \text{Grass} (n,b)\), \(a,b,n \in {\mathbb N}\), \(a,b \leq n\), are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type~\({\mathbb D}\). We also show that the orbit closures in representation varieties of Dynkin quivers of type~\({\mathbb D}\) are normal and Cohen-Macaulay varieties. singularities of Schubert varieties; flag varieties; Dynkin quivers; Cohen-Macaulay varieties Bobiński, Grzegorz; Zwara, Grzegorz, Schubert varieties and representations of Dynkin quivers, Colloq. Math., 94, 2, 285-309, (2002) Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Schubert varieties and representations of Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0721.00007.]
In J. Algebra 88, 89-133 (1984; Zbl 0531.13015), \textit{D. Eisenbud} and the author proved that regular local rings have finite Buchsbaum- representation type, i.e. they have just a finite number of nonisomorphic maximal indecomposable Buchsbaum modules. -- Here are determined the surface singularities of finite Buchsbaum-representation type. Let \(R\) be a Noetherian complete local ring with \(\dim(R)=2\) whose residue field is algebraically closed. Then \(R\) has finite Buchsbaum-representation type iff \(R\cong P/\kappa I\), where \((P,m)\) is a complete regular local ring with \(\dim(P)=3\), \(I\subset P\) is an ideal with \(\hbox{ht}(I)\geq 2\) and \(\kappa\) is an element from \(m\backslash m^ 2\). In particular \(R\) is regular iff \(R\) is unmixed of finite Buchsbaum-representation type. surface singularities of finite Buchsbaum-representation type Shiro Goto, Surface singularities of finite Buchsbaum-representation type, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 247 -- 263. Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of surfaces or higher-dimensional varieties Surface singularities of finite Buchsbaum-representation 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 Let G be a connected, affine algebraic group over an algebraically closed field and B a Borel subgroup of G. For each one dimensional rational B- module L the induced G-module \(Ind^ G_ BL\) is defined. These modules are of fundamental importance in the representation theory of G; in characteristic 0 one obtains every simple rational G-module in this way and in arbitrary characteristic \(Ind^ G_ BL\), when non-zero, has a simple socle and each simple G-module occurs as the socle of some such induced module. The formal character of \(Ind^ G_ BL\) is independent of the characteristic, being given by Weyl's character formula, but the submodule structure depends very heavily upon the characteristic and very little is known about this structure in characteristic p. For G semisimple, the induced modules also have an interpretation as the global sections of line bundles on the quotient variety G/B and so provide a bridge between the representation theory of G and the geometry of G/B.
A good filtration of a rational G-module V is an ascending chain of submodules \(O=V_ 0,V_ 1,V_ 2,..\). of V such that V is the union of the \(V_ i\) and, for each \(i>0\), \(V_ i/V_{i-1}\) is either 0 or isomorphic to \(Ind^ G_ BL\) for some rational one dimensional B-module L. It is known that for G semisimple and simply connected each rationally injective indecomposable G-module has a good filtration. In the monograph under review the author studies for a connected, affine algebraic group G over an algebraically closed field k, the following hypotheses.
Hypothesis 1. For all rational G-modules V, V' which have a good filtration the tensor product \(V\otimes V'\) has a good filtration.
Hypothesis 2. For every rational G-module V which has a good filtration and every parabolic subgroup P of G the restriction of V to P has a good filtration.
In chapters 1, 2 and 5 the author considers some general results on group cohomology and the derived functors of induction which are needed for the specific calculations in chapters 4, 6, 7, 8, 9 and 10. The main purpose of chapter 1 is to establish the notation and explain the relationship between various left exact functors. Chapter 2 contains results computing the cohomology of some modules for parabolic subgroups. This chapter also contains a deduction from Kempf's vanishing theorem of Weyl's character formula for the character of \(Ind^ G_ BL\) for a reductive group G and one-dimensional B-module L.
In chapter 3 the author makes various reductions to the hypotheses so that they become susceptible to the case by case analysis which follows.
In chapter 4 he proves the hypotheses for the classical groups. The argument here is independent of the characteristic and it has been possible to treat the groups of type B, C and D in a unified manner. The restriction of \(Y(\lambda_ i)\) to a proper parabolic subgroup of maximal dimension has only 4 successive quotients in a good filtration (for i in ''general position'') and the module structure is much the same in all three types B, C, and D.
Some additional homological algebra is needed to deal with the exceptional groups and this is given in chapter 5. The hypotheses are proved for \(G_ 2\) in chapter 6 however it would not be difficult to treat this case without the benefit of chapter 5. In chapter 7, treating \(F_ 4\), he found it necessary to consider separately the cases of odd and even characteristic.
The subject, \(E_ 6\), of chapter 8 is altogether easier but again there is a division in the proof into odd and even characteristics. Chapters 9 and 10 are devoted to the remaining exceptional groups \(E_ 7\) and \(E_ 8\). The procedure here is to analyse first the modules \(Y(\lambda_ i)\), corresponding to the terminal vertices \(\alpha_ i\) of the Dynkin diagram, and then exterior powers of these modules are used to deal with \(Y(\lambda_ r)\) for an arbitrary fundamental dominant weight \(\lambda_ r.\)
Chapter 11 opens with an example of a reductive subgroup H of a reductive group G and a G-module V such that V has a good filtration but the restriction of V to H does not. The remainder of the chapter is devoted to applications of the hypotheses to rational cohomology, homomorphisms between Weyl modules, canonical products on induced modules and filtrations over \({\mathbb{Z}}\) of Weyl modules for Kostant's \({\mathbb{Z}}\)-form \(U_{{\mathbb{Z}}}\) of the enveloping algebra U(\({\mathfrak g})\) of a complex semisimple Lie algebra \({\mathfrak g}.\)
The final chapter is devoted to a number of observations on issues not directly concerned with the hypotheses but which nevertheless have mainly arisen in the course of the work on the hypotheses. The issues discussed are the injective indecomposable modules for a parabolic subgroup of a reductive group, Kempf's vanishing theorem for rank 1 groups, Kempf's vanishing theorem in characteristic zero and the exactness of induction. connected, affine algebraic group; Borel subgroup; Weyl's character formula; induced modules; global sections of line bundles on the quotient variety; good filtration; rational G-module; group cohomology; derived functors of induction; parabolic subgroups; Kempf's vanishing theorem; exceptional groups; Dynkin diagram; dominant weight; reductive group S. Donkin, \textit{Rational Representations of Algebraic Groups}, Lecture Notes in Mathematics, Vol. 1140, Springer-Verlag, Berlin, 1985. Representation theory for linear algebraic groups, Affine algebraic groups, hyperalgebra constructions, Cohomology theory for linear algebraic groups, Research exposition (monographs, survey articles) pertaining to group theory, Homogeneous spaces and generalizations Rational representations of algebraic groups: Tensor products and filtrations | 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 projective scheme over a field \(K\) and let \(\mathcal F\) be a coherent sheaf of \(\mathcal O_X\)-modules. We show that the cohomological postulation numbers \(\nu^i_{\mathcal F}\) of \(\mathcal F\), e.g., the ultimate places at which the cohomological Hilbert functions \(n\mapsto h^i(X,\mathcal F(n))\) start to be polynomial for \(n<<0\), are (polynomially) bounded in terms of the cohomology diagonal \((h^i(X, \mathcal F(-i))_{i=0}^{\dim(\mathcal F)}\) of \(\mathcal F\). As a consequence, we obtain that there are only finitely many different cohomological Hilbert functions if \(\mathcal F\) runs through all coherent sheaves of \(\mathcal O_X\)-modules with fixed cohomology diagonal. In order to prove these results, we extend the regularity bound of \textit{D. Bayer} and \textit{D. Mumford} [in: Computational Algebraic Geometry and Commutative Algebra, Proc. Conf. Cortona 1991, Press, 1--48 (1993; Zbl 0846.13017)] from graded ideals to graded modules. Moreover, we prove that the Castelnuovo--Mumford regularity of the dual of \(\mathcal F\) is (polynomially) bounded in terms of the cohomology diagonal of \(\mathcal F\). cohomology of projective schemes; cohomological Hilbert functions; Castelnuovo--Mumford regularity Brodmann, M.; Lashgari, A. F.: A diagonal bound for cohomological postulation numbers of projective schemes. J. algebra 265, 631-650 (2003) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vanishing theorems in algebraic geometry A diagonal bound for cohomological postulation numbers of projective schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be an algebraically closed field. It is well-known that the \(K\)-algebras of finite representation type define an open subset in the variety of finite-dimensional \(K\)-algebras. Unfortunately it is still unknown if an analogous result holds for algebras of tame representation type, i.e. if tame is open. It is shown by \textit{S. Kasjan} [in Fundam. Math. 171, No. 1, 53-67 (2002; Zbl 0997.16008)] that ``tame is open'' is equivalent to finite axiomatizability of the class of tame algebras (of any fixed dimension, over algebraically closed field of fixed characteristic). On the other hand, \textit{Y. Han} [J. Algebra 284, No. 2, 801-810 (2005; Zbl 1134.16302)] formulates a wild rank conjecture and proves that it implies that tame is open. He also proves the conjecture for a relatively large class of algebras.
It is shown in the paper under review that the conjecture is equivalent to finite axiomatizability of tameness. The author also modifies the definition of the rank of a wild \(K\)-algebra introduced by Han in such a way that any wild algebra has finite wild rank. Moreover, he proves by applying Tarski's quantifier elimination theorem that the (modified) wild rank conjecture implies that tame is open. It is also shown that this technique allows to prove that if \(\{A_\xi\}\) is a regular one-parameter family of \(d\)-dimensional \(K\)-algebras such that an uncountable subfamily consists of wild algebras then all algebras in the family are wild. tame representation type; wild algebras; varieties of algebras; finite axiomatizability; quantifier elimination Representation type (finite, tame, wild, etc.) of associative algebras, Model-theoretic algebra, Representations of orders, lattices, algebras over commutative rings, Applications of logic in associative algebras, Foundations of algebraic geometry On the wild rank conjecture of Han. | 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 (in characteristic \(p>0\)) that the algebra of invariants, under certain actions of finite abelian Hopf algebras and one-dimensional formal groups, is a regular local ring. algebras of invariants; actions of finite Abelian Hopf algebras; formal groups; regular local rings Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Formal groups, \(p\)-divisible groups, Actions of groups on commutative rings; invariant theory On the regularity of the algebra of invariants of (co)actions of certain finite Hopf algebras and of formal groups in characteristic \(p\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a finite dimensional associative \(K\)-algebra with unity over an algebraically closed field \(K\), and let \(\text{mod}_A(d)\) be the affine variety of \(d\)-dimensional \(A\)-modules. Then the general linear group \(\text{GL}_d(K)\) acts on this variety by conjugation. An \(A\)-module \(N\in\text{mod}_A(d)\) is called a degeneration of \(M\in\text{mod}_A(d)\) if \(N\) belongs to the closure of the \(\text{GL}_d(K)\)-orbit of \(M\). In fact, this defines a partial order on \(\text{mod}_A(d)\) and the authors investigate how it relates with other types of partial orders appearing in several earlier works [\textit{S. Abeasis, A. del Fra}, J. Algebra 93, 376-412 (1985; Zbl 0598.16030), \textit{K. Bongartz}, Adv. Math. 121, No. 2, 245-287 (1996; Zbl 0862.16007), and others]. In particular, they prove that their partial order coincides with known ones on modules from the additive categories of quasi-tubes [\textit{A. Skowroński}, Topics in algebra, Banach Cent. Publ. 26, Pt. 1, 535-568 (1990; Zbl 0729.16005)], and describe minimal degenerations of such modules. finite dimensional algebras; affine varieties; general linear groups; degenerations of modules; partial orders; categories of quasi-tubes; minimal degenerations Skowroński, A.; Zwara, G.: On degenerations of modules with nondirecting indecomposable summands. Canad. J. Math. 48, 1091-1120 (1996) Representations of associative Artinian rings, Group actions on varieties or schemes (quotients), Formal methods and deformations in algebraic geometry, Module categories in associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Finite rings and finite-dimensional associative algebras On degenerations of modules with nondirecting indecomposable summands | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a construction of nonsmooth self-dual projective algebraic varieties. They appear as certain projectivized orbit closures for some linear actions of reductive algebraic groups. Applying this construction to adjoint representations, a geometric characterization is obtained for distinguished nilpotent elements of semisimple Lie algebras [\textit{P. Bala} and \textit{R. W. Carter}, Math. Proc. Camb. Phil. Soc. 79, 401--425 (1976; Zbl 0364.22006); ibid. 80, 1--18 (1976; Zbl 0364.22007)] (i.e., nilpotent elements whose centralizer contains no nonzero semisimple elements) as nilpotent elements whose projectivized orbit closures are self-dual projective algebraic varieties. In particular, the author shows that the projectivized nilpotent cone of every semisimple Lie algebra is a self-dual projective algebraic variety. He also applies this construction to isotropy representations of symmetric spaces and introduces the notion of a \((-1)\)-distinguished nilpotent element, the counterpart of the notion of distinguished element. The projectivized orbit closures of \((-1)\)-distinguished elements are self-dual projective algebraic varieties as well. linear actions of reductive algebraic groups; semisimple Lie algebras Popov, V.: Self-dual algebraic varieties and nilpotent orbits, Tata inst. Fund. res., 509-533 (2001) Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions, Simple, semisimple, reductive (super)algebras, Exceptional (super)algebras, Projective techniques in algebraic geometry, Nilpotent and solvable Lie groups Self-dual algebraic varieties and nilpotent orbits | 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 0632.00004.]
Let A be a simple complex abelian variety, D be the \({\mathbb{Q}}\)-algebra of its homomorphisms, i.e., a central simple algebra (skew field) of degree \( m^ 2\) over the number field (center) E of degree \( r.\) The author proves that the Hodge group Hdg(A) [see \textit{D. Mumford}, Math. Ann. 181, 345-351 (1969; Zbl 0169.233)] is an algebraic E-group in case E is a totally real number field and the number dim(A)/mr is odd. If D is a skew field of the 1st or 2nd type in the classification of Albert [\textit{D. Mumford}, ``Abelian varieties'' (1970; Zbl 0223.14022)], then Hdg(A) turns out to be ``maximally possible'' and coincides with the symplectic E-group of the E-space \(H_ 1(A,{\mathbb{Q}})\). algebraic groups of Mumford-Tate type; Hodge cycles; simple complex abelian variety; Hodge group Abelian varieties and schemes, Arithmetic ground fields for abelian varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Local ground fields in algebraic geometry, Totally real fields On algebraic groups of Mumford-Tate 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 new construction for the coinvariant algebra \(S_W\) of a Coxeter group \(W\) is presented. \(S_W\) is realized as a graded commutative subalgebra in a Nichols-Woronowicz algebra \(B_W\), which is a braided Hopf algebra over \(W\). The nilCoxeter algebra and its representation on \(S_W\) by divided difference operators are identified with subalgebras of \(B_W\). Connections to the Fomin-Kirillov quadratic algebra and to the Kirillov-Maeno bracket algebras are given. braided Hopf algebras; Nichols-Woronowicz algebras; Schubert calculus; cohomology rings of flag manifolds; coinvariant algebras; finite Coxeter groups; Yetter-Drinfeld categories Y. Bazlov, Nichols--Woronowicz algebra model for Schubert calculus on Coxeter groups, J. Algebra 297 (2006), no. 2, 372--399. Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Reflection and Coxeter groups (group-theoretic aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book under review contains six chapters that can be read independently, each one surveying one mathematical topic. The authors present in a clear manner the basic notions, the basic results, examples, and the chapters also contains exercises left to the reader. The topics presented are the following: {\parindent=6mm \begin{itemize} \item[--] Measure and integral. The chapter also contains a section on the foundations of probability theory. \item [--] Geometry in higher dimensions. The authors concentrate on phenomena which occur in dimensions \(\geq 4\) which contradict our geometric intuition in dimensions 2 and 3. Examples arise in discretization and convex geometry. The authors also make relations with the topic of the first chapter, and in particular with probability theory. Among others, the present chapter includes sections on Gaussian measure, measure concentration, the Brunn-Minkowski inequality and isoperimetry. \item [--] Fourier analysis, including the Fourier transform and the Poisson summation formula. In particular, the authors discuss unexpected applications of the Fourier transform concerning linearity testing and arithmetic progressions. \item [-] Representations of finite groups with an application in communication complexity and other fields. \item [--] Polynomials in several variables. This chapter includes a short introduction to some basic results in algebraic geometry: the Schwartz-Zippel theorem with applications to polynomial identity testing, ideals and the Hilbert basis theorem, Bézout's inequality in all dimensions and the Nullstellensatz \item [--] Topology. The topics include the Borsuk-Ulam theorem, non-embeddability results, homotopy and homology groups of simplicial complexes, simplicial approximations, and other theorems in algebraic geometry.
\end{itemize}}
The book arose from a course taught by the authors to PhD students in computer science and discrete mathematics. It is carefully written, and it is better than a collection of lecture notes. Such books are needed for students, as a complement to the standard textbooks and to present more specialized applications of classical mathematics. The reviewer wishes there were many more such books. measure; integral; geometry concentration; representations of finite groups; varieties, Nullstellensatz; Bézout inequality; Hilbert basis; Borsuk-Ulam theorem; homotopy; homology Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology Mathematics++. Selected topics beyond the basic courses | 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 quote the author's abstract. ``We consider the curve defined by \(y^2= x^p- x+ 1\) over a finite field of characteristic \(p\). Over the field \(\mathbb{F}_{p^2}\), the numerator of the zeta-function is given by \(\Phi_p(p^* t)\), with \(\Phi_p\) the \(p\)-th cyclotomic polynomial, \(p^*= (-1/p) p\), and \((-1/p)\) the Legendre symbol. Determination of the zeta-function over \(\mathbb{F}_p\) comes down to a factorization of \(\Phi_p(p^* t^2)\), and we recover an Aurifeuillian identity. The factors \(h^+\) and \(h^-\) in the factorization of \(h= \Phi_p(p^*)\) satisfy \(h^\pm/p^{(p-1)/2}= \exp(\pm 1)+ O(1/p)\).''
The factorization of \(\Phi_p(p^* t^2)\) is well known (as an `Aurifeuillian identity'). The paper gives a nice geometric explanation for it. The factors \(h^\pm\) are the orders of the groups of rational points on the Jacobians of the curves \(C^+: y^2= x^p- x+ 1\) and \(C^-: y^2= x^p- x+ a\) over \(\mathbb{F}_p\), where \(a\) is a non-square in \(\mathbb{F}_p\). zeta functions of curves over finite fields; factorization of polynomials; cyclotomic polynomials; Aurifeuillian identity; groups of rational points on Jacobians I. Duursma, Class numbers for some hyperelliptic curves, in Arithmetic, Geometry and Coding Theory, Luminy, 1993 (de Gruyter, Berlin, 1996), pp. 45--52 Curves over finite and local fields, Jacobians, Prym varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Factorization; primality, Finite ground fields in algebraic geometry Class numbers for some hyperelliptic 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 the article under review is to prove that for certain homogeneous spaces \(X\) of simple algebraic groups of the classical types \(A_n, B_n, C_n, D_n\) the group \(A_0(X)\) of zero-dimensional cycles of degree zero modulo rational equivalence is trivial. For the type \(A_n\) this is proved for a Severi-Brauer variety (recovering a result of Panin) and for some Severi-Brauer flag varieties. Here the ground field is either perfect or its characteristic does not divide the index of the underlying central simple algebra. In the \(B_n\) and \(D_n\) cases the result is obtained for any orthogonal involution variety, assuming that the characteristic is not 2. In the \(C_n\) case the author takes \(X=V_2(A,\sigma)\), a second generalized involution variety for a central simple algebra \(A\) with symplectic involution \(\sigma\).
The idea of the proof is to translate the rational equivalence of zero-dimensional cycles on a projective variety into R-equivalence (i.e. connecting points with rational curves) on symmetric powers of the original variety, and then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras. The author proposes a construction of such moduli spaces. homogeneous spaces; the Chow group; rational equivalence; simple algebras; Hilbert schemes Daniel Krashen, ``Zero cycles on homogeneous varieties'', Adv. Math.223 (2010) no. 6, p. 2022-2048 Grassmannians, Schubert varieties, flag manifolds, Finite-dimensional division rings Zero cycles on homogeneous 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 \(G\) be a connected reductive group over a finite field \(\mathbb{F}_q\). The corresponding group \(G(K)\) over the local field \(K= \mathbb{F}_q((t))\) contains \(G({\mathcal O})\) as a maximal compact subgroup, where \({\mathcal O}:= \mathbb{F}_q[[t]]\), and the latter group gives rise to the Hecke algebra \(H\) of compactly supported bi-\(G({\mathcal O})\)-invariant functions \(f: G(K)\to\overline{\mathbb{Q}}_\ell\), equipped with the usual convolution product. It is known that the Hecke algebra \(H\) is commutative, whereas the Hecke algebra \(H_I\) associated with the Iwahori subgroup \(I\subset G({\mathcal O})\) is noncommutative. Moreover, if \(G\) is a split group, then \(H\) is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of the Langlands dual group \(\check{G}\). The existence of such an isomorphism has been established by I. Satake many years ago. On the other hand, a result by \textit{I. N. Bernstein} [in: Représentations des groupes réductifs sur un corps local, 1--32 (1984; Zbl 0599.22016)] states that the representation ring of the Langlands dual group \(\check{G}\) is also isomorphic to the center of the Hecke algebra \(H_I\).
Therefore there is a well-defined isomorphism \(\pi: Z(H_I)\widetilde\to H\), the so-called Satake-Bernstein isomorphism, and the main goal of the paper under review is to describe explicitely the inverse of this isomorphism. This is done by giving geometric interpretations of both \(H\) and \(H_I\) in terms of certain perverse sheaves. More precisely, the author considers the quotients \(Gr:= G(K)/G({\mathcal O})\) and \(Fl:= G(K)/I\) as group ind-schemes over \(\mathbb{F}_q\) and studies their associated categories \(P_{G({\mathcal O})}(Gr)\) and \(P_I(Fl)\) of equivariant perverse sheaves (on \(Gr\)) or \(I\)-equivariant perverse sheaves (on \(Fl\)), respectively. The better part of the paper is then devoted to the construction of a functor \(Z: P_{G({\mathcal O})}(Gr)\to P_I(Fl)\) which, the level of Grothendieck groups, induces the inverse Satake-Bernstein map \(\pi^{-1}\). This crucial functor, whose subtle and rather involved construction uses the operation of taking ``nearby cycles'' of perverse sheaves on the affine Grassmannian \(Gr= G(K)/G({\mathcal O})\), is shown to have extremely favorable properties, and supposedly it encodes a deeper representation-theoretic meaning to be explored in the future.
As for a different approach toward a geometric interpretation of the Satake-Bernstein isomorphism, the reader is referred to the just as recent paper by \textit{I. Mirković} and \textit{K. Vilonen} [Math. Res. Lett. 7, No. 1, 13--24 (2000; Zbl 0987.14015)]. group schemes; Hecke algebras; Grassmannians; flag manifolds; preserve sheaves; reductive groups; finite ground fields; local ground fields Gaitsgory, D., \textit{construction of central elements in the affine Hecke algebra via nearby cycles}, Invent. Math., 144, 253-280, (2001) Group schemes, Algebraic cycles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Finite ground fields in algebraic geometry, Local ground fields in algebraic geometry Construction of central elements in the affine Hecke algebra via nearby cycles | 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 inverse problem of Galois is to find which finite groups can appear as Galois groups of certain spaces. If \(U\) is an affine smooth connected curve over an algebraically closed field \(k\) of positive characteristic \(p\), Abhyankar formulates a conjecture in [\textit{S. Abhyankar}, Am. J. Math. 79, 825--856 (1957; Zbl 0087.03603)] about how the set \(\pi_A^{\text{ét}}(U)\) of groups which appear as quotients of the fundamental étale group \(\pi_1^{\text{ét}}(U)\) of \(U\) looks like. A complete description of the problem for the affine line \(\mathbb{A}_k^1\) was given by \textit{M. Raynaud} [Invent. Math. 116, No. 1--3, 425--462 (1994; Zbl 0798.14013)] stating that \(\pi_A^{\text{ét}}(\mathbb{A}_k^1)\) consists of the set of groupts generated by all their \(p\)-Sylow groups, and by \textit{D. Harbater} [Invent. Math. 117, No. 1, 1--25 (1994; Zbl 0805.14014)] for general \(U\).
In [\textit{S. Otabe}, Compos. Math. 154, No. 8, 1633--1658 (2018; Zbl 1402.14063)], a purely inseparable analogue of Abhyankar's conjecture is set, about describing \(\pi_A^{\mathrm{loc}}(U)\), the set of local \(k\)-group schemes appearing as quotients of \(\pi^N(U)\), Nori's fundamental group scheme, settled in the solvable case in [\textit{S. Otabe}, ``An embedding problem for finite local torsors over twisted curves'', Math. Nachr. 294, No. 7, 1384--1427 (2021; \url{doi:10.1002/mana.201900091})]. This paper shows (c.f. Theorem 1.4, Corollary 5.7) that, when the characteristic is \(p>5\), \(\pi_A^{\mathrm{loc}}(\mathbb{A}_k^1)\) consists on the set of finite local non-abelian simple \(k\)-group schemes.
The strategy of the proof goes over the classification of finite local non-abelian simple group \(k\)-schemes, which turns out to be equivalent to the classification of simple LIe algebras over \(k\) that is completely understood for \(p>3\) and asserts that every simple Lie algebra is either a classical one of types A, B, C, D, E, F, G or of Cartan (Definition 5.1) or Melikian (Section 4.4) type. Abhyankar conjecture; simple Lie algebras; inverse Galois problem; étale fundamental groups; étale covers Coverings of curves, fundamental group, Group schemes, Modular Lie (super)algebras A generalized Abhyankar's conjecture for simple Lie algebras in characteristic \(p>5\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For a review of the Russian original (1988) cf. Zbl 0648.22009.]
The book under review is based on notes of the authors' seminar on algebraic and Lie groups held at Moscow in the late sixties. The central theme is the structure theory of semisimple Lie groups which the authors approach via the theory of algebraic groups. The necessary prerequisites from algebraic geometry are provided in Chapter 2 which is an independent introduction to affine and projective algebraic varieties and also covers some dimension theory. Chapter 3 deals with the basics of algebraic groups such as algebraic tori, Jordan decomposition, Borel groups and the tangent algebra. The last section of this chapter is devoted to compact linear groups and complete reducibility. Chapter 4 is the heart of the book. Here one finds all the standard theory of complex semisimple Lie groups up to the classification via root systems and Dynkin diagrams. Moreover there is a classification of those automorphisms of a complex simple Lie algebra whose spectrum is contained in the unit circle.
The part of the book dealing with real Lie groups has been added later to the seminar notes. Chapter 1 is a general introduction to Lie groups in terms of differentiable manifolds. The results of this chapter are only very occasionally used throughout the text up to Chapter 5 which is devoted to a classification of the real semisimple Lie groups. On the way the authors prove the existence of compact real forms for complex semisimple groups, the Cartan decomposition and the standard conjugacy theorems on maximal tori and maximal compact subgroups. In addition there is a section on restricted root systems and the Iwasawa decomposition. Finally in Chapter 6 Levi's and Malcev's theorems are proven and used to show the existence of a Lie group for any given finite dimensional Lie algebra.
The book ends with a reference chapter with a lot of useful formulae and tables concerning root systems and weight lattices for certain representations.
As one can see from the above description this book covers far more material than one would expect in a book of roughly 290 pages (without the reference chapter). The way to achieve this is to formulate almost every statement as a problem, i.e. as an exercise for the reader. The reviewer is convinced that a student who has worked his way through this book has learnt a lot about the subject, but it takes a very energetic and persevering student to do that.
The book considerably underscores the interplay between Lie groups and Lie algebras choosing on various occasions methods of proof from the theory of algebraic groups rather than Lie theoretic ones. As long as one is primarily concerned with semisimple groups this is certainly just a matter of taste and the authors make quite clear where they see the importance of Lie theory: ``Confirming ourselves to algebraic Lie groups and their algebraic actions we may get rid of various nuisances without substantially impoverishing the Lie group theory'' (p. 104). semisimple Lie groups; algebraic groups; affine and projective algebraic varieties; algebraic tori; Jordan decomposition; Borel groups; compact linear groups; complete reducibility; root systems; Dynkin diagrams; Cartan decomposition; Iwasawa decomposition; finite dimensional Lie algebra; weight lattices A. Onishchik and E. Vinberg \textit{Lie groups and algebraic groups. }Translated from the Russian and with a preface by D. A. Leites. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990.Zbl 0722.22004 MR 1064110 Lie groups, Research exposition (monographs, survey articles) pertaining to topological groups, Linear algebraic groups and related topics, Linear algebraic groups over the reals, the complexes, the quaternions, Simple, semisimple, reductive (super)algebras, Semisimple Lie groups and their representations, Classical groups (algebro-geometric aspects), Lie algebras of linear algebraic groups Lie groups and algebraic groups. Translated from the Russian by D. A. Leites | 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 0672.00003.]
1968 gab Tate eine Definition für Residuen von Differentialen auf Kurven mit Hilfe der Spur linearer Operatoren auf Vektorräumen von Adelen und folgerte den Residuensatz direkt aus der Endlichkeit gewisser Kohomologiegruppen. In der vorliegenden Arbeit wird zunächst gezeigt, wie sich der Zugang von Tate mittels zentraler Erweiterungen unendlichdimensionaler Liealgebren interpretieren läßt. Hauptergebnis ist dann, daß sich bei Betrachtung der zentralen Erweiterung der zugehörigen unendlichdimensionalen Gruppen ein Resultat über die Symmetrie der Weilpaarung auf Kurven aus dem Jahre 1940 ergibt. In beiden Fällen wird die von Peterson und Kac entwickelte unendliche Keil-Darstellung verwendet. infinite wedge representation; residues of differentials on curves; traces of linear operators; vector spaces of adeles; residue; theorem; cohomology groups; infinite-dimensional Lie algebras; infinite- dimensional groups; Weil pairing on curves; infinite; wedge representations; reciprocity law Arbarello, E., de Concini, C., Kac, V.G.: The infinite wedge representation and the reciprocity law for algebraic curves. In: Proceeding of Symposia in pure mathematics, \textbf{49}, 171-190 (1989) Infinite-dimensional Lie groups and their Lie algebras: general properties, Algebraic functions and function fields in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Adèle rings and groups, Infinite-dimensional Lie (super)algebras The infinite wedge representation and the reciprocity law for algebraic 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 topic of the book is the study of the interaction between two major subjects of modern mathematics, namely the theory of Lie groups with its specific methods and ways of thinking on the one hand and complex analysis with all its analytic, algebraic and geometric aspects on the other.'' The book is organized as follows: Chapter 1. Lie theory. The local and global Lie group actions on complex spaces are defined. A local action of a Lie group \(G\) on a complex space \(X\) is shown to be real analytic and to give rise to the Lie homomorphism which is a map from the Lie algebra of \(G\) into the Lie algebra of vector fields on \(X\). In accordance with the second fundamental theorem of Sophus Lie the local action can be recovered from this homomorphism. The proof of this theorem is given and some sufficient conditions for a local action to extend to a global one are established. Chapter 2. Automorphism groups. It is shown that there are two important classes of complex spaces \(X\) for which the automorphism group \(\Aut(X)\) has Lie group structure: the first class consists of all (not necessarily reduced) compact spaces, the second one is the class of all bounded domains in \(\mathbb{C}^n\). Chapter 3. Compact homogeneous manifolds. The geometric properties of two kinds of compact homogeneous complex manifolds, namely flag manifolds and parallelizable ones, are studied. The role of these two classes is explained by the normalizer theorem. This theorem states that if \(H\) is a closed complex subgroup of a connected complex Lie group \(G\) and if \(X = G/H\) is compact, then \(X\) admits a fibration, called the Tits fibration, whose base is a flag manifold and whose fiber is a parallelizable manifold. Chapter 4. Homogeneous vector bundles. The proof and some applications are given of a theorem of R. Bott which determines the induced representations \(I^q \varphi\) in the case when \(H\) is a parabolic subgroup of a semisimple group \(G\) and \(\varphi : H \to GL (V)\) is an irreducible representation. As an application the representations induced by the characters of maximal parabolic subgroups are considered. Chapter 5. Function theory of homogeneous manifolds. Holomorphic functions in \(K\)-invariant domains \(\Omega \subset G/H\) are studied. Here \(K\) is a connected compact group and \(G = K_C\) a reductive linear algebraic group obtained by complexification and \(H \subset G\) is a closed complex Lie subgroup. As starting point a theorem of Harish-Chandra is chosen which extends the classical Fourier expansion to the representation theory of compact Lie groups on Fréchet vector spaces \({\mathcal O} (\Omega)\). As a consequence a description of the class of so-called observable subgroups is obtained by using methods of geometric invariant theory. homogeneous vector bundles; holomorphic functions; Lie groups; complex analysis; Lie group actions; complex spaces; Lie homomorphism; Lie algebra; automorphism group; compact homogeneous complex manifolds; flag manifolds; normalizer theorem; Tits fibration; induced representations; characters; linear algebraic group; complexification; Fourier expansion; Fréchet vector spaces; geometric invariant theory D. Akhiezer, \textit{Lie Group Actions in Complex Analysis}, Aspects in Math., Vol. 27, Vieweg, 1995. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry Lie group actions in complex analysis | 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 Kazhdan-Lusztig polynomials \(P_{x,w}(q)\) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values \(P_{x,w}(1)\) in terms of ``patterns''. A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical
definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group \(G\). Our lower bound comes from applying a decomposition theorem for ``hyperbolic localization'' [\textit{T. Braden}, Transform. Groups 8, No. 3, 209-216 (2003; Zbl 1026.14005)] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; representations of semisimple Lie algebras; Weyl groups; symmetric groups; flag varieties; semisimple groups; torus actions \beginbarticle \bauthor\binitsS. C. \bsnmBilley and \bauthor\binitsT. \bsnmBraden, \batitleLower bounds for Kazhdan-Lusztig polynomials from patterns, \bjtitleTransform. Groups \bvolume8 (\byear2003), no. \bissue4, page 321-\blpage332. \endbarticle \OrigBibText Sara C. Billey and Tom Braden, Lower bounds for Kazhdan-Lusztig polynomials from patterns , Transform. Groups 8 (2003), no. 4, 321-332. \endOrigBibText \bptokstructpyb \endbibitem Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Representation theory for linear algebraic groups Lower bounds for Kazhdan-Lusztig polynomials from patterns. | 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 Every finite-dimensional associative simple algebra \(D\) over a field is isomorphic to a matrix algebra over a division algebra \(B\). The index of \(D\) is the square root of the dimension of \(B\) over its center \(F\). The index reduction formula describes what happens with the index under field extensions of \(F\). The paper deals with the case when the field extension is the function field \(F(X)\), where \(X\) is a homogeneous variety of a reductive algebraic group over \(F\). finite-dimensional simple algebras; index changes under field extensions; function fields; homogeneous varieties; reductive algebraic groups Merkurjev, A.: Degree formula. Available at http://www.mathematik.uni-bielefeld.de/rost/degree-formula.html Finite-dimensional division rings, Linear algebraic groups over arbitrary fields, Homogeneous spaces and generalizations, Arithmetic theory of algebraic function fields Index reduction formula | 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 Reductive groups over finite fields are classified by root systems in a lattice with an action of the Frobenius; a way to get these data from the connected reductive algebraic group \(G\) is to take the projective variety \({\mathcal B}\) of Borel subgroups of \(G\), the set of isomorphism classes of G- equivariant line bundles over \({\mathcal B}\) forms a lattice \(X\); the subgroups containing a given \(B\in {\mathcal B}\) are not conjugate under \(G\), and the orbits of the minimal ones by conjugation under \(G\) form a finite set, each one giving \({\mathcal B}\) as the total space of a \({\mathbb{P}}_ 1\)-fibration over it, hence by taking the tangent bundle along the projections we get elements of \(X\): this is the basis of the root system. Now, each orbit of \(G\) in \({\mathcal B}\times {\mathcal B}\) defines an automorhism of X using the two projections on \({\mathcal B}\), and these orbits form the Weyl group \(W\) of \(G\). The Galois group of the algebraic closure of \(F_ q\) is naturally \({\bar {\mathbb{Z}}}\), generated topologically by the Frobenius F, and the set \(H^ 1({\bar {\mathbb{Z}}},W)\) classifies the maximal tori defined over \(F_ q\) in \(G\).
For each \(w\in W\), let \({\mathcal B}_ w\) be the set of Borel subgroups \(B\in {\mathcal B}\) for which \({}^ FB\) is in position w with respect to B; with a maximal tori \(T\subset B\) defined over \(F\), \textit{P. Deligne} and \textit{G. Lusztig} constructed a variety \({\mathcal B}^ T_ w\) projecting over \({\mathcal B}_ w\) with fibers \(T(F_ q)\) and compatible action of \(G(F_ q)\) so the alternate sum of the \(\ell\)-adic cohomology groups give a virtual representation of \(G(F_ q)\) commuting with the action of \(T(F_ q)\): this leads to the representations \(R^ T_{\theta}\) for the characters \(\theta\) of \(T(F_ q)\) in \({\bar {\mathbb{Q}}}^ x_{\ell}\) [Ann. Math., II. Ser. 103, 103-171 (1976; Zbl 0336.20029)]. Up to equivalence, all the irreducible representations of \(G(F_ q)\) occur in these \(R^ T_{\theta}\), when T and \(\theta\) vary. What was not given in this fundamental article, is an explicit formula for the multiplicities of the irreducible components in the \(R^ T_{\theta}\)'s. The book answers this question, completely in the case \(G\) has a connected center (since, Lusztig obtained the general case).
One of the main tools is the étale intersection cohomology of \textit{P. Deligne, A. A. Beilinson} and \textit{J. Bernstein} [Astérisque 100 (1982; Zbl 0536.14011)], applied to the closures of the varieties \({\mathcal B}_ w\), the Schubert cells. Another one is a deep understanding of the Weyl groups and their Hecke algebras; some properties on them are obtained through the theory of primitive ideals of enveloping algebras of complex reductive Lie algebras; the book uses systematically the results obtained by its author and by D. Kazhdan and its author in the theory of Weyl and Coxeter groups. He shows how the classification of irreducible representations reduces to the classification of unipotent representations of the ``endoscopic'' groups, where the solution comes from the Hecke algebra of the corresponding Weyl group. Also, the author, using the Springer correspondence [\textit{T. A. Springer}, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)], gives a parametrisation of the irreducible representations in terms of the special conjugacy classes of the dual group of \(G\). root systems; connected reductive algebraic group; projective variety; Borel subgroups; line bundles; orbits; tangent bundle; Weyl group; maximal tori; \(\ell\)-adic cohomology groups; virtual representation; characters; irreducible representations; multiplicities; irreducible components; intersection cohomology; Schubert cells; Weyl groups; Hecke algebras; enveloping algebras; complex reductive Lie algebras; unipotent representations G. Lusztig. Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton University Press, 1984. ''BN13N22'' -- 2018/1/30 -- 14:57 -- page 225 -- #27 2018] QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS 225 Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Research exposition (monographs, survey articles) pertaining to group theory, Cohomology theory for linear algebraic groups, Universal enveloping (super)algebras, Étale and other Grothendieck topologies and (co)homologies, Group actions on varieties or schemes (quotients) Characters of reductive groups over a finite field | 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 \(X\) and \(Y\) be smooth projective varieties over \(\mathbb{C}\) of dimension \(n \geq 2\) and \(p : Y \to X\) be a finite abelian cover, i.e. a proper finite map with a faithful action of a finite abelian group \(G\). The aim of this article is to compare the topological fundamental groups \(\pi_1 (X)\) and \(\pi_1 (Y)\). We can define a geometrical datum which characterizes the cover \(p : Y \to X\):
To any irreducible component \(D_j\) of the branch locus \(D\) of \(p\) we can associate a cyclic subgroup \(G_j\) of \(G\), the inertia subgroup, and a faithful representation \(\psi_j\) of \(G_j\);
the action of \(G\) on \(p_*({\mathcal O}_Y)\) induces a splitting: \(p_* ({\mathcal O}_Y) = \bigoplus_{\chi\in G^*} L^{- 1}_\chi\), where \(G^*\) is the group of characters of \(G\) and where \(L_\chi^{- 1}\) is a line bundle on \(X\) such that \(G\) acts on \(L_\chi^{- 1}\) via the character \(\chi\).
We say that the \(G\)-cover \(p : Y \to X\) is totally ramified if the inertia subgroups of all the components of \(D\) generate \(G\). We consider the totally ramified abelian cover \(p : Y \to X\), such that the components \(D_1, \ldots, D_k\) of the branch locus \(D\) are ample and flexible. Then: The natural map \(p_* : \pi_1 (Y) \to \pi_1 (X)\) is surjective, the kernel \(K\) is a finite abelian group and \(0 \to K \to \pi_1 (Y) @>p_*>> \pi_1 (X) \to 1\) is a central extension.
Let \(\pi : \widetilde X \to X\) be the universal covering of \(X\) and \(\widetilde D = \pi^{- 1} (D)\), then \(\widetilde D_j = \pi^{- 1} (D_j)\) is connected for every \(j\); then \(K \simeq \ker (\bigoplus G_j \to G)/ \text{Im} (\sigma \circ \rho) \subset (\bigoplus G_j)/ \text{Im} (\sigma \circ \rho)\), where \(\rho\) is the restriction map \(H_c^{2n - 2} (\widetilde X) \to H_c^{2n - 2} (\widetilde D)\) and \(\sigma\) is the map \(H_c^{2n - 2} (\widetilde D) \simeq \bigoplus \mathbb{Z} \widetilde D_j \to \bigoplus G_j\). The cohomology class of the central extension can be computed in terms of the Chern classes of the \(D_j\)'s and the \(L_\chi\)'s.
Let \(c(p) \in H^2 (\pi_1 (X), K) \subset H^2 (X,K)\) be the cohomology class of the extension and \(i_* (c(p)) \in H^2 (X, \widetilde G)\) its image induced by the inclusion \(K \subset \widetilde G = (\bigoplus G_j)/ \text{Im} (\sigma \circ \rho)\), then we have: \(i_* (c(p)) = \Phi_* ([D_1], \ldots, [D_k])\), where \(\Phi\) is the map \(\mathbb{Z}^k \to \widetilde G\).
The authors give some applications of this result. They give a class of examples of nonhomeomorphic covers with the same branch divisors \(D_j\), inertia subgroups \(G_j\) and Galois groups \(G\). In fact the cohomology class \(c(p)\) of the extension of the fundamental groups depends on the choice of the solution \(\{L_\chi\}\) of the characteristic relations of the cover \(p\). Covers corresponding to different solutions may give different extensions and varieties \(Y\) with different fundamental group. finite abelian cover; cohomology class; extension of the fundamental groups Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Projective techniques in algebraic geometry A note on the topology of a totally ramified abelian cover | 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 the real-quadratic number field with discriminant \(D\) and \(\mathfrak o\) its ring of integers. For an ideal \(\mathfrak b\) of \(\mathfrak o\) with Norm \(B\) let \(G(\mathfrak o,\mathfrak b)\) be the Hilbert modular group
\[
G(\mathfrak o,\mathfrak b)= \mathrm{PSL}_2(\mathfrak o,\mathfrak b)=\Bigl\{\left( \begin{matrix} a &b \\ c &d\end{matrix} \right) \mid a,d\in \mathfrak o,\;c\in\mathfrak b,\;b\in\mathfrak b^{-1}\Bigr\}/\{\pm \text{id}\}.
\]
The group \(G(\mathfrak o,\mathfrak b)\) acts on \(\mathbb H^2\) by the modular substitution and admits a unique maximal discrete extension in \((\mathrm{PSL}^+_2(\mathbb R))^2\), the Hurwitz-Maaß extension \(G_m(\mathfrak o,\mathfrak b)\). The map \(\tau: (z_1,z_2)\mapsto (-1/Bz_2,-1/Bz_1)\) induces a holomorphic involution on \(\mathbb H^2/G(\mathfrak o,\mathfrak b)\) and commutes with the elements of \(G_m(\mathfrak o,\mathfrak b)/G(\mathfrak o,\mathfrak b)\). Define \(G_*(\mathfrak o,\mathfrak b):= G_m(\mathfrak o,\mathfrak b)\cup \tau \cdot G_m(\mathfrak o,\mathfrak b)\) and denote by \(Y_*(D,\mathfrak b)\) the minimal nonsingular compactification of the quotient \(\mathbb H^2/G_*(\mathfrak o,\mathfrak b)\).
The author classifies the smooth projective surfaces \(Y_*(D,\mathfrak b)\) for \(D\equiv 1\pmod 4\) and arbitrary \(\mathfrak b\). By estimation of \(K^2_*\), the self-intersection of the canonical divisor \(K_*\) of \(Y_*(D,\mathfrak b)\), and \(\chi_*\), the arithmetic genus of \(Y_*(D,\mathfrak b)\), he shows, that all but a finite number of the surfaces \(Y_*(D,\mathfrak b)\) are of general type. In the remaining cases he constructs characteristic configurations of curves on \(Y_*(D,\mathfrak b)\) to show, that these surfaces are rational, of \(K3\)-type or honestly elliptic. With these investigations he continues the classification of Hilbert modular surfaces initiated by \textit{F. Hirzebruch} in [Enseign. Math., II. Sér. 19, 183--281 (1973; Zbl 0285.14007)]. modular curves; classification of Hilbert modular surfaces Bassendowski, D.: Klassifikation Hilbertscher Modulflächen zur symmetrischen Hurwitz-Maass-Erweiterung. Dissertation, Bonn, 1984 Special surfaces, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces Classification of Hilbert modular surfaces corresponding to the symmetric Hurwitz-Maaß-extension | 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 \(f\in {\mathbb{Z}}[X_ 1,X_ 2,X_ 3]\) be a primitive quadratic form and \(M(f)=\Pr oj({\mathbb{Z}}[X_ 1,X_ 2,X_ 3]/(f)).\) The paper contains proofs of two results about resolution of singularities of the conic bundle surfaces M(f) over Spec \({\mathbb{Z}}\). The first says that the normalization of M(f) is M(B(f)), where B(f) is a suitable quadratic form, whose equivalence class is uniquely determined by f. The second says that if M(f) is normal, then there exists a sequence of conic bundle surfaces \(M_ 0=M(f),M_ 1=M(f_ 1),...,M_ n=M(f_ n)\) such that \(M_{i+1}\) is an elementary transform of \(M_ i\) at the singular points (on \(M_ i)\) in one of the fibers of \(M_ i\) for \(i=0,1,...,n-1\) and \(M_ n\) is regular. These results are presented in more general context of conic bundle surfaces corresponding to lattices on quadratic spaces over Dedekind schemes. The proofs depend on the relations between lattices, conic bundle surfaces and orders in quaternion algebras developed in the first part of the paper [Math. Scand. 46, 183-208 (1980; Zbl 0505.14006)] and extended in the present part: M(f) is normal iff the corresponding order O(f) is Bass and O(B(f)) is the Bass closure of the Gorenstein order O(f). The chain of the \(M_ i's\) corresponds to a minimal chain of Bass orders \(O(f)\subset O(f_ 1)\subset...\subset O(f_ n)\) ending with a hereditary order. quadratic space; conic bundle surface; resolution of singularities; orders in quaternion algebras Singularities of surfaces or higher-dimensional varieties, General binary quadratic forms, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Polynomial rings and ideals; rings of integer-valued polynomials, Global theory and resolution of singularities (algebro-geometric aspects) Arithmetical quadratic surfaces of genus 0. 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 The book has three parts. The first is an elementary introduction to theory of representations of quivers. In this part, many aspects of representation theory of Dynkin quivers are described: Euler forms, root lattices, projective and injective modules, geometry of orbits of certain group actions on the representation space, Gabriel theorem, Hall algebras and preprojective algebras.
In the second part, representation theory of infinite type quivers is discussed. In particular, the author describes preprojective and preinjective representations, Auslander-Reiten quivers, tame and wild quivers, tame-wild dichotomy, Kac's theorem.
The third part is an excursion in quiver varieties theory. This part is more advanced. It contains for example: Hamiltonian reduction and geometric invariant theory, Hilbert schemes, Kleinian singularities, geometric McKay correspondence, geometric realization of Kac-Moody algebras.
The book can be a concise guide to representation theory of quiver representations for beginner and advanced researchers. quiver representations; finite representation type; infinite representation type; quiver varieties; Hall algebras Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Parametrization (Chow and Hilbert schemes) Quiver representations and quiver 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 For any natural number \(r\), the bounded symmetric domain \(\mathbb{D}^r\) of type \(I_{r,r}\) is defined as the set of all complex matrices \(\tau\in M_{r,r}(\mathbb{C})\) such that \((2i)^{-1}\cdot(\tau- {^t\overline\tau})\) is positive definite. This space is acted on by the unitary group \(U_{rr}(\mathbb{C})\subset\text{GL}_{2r}(\mathbb{C})\) in the usual way, and it is invariant under the operation of transposing matrices. Moreover, if \(d\) is a square-free positive integer, then the ring of integers \(\mathbb{Z}[\delta]\) of the imaginary quadratic number field \(\mathbb{Q}(\sqrt{-d})\) defines the lattice \(\mathbb{Z}[\delta]^r\), together with a canonical integral inner product on it. This allows to construct theta functions with characteristics in \(\mathbb{Q}(\sqrt{-d})\) over the ring \(\mathbb{Z}[\delta]\), which appear as quasi-periodic functions \(\Theta_\delta{a\choose b}: \mathbb{D}^r\to \mathbb{C}\), where \(a\) and \(b\) are vectors in \(\mathbb{Q}(\sqrt{-d})^r\), with respect to the lattice \(\mathbb{Z}[\delta]^r\).
These particular theta functions have been used, in the past, to construct modular forms on the symmetric domain \(\mathbb{D}^r\) in special cases [cf. \textit{E. Freitag}, Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1967, 1. Abh. 1--50 (1967; Zbl 0156.09203)], and the possible algebraic relations among them are of highest geometric interest and significance.
In the paper under review, the author derives new quadratic relations among certain theta functions on \(\mathbb{D}^r\) over the rings \(\mathbb{Z}[i]\) and \(\mathbb{Z}[\sqrt{-2}]\), as well as some new cubic relations among certain theta functions on \(\mathbb{D}^r\) with respect to the ring \(\mathbb{Z}[\omega]\) of Eisenstein integers.
This computational effort leads to interesting identities that were already applied to the author's study of the period map for families of \(K3\) surfaces [cf. \textit{K. Matsumoto}, Math. Ann. 295, No. 3, 383--409 (1993; Zbl 0791.32008)] some years ago, and to his (and his co-authors') more recent work on hyperbolic structures on certain complex algebraic varieties [cf. \textit{K. Matsumoto} and \textit{M. Yoshida}, Int. J. Math. 13, 415--443 (2002; Zbl 1052.12030)]. The new cubic relations among theta functions on \(\mathbb{D}^r\) over the ring of Eisenstein integers are useful in the study of periods of some families of algebraic varieties, on the one hand, and of the hyperbolic geometry of complements of certain knots on the other. theta functions; bounded symmetric domains; imaginary quadratic number fields; rings of integers; lattices; modular forms K. Matsumoto: Algebraic relations among some theta functions on the bounded symmetric domain of type \(I_r,r\) , Kyushu J. Math. 60 (2006), 63--77. Other groups and their modular and automorphic forms (several variables), Theta series; Weil representation; theta correspondences, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Theta functions and abelian varieties, Algebraic numbers; rings of algebraic integers Algebraic relations among some theta functions on the bounded symmetric domain of type \(I_{r,r}\) | 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 eigencurve \({\mathcal C}\) (the Coleman-Mazur eigencurve) is a rigid analytic space parameterizing overconvergent -- and hence classical -- modular eigenforms of finite slope. \textit{R. Coleman} and \textit{B. Mazur} [London Math. Soc. Lecture Note Ser. 254, 1--113 (1998; Zbl 0932.11030)] raised the following question: does there exist a \(p\)-adic family of finite slope overconvergent eigenforms over a punctured disc, and converging, at the puncture, to an overconvergent eigenform of infinite slope? In \textit{K. Buzzard} and \textit{F. Calegari} [Doc. Math., Extra Vol., 211--232 (2006; Zbl 1138.11015)], this was proved in the affirmative for the particular case of tame level \(N=1\) and \(p=2\). In this paper the author proves that \({\mathcal C}\) is proper (over the weight space) at the integral weights in the center of weight space. He works with general \(p\) and arbitrary tame level, although the result only applies at certain arithmetic weights. Hasse invariants; Eisenstein series; Coleman-Mazur eigencurve; overconvergent modular eigenforms of finite slope DOI: 10.2140/ant.2008.2.209 Congruences for modular and \(p\)-adic modular forms, Rigid analytic geometry The Coleman-Mazur eigencurve is proper at integral weights | 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 his paper [Invent. Math. 109, 307-327 (1992; Zbl 0781.14022)] the author has shown that the Selmer group attached to a symmetric square of the Tate module of a modular elliptic curve can be finite under some additional restrictions. In particular, he defined some integer which annihilates the Selmer group. In this note the result is extended to a much wider class of modular forms of weight 2. annihilation of Selmer groups; adjoint representation; modular forms of weight 2 Arithmetic aspects of modular and Shimura varieties, Arithmetic varieties and schemes; Arakelov theory; heights, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Annihilation of Selmer groups for the adjoint representation of a modular form | 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 0691.58002. singularities of maps; critical points of functions; monodromy; discriminants; stability; normal forms; mixed Hodge structure; characteristic classes Arnol'd, V. I.; Vasil'ev, V. A.; Goryunov, V. V.; Lyashko, O. V.: Singularities local and global theory in dynamical systems. Enc. math. Sc. 6 (1991) Research exposition (monographs, survey articles) pertaining to global analysis, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Characteristic classes and numbers in differential topology, Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities of differentiable mappings in differential topology Singularities. Local and global 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 In geometric invariant theory actions of algebraic groups on projective varieties are studied. Mumford's methods for the construction of quotients of such group actions do not in general lead to a unique or canonical quotient, rather to an inverse system of quotients. To remedy this situation the authors introduce the --- canonical --- Chow quotient. This quotient is thoroughly explored for the case that the variety under consideration is toric. Connections with the quotients obtained from geometric invariant theory are discussed. toric variety; Chow variety; geometric invariant theory; actions of algebraic groups; Chow quotient M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, ''Quotients of toric varieties,'' Math. Ann., vol. 290, iss. 4, pp. 643-655, 1991. Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Geometric invariant theory Quotients of toric 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 \(f : X \rightarrow \Delta,\) \(\Delta = \{z\in {\mathbb C}: | z| < 1\},\) be a projective and semi-stable family of analytic spaces. Due to \textit{J. Steenbrink} [Invent. Math. 31, 229--257 (1976; Zbl 0303.14002)] a limit Hodge structure is defined for such a family as the limit of Hodge structures \(H^m(X_t, {\mathbb Z}),\) where \(m \in {\mathbb Z}\), \(t\in \Delta\setminus\{0\}.\) In his previous work [\textit{T. Matsubara}, Kodai Math. J. 21, 81--101 (1998; Zbl 1017.32017)], the second author gave an interpretation of the theory of Steenbrink in the framework of the log Hodge theory in the following manner: ``The higher direct images of \({\mathbb Z}_X\) on \(\Delta\) carry the natural variations of polarized log Hodge structure''.
The paper under review is devoted to a generalization of the theory of Steenbrink to log Hodge theory with coefficients. More precisely, general variations of polarized log Hodge structure \({\mathcal H}_{{\mathbb Z}}\) on \(X\) instead of \({\mathbb Z}_X\) are considered. In order to investigate this case the authors introduce and analyze log versions of basic notions of the theory including \(C^\infty\)-functions, degenerations of Hodge decompositions; \(\bar{\partial}\)-Poincaré lemma, Kähler metrics, harmonic forms, etc. [see also \textit{M. Harris} and \textit{D. H. Phong}, C. R. Acad. Sci., Paris, Sér. I 302, 307--310 (1986; Zbl 0597.32025)].
The authors underline that their main theorem gives new proofs of earlier results [\textit{T. Fujisawa}, Compos. Math. 115, No. 2, 129--183 (1999; Zbl 0940.14007); \textit{L. Illusie}, Duke Math. J. 60, No. 1, 139--185 (1990; Zbl 0708.14014); \textit{M. Cailotto}, C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 12, 1089--1094 (2001; Zbl 1074.14513)]. In addition, it is proved that the log Riemann-Hilbert correspondence [\textit{K. Kato} and \textit{C. Nakayama}, Kodai Math. J. 22, No. 2, 161--186 (1999; Zbl 0957.14015)] is, in fact, functorial relative to Hodge filtrations. variation of Hodge structures; limit of Hodge structures; nilpotent orbit; log geometry; log Riemann-Hilbert correspondence Kazuya Kato, Toshiharu Matsubara, and Chikara Nakayama, Log \?^{\infty }-functions and degenerations of Hodge structures, Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 269 -- 320. Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations Log \(C^\infty\)-functions and degenerations of Hodge structures. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In their fundamental 1960 paper on Brauer groups [Trans. Am. Math. Soc. 97, 367-409 (1961; Zbl 0100.26304)], \textit{M. Auslander} and \textit{O. Goldman} prove that over a regular domain \(R\) of Krull dimension at most two, the Brauer group satisfies \(B(R)=\bigcap B(R_{\mathfrak p})\), where \(\mathfrak p\) ranges over the prime ideals of height one. Having shown that a maximal order \(\Delta\) is reflexive, they need the condition \(\dim R\leq 2\) just to ensure that \(\Delta\) is projective. The question remained open whether the theorem remains true in higher dimensions.
In the present paper, the authors confirm the necessity of \(\dim R\leq 2\) by viewing the existence problem of Azumaya algebras in the broader context where \(\mathrm{Spec\,}R\) is replaced by a regular Noetherian integral scheme. As a result, they exhibit a 6-dimensional smooth complex affine variety \(X\) and a Brauer class in \(\mathrm{Br}(X)\) such that the skew-field over the generic point of \(X\) has no Azumaya maximal order over \(X\). Brauer groups; maximal orders; Azumaya algebras; regular Noetherian integral schemes; smooth complex affine varieties Antieau, B.; Williams, B.: On the non-existence of Azumaya maximal orders. Invent. math. 197, No. 1, 47-56 (2014) Brauer groups (algebraic aspects), Brauer groups of schemes, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Orders in separable algebras Unramified division algebras do not always contain Azumaya maximal orders. | 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 class of non-isolated hyperplane singularities which have the transversal type \(A_ k\) is introduced. Special deformations are constructed which preserve the topology of the Milnor fibre, and their analytic properties are investigated. This enables one to compute the number of nondegenerate singular points and to describe the homotopy type of the Milnor fibre. non-isolated hyperplane singularities; topology of the Milnor fibre; homotopy type of the Milnor fibre Topology of vector bundles and fiber bundles, Algebraic topology on manifolds and differential topology, Deformations of singularities, Singularities of differentiable mappings in differential topology, Singularities in algebraic geometry, Local complex singularities On hyperplanar singularities of the transversal type \(A_ k\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a method for computing the Artin-Mumford group of the field of rational functions on a complex algebraic variety which are invariant relative to the action of an affine algebraic group. The Artin-Mumford group, which we shall denote by \(Br_ v(X/G)\), is defined as \(H^ 2(\widetilde{X/G},\mathbb{Q}/\mathbb{Z})/\widehat{\text{Pic}}\tilde X/G\), where \(\widetilde{X/G}\) is a smooth projective model of the field \(\mathbb{C}(X)^ G\). In this paper we show that it is not necessary to construct a projective model in order to compute \(Br_ v(X/G)\). It is enough to be able to control the action on \(X\) of the finite rank-2 abelian subgroups of \(G\). This result was proved by the author in Math. USSR, Izv. 30, No. 3, 455-485 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Math. 51, No. 3, 485-516 (1987; Zbl 0679.14025) and by \textit{D. J. Saltman} in Invent. Math. 77, 71-84 (1984; Zbl 0546.14014) in the special case when \(X\) is a linear space over \(\mathbb{C}\) and the group \(G\) acts linearly and almost freely on \(X\). Here we shall remove these restrictions on \(X\) and \(G\).
The general result in \(\S1\) (theorem 1.3) gives a complete answer to the question of the structure of \(Br_ v\) in many cases, and it makes the computation of the group routine in a broad class of specific examples. In particular, if \(G\) is connected and simply connected, then we are in a position to compute \(Br_ v(H\backslash G/K)\), where \(H\) and \(K\) are subgroups of \(G\), with rather weak assumptions on \(H\) and \(K\). Another aspects of our basic result is a description of \(Br_ v\mathbb{C}(X)\) for any algebraic variety in terms of the Galois group \(\text{Gal}(X)\) of the algebraic closure of the field of rational functions on \(X\). Namely, \(Br_ v\mathbb{C}(X)\approx Br_ v(\text{Gal}(X))\), where \(Br_ v(G)\) for a finite group \(G\) is defined to be \(Br_ v(V/G)\) with \(V\) a faithful linear representation of \(G\), and \(Br_ v(\text{Gal}(X))\) for the profinite group \(\text{Gal}(X)\) is the inductive limit of \(Br_ v(G)\) over all finite quotient groups of \(\text{Gal}(X)\). Brauer groups of fields of invariants; Galois cohomology; Artin-Mumford group of the field of rational functions Bogomolov F.A., Brauer groups of fields of invariants of algebraic groups, Math. USSR-Sb., 1990, 66(1), 285--299 Geometric invariant theory, Galois cohomology, Brauer groups of schemes, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory Brauer groups of fields of invariants of algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for \(n\neq 6\) and all 26 sporadic simple groups. We prove that, if \(K\) is a perfect field and \(X\) is a homogeneous space of a smooth algebraic \(K\)-group \(G\) with finite geometric stabilizers lying in this family, then \(X\) is dominated by a \(G\)-torsor. In particular, if \(G=\mathrm{SL}_n\), all such homogeneous spaces have rational points. homogeneous spaces; rational points; non-abelian cohomology; finite simple groups Homogeneous spaces and generalizations, Galois cohomology, Rational points On homogeneous spaces with finite anti-solvable stabilizers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book under review is the companion volume of the author's and \textit{R. Busam's} well-known, meanwhile utmost popular and widespread introductory text ``Complex Analysis'' [Universitext. Berlin: Springer (2009; Zbl 1167.30001)]. The German original of the first volume entitled ``Funktionentheorie 1'' [Springer-Lehrbuch. Berlin: Springer-Verlag (1993; Zbl 0783.30001)]. Providing a fairly detailed and comprehensive introduction to the classical main topics of the theory of functions of one complex variable, including elliptic functions and elliptic modular forms, the text of the first volume has successively been revisited and enlarged over the past fifteen years, culminating so far in its fourth German and its second English edition, respectively.
Also, in the preface to the first volume, the authors incidentally indicated that a subsequent companion volume was planned, and that promised second part of their popular primer of complex analysis has now been brought about by the first author. Based on courses taught by him at the University of Heidelberg, Germany, the notes of which could be found on his web site for some time, and building upon the foundations laid in the first volume, the current text turns to a number of more advanced topics in classical complex analysis. The main themes are indicated in the subtitle of book: Riemann surfaces, several complex variables, abelian functions, and modular forms of several variables. As the author points out, these topics have been chosen deliberately, since they actually represent consistent further developments of the basic fundamentals covered in the first volume of the entire treatise.
As for the contents, the present second volume contains eight chapters, each of which is subdivided into several sections.
The first four chapters are devoted to one of the main parts of the text, namely to the theory of Riemann surfaces.
Chapter I treats the elementary theory of Riemann surfaces, including the basic prerequisites from general topology, the concept of Riemann surface, and first important examples of Riemann surfaces.
Chapter II is titled ``Harmonic Functions on Riemann Surfaces'' and discusses a series of constructive methods in the study of Riemann surfaces. In the course of this chapter, the author describes several approaches to the construction of harmonic functions on Riemann surfaces with prescribed properties, with particular emphasis laid on boundary value problems, singularities, and stability problems.
Chapter III provides an introduction to the uniformization theory of Riemann surfaces in both its topological and its analytical aspects. The highlight of this chapter is a proof of the Uniformization Theorem due to P. Koebe and H. Poincaré, accompanied by proofs of the famous Picard Theorems as illustrating applications.
Chapter IV, the most extensive part of the discussion of Riemann surfaces in the present book, is exclusively devoted to compact Riemann surfaces. The author thoroughly explains meromorphic differentials, the interrelation between compact Riemann surfaces and algebraic functions, triangulations of compact Rienann surfaces, normal forms, coverings and the Hurwitz Ramification Formula, the Riemann period relations, the Riemann-Roch Theorem, Abel's Theorem, Jacobi's Inversion Problem, and multicanonical modular forms on compact Riemann surfaces in full detail. The treatment of Jacobi's Inversion Theorem leads the reader in a very natural way to the problem of creating a theory of functions of several complex variables, the basics of which are then developed in the subsequent Chapter V. The material covered in this comparatively brief chapter includes the elementary properties of holomorphic functions of several variables, the allied theory of complex power series in several variables, the concept of analytic maps, the Weierstrass Preparation Theorem, the representation of meronorphic functions as quotients of analytic functions, and an introduction to alternating differential forms. The leading theme, namely the inversion of algebraic integrals in general, is then taken up again in Chapter VI. Building upon the framework of Riemann surfaces, and on the necessary background material from higher-dimensional complex function theory, as having been made available so far, this chapter provides an introduction to the classical theory of abelian functions. Accordingly, the reader becomes here acquainted with lattices and tori, the Hodge theory of the latter, factors of automorphy, Riemann forms on lattices, the higher-dimensional period relations, and with general theta series as generalizations of the Jacobi theta function in the elliptic case. The discussion culminates in the description of the field of abelian functions in arbitrary genus, together with an outlook to polarized abelian varieties and general complex manifolds.
Chapter VII consistently turns then to the allied theory of modular forms of several variables, thereby generalizing the theory of elliptic modular forms as developed in the foregoing first volume of the treatise. The aim of this concluding chapter is to deliver a rather elementary and self-contained introduction to the subject, with special emphasis on the next simple case of dimension two. Siegel modular groups, modular forms of degree \(n\), the Koecher principle, modular congruence subgroups and their generators, theta functions associated to modular groups, and fundamental domains of modular groups are the main objects of study in Chapter VII. The main goal and highlight of this part of the book is to derive an elementary proof of J. Igusa's structure theorem describing the ring of modular forms for the modular group \(\Gamma_2[4,8]\) by means of the ten classical ``theta nullwerte'' of degree 2. The analogous statement in the elliptic case had been proved by quite similar methods in Volume 1 which shows once more to what great extent the two volumes together actually form a unit, both epistomologically and didactically.
Chapter VIII is an appendix gathering some purely algebraic tools from the theory of rings and fields, including factorial rings, discriminants, and algebraic function fields, and that as far as those are utilized in the course of the text.
Each single section comes with a number of related exercises complementing the respective material, where many of them are equipped with directing hints and remarks. However, solutions to the exercises (like in Volume 1) are not yet provided in the current first edition of this companion volume.
All together, this long anticipated second volume of the author's introduction to various topics in complex analysis stands out by all the features that already characterized the first part of the overall treatise. Those have been extensively summarized and appraised in our review of the second English edition of the first volume (Zbl 1167.30001), and we may refer to this very judgement in every regard and detail, as the present second volume breathes exactly the same individual spirit, the same refined cultural viewpoint, and the same didactic mastery as its predecessor. Together with the latter, the current book is largely self-contained, pleasantly down-to-earth, remarkably versatile, and utmost educating simultaneously.
No doubt, this textbook provides an excellent source for the further study of more advanced topics in the theory of Riemann surfaces, their Jacobians and moduli aspects, and in the general theory of complex abelian varieties and modular forms likewise.
Hopefully, an English edition of this second volume of a masterful primer of complex analysis will be made available in the near future as well. textbook (functions of complex variables); Riemann surfaces; harmonic functions; uniformization; functions of several complex variables; abelian functions; modular forms Freitag B., Complex Analysis (2009) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable, Riemann surfaces, Holomorphic functions of several complex variables, Analytic theory of abelian varieties; abelian integrals and differentials, Modular and automorphic functions, Holomorphic modular forms of integral weight, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Theta series; Weil representation; theta correspondences Function theory 2. Riemann surfaces, several complex variables, abelian functions, higher 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 \(X\subset \mathbb{P}^n\) be a general complex hypersurface of degree \(d\). It is known that if \(d\) is sufficiently large, \(d\geq 2n - 2\), then \(X\) does not contain rational curves [see \textit{H. Clemens}, Ann. Sci. Éc. Norm. Supér., IV Sér. 19, No.~4, 629--636 (1986; Zbl 0611.14024); \textit{C. Voisin}, J. Differ. Geom. 44, 200--213 (1996; Zbl 0883.14022) and 49, No. 3, 601--611 (1998; Zbl 0994.14026)]. For \(d = 2n - 3\), \(n\geq 6\), \(X\) contains a finite number of lines but does not contain rational curves of degree \(e \geq 2\) [\textit{G. Pacienza}, J. Algebr. Geom. 12, 245--267 (2003; Zbl 1054.14057)]. In this paper the authors prove the following relevant result for low degree: For \(n\geq 2\) and \(d < (n + 1)/2\), a general complex hypersurface \(X\) in \(\mathbb P^n\) of degree \(d\) has the property that for each integer \(e\geq 1\) the scheme \(R_e(X)\) parametrizing degree \(e\) smooth rational curves on \(X\) is an integral local complete intersection scheme of ``expected'' dimension \((n+1-d)e +(n-4)\). The scheme \(R_e(X)\) is embedded as an open subscheme in the Kontsevich moduli space \(\overline{\mathcal M}_{0,0}(X,e)\) parametrizing stable maps to \(X\) and a partition of \(\overline{\mathcal M}_{0,0}(X,e)\) into locally closed subsets is used as in \textit{K. Behrend} and \textit{Yu. Manin} [Duke Math. J. 85, 1--60 (1996; Zbl 0872.14019)]. The authors use classical results about lines on hypersurfaces and include a new result about flatness of the projection map from the space of pointed lines. Moreover they use the deformation theory of stable maps, properness of the stack \(\overline{\mathcal M}_{0,r}(X,e)\) and a version of Mori's bend-and-break lemma. The authors use dual graphs associated to pointed curves and stable \(A\)-graphs as in the above article of K. Behrend and Yu. Manin (loc. cit.). dual graphs; Hilbert schemes; Kontsevich moduli spaces of stable maps; stacks Harris, J; Roth, M; Starr, J, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math., 571, 73-106, (2004) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Hypersurfaces and algebraic geometry, Parametrization (Chow and Hilbert schemes) Rational curves on hypersurfaces of low degree | 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 paper we construct the Shimura canonical model for the quaternion algebra \(B\) of discriminant 6 over \(Q\) in the sense of the original paper [\textit{G. Shimura}, Ann. Math. (2) 85, 58--159 (1967; Zbl 0204.07201)] (see also [the author, in: Journées arithmétiques. Exposés présentés aux dix-septièmes congrès à Genève, Suisse, 9-13 septembre 1991. Paris: Société Mathématique de France. 293--305 (1992; Zbl 0862.11046)]). The ``unit group'' of \(B\) becomes to be a quadrangle group \({\Gamma} = \square(0; 2, 2, 3, 3)\) acting on the complex upper half space \(H\). For our purpose we use the defining equation of the Shimura curve \(V = \textbf{H} / \square(0; 2, 2, 3, 3)\) given by Y. Ihara which found in the work of \textit{A. Kurihara} [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 277--300 (1978; Zbl 0428.14012)]. We are requested to find a modular function \({\psi}\) defined on \(\textbf{H}\) with respect to \({\Gamma}\) which realizes \(V\) satisfying several arithmetic conditions stated later. Basically our \({\psi}\) is the modular function for the family of abelian surfaces with quaternion multiplication by \(B\), so called false elliptic curves, with some arithmetic conditions. Our main tool is the hypergeometric modular function given by \textit{M. Petkova} and the author [Arch. Math. 96, No. 4, 335--348 (2011; Zbl 1219.11092)]. This is the full paper of the survey report [the author, loc. cit.] Shimura curves; QM type abelian surfaces; Picard modular forms; hypergeometric functions; false elliptic curves; complex multiplication Complex multiplication and moduli of abelian varieties, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Classical hypergeometric functions, \({}_2F_1\) The Shimura canonical model for the quaternion algebra of discriminant 6 on the Ihara-Kurihara conic | 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 known that the supermultiplet of beta-deformations of \(\mathcal{N} = 4\) supersymmetric Yang-Mills theory can be described in terms of the exterior product of two adjoint representations of the superconformal algebra. We present a supergeometrical interpretation of this fact, by evaluating the deforming operator on some special coherent states in the space of supersingletons. We also discuss generalization of this approach to other finite-dimensional deformations of the \(\mathcal{N} = 4\) supersymmetric Yang-Mills theory. supergeometry; AdS/CFT correspondence; representations of Lie superalgebras; twistor theory; scattering amplitudes Spinor and twistor methods applied to problems in quantum theory, Supersymmetry and quantum mechanics, Yang-Mills and other gauge theories in quantum field theory, Supermanifolds and graded manifolds, Graded Lie (super)algebras, Applications of Lie (super)algebras to physics, etc., Coherent states, Formal methods and deformations in algebraic geometry A geometrical point of view on linearized beta-deformations | 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 issue is an English translation of [\textit{F. Mangolte}, Variétés algébriques réelles. Cours Spécialisés (Paris) 24. Paris: Société Mathématique de France (SMF) (2017; Zbl 1375.14003)]. The author of this book is an active mathematician who fruitfully works in real algebraic geometry. The monograph is interesting for experts in this subject because it exposes major developments of the last twenty five years in the field. On the other hand, the monograph gives a good introduction to geometry and topology of real algebraic varieties for those who begin to study these objects. From this point of view it is important that in the first two chapters the author builds bridges between the languages of description of real algebraic varieties used in general sources on this subject: the language of germs of real algebraic varieties in [\textit{J. Bochnak et al.}, Real algebraic geometry. Transl. from the French. Rev. and updated ed. Berlin: Springer (1998; Zbl 0912.14023)], the language of schemes in [\textit{R. Silhol}, Real algebraic surfaces. Lecture Notes in Mathematics, 1392. Berlin etc.: Springer (1989; Zbl 0691.14010)] and the language of complex algebraic varieties equipped with an antiholomorphic involution (see [\textit{A. Degtyarev} and \textit{V. Kharlamov}, Russ. Math. Surv. 55, No. 4, 735--814 (2000); Transl. from Usp. Mat. Nauk 55, No. 4, 129--212 (2000; Zbl 1014.14030)] and [\textit{A. Degtyarev et al.}, Real Enriques surfaces. Lecture Notes in Mathematics, 1746. Berlin: Springer (2000; Zbl 0963.14033)]).
The topology of real algebraic varieties is developing in two directions: prohibitions and constructions. The author only slightly concerns the latter and concentrates on the prohibition type results. The third chapter presents general tools for the study of the topology of real algebraic varieties and observes the results connected with the famous Hilbert's 16th problem. The fourth chapter describes in detail the classification of real algebraic surfaces including the case of their minimal real models, the fifth one studies nonsingular real algebraic varieties as smooth manifolds and is devoted, in particular, to approximations of smooth manifolds by real algebraic varieties. The sixth chapter considers real algebraic threefolds from the point of view of Nash conjecture on presentation of diffeomorphism classes of closed manifolds by real rational models. Six appendices to the main chapters supply the reader with necessary material from Nullstellensatz to Riemann-Roch and more. Many proofs are replaced with references to other sources that helps to reduce the volume of the book. The text is supplied with various level exercises, and the solutions of many of them are given at the end of each chapter.
Reviewer's remark: Unfortunately in References, there are 13 misprints in the names of some authors: J. Kollár instead of V.A. Krasnov in [Kra06], [Kra09], and instead of W. Kucharz in [Kuc99]--[Kuc16b]; J.-P. Serre instead of G. Shimura in [Shi72a]; T. Shioda instead of R. Silhol in [Sil84]--[Sil92]. real algebraic geometry; real algebraic varieties; complexification; Smith's theory; Galois-Maximal varieties; algebraic cycles; real algebraic models; algebraic curves; algebraic surfaces; topology of algebraic varieties; regular maps; rational maps; singularities; algebraic approximation; Comessatti theorem; Rokhlin theorem; Nash conjecture; Hilbert's XVI problem; Cremona group; real fake planes Topology of real algebraic varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Real algebraic and real-analytic geometry, Surfaces and higher-dimensional varieties, Curves in algebraic geometry, Varieties and morphisms, Special varieties Real 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 algebraic groups; anisotropic groups; projective spaces; simple connected groups; Zariski topology; maximal torus; root system; group schemes B. Weisfeiler, ''On abstract homomorphisms of anisotropic algebraic groups over real-closed fields,'' J. Algebra,60, No. 2, 485--519 (1979). Linear algebraic groups over arbitrary fields, Other algebraic groups (geometric aspects), Group schemes On abstract homomorphisms of anisotropic algebraic groups over real-closed 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 A complete intersection curve \(C\) contained in a general hypersurface \(X\) of degree \(d>n\geq 1\) in \(\mathbb{P}^{n+1}\) is called nontrivial, if the equation of \(X\) is not part of a regular sequence defining \(C\), i.e. if \(C\) is not cut out by \(n-1\) hypersurfaces on \(X\). The main result of this paper is a classification of the possible types (i.e. the degrees of the equations defining \(C\)) of nontrivial complete intersection curves contained in a general hypersurface \(X\) of degree \(d\geq 6\). Furthermore, the author shows that
(1) there are no nontrivial complete intersection surfaces on \(X\),
(2) for \(d>2n-1\), there are no nontrivial complete intersection curves on \(X\),
(3) for \(d>\frac32n+ \frac12\), all nontrivial complete intersection curves on \(X\) are lines, and
(4) for \(d>n+2\), all nontrivial complete intersection curves on \(X\) are lines or conics.
The proof is divided into two parts. In the first part, results about the regularity and the components of Hilbert schemes of projective spaces are used to reduce the questions to a certain inequality for Hilbert functions of complete intersections. Then the author gives a rather long and tedious inductive proof of that inequality. Noether-Lefschetz theorem; nontrivial complete intersection curves contained in a general hypersurface; Hilbert schemes; Hilbert functions Szabó, E.: Complete intersection subvarieties of general hypersurfaces, Pacific J. Math. 175, No. 1, 271-294 (1996) Complete intersections, Hypersurfaces and algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Complete intersection subvarieties of general 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 In this paper, we study the following problem: ``Find the projective manifolds V/G with trivial canonical bundle and zero first Betti number by the construction of resolving singularities of the quotient of a complex torus V by a finite abelian group G. Computer their Euler numbers.''
By the toroidal desingularization, we find a criterion for the existence of such V/G for those (V,G) with trivial dualizing sheaf \(\omega_{V/G}\). Because the dimension of V is 2 or 3, such V/G can always be constructed. We also derive a formula of the Euler number of V/G in terms of the Euler numbers of fixed points of elements of G, which was suggested by string theorists. Kummer surfaces; resolving singularities of the quotient of a complex torus by a finite abelian group; Euler numbers; toroidal desingularization; string J. Halverson, C. Long and B. Sung, \textit{Algorithmic universality in F-theory compactifications}, \textit{Phys. Rev.}\textbf{D 96} (2017) 126006 [arXiv:1706.02299] [INSPIRE]. Special surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Topological properties in algebraic geometry On the generalization of Kummer surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The notion of gluing of Abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied further by Polishchuk. We observe that this notion is a particular case of a general categorical construction.
We then apply this general notion to the study of the ring of global differential operators \(\mathcal D\) on the basic affine space \(G/U\) (here \(G\) is a semi-simple simply connected algebraic group over \(\mathbb{C}\) and \(U\subset G\) is a maximal unipotent subgroup). We show that the category of \(\mathcal D\)-modules is glued from \(|W|\) copies of the category of \(D\)-modules on \(G/U\) where \(W\) is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that the algebra \(\mathcal D\) is Noetherian, and get some information on its homological properties. rings of differential operators; semisimple Lie algebras; gluing of categories DOI: 10.1017/S1474748002000154 Rings of differential operators (associative algebraic aspects), Homogeneous spaces and generalizations, Eilenberg-Moore and Kleisli constructions for monads Gluing of Abelian categories and differential operators on the basic affine space. | 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 \(f : X \to S\) be a morphism of schemes and \(\mathcal{F}\) an \(\mathcal{O}_X\)-module. The Quot functor associates to a scheme \(T\) over \(S\), the \(\mathcal{O}_{X_T}\)-module quotient \(\mathcal{F}_T\to \mathcal{G}\) of the pullback \(\mathcal{F}_T\) of \(\mathcal{F}\) to \(X_T = X\times_S T\) such that \(\mathcal{G}\) is flat over \(S\) and where two quotients are equivalent if they have the same kernel. Of particular interest is when \(X \to S\) is a morphism of affine schemes associated to an essentially finite \(A\)-algebra \(B\) and \(\mathcal{F}\) corresponds to a \(B\)-module, \(M\). Although other situations yield a Hilbert polynomial, this is not the case here. When the dimension \(\dim_{\kappa(P)}(\kappa(P)\otimes_B M)\) is finite for all primes \(P\) of \(A\) with \(\kappa(P) = A_P/PA_P\), it is possible to use this dimension as a substitute for the Hilbert polynomial. This paper studies the similarities between three statements:
\(M\) is a locally free \(A\)-module of rank \(d\);
\(M_P\) is a free \(A_P\)-module of rank \(d\) for all primes \(P\) of \(A\);
The \(A\)-module M is flat and \(\dim_{\kappa(P)}(\kappa(P)\otimes_B M) = d\) for all primes \(P\) of \(A\). morphism of affine schemes; Hilbert polynomial; locally free module Laksov, D., Pitteloud, Y. andSkjelnes, R. M., Notes on flatness and the Quot functor on rings,Comm. Algebra 28 (2000), 5613--5627. Injective and flat modules and ideals in commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Schemes and morphisms Notes on flatness and the Quot functor on rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities By the result of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], there exists an equivalence between the derived category of \(G-\)equivariant coherent sheaves on a quasiprojective variety \(M\) with the derived category of coherent sheaves on the irreducible component \(Y\) of the \(G-\)Hilbert scheme of \(M\) that contains free orbits. The equivalence holds under the assumption that \(G\) is a finite group acting on \(M\) such that the canonical bundle on \(M\) is locally trivial as a \(G-\)sheaf and \(\text{dim} Y\times_{(M/G)} Y \leq \dim M +1.\) The article under review generalizes this result to the case of any smooth Deligne-Mumford stack \(\mathcal X\) with coarse moduli space \(X\) which is a quasiprojective Gorenstein variety. Denote by \(\text{Hilb}({\mathcal X})\) a scheme representing the Hilbert functor studied by \textit{M. Olsson} and \textit{J. Starr} [Commun. Algebra 31, No. 8, 4069--4096 (2003; Zbl 1071.14002)]. Then, the role of scheme \(Y\) is played by the component \(\text{Hilb}'({\mathcal X})\subset \text{Hilb}({\mathcal X})\) containing non-stacky points in \(\mathcal X.\)
In the above setting, the main theorem asserts that if \(\text{dim} \text{Hilb}'({\mathcal X})\times_X \text{Hilb}'({\mathcal X}) \leq \dim X +1,\) then \(\text{Hilb}'({\mathcal X})\) is smooth and there is an equivalence between the categories \(D^b({\mathcal X})\) and \(D^b(\text{Hilb}'({\mathcal X}))\) given by the integral functor with universal object over \(\text{Hilb}'({\mathcal X})\) as a kernel.
Moreover, in the setting of Bridgeland, King and Reid, the authors prove the twisted version of equivalence in the sense of \textit{V. Baranovsky} and \textit{T. Petrov} [Adv. Math. 209, No. 2, 547--560 (2007; Zbl 1113.14033)]. McKay correspondence; derived categories; stacks; Hilbert scheme; Brauer group Chen, J-C; Tseng, H-H, A note on derived mckay correspondence, Math. Res. Lett., 15, 435-445, (2008) McKay correspondence, Stacks and moduli problems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), Brauer groups of schemes A note on derived McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper, the author gives the definitions of totally symmetric algebras and polarization algebras in section 2 and section 3, respectively. He proves that a finite dimensional algebra \(u\) has a non-zero invariant bilinear form iff the \(L\)-deformation of \(u\) is totally symmetric with respect to some basis, where \(L\) is a non-zero linear map of \(u\). In section 3, the author studies the homogeneous polynomials of degree 2 and proves the standard polynomial \(st_n(X)\in \mathrm{Inv}(F)\) iff \(J_XF\) is symmetric. He also proves the polarization algebra \(B_F\) is associative iff \(f_i\in \mathrm{Inv}(F)\) for all \(1\leq i\leq n\), where \(F=(f_1,\ldots,f_n)\) is a homogeneous polynomial endomorphism of degree 2 in section 4. Finally the author gives the relation between invariant bilinear forms and the Cartan's exterior differential for polynomials. non-associative algebras; problem of Albert; invariant bilinear forms; nil-algebras; solvable algebras; polynomial endomorphisms; Jacobian conjecture Power-associative rings, Structure theory for nonassociative algebras, Automorphisms, derivations, other operators (nonassociative rings and algebras), Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Jacobian problem The space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism: an approach to the problem of Albert | 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 note, we construct curves over finite fields which have, in a certain sense, a ``lot'' of points, and give some applications to the zeta-functions of curves and Abelian varieties over finite fields. In fact, we found the basic construction of curves \(\mathbb{Z}^n\) which go through every rational point, as part of an unsuccessful attempt to find curves of growing genus over a fixed finite field with lots of points in the sense of the Drinfeld-Vladut bound.
Theorem. Let \(k\) be a finite field, \(X/k\) smooth and quasi-projective and geometrically connected, of dimension \(n\geq 1\). Let \(E/k\) be a finite extension. There exists a smooth, geometrically connected curve \(C_0/k\), and an immersion \(\pi:C_0\to X\) which is bijective on \(E\)-valued points.
(From the correction:) ``O. Gabber has kindly pointed out to me that the proof of Lemma 5 of the original paper is wrong. The error occurs in the last paragraph of the proof. The effect of correcting this error is that in Lemmas 4, 5, and 6, and in Corollary 7, what is asserted to hold for \(r\) sufficiently large holds only for \(r\) sufficiently large and sufficiently divisible. Indeed, Gabber has constructed examples to show that Lemma 6 and Corollary 7 can be false without this extra proviso. In the corrections, we also modify the statement of Lemma 5, so that its new, weaker conclusion applies in a more general setting''. curves over finite fields; zeta-functions of curves; Abelian varieties over finite fields Katz, N.: Spacefilling curves over finite fields. Mrl 6, 613-624 (1999) Curves over finite and local fields, Finite ground fields in algebraic geometry, Abelian varieties of dimension \(> 1\), Varieties over finite and local fields Space filling curves over finite fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The objective of this work is to provide a conceptual reformulation of the Siegel series as well as an inductive formula. This explains that the local intersection multiplicities of Gross \& Keating and the Siegel series have the same structure, going further in the comprehension of the matching of the values computed independently.
Let \(\mathfrak{o}\) be the ring of integers of a finite field extension of \(\mathbb{Q}_p\). We denote by \(f\) the cardinality of its residue field. Let \((L,q_L)\) a quadratic lattice over \(\mathfrak{o}\). If \(n\) is the rank of \(L\), let \((a_1,\dotsc,a_n)\) be its Gross-Keating invariant where \(a_1\leq\dotsc\leq a_n\) are integers. We denote by \(n_0\) the number of zeros in the sequence \(a_1\leq\dotsc\leq a_n\). The shape of the formula for the Siegel series proven by the authors is the following ploynomial whose coefficients are expressed as sum of numbers of sets of lattices
\begin{multline*}
\mathcal{F}_L(X)=(1-X)\sum_{\substack{0\leq b\leq (a_1+\dotsc+a_n)/2\\ n_0\leq a\leq n}}\Bigg(\#\mathcal{S}_{(L,a^\pm,b)}f^{b(n+1)}X^{2b}\\\times\left(1+\chi(a^\pm)f^{n-a/2}X\right)\prod_{1\leq j<n-a/2}\left(1-f^{2j}X^2\right)\Bigg)
\end{multline*}
where \(\mathcal{S}_{(L,a^\pm,b)}\) is an explicit set of quadratic lattices containing \(L\) and \(\chi(a^\pm)\) is \(-1\), \(0\) or \(1\) accordingly to an explicit formula described in the text.
From this expression the authors derive an inductive formula, assuming a conjecture that they prove to be true if \(p\) is odd or if \((L,q_L)\) is an anistropic \(\mathbb{Z}_2\) lattice.
This inductive formula is used to prove that the local intersection multiplicity \(\alpha_p(a_1,a_2,a_3)\) associated to \((L,q_L)\) is, up to an explicit multiplicative constant, the derivative of \(\mathcal{F}_L\) evaluated at \(1/p^2\).
An application is given to prove a new identity between the intersection number of two modular correspondences over \(\mathcal{F}_p\) and Fourier coefficients of the Siegel-Eisenstein series ow weight \(2\) on \(\mathrm{Sp}_4/\mathbb{Q}\). Siegel series; local intersection multiplicity; Gross-Keating invariant; modular correspondence; Siegel-Eisenstein series Fourier coefficients of automorphic forms, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Arithmetic aspects of modular and Shimura varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Modular and Shimura varieties, Moduli, classification: analytic theory; relations with modular forms A reformulation of the Siegel series and intersection numbers | 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 automorphisms of a complex smooth manifold which act trivially in cohomology have been classified for several classes of surfaces, but the problem is still open for the surfaces of general type (unless some additional hypotesis is considered).
The author of this note had previously shown [Tohoku Math. J. (2) 64, No. 4, 593--605 (2012; Zbl 1259.14045)] that, if the Euler characteristic of a surface of general type \(S\) is big enough, then the group \(G\) of the automorphisms which act trivially in cohomology has order at most 4.
In this note the author classifies the boundary case under the additional assumption that the irregularity is at least 3. More precisely, he shows that, if \(G\) has order 4, the irregularity is at least 3 and the geometric genus is at least 35, then the surface belongs to the family constructed in the first section of the paper. These are surfaces with irregularity 3 isogenous to a product of two curves: more precisely the quotients of the product of any hyperelliptic curve with a very special curve of genus 5 having three pairwise commuting involutions, by an involution acting separately on the two factors, and as the hyperelliptic involution on the first factor, and as one of the three given involution (without fixed points) on the second factor. The two further involutions on the second factor induce the required cohomologically trivial action of a (noncyclic) group of order 4.
About the assumptions: the hypothesis on the geometric genus is technical and not expected to be sharp; the assumption on the irregularity is necessary, by some constructions in smaller irregularity in the Ph.D. thesis of Wenfei Liu.
The strategy of the proof is the following. The author first uses some Hodge theory to show that under his assumptions, the canonical map of \(S\) factors through the quotient map onto \(S/G\). Then, by an argument of \textit{A. Beauville} [Manuscr. Math. 110, No. 3, 343--349 (2003; Zbl 1016.14016)], he shows that the canonical map has order at most 9, and therefore 4 or 8. He then excludes the case 4 by using the Castelnuovo inequality. So the canonical map has degree 8 and by a result of \textit{G. Xiao} [Compos. Math. 56, 251--257 (1985; Zbl 0594.14029)] the canonical image is ruled. The author concludes by showing that the pull-back of the ruling is an isotrivial fibration of genus 5, and then by using it to prove that the surface is one of surfaces described above. surfaces of general type; automorphism groups; cohomology Cai, J.-X., Automorphisms of an irregular surface of general type acting trivially in cohomology, J. Algebra, 367, 95-104, (2012) Automorphisms of surfaces and higher-dimensional varieties, Surfaces of general type Automorphisms of an irregular surface of general type acting trivially in 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 construct the minimal integral model of a finite type of an algebraic torus defined over a complete non-Archimedean extension of an algebraic number field. The main problem is to study this model in the case of the ramified splitting field. We describe the models of quasi-split tori, norm tori and tori of low dimension. We also calculate the reduction of these models. algebraic group schemes; non-reduced group schemes; minimal splitting fields; Galois groups; coordinate rings; groups of rational characters; maximal tori; connected unipotent groups; products of reductions Linear algebraic groups over local fields and their integers, Varieties over global fields, Group schemes Galois lattices and reductions of algebraic tori. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the first of a series of two papers laying the foundations for the study of algebraic families of Harish-Chandra pairs and the attached modules. See [\textit{J. Bernstein} et al., Int. Math. Res. Not. 2020, No. 11, 3494--3520 (2020; Zbl 1484.22010)] for the second paper in the series.
The authors introduce a general algebraic framework, using sheaf theory, to study some phenomena that occur in relation with a \textit{variation of parameters} in either (a) the structure of a reductive Lie group, or (b) the structure of a representation of a given reductive group, or (c) both at the same time.
This is inspired by the notion of `contraction' from mathematical physics; see [\textit{E. Inönü} and \textit{E. P. Wigner}, Proc. Natl. Acad. Sci. USA 39, 510--524 (1953; Zbl 0050.02601)] and [\textit{I. E. Segal}, Duke Math. J. 18, 221--265 (1951; Zbl 0045.38601)]. In particular, the notion allows for variations of reductive group structures where non-reductive groups appear at singular parameter values.
In the second paper of the series [loc. cit.], the authors prove that their purely algebraic framework naturally produces families of unitary representations, thereby connecting several classical families from mathematical physics which may appear unrelated at first sight.
Section 2 introduces the general framework. Let \(X\) be a complex algebraic variety. Then the authors introduce notions of \textit{algebraic family over} \(X\) for several familiar Lie-theoretic and representation-theoretic objects. They assume \(X\) is an irreducible, nonsingular, quasi-projective complex algebraic variety, and define algebraic families over \(X\) of Lie algebras, algebraic groups, Harish-Chandra pairs \((\mathfrak{g},K)\), of Harish-Chandra modules... The language is that of algebraic geometry: all objects introduced by the authors the authors are sheaves on \(X\), and the variation of objects over \(X\) is encoded by algebraic notions.
The two main examples, given in Sections 2.1.2 and 2.1.3, are over \(X=\mathbb{C}\). Both are based upon the Inonu-Wigner contraction of Lie algebras, and to the deformation to the normal cone construction in algebraic geometry.\\
The other two sections focus on examples. A short Section 3 outlines a construction of algebraic families of real algebraic groups, and specializes to the classical groups. The more detailed Section 4 focuses on a particular family of Harish-Chandra pairs, related to the contraction which connects \(\mathrm{SL}(2,\mathbb{R})\), \(\mathrm{SU}(2)\) and \(\mathrm{SO}(2) \ltimes \mathbb{R}^2\). The main theme is representation theory: the authors outline a classification for the corresponding ``generically irreducible'' families of Harish-Chandra modules. Along the way, they introduce natural invariants for such families of Harish-Chandra modules, which are (new, and nontrivial) family versions of some of the usual representation-theoretic invariants: infinitesimal characters, Casimir eigenvalues, \(K\)-types. contractions of Lie groups; Harish-Chandra modules; algebraic families Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Linear algebraic groups over the reals, the complexes, the quaternions, Group schemes Algebraic families of Harish-Chandra pairs | 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 determines the Mordell-Weil groups over \({\mathbb{Q}}\) of the abelian variety \(J_ 0(81)\), and of each of its \({\mathbb{Q}}\)-simple factors. Ligozat has shown that the two elliptic curve factors each have rank zero. Utilizing Mazur's descent techniques, the author shows that the simple two-dimensional factor \(J_ 0(81)^{new}\) also has rank zero. The computation of the torsion subgroups of the Mordell-Weil groups follows the approach taken by \textit{A. P. Ogg} [in Analytic Number Theory,Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 221-231 (1973; Zbl 0273.14008)]. This involves counting rational points on the abelian varieties over various finite fields, and then using the fact that reduction modulo p is injective on the prime-to-p torsion points (for \(p\neq 3)\). Finally, the author notes that the finiteness of \(J_ 0(81)^{new}({\mathbb{Q}})\) is predicted by the conjecture of Birch and Swinnerton-Dyer. modular curve; jacobian; abelian variety; \({\mathbb{Q}}\)-simple factors; torsion subgroups of the Mordell-Weil groups; conjecture of Birch and Swinnerton-Dyer Rational points, Holomorphic modular forms of integral weight, Jacobians, Prym varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Special algebraic curves and curves of low genus, Elliptic curves, Arithmetic ground fields for abelian varieties Points rationnels sur \({\mathbb{Q}}\) de la jacobienne de la courbe modulaire \(X_ 0(3^ 4)\). (Rational points over \({\mathbb{Q}}\) of the jacobian of the modular curve \(X_ 0(3^ 4))\) | 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 reductive algebraic group and \(P_1,\dots,P_k\) be parabolic subgroups containing a fixed Borel subgroup. If \(G\) has finitely many orbits under the diagonal action in
\[
X=G/P_1 \times\cdots \times G/P_k,
\]
then \(X\) is called a multiple flag variety of finite type. In the paper under review, the authors classify multiple flag varieties of finite type for \(G=\text{Sp}_{2n}\) continuing their earlier work for \(G=\text{GL}_n\). They also give a complete enumeration of the orbits and explicit representatives for them. Their main tool (as in their earlier paper) is the algebraic theory of quiver representations. reductive algebraic group; Borel subgroup; multiple flag variety of finite type; quiver representations Magyar, P.; Weyman, J.; Zelevinsky, A., Symplectic multiple flag varieties of finite type, J. Algebra, 230, 1, 245-265, (2000) Group actions on varieties or schemes (quotients), Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds Symplectic multiple flag varieties of finite 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 We show that any toroidal DM stack \(X\) with finite diagonalizable inertia possesses a maximal toroidal coarsening \(X_{\operatorname{tcs}}\) such that the morphism \(X\to X_{\operatorname{tcs}}\) is logarithmically smooth.
Further, we use torification results of the first two authors [J. Algebra 472, 279--338 (2017; Zbl 1376.14051)] to construct a destackification functor, a variant of the main result of \textit{D. Bergh} [Compos. Math. 153, No. 6, 1257--1315 (2017; Zbl 1372.14002)], on the category of such toroidal stacks \(X\). Namely, we associate to \(X\) a sequence of blowings up of toroidal stacks \(\widetilde{\mathcal F}_X\:Y\to X\) such that \(Y_{\operatorname{tcs}}\) coincides with the usual coarse moduli space \(Y_{\operatorname{cs}}\). In particular, this provides a toroidal resolution of the algebraic~space~\(X_{\operatorname{cs}}\). Both \(X_{\operatorname{tcs}}\) and \(\widetilde{\mathcal F}_X\) are functorial with respect to strict inertia preserving morphisms \(X'\to X\).
Finally, we use coarsening morphisms to introduce a class of nonrepresentable birational modifications of toroidal stacks called Kummer blowings up.
These modifications, as well as our version of destackification, are used in our work on functorial toroidal resolution of singularities. algebraic stacks; toroidal geometry; logarithmic schemes; birational geometry; resolution of singularities Generalizations (algebraic spaces, stacks), Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects) Toroidal orbifolds, destackification, and Kummer blowings up | 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 survey article the author studies in detail the actions of a semisimple Lie or algebraic group on its Lie algebra by the adjoint representation and on itself by the adjoint action. The main focus is on orbits through nilpotent elements in the Lie algebra. The author collects some of the most significant results on nilpotent orbits that have found wide application to representation theory. The exposition is primary in the setting of a semisimple Lie algebra and its adjoint group over an algebraically closed field of characteristic 0, but much of the presentation covers also semisimple Lie algebras over the reals or an algebraically closed field of prime characteristic, and conjugacy classes in semisimple algebraic groups. Many results are given with detailed proofs, others are summarized with reasonably complete references. The reader is expected to be familiar with the structure and classification of complex semisimple Lie algebras and with basic definitions and theorems on the level of a first course on that subject.
Chapter 1 collects the background material on the structure and classification of complex and real semisimple Lie algebras and related objects.
Chapter 2 contains basic facts about orbits, conjugacy classes and centralizers. The author considers in detail the general and the special linear groups as a ready source of examples, as well as the other classical groups and Lie algebras. It is surprising that very often the text contains very explicit results on such groups and algebras derived using nothing more than linear algebra.
Chapter 3 is devoted to the first major result in the theory of nilpotent orbits and unipotent classes, that there are only finitely many of them in any semisimple Lie algebra or algebraic group.
The goal of Chapter 4 is to study the adjoint orbits of maximal dimension in semisimple Lie algebras. As an application, the behaviour of the adjoint quotient is studied in more detail.
Chapter 5, Induction of Orbits, deals with a fundamental construction which starts from a nilpotent orbit in a Levi subalgebra of a reductive Lie algebra and produces one in the whole algebra. The applications concern subregular orbits, sheets in semisimple Lie algebras and the classification of nilpotent orbits, including as an appendix tables of the exceptional orbits.
Chapter 6 studies the partial order on nilpotent orbits given by containment of their closures. The author characterizes this order in terms of partitions in the classical case and shows that, if some orbits are discarded, the remaining ones admit an order-reversing involution. He also gives the Hasse diagrams with respect to the order by closure of nilpotent orbits in exceptional simple Lie algebras.
Chapter 7 deals with some important and surprising relationships between the nilpotent variety and the flag variety of a semisimple Lie algebra. The basic object to study is the subvariety of the flag variety consisting of all Borel subalgebras containing a fixed nilpotent element.
Chapter 8 continues the study of the fiber of Borel subalgebras containing a fixed nilpotent element and its relation with the representations of the Weyl group.
Finally, in Chapter 9 the author summarizes some of the most recent work being done in the considered topics and indicates some directions of current research. semisimple Lie algebras; semisimple algebraic groups; adjoint representation; adjoint action; nilpotent orbits W.M. McGovern, \textit{The Adjoint Representation and the Adjoint Action}, in \textit{Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action}, Springer-Verlag Berlin Heidelberg, Germany (2002). Simple, semisimple, reductive (super)algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras The adjoint representation and the adjoint action. | 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 \(R=\oplus_{n\geq 0}R_ n\) be a noetherian graded ring, with \(R_ 0=k\) an infinite field, \(R=k[R_ 1]\) and suppose \(R\) Gorenstein. In this paper one proves facts supporting the conjecture that, for a homogeneous Gorenstein ideal \(I\) in the linkage class of a complete intersection, \(h(R/I)\) is unimodal. For instance, one shows that the unimodal property is preserved under tight double linkage, when the degree of the socle does not diminish. Gorenstein algebras; unimodal \(h\)-sequences; Hilbert series; graded ring; linkage class of a complete intersection Beintema, M. B.: Gorenstein algebras with unimodal h-sequences. Comm. algebra 20, 979-997 (1992) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Linkage, complete intersections and determinantal ideals, Linkage Gorenstein algebras with unimodal \(h\)-sequences | 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 \(M_ n\) be the space of \(n\times n\)-matrices over an algebraically closed field k, the group \(GL_ n\) of invertible matrices acts on \((M_ n)^ r\) diagonally via conjugation. Only in a few cases is it known that the corresponding field of invariant rational functions is rational [cf. \textit{E. Formanek}, J. Algebra 62, 304-319 (1980; Zbl 0437.16013)]. The author gives a short elegant proof of the fact that, if one replaces a copy of \(M_ n\) by its subvariety of singular matrices, the field of invariants is rational. invariants; conjugation; field of invariant rational functions; subvariety of singular matrices Vector and tensor algebra, theory of invariants, Trace rings and invariant theory (associative rings and algebras), Rational and unirational varieties, Geometric invariant theory Rationality of certain field of invariant 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 aim of the book under review is to present a very powerful approach based on modern methods of monodromy theory to the study of many problems from singularity theory, algebraic geometry and theory of differential equations.
In the preface the author outlines the historical background of development of the monodromy theory from B. Riemann till present time and briefly describes the contents of his book. He also underlines that essential parts of the book are based on a two-year course at Warsaw University given by the author more than ten years ago.
In the first four chapters the author gives an introduction to the notion of monodromy in applications to multi-valued holomorphic functions and their Riemann surfaces, to the Morse theory in the real domain and the theory of normal forms for functions, he also describes some basic facts from the algebraic topology of manifolds and fibre bundles as well as from the topology and monodromy of analytic and algebraic functions including the Milnor theorem, Picard-Lefschetz formula, root systems, Coxeter groups, resolution and normalization of singularities, etc.
The next chapter 5 is devoted to the study of integrals along vanishing cycles, including basic results from the theory of Gauss-Manin connection, Picard-Fuchs systems, oscillating integrals and their relations with singularity theory. Then a method of abelian integrals is explained in detail; this chapter contains a few good examples and description of results Khovanski, Gabrielov, Petrov, and others.
It should be remarked that most of the above materials can also be found in the book [\textit{V. I. Arnol'd}, et al., Singularities of differentiable maps. Volumes I, II. Monographs in Mathematics, Vol. 82, 83. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001, 1988; Zbl 0659.58002)] .
In chapter 7 general ideas of the theory of Hodge structures and period mappings are described. Starting from classical results on the Hodge structure on algebraic manifolds, the author goes into the theory of mixed Hodge structures on incomplete manifolds and on the cohomological Milnor bundle, he also discusses the notions of limit Hodge structures in the sense of Schmid and Steenbrink, relations with the monodromy theory, etc. In conclusion, some basic constructions due to D. Mumford and Ph. Griffiths from the theory of period mappings in algebraic geometry are considered [see, for example, \textit{P. A. Griffiths}, Bull. Am. Math. Soc. 76, 228--296 (1970; Zbl 0214.19802)].
The subject of the next chapter 8 are the non-autonomous linear differential systems \(\dot z = A(t)z\), \( z\in \mathbb{C}^m,\) and the linear higher order differential equations \(x^{(n)} + a_1(t)x^{(n-1)} + \cdots + a_n(t)x = 0, \, x\in \mathbb{C},\) where \(t\in {\mathbb C}\) and all the entries of the matrix \(A(t)\) as well as the coefficients \(a_i(t)\) are meromorphic functions. The author discusses the notion of regular and irregular singularities, some aspects of the global theory of linear differential equations, the Riemann-Hilbert problem with a detailed analysis of Bolibruch's counter-example, the notion of isomonodromic deformations and some applications and relations with quantum field theory.
The chapters 9 and 10 are mainly concentrated on the local and global theory of holomorphic foliations in \(\mathbb{C P}^2,\) respectively. They contain the theory of resolution of vector fields, the theory of resurgent functions in the sense of Ecalle, the theory of Martinet-Ramis modules, theorems of Bryuno and Yoccoz, some results concerning the nonlinear Riemann-Hilbert problem [\textit{P. M. Elizarov}, et al., Nonlinear Stokes phenomena. Providence, RI: American Mathematical Society. Adv. Sov. Math. 14, 57--105 (1993; Zbl 1010.32501)], Ziglin theory, and a lot of other very interesting material with a number of useful examples.
In chapter 11 the author presents the basic notions and tools of differential Galois theory including the theory of Picard-Vessiot extensions with applications to the problem of integration of polynomial vector fields (Singer's theorem). He then discusses the monodromy theory of algebraic functions including the topological proof of Abel-Ruffini theorem, Khovanski's generalization of the monodromy group for large class of functions, the monodromy properties of Singer's first integrals.
The concluding chapter 12 is mainly based on the famous books of \textit{F. Klein} [see `Vorlesungen über die hypergeometrische Funktion'. Berlin: Springer (1933; Zbl 0007.12202 and JFM 59.0375.11), and so on.] It contains classical results from the theory of hypergeometric functions and explicit calculations of the monodromy and solutions of hypergeometric equations in quadratures. In conclusion two kinds of generalizations of hypergeometric functions are described. The first one is based on the Picard-Deligne-Mostow approach while the second -- on the approach of I. Gelfand, A. Varchenko, and others.
The book under review is written in a very clear and concise style, almost all key topics are followed by carefully chosen examples, non-formal remarks and comments. It contains a large number of pictures which may be considered as real visualizations of the discussed ideas. The bibliography includes 359 selected references. This makes the exposition not only accessible to beginning graduate students but highly interesting and useful for advanced research workers in algebraic and differential topology, algebraic geometry, complex analysis, differential equations and related fields of pure and applied mathematics and mathematical physics. monodromy theory; isolated singularities; critical points; Picard-Lefschetz theory; Gauss-Manin connection; Picard-Fuchs systems; oscillating integrals; abelian integrals; Hodge structures; period mappings; regular and irregular singularities; Riemann-Hilbert problem; isomonodromic deformations; holomorphic foliations; differential Galois theory; hypergeometric functions \.Zoł\polhk adek, Henryk, The monodromy group, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)] 67, xii+580 pp., (2006), Birkhäuser Verlag, Basel Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Monodromy on manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.), Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to global analysis, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Period matrices, variation of Hodge structure; degenerations, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Deformations of complex singularities; vanishing cycles, Mixed Hodge theory of singular varieties (complex-analytic aspects), Milnor fibration; relations with knot theory, Critical points of functions and mappings on manifolds, Singularities of vector fields, topological aspects, Topological invariants on manifolds, Differential algebra The monodromy group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0534.14014. moduli space of stable curves; homotopy type of moduli space; Satake compactification of the Siegel modular varieties; period mapping; Riemann surfaces; period matrices Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Classification of homotopy type, Period matrices, variation of Hodge structure; degenerations, Compactification of analytic spaces Moduli space of stable curves from a homotopy viewpoint | 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 comprises the notes of three lectures delivered by the author at the Summer School ``Algebraic Groups'', which took place at the University of Göttingen, Germany, in June/July 2005. Apparently geared toward a wider audience, these lectures were to provide a concise introduction to loop groups of compact Lie groups and some aspects of their representation theory. In view of their introductory character, these notes are organized as follows.
Lecture 1: Review of compact Lie groups and their representations, basics of loop groups of compact Lie groups, and illustrating examples.
Lecture 2: Finer properties of loop groups, root systems and Weyl groups of loop groups, and central extensions of loop groups.
Lecture 3: Representations and homogeneous spaces of loop groups, with a view toward infinite Grassmannians.
The mostly explanatory and survey-like notes are largely based on the excellent standard monograph ``Loop Groups'' by \textit{A. Pressley} and \textit{G. Segal} (1986; Zbl 0618.22011), which the author generally refers to. Apart from providing a quick and lucid introduction to the conceptual framework of loop groups of compact Lie groups, these lectures also touch upon the physical background and significance of the subject, mainly by paying special attention to the so-called representations of positive energy for loop groups. However, the relations of loop groups to what is nowadays understood by the notion of string topology [cf. \textit{M. Chas} and \textit{D. Sullivan}, String Topology, math.GT/9911159] are barely mentioned in these notes, in contrast to what their title might suggest. As for the style of exposition, the author naturally focusses on explaining the basic constructions and fundamental theorems within his theme of discussion, with numerous concrete examples, outlines of proofs of major theorems, and a few related exercises included. In this regard, the present notes provide an inspiring first reading for beginners in the rapidly growing field of loop groups and their applications to various theories in contemporary mathematics and theoretical physics. loop groups; representations of loop groups; infinite-dimensional Lie groups; compact Lie groups; homogeneous spaces; Grassmannians Loop groups and related constructions, group-theoretic treatment, Infinite-dimensional Lie groups and their Lie algebras: general properties, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Compact groups, Homogeneous complex manifolds, Grassmannians, Schubert varieties, flag manifolds Loop groups and string topology | 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 Brauer group \(\mathrm{Br}(X)\) of a complex Enriques surface \(X\) is isomorphic to \(\mathbb{Z}/2\), and hence \(X\) admits a unique non-trivial Brauer class \(b_X\). Since \(b_X\) has order \(2\), \(b_X\) can be represented by a quaternion algebra \(\mathcal{A}\) on \(X\).
The paper under review studies the properties of the projective moduli scheme \(\mathcal{M}_{\mathcal{A}/X}\) of torsion-free \(\mathcal{A}\)-modules generically of rank \(1\), in the case when the Enriques surface \(X\) is very general. The author shows that in this case \(\mathcal{M}_{\mathcal{A}/X}\) is smooth, and that the locus parametrizing locally projective \(\mathcal{A}\)-modules is dense in \(\mathcal{M}_{\mathcal{A}/X}\). Finally, for fixed Chern classes \(c_1\) and \(c_2\), the moduli scheme \(\mathcal{M}_{\mathcal{A}/X,c_1,c_2}\) is realized as an étale double cover of a Lagrangian submanifold \(\mathcal{L}\) of \(\mathcal{M}_{\overline{\mathcal{A}}/\overline{X},\overline{c_1},\overline{c_2}}\), where \(\overline{X}\) is the \(K3\) cover of \(X\) and \(\overline{\mathcal{A}}\) is the pullback of the quaternion algebra \(\mathcal{A}\) to \(\overline{X}\). Enriques surfaces; quaternion algebras; moduli schemes of sheaves \(K3\) surfaces and Enriques surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Rank one sheaves over quaternion algebras on Enriques surfaces | 0 |
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