text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present book about invariant theory is centered around the study of finite groups and their polynomial rings. As the author indicates the material grows out of his elaboration on \textit{H. Weyl}'s ``The classical groups, their invariants and representations'' (1939; Zbl 0020.20601), and \textit{N. Bourbaki}'s ``Groupes et algèbres de Lie'', Chapitres 4, 5 et 6 (1981; Zbl 0483.22001), presented at several university courses on invariant theory. Because of the impossibility to encompass all of his notes and musings on the subject he confined to finite groups and polynomial invariants. Because invariant theory served as one of the major inputs for the development of commutative algebra the author includes several chapters on commutative and homological algebra, see chapter 5, `Dimension theoretic properties of rings of invariants', and chapter 6, `Homological properties of invariants'. They include Noether's normalization theorem and Hilbert's syzygy theorem, one of the origins of homological algebra. These parts make the presentation of the book self-contained and independent on corresponding textbooks. This might be of some interest for readers mainly interested in invariant theory. The last two chapters, chapter 10, `Invariant theory and algebraic topology' and chapter 11, `The Steenrod algebra and modular invariants' are concerned with the influence of invariant theory on algebraic topology and vice verse a special research interest of the author for several years among others motivated by a paper by \textit{A. Clark} and \textit{J. Ewing} [Pac. J. Math. 50, 425-434 (1974; Zbl 0333.55002)].
In chapter 1, `Invariants and relative invariants' there are the basic definitions and a few instructive examples. The second chapter, `Finite generation of invariants' is devoted to the so-called first main theorem of invariant theory (in the case of finite groups), i.e. the description of algebra generators of the ring of invariants. It contains Hilbert's finiteness result and Noether's bound on the degree of the generators for a field of characteristic zero. This is continued in chapter 3, `Construction of invariants', where Noether's proof is made more explicit based on an idea of Weyl [see loc. cit.]. In chapter 4, `Poincaré series', Molien's theorem is proved and several examples are discussed. In particular it is shown that Molien's formula is independent of the ground field in the case of a finite permutation group. The interesting chapter 7, `Groups generated by reflections', centers around the Shepard-Todd theorem characterizing groups with polynomial rings of invariants as those generated by pseudoreflections. Besides of the proof of that theorem the chapter contains a discussion of Coxeter groups, an account of the relative invariants of pseudoreflections in the non-modular case and deals with automorphisms of the ring of covariants.
A large part of the book is concerned with the modular case of invariant theory, i.e. when the characteristic of the ground field is a divisor of the order of the group. In chapter 8, `Modular invariants', there is a proof of Dickson's result on the ring of invariants of \(GL(n,\mathbb{F}_q)\) acting on \(\mathbb{F}^n_q\), where \(\mathbb{F}_q\) denotes the field with \(q=p^s\) elements, \(p\) a prime number. This is continued in chapter 11, `The Steenrod algebra and modular invariant theory', where the interplay between Steenrod operations, the Dickson invariants, and the ideal structure of the rings of invariants is examined and applied to the transfer homomorphism. There is a completely algebraic construction of the Steenrod algebra, a tool for a field of prime characteristic depending on the Frobenius homomorphism. In chapter 9, `Polynomial tensor exterior algebras', the author deviates from considering only polynomial invariants. He studies the invariants of pseudoreflection groups acting on polynomial tensor exterior algebras based on \textit{L. Solomon}'s paper [Nagoya Math. J. 22, 57-64 (1963; Zbl 0117.27104)]. In particular there is a variant of Molien's theorem in this case. Chapter 10, `Invariant theory and algebraic topology', firstly presents the construction of the Steenrod algebra (in a manner independent of algebraic topology) as another feature of invariant theory over a finite field, derived from the Frobenius homomorphism, as well as its application in invariant theory: a certain converse of the Shephard-Todd theorem, a classification of groups generated by pseudoreflections in characteristic \(p\) when the group order is prime to \(p\), another proof of Dickson's theorem. A large part of this chapter appears for the first time in publication.
During the presentation of his material the author is interested to include as much as possible illustrating examples to concrete cases. This could be very good for beginners in the subject, sometimes also in order to stimulate further questions. -- On the other side the author tried to extend characteristic zero results to arbitrary characteristics, or at least to the modular case. Often this requires completely different techniques and often corresponding extensions are false in general, as the Cohen-Macaulayness of rings of invariants. These efforts make the book valuable for everybody interested in these problems. The exposition starts at a basic level with illuminating examples. Moreover the author leads the reader to current areas of active research. So the book is not primarily a textbook but also a reference for recent research on the subject. Moreover the reviewer expects also some further inputs of research motivated by the book.
Besides of the selfcontained account on Cohen-Macaulay rings there is no discussion about the Gorensteinness of rings of invariants. Zero-dimensional Gorenstein rings are introduced by the unusual name Poincaré duality algebras. Besides of a very few statements there is no discussion of factoriality of rings of invariants.
In his introduction the author writes: `After an abortive attempt to write a jointly authored book with David Benson I finally decided to take the plunge and write about the polynomial invariants on finite groups on my own.' He motivates this by the impossibility to enclose in a single volume all aspects of the theory. So one might consult \textit{D. J. Benson}'s book [``Polynomial invariants of finite groups'', Lond. Math. Soc. Lect. Note Ser. 190 (1993; see the preceding review)] for recent results about factoriality and divisor class groups in invariant theory and furthermore for a different point of view on the same subject and comparing similar arguments as certain places. The author tried to make the results constructive for explicit calculations. People interested in this branch might consult also the book by \textit{B. Sturmfels}: ``Algorithms in invariant theory'' (1993; Zbl 0802.13002) not listed in the extensive bibliography of 288 items covering most of the relevant textbooks and research articles. ring of invariants; action of finite group; prime characteristic; Steenrod algebra; modular invariants; Molien's theorem; Shepard-Todd theorem; polynomial rings; Cohen-Macaulay rings L. Smith, Polynomial Invariants of Finite Groups, A.\ K. Peters, Wellesley, 1995. Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Research exposition (monographs, survey articles) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Polynomial invariants of finite groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal B^*(A)\) denote the Hodge ring of an Abelian variety \(A\), \(\mathcal D^*(A)\) denote the subring of \(\mathcal B^*(A)\) generated by divisor classes. It is proved that \(\mathcal B^*(A^n)=\mathcal D^*(A^n)\) for all \(n\geq 1\) if and only if the rank of the Hodge group of \(A\) is equal to its reduced dimension. From this result the author deduces the Hodge conjecture for some abelian varieties and their powers. The Hodge conjecture is also proved for the square of the minimal resolution of the compactification of a Hilbert modular surface satisfying some conditions on its Hodge structure. Hodge ring of an Abelian variety; divisor classes; Hodge conjecture; Hilbert modular surface Fumio Hazama, ''Algebraic cycles on certain abelian varieties and powers of special surfaces'', J. Fac. Sci. Univ. Tokyo Sect. IA Math.31 (1985) no. 3, p. 487-520 Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic theory of abelian varieties, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Cycles and subschemes, Special surfaces Algebraic cycles on certain abelian varieties and powers of special 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 paper deals with the problem of \(K\)-unirationality for conic bundles with a \(K\)-rational point. A rational \(K\)-surface \(X\) is called a conic bundle over a rational curve \(C\) if there exists a \(K\)-morphism \(f : X \to C\) whose generic fibre is a rational curve. The author puts the question: Are rational conic bundle surfaces with a \(K\)-rational point \(K\)-unirational? The problem has some algebraic significance. A conic bundle is \(K\)-unirational if and only if the corresponding quaternion algebra over a \(K\)-rational field has a \(K\)-rational splitting field [\textit{V. A. Iskovskikh}, Math. USSR, Sb. 3(1967), 563-587 (1969); translation from Mat. Sb., Nov. Ser. 74(116), 608-638 (1967; Zbl 0181.240)].
So the problem of \(K\)-unirationality can be formulated as a problem in the theory of central simple algebras. This problem is generalized to arbitrary finite dimensional central simple algebras. Let \(A\) be a finite-dimensional central simple algebra over a \(K\)-rational field, such that \(K\) has a \(K\)-rational point. Does \(A\) have a \(K\)-rational splitting field? The author proved that the answer is positive for Henselian fields [see Dokl. Akad. Nauk BSSR 29, 1061-1064 (1985; Zbl 0609.14030)]. The present paper contains results in the case of pseudo-closed \(K\) and similar results for so-called large arithmetic fields. These results were obtained by the author and \textit{Yu. Drakokhrust}. unirationality for conic bundles; rational point; rational conic bundle surfaces; finite dimensional central simple algebras; large arithmetic fields Yanchevskiĭ, V. I., Astérisque, 209, 311-320, (1992), Soc. Math. France, Paris Rational and unirational varieties, Rational points, Global ground fields in algebraic geometry, Varieties over finite and local fields \(K\)-unirationality of conic bundles over large arithmetic fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I see ibid. 38, 321-341 (1986; Zbl 0618.14004).]
Let S be a noetherian scheme and let \(f: X\to Y\) be a separated morphism of S-schemes of finite type. One natural problem is whether the properness of f is equivalent to the fact that every closed one- dimensional subscheme Z of X is proper over Y. In general the answer to this problem is negative, as the author shows by a counterexample. Then he defines a good property for a scheme S such that the above statements are equivalent for every such morphism f, namely the property of being universally one-equicodimensional. If \(S=Spec(A)\) is affine, S is said to be universally one-equicodimensional if every finitely generated integral A-algebra that has a maximal ideal of height one, is necessarily of dimension one. A thorough study of these schemes is undertaken. For example, if \(u: S\to T\) is a morphism of schemes, one finds conditions on u ensuring that T is universally one-equicodimensional if S is. This concept turns out to be very useful in the study of finite generation of subalgebras of a k-algebra of finite type (with k a field).
[For part III see the following review.] proper morphisms; universally one-equicodimensional schemes; finite generation of subalgebras Adrian Constantinescu. Proper morphisms and finite generation of subalgebras. II. Universally 1-equicodimensional rings and schemes. Stud. Cerc. Mat., 38(5):438--454, 1986. Schemes and morphisms, Rational and birational maps, Commutative Artinian rings and modules, finite-dimensional algebras Proper morphisms of schemes and finite generation of subalgebras. II: Universal 1-equicodimensional rings and 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 It is known, if not well-known, that the classical Cayley-Bacharach theorem for complete intersections in \({\mathbb{P}}^ 2\) is valid for 0- dimensional arithmetically Gorenstein subschemes of \({\mathbb{P}}^ n\). It is shown, more generally, that the result can be extended to 0-dimensional arithmetically Cohen-Macaulay subschemes of \({\mathbb{P}}^ n\) whose minimal Cohen-Macauley type is compatible with their Hilbert functions. The Gorenstein version of the Cayley-Bacharach theorem is deduced from a technical result which relates the Hilbert functions of linked subschemes. The note ends by showing that the 0-dimensional, arithmetically Gorenstein, reduced subschemes of \({\mathbb{P}}^ n\) are characterized by the validity of the Cayley-Bacharach theorem and the symmetry of the Hilbert function. liaison; arithmetically Cohen-Macaulay subschemes; Cayley-Bacharach theorem; Hilbert functions of linked subschemes E. D. Davis, A. V. Geramita, and F. Orecchia, ''Gorenstein Algebras and the Cayley-Bacharach Theorem,'' Proc. Am. Math. Soc. 93(4), 593--597 (1985). Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Gorenstein algebras and the Cayley-Bacharach theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 scheme defined over the finite field \(\mathbb F_p\), with Frobenius kernel \(G_1\). The tilting modules of \(G\) are defined as rational \(G\)-modules for which both the module itself and its dual have good filtrations. In 1997, J.~E.~Humphreys conjectured that the support varieties of certain tilting modules for regular weights should be given by the Lusztig bijection between cells of the affine Weyl group and nilpotent orbits of \(G\), when \(p>h\), where \(h\) is the Coxeter number. We present a conjecture for the support varieties of tilting modules when \(G=\text{GL}_n\). Our conjecture is equivalent to Humphreys' conjecture for \(p\geq h\) and regular weights, but our formulation allows us to consider small \(p\) or singular weights as well. We obtain results for several infinite classes of tilting modules, including the case \(p=2\), and tilting modules whose support variety corresponds to a hook partition. In the case \(p=2\), we prove the conjecture by S. Donkin for the support varieties of tilting modules. reductive algebraic group schemes; tilting modules; good filtrations; support varieties; cells of affine Weyl groups; nilpotent orbits Cooper, B. J., On the support varieties of tilting modules, J. Pure Appl. Algebra, 214, 1907-1921, (2010) Representation theory for linear algebraic groups, Group schemes, Cohomology theory for linear algebraic groups On the support varieties of tilting modules. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities On a non-singular projective algebraic variety \(X\) over \(\mathbb C\), using the existence theorem of Cauchy-Kovalevska and Serre's GAGA, we have the Riemann-Hilbert correspondence: There exists a unique algebraic vector bundle with a connection for a given representation of the fundamental group \(\pi_1(X)\) and vice versa. On a non-complete variety, the uniqueness no longer holds: There can exist several vector bundles with connection with a given monodromy. Deligne showed the existence and uniqueness when a regularity condition at infinity is assumed.
Instead of regularity, in this article, we think about a restriction of the underlying vector bundle of the connection. Deligne showed one can realize any character of the fundamental group of an affine curve \(U\) as the underlying monodromy of a connection \(\nabla = d-\omega\) on the trivial bundle of rank 1 on \(U\) (i.e., \({\mathcal O}_U)\).
The main theorem (theorem 4) generalizes Deligne's result (theorem 1): On an affine curve, there is a connection on the trivial bundle with a prescribed monodromy.
The aim of this note is to introduce Deligne's result on the rank 1 case and to generalize it in two directions: When \(U\) is of higher dimension and the case of higher rank representation of the fundamental group of an affine curve. affine curve; algebraic connection on the trivial bundle; Riemann-Hilbert correspondence; representation of the fundamental group; prescribed monodromy Vector bundles on curves and their moduli, Structure of families (Picard-Lefschetz, monodromy, etc.), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coverings of curves, fundamental group Monodromies of algebraic connections on the trivial bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 manuscript under review is an expanded version of a course of lectures given by B. Teissier in the framework of the ``2ndo Congreso Latinoamericano de Matemáticos'' held in Cancun, Mexico, on July 20--26, 2004. The aim of this work, which contains a large amount of various information, is to acquaint people in the mathematical community with the ideas and methods of the theory of polar varieties developed in the past 5-6 decades. Above all, in the special ``dedicatory'' the authors also address their article ``to those administrators of research who realize how much damage is done by the evaluation of mathematical research solely by the rankings of the journals in which it is published, or more generally by bibliometric indices''.
The study of polar varieties has its origin in a quite general observation, appeared in the late 1960s, that basic properties of complex spaces or varieties with singularities can be properly explained with the use of suitable partitions of singular varieties into finitely many nonsingular complex analytic manifolds. These manifolds, or strata, have the property that the local geometry of the space under consideration is constant on each stratum. Thus, in his earliest works \textit{H. Whitney} (see, e.g., [Ann. Math. 81, No. 3, 496--549 (1965; Zbl 0152.27701)]) initiated the study in complex analytic geometry of limits of tangent spaces at nonsingular points, the theory of stratifications and some related topological and differential-geometric constructions (see also [\textit{R. Thom}, Bull. Am. Math. Soc. 75, 240--284 (1969; Zbl 0197.20502)], [\textit{J. N. Mather}, Proc. Sympos. Univ. 1971, 195--232 (1973; Zbl 0286.58003)]).
The authors emphasize that one of the main points of the present article is to describe such partitions algebraically using polar varieties. Of course, for evident reasons, this approach can be applied in the case of reduced spaces only. As usually, the term ``local'' means that ``sufficiently small'' representatives of germs of the corresponding space are considered. In addition, the authors also assume that the spaces under consideration are equidimensional, i.e., all their irreducible components have the same dimension.
At the beginning of the manuscript, in the introduction, a very interesting and detailed historical background is given; the authors' attention focusses on those ideas and results which are closely related to the birth of the concept of polar varieties and its further development (see, e.g., [\textit{B. Teissier}, Apparent contours from Monge to Todd, 1830--1930: A century of geometry, Lect. Notes Phys. 402, 55--62 (1992)], [\textit{R. Piene}, Lect. Notes Comput. Sci. 8942, 139--150 (2015; Zbl 1439.14141)]). Then they discuss the basic properties of limits of tangent spaces, the notions of conormal spaces, projective duality, tangent cones, multiplicity, and describe many other standard and useful constructions in a quite elementary manner. The most important properties of normal cones and polar varieties, the basic relations between the conormal space and its Semple-Nash modification, are discussed in the third section. Then the authors prove that for a given complex analytic space, there is a unique minimal (coarsest) Whitney stratification. Any other Whitney stratification of the space is obtained by adding strata inside the strata of the minimal one (see [\textit{B. Teissier}, Lect. Notes Math. 961, 314--491 (1982; Zbl 0585.14008)]). Among other things, it is explained in an explicit form and formulas how the multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler-Poincaré characteristics, the degree of the dual of a projective variety, etc.
It should be noted that the authors' expressive exposition of highly nontrivial materials contains many nice pictures, examples, comments and remarks, as well as the list of bibliographies, including 127 items. Without any doubt, all this inspires enthusiastic readers to begin studying of the subject or to continue their own research at a higher level than before. polar varieties; equidimensional varieties; singularities; stratifications; Whitney stratifications; Whitney conditions; tubular neighborhoods; tangent cones; limits of tangent spaces; conormal spaces; projective duality, multiplicity; Nash modifications; Plücker-type formulas; Todd's formulas; characteristic classes; Chern classes Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, History of several complex variables and analytic spaces, History of mathematics in the 20th century, History of mathematics in the 21st century, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Stratified sets, Germs of analytic sets, local parametrization, Characteristic classes and numbers in differential topology, Families, moduli of curves (analytic), Local complex singularities, Singularities in algebraic geometry, History of algebraic geometry Local polar varieties in the geometric study of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for \({\mathfrak{sl}_2}\). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra \({\mathfrak{gl}_n}\). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules. Categorification; quantum groups; Lie algebras; canonical bases; flag varieties I. Frenkel, M. Khovanov and C. Stroppel, \textit{A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products}, \textit{Selecta Math. (N.S.)}\textbf{12} (2006) 379431 [math/0511467]. Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Representations of associative Artinian rings A categorification of finite-dimensional irreducible representations of quantum \({\mathfrak{sl}_2}\) and their tensor products | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(D_1\) and \(D_2\) be (associative) algebras finite-dimensional over their centres \(F_1\) and \(F_2\), respectively. Suppose that \(F_1\) and \(F_2\) contain as a subfield an algebraically closed field \(F\). As noted by the authors, the research presented in the paper under review has initially been motivated by the question of whether the tensor product \(D_1\otimes_FD_2\) is a domain. This question was posed by Schacher and studied, for instance, by \textit{G. M. Bergman} [Lect. Notes Math. 545, 32-82 (1976; Zbl 0331.16015)].
The paper gives an example where the answer is ``no'', as a part of which it obtains results of independent interest about division algebras and Brauer groups over curves; specifically, this includes a splitting criterion for certain Brauer group elements on the product of two curves over \(F\). This is preceded by a study of Picard group and Brauer group properties of \(F_1\otimes_FF_2\). At the same time, the authors show that the answer is ``yes'', provided that \(D_1=F_1\). Assuming in addition that \(F_1\) is endowed with a discrete valuation \(v\) and \(R\) is the valuation ring of \((F_1,v)\), they prove that \(D_1\otimes_FD_2\) is a domain whenever \(D_1\) is totally ramified at \(R\); in particular, this holds in case \(D_1/F_1\) has prime degree \(p\) different from the characteristic of \(F\), and \(D_1\) is ramified at \(R\). division algebras; tensor products; Schur indices; ramification; Picard groups; Brauer groups; products of curves; domains Louis Rowen and David J. Saltman, Tensor products of division algebras and fields, J. Algebra 394 (2013), 296 -- 309. Finite-dimensional division rings, Brauer groups of schemes, Picard groups, Brauer groups (algebraic aspects), Galois cohomology Tensor products of division algebras and fields. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the rational representations \(\phi\) : \(G\to GL(V)\) are studied, where V is a finite dimensional vector space and G is a semisimple group over \({\mathbb{C}}\) satisfying the following condition: the algebra \({\mathbb{C}}[V]^ U\) of U-invariant functions on V, where U is a maximal unipotent subgroup of G, is free. In this case V and the corresponding representation are called exceptional.
In the first part of the paper the closure of G-orbits in V are characterized. The second part of the paper is devoted to the classification of exceptional representations of simple groups. In the third part it is established that all singularities of the variety \(\overline{Gx}\), \(x\in V\), lie in its boundary.
[The results of this paper were announced in C. R. Acad. Sci., Paris, Sér. I 296, 5-6 (1983; Zbl 0538.14007)]. rational representations; semisimple group; exceptional representations; simple groups; singularities Brion, M., Représentations exceptionnelles des groupes semi-simples, Ann. sci. école. norm. sup., 18, 4, 345-387, (1985) Semisimple Lie groups and their representations, Geometric invariant theory, Representation theory for linear algebraic groups, Singularities of surfaces or higher-dimensional varieties Représentations exceptionnelles des groupes semi-simples | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Identical with the paper reviewed above. discriminant quadratic form; weakly distinguished basis; Milnor lattices of hypersurface singularities; Dynkin diagrams; elliptic hypersurface singularities; action of the braid group; bimodular singularities W. Ebeling : Milnor lattices and geometric bases of some special singularities . L'Enseignement Math. 29 (1983) 263-280. Singularities in algebraic geometry, Braid groups; Artin groups, Group actions on varieties or schemes (quotients), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex singularities, Quadratic forms over global rings and fields Milnor lattices and geometric bases of some special singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a singular algebraic variety defined over a perfect field \(k\), with quotient field \(K(X)\). Let \(s\geq 2\) be the highest multiplicity of \(X\) and let \(F_s(X)\) be the set of points of multiplicity \(s\). If \(Y\subset F_s(X)\) is a regular center and \(X\leftarrow X_1\) is the blow up at \(Y\), then the highest multiplicity of \(X_1\) is less than or equal to \(s\). A sequence of blow ups at regular centers \(Y_i\subset F_s (X_i)\), say \(X\leftarrow X_1\leftarrow\cdots\leftarrow X_n\), is said to be a \textit{simplification} of the multiplicity if the maximum multiplicity of \(X_n\) is strictly lower than that of \(X\), that is, if \(F_s(X_n)\) is empty. In characteristic zero there is an algorithm which assigns to each \(X\) a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties \(\beta : X^\prime \to X\) of generic rank \(r\geq 1\) (i.e., \([K( X^\prime):K(X)]=r)\). We will see that, when imposing suitable conditions on \(\beta\), there is a strong link between the strata of maximum multiplicity of \(X\) and \(X^\prime\), say \(F_s(X)\) and \(F_{rs}(X^\prime)\) respectively. In such case, we will say that the morphism is strongly transversal. When \(\beta :X^\prime\to X\) is strongly transversal one can obtain information about the simplification of the multiplicity of \(X\) from that of \(X^\prime\) and vice versa. Finally, we will see that given a singular variety \(X\) and a finite field extension \(L\) of \(K(X)\) of rank \(r\geq 1\), one can construct (at least locally, in étale topology) a strongly transversal morphism \(\beta :X^\prime\to X\), where \(X^\prime\) has quotient field \(L\). finite morphisms; multiplicity; Rees algebras; singularities Integral closure of commutative rings and ideals, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Finite morphisms and simultaneous reduction of the multiplicity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 considers the relation between deformations of a rational surface singularity with a reflexive module, and deformations of a partial resolution of the singularity with the locally free strict transform of the module. The results indicate how a family of small resolutions of a 3-dimensional index one terminal singularity and its flop are obtained by blowing up in a maximal Cohen-Macaulay module and its syzygy.
The article is motivated by the work on the geometrical McKay correspondence which can be said to give a one-to-one correspondence between the isomorphism classes of indecomposable reflexive modules \(\{M_i\}\) and the prime components \(\{E_j\}\) of the exceptional divisor in the minimal resolution \(\tilde X\rightarrow X\) of a rational double point (RDP), that is \(A_n\), \(D_n\), \(E_{6 - 8}.\)
For a natural class of special reflexive modules named Wunram modules after its inventor, the correspondence holds for any rational surface singularity. \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] used the endomorphism ring of a higher dimensional Wunram module to prove derived equivalences for flops, and this again led to attention to the \(2\)-dimensional case with interesting results by \textit{O. Iyama} and \textit{M. Wemyss} [Math. Z. 265, No. 1, 41--83 (2010; Zbl 1192.13012); Ill. J. Math. 55, No. 1, 325--341 (2011; Zbl 1258.13015)] and \textit{M. Wemyss} [Math. Ann. 350, No. 3, 631--659 (2011; Zbl 1233.14012)].
In this article, the authors prove that blowing up a rational surface singularity \(X\) in a reflexive module \(M\) gives a partial resolution \(f:Y\rightarrow X\) where \(Y\) is normal, dominated by the minimal resolution , and where the strict transform \(\mathcal M=f^\Delta(M)\) is locally free. This partial resolution is determined by the first Chern class \(c_1(\mathcal F)\) of the strict transform \(\mathcal F\) of \(M\) to \(\tilde X.\) Thus, in particular, any partial resolution dominated by the minimal resolution is given by blowing up in a Wunram module, and the authors mention the RDP-resolution obtained by contracting the \((-2)\)-curves in the minimal resolution in particular; this is given by blowing up in the canonical module \(\omega_X.\)
Consider the category of deformations \(\text{Def}_{Y,\mathcal M}\) of the pair \((Y,\mathcal M)\) blowing down to \((X,M)\). The main result in the present article says that the blowing down map \(\alpha:\text{Def}_{Y,\mathcal M}\rightarrow\text{Def}_{X,M}\) is injective and that it commutes with the forgetful map \(\beta:\text{Def}_{Y,\mathcal M}\rightarrow\text{Def}_{Y}\) and the blowing down map \(\delta:\text{Def}_Y\rightarrow\text{Def}_X.\) Furthermore, the forgetful map \(\beta\) is smooth and an isomorphism in many situations. The blowing down map \(\delta\) is a Galois covering onto the Artin component \(A\) on spaces, which for RDPs equals \(\text{Def}_X\). In general, it not injective, making the injectivity of \(\alpha\) surprising. The authors prove that \(\beta\) is an isomorphism if \(M\) is Wunram, implying that \(\delta\) factors through a closed embedding \(\alpha\beta^{-1}:\text{Def}_Y\subseteq\text{Def}_{(X,M)}\) realizing deformations of the pair as conjectured by \textit{C. Curto} and \textit{D. R. Morrison} [J. Algebr. Geom. 22, No. 4, 599--627 (2013; Zbl 1360.14053)] in the RDP case.
A deformation of the pair \((X,M)\) in the geometric image of \(\text{Def}_{(Y,\mathcal M)}\) lifts to a deformation of \((Y,\mathcal M)\) without any base change. In general, \(\text{Def}_{X,M}\) is not dominated by \(\text{Def}_{(Y,\mathcal M)}\) even for RDPs. A main ingredient in Wahl's proof that the covering \(\text{Def}_{\tilde X}\rightarrow A\) has Galois action by a product of Weyl groups is the injectivity of \(\delta\) in the case that \(Y\) is the RDP resolution. This follows directly from the authors result because \(\text{Def}_{X,\omega_X}\cong\text{Def}_{X}\). The results indicate that there are interesting relations to \(\text{Def}_{X}\), for instance the component structure.
The main application of the result above is a generalization of three conjectures of Curto and Morrison [loc. cit.] concerning the nature of small partial resolutions of \(3\)-dimensional index one terminal singularities and their flops. When \(g:W\rightarrow Z\) is a small partial resolution and \(X\subseteq Z\) is a sufficiently generic hyperplane section with strict transform \(f:Y\rightarrow X\), a result of Reid states that \(f\) is a partial resolution of an RDP. Thus \(g\) is a \(1\)-parameter deformation of \(f\) and so an element in \(\text{Def}_Y\). The authors prove that \(Y\) then is the blowing up of \(X\) in a reflexive module \(M\). As a consequence, \(\alpha\beta^{-1}\) takes this \(g\) to a \(1\)-parameter deformation of \((Z,N)\) of the pair \((X,M).\) Verbatim:
Corollary. There is a maximal Cohen-Macaulay \(\mathcal O_Z\)-module \(N\) such that (i) The small partial resoltuion \(W\rightarrow Z\) is given by blowing up \(Z\) in \(N.\) (ii) Blowing up \(Z\) in the syzygy module \(N^+\) of \(N\) gives the unique flop \(W^+\rightarrow Z\). (iii) The length of the flop equals the rank of \(N\) if the flop is simple.
A version of this statement is given for flat families of small partial resolutions and flops. It is proved that there is a family of pairs \((\mathbf{X},\mathbf{M})\) in \(\text{Def}_{(X,M)}\) such that the blowing up of \(\mathbf{X}\) in \(\mathbf{M}\) and in the syzygy \(\mathbf{X}^+\) give two simultaneous partial resolutions \(\mathbf{Y}\rightarrow\mathbf{X}\leftarrow\mathbf{Y}^+\) inducing any local family of flops \(g\) by pullback, for any \(g\) with hyperplane section \(f\). In fact, \(\mathbf{X}\) equals \(A_1, D_4, E_6, E_7, E_8\) for \(l=1,2,3,4,5,6\) respectively, so that the result above proves that there is in each case a unique reflexive module \(M\) of rank \(l\) such that any simple flop of length \(l\) is obtained by pullback from the \(\mathbf{Y}\rightarrow\mathbf{X}\leftarrow\mathbf{Y}^+.\) This gives the universal simple flop of length \(l\) realized as blowing-ups in families of reflexive modules (as suggested by Curto and Morrison [loc. cit.]).
The RDPs are hypersurfaces, and any maximal Cohen-Macaulay module is given by a matrix factorization. The conjectures of Curto and Morrison [loc. cit.] are stated by matrix factorizations, and they are verified for \(A_n\) and \(D_n\) by brute force computations. The present argument is conceptual, coordinate-free, and makes the conjectures transparent.
The singularities considered in this work will all be henselisations of finite algebras and the results will therefore have finite type representations locally in the étale topology.
Work by Donovan, Wemyss et.al. links properties of various noncommutative algebras to flops. This involves quiver algebras, mutations, tilting theory and GIT-constructions with endomorphism algebras as input. This work offers a direct proof of the original Curto-Morrison conjectures using deformation theory where the blowing up ideal for the small, partial resolution is obtained directly from the \(2\)-dimensional Wunram module. Any flop with fixed RDP hyperplane section and Dynkin diagram is a pullback from a pair of universal blowing ups.
The article is impressing. The deformation functors are studied conceptually, more as fibred categories than as functors, and natural transformations are transformed into maps between the categories of resolutions of singularities. The article gives a lot of techniques that that can be used in the study of more general contractions, and shows a brilliant use of deformation theory in general. Also, the article is self contained with respect to the deformation theory, and contains all preliminaries needed for understanding the importance of the results. flatifying blowing-up; maximal Cohen Macaulay module; simultaneous partial resolution; small resolution; rational double point; RDP; matrix factorization; deformation of algebras; deformation of rational singularities; deformations of exceptional module; partial resolution; domination of resolution; contracting curves; strict transform; Wunram module; blowing up Deformations of singularities, Minimal model program (Mori theory, extremal rays), Stacks and moduli problems, McKay correspondence Deformations of rational surface singularities and reflexive modules with an application to flops | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(X=G/K\) be a connected Riemannian space with isometry group \(G\) and denote by \({\mathcal D}(X)^G\) the algebra of \(G\)-invariant differential operators. If \({\mathcal D}(X)^G\) is commutative then \(X\) is also called commutative and the pair \((G,K)\) is called a Gelfand pair. Several other characterizations of commutative spaces, for instance in terms of the Poisson bracket operation on the universal enveloping algebra, are discussed.
\(X\) is said to be of Heisenberg type if in the Levi decomposition \(G=N\rtimes L\) one has \(L=K\). It is called saturated if \(N_F(K)^0=K\). A list of all saturated commutative spaces of Heisenberg type is given, for which \(\mathfrak{n}/\left[\mathfrak{n},\mathfrak{n}\right]\) is reducible. Gel'fand pair; Lie groups; Poisson algebras O. S. Yakimova, Saturated commutative homogeneous spaces of Heisenberg type, Acta Applicandae Math. 81 (2004), 339--345. Homogeneous spaces, Group actions on varieties or schemes (quotients), Differential geometry of homogeneous manifolds Saturated commutative spaces of Heisenberg 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 In this paper the authors classify all surfaces with \(p_g=q=0\) which are isogenous to a product of curves. A surface \(S\) is said to be isogenous to a product of curves if \(S\) has a finite étale cover which is isomorphic to the product of two curves. \(S\) is said to be isogenous to a higher product if both curves have genus bigger or equal to 2.
These definitions are due to the second author [Am. J. Math. 122, No. 1, 1--44 (2000; Zbl 0983.14013)]. In the same paper it is proved that \(S\) is isogenous to a higher product if and only if there is a finite étale Galois cover of \(S\) isomorphic to a product of two curves of genera at least two, i.e. \(S \cong (C_1 \times C_2)/G\), where \(G\) is a finite group acting freely on the product \(C_1 \times C_2\). Moreover there exists a unique minimal such Galois realization \(S \cong (C_1 \times C_2)/G\). There are two cases: the unmixed case where \(G\) acts via diagonal action, and the mixed case where the action of \(G\) exchanges the two factors. In the mixed case the curves \(C_1\) and \(C_2\) have to be isomorphic to each other. It turns out that any surface with \(p_g=q=0\) and isogenous to a product is either \(\mathbb{P}^1 \times \mathbb{P}^1\) or it is isogenous to a higher product, and this last condition is equivalently to \(S\) being of general type. Thus in order to classify all smooth projective surfaces \(S\) isogenous to a product of curves with \(p_g=q=0\), the authors assume without loss of generality that \(S\ncong \mathbb{P}^1 \times \mathbb{P}^1\) and therefore that \(S\) is of general type.
Previously [in: The Fano Conference, 123--142, Univ. Torino, Turin (2004; Zbl 1078.14051)], the first two authors studied smooth projective surfaces isogenous to a product of curves and of unmixed type with \(p_g=q=0\), and with \(G\) a finite abelian group. They showed that only four abelian groups \(G\) are possible in this case. Moreover, they showed that these groups really occur and gave a complete description of the connected components of the moduli space formed by the corresponding surfaces \((C_1\times C_2)/G\).
In the article under review, the authors complete this classification admitting arbitrary finite groups and considering also the mixed case. They find all finite groups occurring as groups \(G\) associated to a surface isogenous to a product of curves with \(p_g=q=0\), show that these groups really occur and describe the components formed by the corresponding surfaces \((C_1\times C_2)/G\) inside the moduli space \(\mathfrak{M}_{(1,8)}\) of minimal smooth complex projective surfaces with \(\chi (S) = 1\) and \(K_S^ 2 = 8\). In total, beside the trivial case \(S\cong \mathbb{P}^1 \times \mathbb{P}^1\), the authors find 17 families of such surfaces of general type. Many of these families contain new and interesting examples of surfaces of general type with \(p_g=q=0\).
The proof relies heavily on the use of the \texttt{MAGMA}-library containing all groups of low order. The authors remark that the main result of the paper can be seen as the solution, in a very special case, to the program advanced by David Mumford at the Montreal Conference in 1980, which says that problem on the existence of surfaces of general type with \(p_g=0\) or the number of components of their moduli space are in principle solvable by computer. surfaces isogenous to a product of curves; surfaces of general type; groups of small order; MAGMA's library Bauer, I.; Catanese, F.; Grunewald, F., The classification of surfaces with \(p_g = q = 0\) isogenous to a product of curves, Special Issue: In Honor of Fedor Bogomolov. Part 1. Special Issue: In Honor of Fedor Bogomolov. Part 1, Pure Appl. Math. Q., 4, 2, 547-586, (2008) Moduli, classification: analytic theory; relations with modular forms, Surfaces of general type The classification of surfaces with \(p_g=q=0\) isogenous to a product of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In connection with the study of certain special actions of semisimple algebraic groups, \textit{V. Popov} [Algebraic groups and homogeneous spaces. Proceedings of the international colloquium, Mumbai, India, 2004. Studies in Mathematics. Tata Inst. Fundam. Res. 19, 481--523 (2007; Zbl 1135.14038)] introduced the notion of the separation index \(\text{sep}(\Delta)\) of an irreducible root system \(\Delta\) and obtained the bounds \(\text{rk}(\Delta) + 1 \leq \text{sep}(\Delta) \leq | W(\Delta)|\), where \(W(\Delta)\) is the Weyl group of \(\Delta\). In further papers by V. \textit{S. Zhgun} and \textit{D. V. Mironov} [Russ. Math. Surv. 62, No. 6, 1228--1230 (2007); translation from Usp. Mat. Nauk 62, No. 6, 171--172 (2007; Zbl 1189.17010); Math. Notes 82, No. 2, 272--276 (2007; Zbl 1287.17023); translation from Mat. Zametki 82, No. 2, 310--314 (2007)] and by \textit{F. V. Petrov} [in: Mathematical Education (MTsNMO, Moscow, 2009), Ser. 3, Vol. 13, 189--190 (2009)], it was shown that the upper bound can be rendered much smaller than the order of the Weyl group. For example, for root systems of types \(B_\ell\), \(C_\ell\), and \(D_\ell\), a bound of order \(2^\ell\) was obtained. The objective of this paper is to sharpen the lower and the upper bound for the separation index. In particular, we prove that the separation index of a classical root system of rank \(\ell\) is at most \(2^{\ell^2}\). separation index of a root system; irreducible root system; Weyl group; Weyl chamber; open dual cone; Dynkin diagram Root systems, Simple, semisimple, reductive (super)algebras, Group actions on varieties or schemes (quotients) Separation indices of irreducible root systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This book is neither a graduate text nor a research monograph, but somewhere in between. It expounds how to compute topological invariants of algebraic varieties (mostly hypersurfaces) and their complements --- homology groups, fundamental groups, Alexander polynomials, and (eventually mixed) Hodge structures, in both global and local cases, with some emphasis on the relation between the two. Inevitably this requires rather an extensive array of prerequisites. This is dealt with in 3 ways: first, there are three introductory chapters explaining respectively Whitney stratifications, the structure of plane curve and normal surface singularities, and the Milnor fibration and lattice for an isolated hypersurface singularity. As well as giving background material on these topics, numerous examples are included, and calculations of topological invariants giving a foundation for subsequent calculations. Secondly, three appendices give digests of information on integral bilinear forms, weighted projective varieties and mixed Hodge structures respectively. Thirdly, the author is always ready to quote results from other sources and refer the reader to them for further information; modulo this, the book is reasonably self-contained.
The three main chapters of the book describe methods of computation of fundamental groups of hypersurface complements, of cohomology of complete intersections (smooth or with isolated singularities), and of de Rham cohomology of complements of hypersurfaces (with critical locus of dimension \(\leq 1\)). The book is well written: the explanations are firmly rooted in detailed treatments of examples, which indeed occupy most of the text.
This is not a book to skip over: without following how the calculations are done, one loses the whole point. There is a steady progression to more sophisticated ideas, and the final chapter culminates with some very nice results of the author calculating Alexander polynomials, where a crucial ingredient is the defect of some linear system.
It is unusual to find a book devoted to explaining how to make calculations. This is not a book to suit everyone, but for those who want to understand how to calculate topological invariants in local and global complex algebraic geometry it is unrivalled. topological invariants of algebraic varieties; homology groups; fundamental groups; Alexander polynomials; Hodge structures; Whitney stratifications; plane curve and normal surface singularities; Milnor fibration; lattice for an isolated hypersurface singularity; integral bilinear forms; weighted projective varieties; mixed Hodge structures; hypersurface complements; cohomology of complete intersections A. Dimca, ''Singularities and Topology of Hypersurfaces'', Universitext, Springer-Verlag, New York, 1992. DOI: 10.1007/978-1-4612-4404-2 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes, Research exposition (monographs, survey articles) pertaining to algebraic topology, Algebraic topology on manifolds and differential topology, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Deformations of complex singularities; vanishing cycles, Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Classical real and complex (co)homology in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), de Rham cohomology and algebraic geometry, Mixed Hodge theory of singular varieties (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Milnor fibration; relations with knot theory, Stratifications in topological manifolds, Complex surface and hypersurface singularities Singularities and topology of 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 We classify coherent modules on \(k [x, y]\) of length at most 4 and supported at the origin. We compare our calculation with the motivic class of the moduli stack parametrizing such modules, extracted from the Feit-Fine formula. We observe that the natural torus action on this stack has finitely many fixed points, corresponding to connected skew Ferrers diagrams. coherent sheaves; finite length modules; Grothendieck ring of varieties; Hilbert scheme of points; torus actions Stacks and moduli problems, Parametrization (Chow and Hilbert schemes) On coherent sheaves of small length on the affine plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves some theorems describing equations of a noncylindrical algebraic surface in an Euclidean space which has infinite set of planes of oblique (in particular orthogonal) symmetry. The main goal of the paper is to give detailed (in individual cases, modified) proofs of the results of two early papers of the author [Differ. Geom. Mnogoobrazij Figur. 7, 34-39 (1976; Zbl 0429.51011) and Ukr. Geom. Sb. 20, 35-46 (1977; Zbl 0429.51012)]. Some results of invariant theory of reflection groups are used in the proofs of the theorems of the reviewed paper. algebraic surface; Euclidean space; symmetry; invariant theory of reflection groups V. F. Ignatenko, ''Algebraic surfaces with an infinite set of skew symmetry planes, I,''Ukr. Geom. Sb., No. 32, 47--60 (1989). Questions of classical algebraic geometry, Projective techniques in algebraic geometry, Reflection groups, reflection geometries Algebraic surfaces with an infinite set of planes of oblique symmetry. 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 Let Y be a K3 surface obtained by resolution of singularities of the two- sheeted covering of \({\mathbb{P}}^ 2_{{\mathbb{C}}}\) branched along six lines. The Schoen construction allows one to describe Y on the other hand as the quotient (by a finite group) of the symmetric square of a curve C of genus 5 having an automorphism J of order 4 such that \(C/J=E\) is an elliptic curve. One proves the
Theorem: Let P be the Prym variety of the four-sheeted covering \(C\to C/J=E\), i.e., the connected component of the unity in ker(Jac(C)\(\to Jac(E))\) \((\dim (P)=4)\). Then there exists an algebraic cycle \(\gamma \in CH^ 2(P\times Y)\) such that the corresponding homomorphism \(H^ 2(Y,{\mathbb{Q}})\to H^ 2(P,{\mathbb{Q}})\) is a monomorphism on the lattice of transcendental cycles.
The algebraic correspondence \(\gamma\) is the Kuga-Satake-Deligne correspondence for K3 surfaces of the considered type up to isogenies and copies of P as a direct summand. K3 surface; resolution of singularities; Prym variety; lattice of transcendental cycles; Kuga-Satake-Deligne correspondence Bonfanti, M.: On the cohomology of regular surfaces isogenous to a product of curves with \(\chi ({\mathcal O}_S)=2\). arXiv:1512.03168v1 Picard schemes, higher Jacobians, \(K3\) surfaces and Enriques surfaces, Transcendental methods, Hodge theory (algebro-geometric aspects), Elliptic curves, Algebraic cycles Abelian varieties associated to certain K3 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 Let g be a fixed integer. Let \(X_ g=Spec(R_ g)\) (where \(R_ g=S.(S_ gV)\), V being a two-dimensional vector space) be the space of binary forms of degree \( g.\) Let \(X_{p,g}\subset X_ g\) be the subset of binary forms having a root of multiplicity \(\geq p\) and \(J_ p\) be the ideal of polynomial functions vanishing on \(X_{p,g}\). The main result of the paper is the following theorem 3:
The generators of the ideal \(J_ p\) have degrees \(\leq 4\) for \(p\geq [g/2]+1.\)
The generators of \(J_ p\) are explicitly described for \(p>[g/2]+1\). The Hilbert function of \(R_ g/J_ p\) and the decomposition of \(R_ g/J_ p\) into representations of SL(V) are also explicitly described. generators of ideal of algebraic set; binary forms; polynomial functions; Hilbert function Weyman, J., The equations of strata for binary forms, J. Algebra, 122, 1, 244-249, (1989) Schemes and morphisms, Rational and unirational varieties The equations of strata for binary 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 \(S\) be an hyperelliptic surface of general type, \(f:S\to C\) a genus \(g\) hyperelliptic fibration. In this paper, we prove that if the maximal \(\mathbb Z_2\)-quotient rank of the vertical part of the fundamental group of \(S\) is \(r\), then its slope
\[
\lambda(f) \geq \begin{cases} 4+ \frac{4r-8}{g(r+1)-r^2},\quad &\text{if \(r\) is even},\\ 4+\frac{4r-8}{g(r+3)-(r+1)^2},\quad &\text{if \(r\) is odd},\end{cases}
\]
with equality only if the branch locus \(R\) of the double cover induced by the hyperelliptic involution on \(S\) has \((r+1\to r+1)\) type singularities (if \(r\) is even), or \((r+2\to r+2)\) type singularities (if \(r\) is odd). double cover; hyperelliptic surface of general type; hyperelliptic fibration; fundamental group; slope; singularities Surfaces of general type, Coverings in algebraic geometry, Fibrations, degenerations in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The slopes of hyperelliptic surfaces with \(Z_2\)-quotient ranks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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, for \(F_{1}\) and \(F_{2}\) subbundles of a vector bundle \(E\), a `Koszul duality' equivalence between derived categories of \(\mathbb G_{\mathbf m}\)-equivalent coherent (dg-)sheaves on the derived intersection \(F_1\overset {R} \cap _EF_2\), and the corresponding derived intersection \(F_1^{\perp}\overset {R} \cap _{E^*}F_2^{\perp}\). We also propose applications to Hecke algebras. dg-schemes; sheaves of dg-algebras; derived intersection; Koszul duality I. Mirkovíc and S. Riche, Linear Koszul duality, Compos. Math. 146 (2010), no. 1, 233-258. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differential graded algebras and applications (associative algebraic aspects), Derived categories, triangulated categories Linear Koszul duality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 investigate the deformations of pairs \((X,L)\), where \(L\) is a line bundle on a smooth projective variety \(X\), defined over an algebraically closed field \(\mathbb{K}\) of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair \((X,L)\) is homotopy abelian whenever \(X\) has trivial canonical bundle, and so these deformations are unobstructed. deformations of manifold and line bundle; differential graded Lie algebras Formal methods and deformations in algebraic geometry, Deformations of complex structures, Deformations and infinitesimal methods in commutative ring theory, Graded Lie (super)algebras, Global differential geometry of Hermitian and Kählerian manifolds, Deformations of general structures on manifolds, Deformations of fiber bundles Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 geometric criterion for Dirichlet \(L\)-functions associated to cyclic characters over the rational function field \(\mathbb{F}_q (t)\) to vanish at the central point \(s = \frac{1}{2}\). The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over \(\mathbb{F}_q\). Using this geometric criterion, we obtain a lower bound on the number of cubic characters over \(\mathbb{F}_q (t)\) whose \(L\)-functions vanish at the central point where \(q = p^{4n}\) for any rational prime \(p \equiv 2 \mod 3\). We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the \(L\)-functions of Dirichlet characters of other orders. Chowla's conjecture; \(L\)-functions; zeta functions of curves; Carlitz extensions; cyclotomic function fields; abelian varieties over finite fields Zeta and \(L\)-functions in characteristic \(p\), Zeta functions and \(L\)-functions, Cyclotomy, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Vanishing of Dirichlet \(L\)-functions at the central point over function fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The notion of a blow analytic homeomorphism introduced in [\textit{T.-C. Kuo}, J. Math. Soc. Jap. 32, 605-614 (1980; Zbl 0509.58007)] and its applications to the classification theory of real analytic function-germs are discussed. In particular, the author explains that for the two blow analytic equivalent functions \(f = z(x^4+y^6)+x^6\) and \(g = z(x^4+y^6)-xy^5+x^6\) there does not exist a blow analytic homeomorphism \(h\) which is Lipschitz such that \(g = f \circ h\). real singularity; blow analytic homeomorphism; bianalytic isomorphisms; classification of real singularities; arc analytic functions; blow-up; modifications; analytic arcs; Lipschitz map Paunescu, L.: An example of blow analytic homeomorphism. Pitman res. Notes math. Ser. 381, 62-63 (1998) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Modifications; resolution of singularities (complex-analytic aspects), Jets in global analysis, Semi-analytic sets, subanalytic sets, and generalizations, Singularities of differentiable mappings in differential topology, Real-analytic and semi-analytic sets, Real-analytic functions, Lipschitz (Hölder) classes An example of blow analytic homeomorphism | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 second part of the authors, ibid., 109-160 (1997; Zbl 0879.58007), see the review above. It presents numerous results about mappings of cusp type or, in other words, mappings \(F\), which in a suitable nonlinear system of ``coordinates'' \(\mathbb{R}^2\times E\) can be represented in the form \(F(s,t,v) =(s^3-ts,t,v)\); some other singularities are also presented in this part. The account is also based on the abstract global characterization of the cusp maps which was obtained by the authors in 1993.
The contents of this part are: (13) Introduction; (14) Critical values of Fredholm mappings; (15) Applications of critical values to nonlinear differential equations; (16) Factorization of differentiable maps; (17) Local structure of cusps; (18) Some local cusp results (Lazzeri-Micheletti cusp study, Cafagna-Tarantello multiplicity results, Lupo-Micheletti cusp, other local cusp results); (19) von Kármán equation; (20) Abstract global characterization of the cusp map; (21) Mandhyan integral operator cusp map; (22) Pseudo-cusp; (23) Cafagna and Donati theorems on ordinary differential equations (Cafagna-Donati global cusp map, Donati pendulum cusp, Cafasgna-Donati generalized Riccati equation); (24) Micheletti cusp-like map; (25) Cafagna Dirichlet example; (26) \(u^3\) Dirichlet map -- initial results; (27) \(u^3\) Dirichlet map -- the singular set and its image; (28) \(u^3\) Dirichlet map -- the global results; (29) Ruf \(u^3\) Neumann cusp map; (30) Ruf's higher order singularities; (31) Damon's work in differential equations.
The second part of this survey is written with the same accuracy and fullness as the first one; the acquaintence with both parts of this survey is undoubtedly useful to all specialists in the field and all who study Nonlinear Analysis. survey; mappings of cusp type; singularities Church, P. T.; Timourian, J. G.: Global structure for nonlinear operators in differential and integral equations. I. folds; II. Cusps: topological nonlinear analysis. (1997) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Equations involving nonlinear operators (general), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Global structure for nonlinear operators in differential and integral equations. II: Cusps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(P\) be an algebra of complex polynomials in \(\lambda_0, \ldots, \lambda_n\) and \(L_P\) the free left \(P\)-module with a basis \(1, l_0, \ldots, l_n\). Define the structure of a Lie algebra \(\mathfrak a(C,V)\) on \(L_P\) by setting
\[
[l_i,l_j]=\sum c_{i,j}^k(\lambda)l_k,\quad [l_i,\lambda_q]=v_{i,q}(\lambda), \quad [\lambda_i,\lambda_j]=0,
\]
where \(c_{i,j}^k, v_{i,q}(\lambda)\in P\) and \(C=\{c_{i,j}^k\}\), \(V=\{v_{i,q}\}\). It is easy to see that the linear span of \( l_0, \ldots, l_n\) is a Lie subalgebra and the matrix \(V\) induces a representation of \(\mathfrak a(C,V)\) into the Lie algebra \(\text{Der}\, P\) of derivations of the algebra \(P\). A Lie algebra \(\mathfrak a(C,V)\) is non-degenerate if \(\Delta=\det V\neq 0\). The algebra \(\mathfrak a(0,1)\) is isomorphic to the canonical Lie algebra on \(\mathcal L=P \oplus P\partial_1\oplus \cdots \oplus P\partial_n.\) A homomorphism of \(P\)-modules \(L_P\to \mathcal L\) defined by
\[
l_j\mapsto L_j=\sum_qv_{j,q}(\lambda)\partial_q
\]
is a homomorphism of Lie algebras. This representation is faithful if and only if \(\mathfrak a(C,V)\) is non-degenerate. One of the main results of the paper shows that if \(\mathfrak a(C,V)\) is non-degenerate then
\[
L_i(\Delta)=\phi_i\Delta,\quad \phi_i\in P,
\]
and the vector fields \(L_j\) are tangent to the hyperspace \(\{(\lambda_0, \ldots, \lambda_n)\in \mathbb C^{n+1}\mid \Delta=0\}\). The authors study changes of \(C\) under actions of invertible linear operators on \(V\) and gradings on \(\mathfrak a(C,V)\). In terms of an operator of a symmetrization a structure of a coassociative coalgebra on an affine commutative algebra is constructed. This construction is applied to a study of convolution matrices of invariants for isolated singularities of functions. Lie algebra of vector fields; polynomial Lie algebras Бухштабер, В. М.; Лейкин, Д. В., Функц. анализ и его прил., 36, 4, 18-34, (2002) Lie algebras of vector fields and related (super) algebras, Applications of Lie algebras and superalgebras to integrable systems, Singularities in algebraic geometry Polynomial Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The arithmetical theory of modular functions has been a central theme in the 19th century mathematics. Many outstanding mathematicians have contributed a wealth of ideas and results to this fascinating area, and it has taken more then 150 years for this theory to reach its contemporary, highly developed state. At present, the theory of modular functions enjoys a new period of great interest, vast activities, and spectacular applications, particularly in algebraic geometry, arithmetical geometry, algebraic number theory, and - recently - in the mathematical framework of quantum field theory.
One of the main sources of the theory of modular functions is the theory of complex multiplication of elliptic functions, whose origin can be traced back to earlier papers by Gauss, Abel, and Eisenstein. However, the theory of complex multiplication is mainly associated with the name of Leopold Kronecker, who not only made decisive contributions to it, but also inspired (and challenged) the following generations of mathematicians by a general problem which he called the most beloved dream of his youth (``liebster Jugendtraum''). This dream of his was to see the formulation of a complete theory of complex multiplication in full comprehension.
In the present book, the author studies Kronecker's ``Jugendtraum'' from its genesis, i.e., from the development of elliptic function theory, up to the present state of modular function theory, including its most recent achievements and applications. According to this pretentious programme, the book is divided into three parts.
Part I is of historical nature and describes the development of complex multiplication theory in its relationship to the arithmetical theory of modular functions. Chapter I is devoted to Kronecker's biography, comments on his main works, and the lasting impact of his ingenious ideas. Chapter II sketches the development of various topics in mathematics (such as elliptic functions, modular equations, theta- functions, elliptic curves, the origins of algebraic number theory from Gauss to Kummer, etc.) that finally led to questions related to Kronecker's ``Jugendtraum''. Chapter III discusses the works of Kronecker's predecessors with regard to complex multiplication theory, namely the papers of Gauss, Abel, and Eisenstein.
Kronecker's synthesis, i.e., his papers devoted to his ``Jugendtraum'', is analyzed in Chapters IV, whilst Chapter V explains the development of complex multiplication theory after Kronecker, i.e., from the research at the end of the 19th century, via M. Deuring's algebraic approach, up to its multidimensional generalization formulated by Weil, Shimura and Taniyama between 1950 and 1960. In an Appendix A, at the end of Part I, the author recalls some definitions and results from algebraic number theory, elliptic function theory, and the theory of elliptic curves, which are helpful to the reader of this historical part of the book.
Part II is just a reprint of L. Kronecker's original and very important paper ``Zur Theorie der elliptischen Funktionen. XI'' published in Berl. Ber. 1886, 701-780 (1986; JFM 18.0396.04), which contains many crucial ideas of the arithmetical theory of modular functions, viewed by Kronecker himself as the fundamentals for his ``Jugendtraum''.
Part III, which takes up nearly one half of the book, is devoted to the current state and various applications of the theory. Chapter I discusses three main approaches to modular function theory from the contemporary point of view, namely the viewpoint of function theory (modular functions and modular forms), the viewpoint of algebraic geometry (modular curves), and the viewpoint of representation theory (automorphic representations). Some topics from the arithmetic theory of modular forms are treated in Chapter II. Here the author focusses on the Eichler-Shimura relation, whose prototype was discovered by Kronecker in his paper reprinted in Part II of the book, and its applications to zeta-functions of modular curves, cusp forms and \(\ell\)-adic representations, and the modern theory of complex multiplication.
Chapter III deals with a very recent development in the theory, namely with the finite-characteristic analogue in the function field case, the so-called Drinfeld theory of modular functions. This includes Drinfeld's theory of elliptic modules over a field and their morphisms, the theory of nonarchimedean modular forms on rigid analytic spaces, the Tate algebra, moduli schemes for elliptic modules and their compactifications, and some fragments from the Langlands program.
The final Chapter IV is devoted to five selected, utmost brilliant and celebrated applications of the theory of modular functions. The first four of them deal with rather subtle problems in number theory, namely with effective methods in the arithmetic of complex quadratic fields, bounds for division points on elliptic curves, the main conjecture in the Iwasawa theory and its proof (by Mazur and Wiles) via the theory of abelian extensions of \({\mathbb{Q}}\) and their class fields, and the very recently discovered link between modular function theory and Fermat's Last Theorem (via the Taniyama-Weil conjecture). The fifth, and concluding, application consists in explaining Goppa's construction of error-correcting codes from algebraic curves over finite fields and its application to different classes of modular curves. This leads to important new results concerning the parameters for error-correcting codes of large length. Chapter III ends with an Appendix B, in which the author recalls the basic facts from the arithmetic theory of elliptic curves, as they are used through the entire Chapter III. The whole treatise is enhanced by some bibliographic remarks to Part III (i.e., to the recent developments), and by a carefully selected bibliography, ranging from the origins to the most recent papers.
Altogether, this book is a brilliant piece of mathematical culture, an invaluable guide and reference book of encyclopedic character, and a welcome source of inspiration for any interested reader, including specialists in the field, students, and historians specialized in the 19th century mathematics. The excellent style of writing and the many (complete or sketched) proofs, the introductory remarks, and the added appendices make this book even largely self-contained, at least for the first, informative reading. The author really succeeded in producing a masterpiece of mathematical writing! arithmetical theory of modular functions; complex multiplication of elliptic functions; FdM 18, 396; modular curves; automorphic representations; \(\ell \)-adic representations; Drinfeld theory; elliptic modules; error-correcting codes; elliptic curves Research exposition (monographs, survey articles) pertaining to number theory, Modular and automorphic functions, Complex multiplication and moduli of abelian varieties, Complex multiplication and abelian varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Modular and Shimura varieties, Families, moduli of curves (algebraic), History of mathematics in the 19th century, Collected or selected works; reprintings or translations of classics, History of number theory Kronecker's Jugendtraum and modular functions. Transl. from the Russian by M. Tsfasman | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 constructs a closed subscheme \(\underline{\text{Hom}}_S^{\text{full}}((\mathbb{Z}/p\mathbb{Z})^2, \mu_p \times \mu_p)\) of \(\underline{\text{Hom}}_S((\mathbb{Z}/p\mathbb{Z})^2, \mu_p \times \mu_p)\) such that \(\underline{\text{Hom}}_S^{\text{full}}((\mathbb{Z}/p\mathbb{Z})^2, \mu_p \times \mu_p)\) is flat over \(\mathbb{Z}\). He futher explains that how this closed subscheme is related to the notion of ``full set of sections'' or ``\(\times\)-homomorphisms'' by \textit{N. M. Katz} and \textit{B. Mazur} [Arithmetic moduli of elliptic curves. Princeton, New Jersey: Princeton University Press (1985; Zbl 0576.14026)]. level structures; finite flat group schemes; roots of unity Wake, P, Full level structures revisited: pairs of roots of unity, J. Number Theory, 168, 81-100, (2016) Group schemes, Modular and Shimura varieties Full level structures revisited: pairs of roots of unity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 smoothness of a morphism between Noetherian schemes can be recognized in terms of the induced mappings between tangent and obstruction spaces. This observation can be effectively applied in the study of schemes parametrizing certain objects of interest in deformation theory. Strong versions of the classical results on rigidity and stability of subalgebras in finite-dimensional Lie algebras are derived as an application. Some special cohomological conditions ensuring rigidity or stability are obtained in case of a field of positive characteristic. smoothness; Noetherian schemes; rigidity; stability; Lie algebras S. M. Skryabin, ''Smoothness, flatness and deformations of subalgebras,''Preprint, Manchester Centre Pure Mathematics (1997). Homological methods in Lie (super)algebras, Formal methods and deformations in algebraic geometry, Infinitesimal methods in algebraic geometry Smoothness, flatness and deformations of subalgebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) denote a regular local ring and \(K\) be its field of fractions. Let \(A_1\) and \(A_2\) be Azumaya algebras with involutions over \(R\). The author proves that if \(A_1\) and \(A_2\), when extended to \(K\), are isomorphic as \(K\)-algebras, then they are isomorphic over \(R\) itself. This shows in particular that two quadratic spaces over \(R\) which are similar over \(K\) are in fact similar over \(R\).
Grothendieck conjectured that for any reductive group scheme \(G\) over \(R\), rationally trivial \(G\)-homogeneous spaces are trivial. Via Weil's description of adjoint algebraic groups of classical type, in terms of algebras with involutions, the results in the paper prove the above mentioned conjecture for adjoint classical groups.
The conjecture of Grothendieck has been settled in most cases when \(R\) is essentially a smooth local \(k\)-algebra and \(G\) is defined over \(k\) (i.e. \(G\) is constant). Colliot-Thélène and Ojanguren proved the conjecture for perfect infinite \(k\) and Raghunathan for any infinite field \(k\). The case of finite base field is open and for non-constant \(G\), only a few cases have been settled. Colliot-Thélène and Sansuc did the case of tori, Panin and Suslin proved it for \(\text{SL}_1(D)\) where \(D\) is an Azumaya algebra over \(R\). The unitary group case was settled by Panin and Ojanguren and the special unitary group case by Zainoulline. Nisnevich has settled the conjecture for semi-simple group schemes \(G\) over discrete valuation rings. Azumaya algebras; purity; adjoint groups; group schemes I. Panin, Purity for multipliers, Algebra and number theory, pages 66-89. Hindustan Book Agency, Delhi, 2005. Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Algebraic theory of quadratic forms; Witt groups and rings, Witt groups of rings, Group schemes, Linear algebraic groups over adèles and other rings and schemes Purity for multipliers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well-known that almost all Hilbert modular varieties of a totally real number field \(k/Q\) are of general type where the exceptions being found in degree \(\leq 6\). In the present paper explicit bounds for the discriminant of a cubic number field are given so as to guarantee non- rationality resp. general type of the variety in question. Concerning non-rationality the starting point is the formula of the arithmetic genus given by Hirzebruch-Vignéras [\textit{M.-F. Vignéras}, Math. Ann. 224, No. 3, 189-215 (1976; Zbl 0325.12004)] and an estimation of the number of elliptic fixed points in terms of relative class numbers following the ideas of \textit{D. Weisser} [Math. Ann. 257, No. 1, 9-22 (1981; Zbl 0467.10021)\ whereas the second result stems on a condition due to \textit{S. Tsuyumine} [Invent. Math. 80, 269-281 (1985; Zbl 0576.14036)] and the reviewer [Math. Ann. 264, 413-422 (1983; Zbl 0504.14028)]. classification of Hilbert modular varieties; explicit bounds; discriminant of cubic number field; non-rationality Grundman, H. G.: On the classification of Hilbert modular threefolds. Manuscripta math. 72, 297-305 (1991) Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, \(3\)-folds, Automorphic forms in several complex variables, Class numbers, class groups, discriminants, Cubic and quartic extensions On the classification of Hilbert modular threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is an expanded version of the lecture delivered at the 97 Seattle summer research conference on finite fields. It gives a quick exposition of Dwork's conjecture about \(p\)-adic meromorphic continuation of his unit root \(L\)-function arising from a family of algebraic varieties defined over a finite field of characteristic \(p\). As a simple illustration, we discuss the classical example of the universal family of elliptic curves where the conjecture is already known to be true and where the conjecture is closely related to arithmetic of modular forms such as the Gouvêa-Mazur conjecture. Special attention is given to questions related to the \(p\)-adic absolute values of the unit root \(L\)-function. In particular, it is observed that an average version of a suitable \(p\)-adic Riemann hypothesis is true for the elliptic family. Following a suggestion of Mike Fried, the author also includes a section describing some of his personal interactions with Dwork. This extra section serves as a dedication to the memory of Dwork who actively attended the conference and died nine months later. Dwork's conjecture; \(p\)-adic meromorphic continuation; unit root \(L\)-function; algebraic varieties; finite field of characteristic \(p\); arithmetic of modular forms; Gouvêa-Mazur conjecture Wan, D, A quick introduction to dwork's conjecture, Contem Math., 245, 147-163, (1999) \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Varieties over global fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) A quick introduction to Dwork's 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 [For part I, cf. ibid. 45, 377-383 (1992; Zbl 0745.12004).]
The author looks at some subgroups of Brauer groups which arise from twists of matrix algebras by some continuous characters, and gives explicit descriptions of them in terms of Gauss sums and Dirichlet characters. subgroups of Brauer groups; twists of matrix algebras; Gauss sums; Dirichlet characters Chi W.C., Bull.Austral.Math.Soc 50 pp 245-- (1994) Galois cohomology, Galois cohomology, Finite-dimensional division rings, Endomorphism rings; matrix rings, Brauer groups of schemes Twists of matrix algebras and some subgroups of Brauer groups. 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 present textbook grew out of an introductory course on algebraic geometry, which the author recently taught at the University of Regensburg (Germany). This course was designed as a direct continuation of \textit{E. Kunz}'s lectures on advanced algebra, which, on their part, had been published beforehand as the textbook ``Algebra'' in the same series (1992; Zbl 0743.12002). Accordingly, the author's new textbook addresses to those students who already have acquired a basic knowledge in the theory of commutative rings and fields, and who are now interested in approaching the deeper-going geometric aspects of the subject. Compared to his former, well-proved textbook ``Introduction to commutative algebra and algebraic geometry'', published more than 15 years ago in its German original form (1980; Zbl 0432.13001) and translated into English a few years later (1985; Zbl 0563.13001), E. Kunz's new introduction to algebraic geometry appears to be of more elementary and purely algebro-geometric character. While the former textbook aimed at developing those methods of commutative algebra and related algebraic geometry which led to the (at that time) current state of knowledge of complete intersection varieties in projective space, which also made this work into an excellent, unique and up-to-date reference book, the new textbook is exclusively to be a primer in modern algebraic geometry. The author, as he points out in the preface of the book under review, also had in mind to provide German students with a new textbook on algebraic geometry that is within their financial means, so much the more as the German version of his former book is out of print, and the English translation of it is rather expensive.
As to the approach to introducing the basic concepts and methods of algebraic geometry, the author has chosen, amongst the various methodological possibilities, the classical course of development from the modern point of view. That means, he starts off from affine varieties, which are motivated by the study of solution sets of systems of polynomial equations over a groundfield, deals with abstract algebraic varieties and, particularly, with projective varieties in the sequel, explains the basic properties of algebraic schemes, and discusses then, in greater detail, the local and global geometry of algebraic varieties as far as this is possible without using sheaf-theoretic and cohomological methods. -- More precisely, the text consists of eight chapters, whose contents are as follows:
Chapter I: Affine algebraic varieties;
Chapter II: Projective algebraic varieties;
Chapter III: The spectrum of a ring;
Chapter IV: Regular and rational functions on algebraic varieties;
Chapter V: Schemes;
Chapter VI: Dimension theory (for topological spaces, rings, affine algebras, varieties, and schemes over a field);
Chapter VII: Regular and singular points of algebraic varieties (including Cohen-Macaulay varieties, complete intersection varieties, and Gorenstein varieties);
Chapter VIII: Systems of algebraic equations with finite solution sets (Intersection theory for zero-dimensional projective complete intersections).
The material is enhanced by an appendix providing some basic results from commutative algebra for the sake of completeness and self- containedness of the book. The topics covered here (with detailed proofs) concern:
A. Graduate rings and modules;
B. Localization and homogeneous localization;
C. Modules over Noetherian rings;
D. Filtred algebras and modules;
E. Regular and quasiregular sequences;
F. Quotients of ideals.
Obviously, the present new textbook has basically the same methodical arrangement as the author's former text ``Introduction to commutative algebra and algebraic geometry''; however, as already said before, it stresses much more the geometric viewpoint and, in addition, offers the fundamental concepts of Grothendieck's theory of algebraic schemes. On the other hand, the portion of commutative algebra is less dominating and, in its depth, more elementary. As for the deeper results on algebraic intersection theory, the reader is prudently advised to consult the author's former textbook, to which the present one under review is a perfect geometric completion.
Altogether, E. Kunz's new introduction to algebraic geometry is a welcome enhancement of the current textbook literature in the field. Being written in the customary Kunzian style, which is characterized by its extreme comprehension, clarity, rigor, detailedness, and thorough motivations, this book will become a standard text, too, especially so for beginners in algebraic geometry, who strive for precision and ultimate strictness. It would certainly be desirable and rewarding to have an English translation of this valuable textbook, too, and that the sooner the better. affine algebraic varieties; projective varieties; algebraic schemes; rational algebraic functions; singularities; intersection theory; complete intersections; Gorenstein varieties Kunz, E., Einführung in die algebraische geometrie, (1997), Vieweg Braunschweig/Wiesbaden Varieties and morphisms, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Relevant commutative algebra, Schemes and morphisms Introduction to algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a certain class of finite dimensional algebras the author shows that certain moduli spaces of finite dimensional modules are isomorphic to certain Grassmannian varieties. From this result it follows that the two different varieties associated to a given quiver by \textit{G. Lusztig} [in Adv. Math. 136, No. 1, 141-182 (1998; Zbl 0915.17008)] are isomorphic. The isomorphism the author constructs induces a bijection between the \(\mathbb C\)-points of these varieties constructed already in the above mentioned paper of Lusztig. schemes; Grassmannians; finite-dimensional algebras; moduli spaces; varieties Shipman B. A., Discrete and Continuous Dynamical Systems Representations of associative Artinian rings, Grassmannians, Schubert varieties, flag manifolds On representation schemes and Grassmanians of finite dimensional algebras and a construction of Lusztig. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 study the automorphism group \(\Aut(D^bA)\) of the bounded derived category of a finite dimensional \(k\)-algebra \(A\) which is derived equivalent to a canonical algebra. Recall that an automorphism of the derived category is, by definition, a \(k\)-linear selfequivalence of categories which is also an exact functor, i.e., preserves distinguished triangles. Assume that the base field \(k\) is algebraically closed. Then a complete description of the group \(\Aut(D^bA)\), or equivalently of the automorphism group \(\Aut(D^b\mathbb{X})\) of the bounded derived category \(D^b\mathbb{X}\) of coherent sheaves on the weighted projective line \(\mathbb{X}\) associated to \(A\), is provided.
More precisely, the authors show that up to a translation each automorphism of the derived category \(D^b\mathbb{X}\) of coherent sheaves on a weighted projective line \(\mathbb{X}\) is a composition of tubular mutations with an automorphism of the curve \(\mathbb{X}\), or equivalently, with an automorphism of the module category of the canonical algebra associated to \(\mathbb{X}\). In case \(\mathbb{X}\) has genus one, the automorphism group is, in fact, a semi-direct product of the braid group on three strands by a finite group. The key discussion in the paper is the so-called tubular case, where the authors introduce a combinatorial object, called the rational helix, whose automorphism group is isomorphic to the braid group on three strands. In this way, the braid group enters. Moreover, the authors point out that most automorphims on the derived level can be obtained from the Grothendieck group level by lifting. automorphism groups; derived categories; canonical algebras; coherent sheaves; finite dimensional algebras; selfequivalences; exact functors; weighted projective lines; tubular mutations; braid groups; translations Lenzing, H.; Meltzer, H., The automorphism group of the derived category for a weighted projective line, Comm. Algebra, 28, 1685, (2000) Representations of orders, lattices, algebras over commutative rings, Vector bundles on curves and their moduli, Derived categories, triangulated categories, Braid groups; Artin groups, Automorphisms and endomorphisms, Module categories in associative algebras The automorphism group of the derived category for a weighted projective line | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. gradings of semisimple Lie algebras; Lax operator algebras; integrable systems; spectral parameter on a Riemann surface; Tyurin parameters; Hamiltonian theory; prequantization Sh_UMN_2015 Sheinman, O.K. \emph Lax operator algebras and integrable systems. Russian Math. Surveys, 71:1 (2016), 109--156. Infinite-dimensional Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Riemann surfaces; Weierstrass points; gap sequences, Applications of Lie algebras and superalgebras to integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Lax operator algebras and integrable systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0504.00008.]
This is a continuation of a previous paper [Journées de géométrie algebrique, Angers/France 1979, 273-310 (1980; Zbl 0451.14014)] here quoted as C.3. These two papers together give new, very interesting, insight in the difficult problems regarding the classification of 3-folds of ''general type''. In the case of surfaces (of general type) it is well known that the canonical model X may have only very simple singularities, the minimal model S is non singular, \(K_ S\) is ''numerically effective and free'' \((=:nef)\) and the morphism \(f: S\to X\) is such that \(f^*\omega_ X=\omega_ S\) (f is ''crepant''). In this paper a definition of minimal model S for a 3-fold X is proposed (X of f. g. general type (C.3.) and \(\kappa_{num}\geq 0)\) which preserves the maximum of the previous properties for surfaces. S may have singularities of a specified simple type called ''quick'' and is obtained by blowing up the canonical models X. Before stating the main result we need some more definitions. \(f: Y\to X\) is a partial resolution if it is a proper birational morphism in which Y is always assumed normal. If f is a partial resolution an exceptional prime divisor of f is any prime divisor \(\Gamma\) \(\subset Y\) such that \(co\dim f(\Gamma)\geq 2.\) Let X be a variety of dimension 3 with canonical singularities (C.3.), \(P\in X\) is a terminal singularity if it has a resolution \(f: Y\to X\) such that (i) f has at least one exceptional prime divisor and (ii) if \(K_ Y=f^*K_ X+\Delta\) every exceptional prime divisor of f appears in \(\Delta\) with strictly positive coefficient.
The main theorem is the following: 1. Let \(P\in X\) a 3-fold point then P is terminal if and only if it is quick. - 2. Let X be a 3-fold with canonical singularities. Then there exists a partial resolution \(f: S\to X\) such that (a) f is crepant, and (b) S has quick singularities. Furthermore this f can be chosen as the composite of certain elementary steps (blow-ups) which are intrinsic to X and is then uniquely determined and projective. - The paper contains many other results of interest in themselves and many appealing conjectures and open problems. classification of 3-folds of general type; numerically effective canonical divisor; crepant resolution; canonical models; partial resolution; exceptional prime divisor; terminal singularity; quick singularities M. Reid, \textit{Minimal models of canonical} 3\textit{-folds}, in \textit{Algebraic varieties and analytic varieties (Tokyo, 1981)}, \textit{Adv. Stud. Pure Math.}\textbf{1} (1983) 131, North-Holland, Amsterdam, The Netherlands. \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry Minimal models of canonical 3-folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(X\) a smooth quasi-projective algebraic surface and \(E\) a line bundle on \(X\). Consider \(X^{[n]}\), the Hilbert scheme of \(n\) points on \(X\), and the tautological bundle \(E^{[n]}\) on it naturally associated to \(E\). In the present paper the author relates the cohomology of \(X^{[n]}\) with values in \((E^{[n]})^{\otimes 2}\) with the cohomologies of \(X\) with values in \(E^{\otimes 2}\), \(E\) and \({\mathcal O}_X\).
This calculation is done using recent results on the McKay correspondence [\textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] adapted to the case of an isospectral Hilbert scheme in [\textit{M. Haiman}, J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)], which give a Fourier-Mukai equivalence \(\Phi\) between the derived category of the Hilbert scheme \(X^{[n]}\) and the \(S_n\)-equivariant derived category of \(X^n\). These results allow indeed to calculate the cohomologies of \(X^{[n]}\) with values in the tensor powers \((E^{[n]})^{\otimes k}\) of the tautological bundle as the hypercohomologies of \(S^n X\) with values in the invariants \(\Phi((E^{[n]})^{\otimes k})^{S_n}\). The latter groups can be calculated using polygraphs and the calculation is explicitely performed for \(k=2\) by means of a spectral sequence in order to obtain the main result. Hilbert scheme; tautological bundles; McKay correspondence; Fourier-Mukai functor Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cohomology of the Hilbert scheme of points on a surface with values in the double tensor power of a tautological bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 classical problem of infinitesimal deformations of a closed subscheme in a fixed smooth variety, defined over an algebraically closed field of characteristic \(0\), is studied using the semicosimplicial language.
Let \(X\) be a smooth variety defined over an algebraically closed field \(K\) of characteristic \(0\) and \(Z\subset X\) a closed subscheme. Let \(\text{Hilb}_X^Z\) be the functor of infinitesimal deformations of \(Z\) in \(X\) and \(\text{Hilb'}^Z_X\) be the subfunctor of locally trivial infinitesimal deformations. Let \(\theta_X\) be the tangent sheaf of \(X\) and \(\theta_X(-\log Z)\) the sheaf of tangent vectors to \(X\) which are tangent to \(Z\). Let \(\mathcal{O}_X(-\log Z) (\mathcal{U})\) resepctively \(\theta_X(\mathcal{U})\) be the \(\check{\text{C}}\)ech semicosimplicial Lie algebras and \(\chi:\theta_X(-\log Z)(\mathcal{U})\to\theta_X(\mathcal{U})\) the corresponding bisemicosimplicial Lie algebra induced by the inclusion \(\theta_X(-\log Z)\hookrightarrow \theta_X\). Using the Thom--Whitney construction a differential graded Lie algebra \(\text{Tot}_{TW}(\chi)\) can be constructed. It is proved that there exists an isomorphism of functors \(\text{Def}_{\text{Tot}_{TW}(\chi)} \cong \text{Hilb'}Z_X\). In particular, if \(Z\subset X\) is smooth then \(\text{Def}_{\text{Tot}_{TW}(\chi)}\cong \text{Hilb'}^Z_X\). Differential graded Lie-algebras; functors of Artin rings Deformations and infinitesimal methods in commutative ring theory, Formal methods and deformations in algebraic geometry Deformations of algebraic subvarieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the determination of the nilpotent classes in the Lie algebra \({\mathfrak g}\) of a reductive algebraic group G defined over an algebraically closed field k in one of the few cases where it was not yet known: G of type \(F_ 4\), \(char(k)=2\). In this case there are 22 nilpotent orbits (and 26 over a finite field). The structure of the stabilizers (dimension, group of components, type of the reductive part of the identity component) is also described, as well as the order relation by inclusion of Zariski closures and the Springer correspondence with representations of the Weyl group. There are also some results on the sheets in \({\mathfrak g}\). The basic tool is a systematic use of the commutation formulae.
[Since this paper was written, D. F. Holt and the author have worked out the remaining cases (''Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic'', to appear in J. Aust. Math. Soc.); the method used there gives less information, and in the case of \(F_ 4\), \(char(k)=2\), would require some modification to handle the class \(\tilde A_ 2+A_ 1.]\) centralizers; exceptional Lie algebras; characteristic two; nilpotent classes; reductive algebraic group; nilpotent orbits; stabilizers; order relation; Springer correspondence; Weyl group; sheets Spaltenstein, N., Nilpotent classes in Lie algebras of type \textit{F}_{4} over fields of characteristic 2, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, 3, 517-524, (1984) Exceptional (super)algebras, Lie algebras of linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Group actions on varieties or schemes (quotients) Nilpotent classes in Lie algebras of type \(F_ 4\) over fields of characteristic 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This monograph is the author's doctoral dissertation. Besides containing the original results, it is a very nicely written survey of many aspects of the Fourier-Mukai transforms and their importance in geometry and mirror symmetry, with lots of relevant examples. Newcomers to the topic are given here a quick overview of the most significant aspects of the theory of integral functors and equivalences of categories, even if some important references are missing. The first part is devoted to the general theory. Most results are reported without proofs. However, one finds here a new and simpler proof of the fact the integral functors whose kernel is a spherical object are equivalences of categories.
In the second chapter, the author deals with autoequivalences of the derived category of a \(K3\) surface, continuing the study by Mukai and Orlov of the Torelli theorem for such surfaces. The main result is that the image of the group of autoequivalences of the derived category in the group of Hodge isometries is a subgroup of index two, so that every other Hodge isometry of the cohomology lattice can be lifted to an autoequivalence of the derived category. Producing explicit examples of nontrivial integral functors on the derived category of a \(K3\) surface which are equivalences is nevertheless a difficult task. The first example is due to \textit{C. Bartocci, U. Bruzzo} and the reviewer [J. Reine Angew. Math. 486, 1--16 (1997; Zbl 0872.14013)]; some others were constructed by \textit{S. Mukai} [in: New trends in algebraic geometry. Selected papers presented at the Euro conference, Warwick, UK, July 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 264, 311--326 (1999; Zbl 0948.14032)] and \textit{K. Yoshioka} [Math. Ann. 321, No. 4, 817--884 (2001; Zbl 1066.14013)]. Chapter 2 also contains a study of the group \(\text{Aut}_0(X)\) of autoequivalences of the derived category whose associated Hodge isometry is the identity, and some of its braid subgroups. Bridgeland's conjecture relating \(\text{Aut}_0(X)\) with the space of stability conditions in the derived category is very clearly described.
The third chapter is devoted to equivariant derived categories and their equivalences. The author considers only actions by finite groups, and actually deals with the derived category of linearised complexes of sheaves, rather than just equivariant complexes. This technical issue is quite significant because the category of linearised coherent sheaves is abelian, and then it has an associated derived category, which is the one considered by the author. On the other hand, the category of equivariant coherent sheaves is not abelian. The theory of linearised derived categories and equivariant integral functors has already proven to have important applications, such as the derived version of the McKay correspondence due to \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)]. The author presents here some general results on the relation between the different groups of autoequivalences of the derived category of a scheme acted on by a finite group and the equivalences of the linearised derived category. An application of this study is the proof that birational Hilbert schemes of points on a \(K3\) surface have equivalent derived categories. This in turn implies that there exist only finitely many non-birational Hilbert schemes of points of \(K3\) surfaces.
The fourth part is the application of equivariant derived categories and integral functors to Kummer surfaces, taking advantage of the fact that they are quotients of abelian varieties by an action of \(\mathbb Z/2\). Mukai and Orlov's results on the autoequivalences of the derived categories of abelian varieties are also reported. autoequivalences of derived categories; integral functors; Fourier-Mukai; linearised derived categories. Hilbert schemes of points; K3 surfaces --------, Groups of autoequivalences of derived categories of smooth projective varieties, Ph.D. dissertation, Freie Universität, Berlin, 2005. \(K3\) surfaces and Enriques surfaces, Derived categories, triangulated categories, Calabi-Yau manifolds (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Groups of autoequivalences of derived categories of smooth projective varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper provides an application of commutative algebra to error-correcting codes. It defines the notion of minimum distance function of a graded ideal \(I\) in a polynomial ring \(S=K[t_1, \dots, t_s]\), \(K\) a field, function which allows to give an algebraic formulation of the minimum distance of a projective Reed-Muller-type code (a projective Reed-Muller-type code of degree \(d\) is the image of a certain evaluation map \(ev_d: S_d\longrightarrow K^m\), where \(K\) is now a finite field) and to find lower bounds for the minimum distance of these codes.
Section 1 defines the minimum distance function \(\delta_I\) of the ideal \(I\) and summarizes the content of the paper. Sections 2 and 3 gather some concepts and results needed in the following. Section 4 studies the properties of \(\delta_I\) and Theorem 4.7 proves that \(\delta_I\) generalizes the minimum distance of projective Reed-Muller-type codes.
Then the paper considers the case of projective nested cartesian codes and a conjecture about the minimum distance of these codes, conjecture due to \textit{C. Carvalho} et al., [``Projective nested Cartesian codes'', Preprint, \url{arXiv:1411.6819}]. The present paper provides some support to that conjecture (Section 6, Theorem 6.6]. Finally Section 7 shows several examples (with procedures for Macaulay2) illustrating the results obtained. graded ideal; minimum distance function; Reed-Muller-type code; Hilbert function; Gröbner bases; Carvalho, Lopez-Neumann and López conjecture Martínez-Bernal, J., Pitones, Y., Villarreal, R.H.: Minimum distance functions of graded ideals and Reed-Muller-type codes. J. Pure Appl. Algebra 221, 251-275 (2017) Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Algebraic coding theory; cryptography (number-theoretic aspects) Minimum distance functions of graded ideals and Reed-Muller-type codes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 and let \(A\) be a finitely generated \(k\)-algebra (that is, \(A\cong k\langle X_1,X_2,\dots,X_n\rangle/I\), with \(I\) a two-sided ideal). It is known that if \(\varphi\colon A\to B\) is a homomorphism of finitely generated algebras, then there are induced regular \(\text{Gl}_d(k)\)-equivariant morphisms of affine varieties \(\varphi^{(d)}\colon\text{mod}_B(d)\to\text{mod}_A(d)\), for all \(d\geq 1\).
The author deals with the relation between epimorphisms in the category of rings and regular morphisms of module varieties which are immersions. He proves that \(\varphi\colon A\to B\) is an epimorphism in the category of rings if and only if \(\varphi^{(d)}\colon\text{mod}_B(d)\to\text{mod}_A(d)\) is an immersion, for all \(d\geq 1\). He applies this result for classifying the types of singularities (and minimal singularities) in the subvarieties of modules from homogeneous standard tubes in the Auslander Reiten quiver of finite-dimensional algebras. module varieties; regular morphisms; immersions; finitely generated algebras; equivariant morphisms; affine varieties; epimorphisms; minimal singularities; Auslander-Reiten quivers Representations of associative Artinian rings, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Embeddings in algebraic geometry, Automorphisms and endomorphisms Immersions of module 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 simply connected simple algebraic group defined over an algebraic number field K, and let S be a finite subset of the set \(V^ K\) of valuations of K which contains the set \(V^ K_{\infty}\) of Archimedean valuations. Let \(G(K)^{\wedge}\) (\(\overline{G(K)}\), resp.) denote the completion of the group of K-rational points G(K) with respect to the S-arithmetic topology (the S-congruence topology, resp.). The kernel of the natural projection \(\pi\) : G(K)\({}^{\wedge}\to \overline{G(K)}\) is denoted by C(S,G), and called the congruence kernel. \textit{J.-P. Serre} [Ann. Math., II. Ser. 92, 489-527 (1970; Zbl 0239.20063)] made the following ``congruence conjecture'': If \(rank_ SG=\sum_{v\in S}rank_{K_ v}G\geq 2\) and if G is isotropic over \(K_ v\) for all \(v\in S\setminus V^ K_{\infty}\), then the congruence kernel C(S,G) is finite. By developing the methods of \textit{M. Kneser} [J. Reine Angew. Math. 311/312, 191-214 (1979; Zbl 0409.20038)] and bringing in some new ideas, the author obtains a proof of the congruence conjecture for all groups of classical types which have a geometrical realization of sufficiently large degree. The geometrical method used there turns out not to be applicable to most of the exceptional groups. By using the inner structure of the group, the author shows that, if G is an anisotropic K-group of one of the types \(E_ 7\), \(E_ 8\), \(F_ 4\), or \(G_ 2\), then the congruence kernel is trivial for \(rank_ SG\geq 2\). Groups of type \(E_ 6\) are not included here. simply connected simple algebraic group; algebraic number field; valuations; completion; group of K-rational points; S-arithmetic topology; S-congruence topology; congruence kernel; congruence conjecture; exceptional groups; anisotropic K-group Rapinchuk A S, On the congruence subgroup problem for algebraic groups,Dokl. Akad. Nauk. SSSR 306 (1989) 1304--1307 Linear algebraic groups over global fields and their integers, Unimodular groups, congruence subgroups (group-theoretic aspects), Classical groups (algebro-geometric aspects) On the congruence problem for 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 The authors define the notion of best approximation in rings of algebraic functions. For algebraic curves of a very restricted type they give, in terms of best approximation, a criterion for certain divisors to be of finite order in the Jacobian. points of finite order; best approximation in rings of algebraic functions; Jacobian Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Special algebraic curves and curves of low genus, Jacobians, Prym varieties Unités de certains sous-anneaux de corps de fonctions algébriques | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 nonsingular surface over complex numbers and \(S_n\) be the symmetric group on \(n\) letters. Denote the Hilbert scheme of \(n\) points on \(X\) by \(X^{[n]}\). By a result of Haiman \(X^{[n]}\) can be identified with the fine moduli space of \(S_n\)-clusters in \(X^n\). If we denote by \(\mathcal Z \subset X^{[n]}\times X^n\) the universal family of \(S_n\)-clusters and \(X^{[n]}\overset{q}{\leftarrow} \mathcal Z \overset{p}\rightarrow X^n\) be the projections then, the derived McKay correspondence of \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] in this set up gives an equivalence of derived categories \(\Phi: D(X^{[n]})\overset{\sim}{\rightarrow} D_{S_n}(X^n)\) of (\(S_n\)-equivariant) coherent sheaves, where \(\Phi=Rp_*\circ q^*\).
Scala showed that for any vector bundle \(F\) on \(X\), the image of the tautological bundle \(F^{[n]}\) on \(X^{[n]}\) under \(\Phi\) is given by an explicit complex \(\mathsf C^\bullet_F\) of (\(S_n\)-equivariant) coherent sheaves concentrated in nonnegative degrees.
The paper under review studies the derived McKay correspondence above in the reverse order by means of \(\Psi=q_*^{S_n}\circ Lp^*\) (which is not the inverse of \(\Phi\)). The main result of the paper is that if one replaces \(\Phi\) by \(\Psi^{-1}\) the images of \(F^{[n]}\) and \(\bigwedge^k L^{[n]}\) where \(L\) is a line bundle are (explicitly given) sheaves (instead of complexes of sheaves).
This enables the author to prove new formulas and also give simpler proofs for existing formulas for homological invariants of tautological bundles and their wedge powers. derived McKay correspondence; Hilbert scheme; tautological bundle McKay correspondence, Parametrization (Chow and Hilbert schemes) Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological 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 Let \(n\) be a positive integer. The Hilbert scheme of \(n\) points in the complex plane has a natural \((\mathbb C^*)\)-action induced by the action of the torus \(T:=(\mathbb C^*)^2\) on \(\mathbb C^2\). Let now \(T:=\left\{(t^a,t^b)\in T\,\,|\,\, t\in \mathbb C^* \right\}\), where \(\gcd(a,b)=1\) and \(a,b\geq 1\) be a one dimensional subtorus of \(T\). The set of fixed points under the action of such a subtorus has the structure of a smooth variety, denoted by \((\mathbb C^2)_{a,b}^{[n]}\).\newline In the paper under review, the author generalizes \textit{A. Iarrobino} and \textit{J. Yaméogo} [Commun. Algebra 31, No. 8, 3863--3916 (2003; Zbl 1048.14003)] by providing a formula for the class of the irreducible components of \((\mathbb C^2)_{1,k}^{[n]}\) in terms of polynomials in \(\mathbb L\), the class of \(\mathbb A_{\mathbb C}^1\) in the Grothendieck ring of complex quasi-projective varieties. Based on computer calculations, the author also makes a conjecture on a possible formula for the Grothendieck ring classes of the more general varieties \((\mathbb C^2)_{a,b}^{[n]}\).\newline Another result in the paper is an interesting relation between the classes of certain open strata of \((\mathbb C^2)^{[n]}\) and the \((q,t)\)-Catalan numbers.\newline Finally, using a well known quiver description of \((\mathbb C^2)^{[n]}\), the author provides sufficient conditions under which a \((1,k)\)-quasi-homogeneous Hilbert scheme of points is isomorphic to a homogeneous nested Hilbert scheme of points. The latter result generalizes \textit{J. Cheah} [Pac. J. Math. 183, No. 1, 39--90 (1998; Zbl 0904.14001)]. Hilbert schemes of points; torus action; \((q,t)\)-Catalan numbers; quiver varieties A. Buryak, ''The classes of the quasihomogeneous Hilbert schemes of points on the plane,'' Mosc. Math. J., 12:1, 21--36; http://arxiv.org/abs/1011.4459 . Parametrization (Chow and Hilbert schemes), Combinatorial aspects of partitions of integers, Representations of quivers and partially ordered sets The classes of the quasihomogeneous Hilbert schemes of points on the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the elliptic operators naturally derived from loop spaces and show that their modularity implies their rigidity. As consequences we first prove the rigidity of the Dirac operator on loop space twisted by loop group representations of positive energy of any level, while the rigidity theorems conjectured by Witten are the special cases of level 1. Then we generalize these rigidity theorems to non-zero anomaly case from which we obtain holomorphic Jacobi forms and many vanishing theorems, especially an \(\widehat {\mathfrak A}\)-vanishing theorem for loop spaces. We also discuss elliptic genera of level 1, mod 2 elliptic genera and the relationships between elliptic genera and the geometry of elliptic modular surfaces, and the classical elliptic modular functions. \(\widehat {A}\)-genus; circle action; Lefschetz number of elliptic operators; elliptic operators; loop spaces; modularity; rigidity; Dirac operator on loop space twisted by loop group representations; \(\widehat {\mathfrak A}\)-vanishing theorem; elliptic genera; elliptic modular surfaces; elliptic modular functions Liu, K.: On modular invariance and rigidity theorems. J. Diff. Geom. 41(2), 343--396 (1995) Compact Lie groups of differentiable transformations, Theta series; Weil representation; theta correspondences, Index theory and related fixed-point theorems on manifolds, Loop groups and related constructions, group-theoretic treatment, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Quantum field theory on curved space or space-time backgrounds, Modular and automorphic functions On modular invariance and rigidity theorems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 discusses a conjecture which relates central critical values of derivatives of twists of modular \(L\)-series to coefficients of the corresponding \(p\)-adic half-integral weight modular forms (see p. 240 for a rough statement of the conjecture).
Relations to class numbers of quadratic fields and to orders of Tate-Shafarevich groups of quadratic twists of a given elliptic curve are discussed (the motivating examples). Another evidence is that the theorem of Rubin [Corollary 10.2 in \textit{K. Rubin}, Invent. Math. 107, 323-350 (1992; Zbl 0770.11033)] is equivalent to a special case of the conjecture (see Example 2).
Despite the highly informal style, this is a very interesting article. derivatives of twists of modular \(L\)-series; \(p\)-adic half-integral weight modular forms; central critical values; class numbers; quadratic fields; orders of Tate-Shafarevich groups; elliptic curve N. Jochnowitz, A \(p\)-adic conjecture about derivatives of \(L\)-series attatched to modular forms, Proceedings of the Boston University Conference on \(p\)-Adic Monodromy and the \(p\)-Adic Birch and Swinnerton-Dyer Conjecture (to appear), CMP 94:13. Congruences for modular and \(p\)-adic modular forms, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Local ground fields in algebraic geometry, Forms of half-integer weight; nonholomorphic modular forms A \(p\)-adic conjecture about derivatives of \(L\)-series attached to 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 See the preview in Zbl 0527.14031. symmetric monodromy groups of singularities; reflection subgroup; complete vanishing lattice; middle homology group of Milnor fibre; vanishing cycles; isolated hypersurface singularities; complete intersection singularities Ebeling W.: An arithmetic characterisation of the symmetric monodromy groups of singularities. Invent. Math. 77(1), 85--99 (1984) Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Cycles and subschemes, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, General binary quadratic forms, Classical groups An arithmetic characterisation of the symmetric monodromy groups of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An old question, raised by A. A. Albert, asks whether every division algebra of prime index is cyclic. It has variously been suggested that counter-examples might be found among division algebras over a function field (assuming an algebraically closed ground field) that are ramified along a cubic divisor. The author shows that this is not so by proving the following Proposition. Let \(D\) be a non-trivial central division algebra over \(K(P^2_k)\), with ramification divisor \(R\) of degree 3 and assume either (i) \(R\) is singular, or (ii) \(R\) is smooth and \(D\) is of odd period in the Brauer group. Then \(D\) is cyclic of period (i.e. exponent) equal to its index. -- Part (i) was proved by by \textit{T. J. Ford} [New York J. Math. 1, 178-183 (1995; Zbl 0886.16017)]\ while \textit{D. J. Saltman} [in Proc. Symp. Pure Math. 58, Pt. I, 189-246 (1995; Zbl 0827.13003)]\ showed that for smooth \(R\), \(D\) is similar to a tensor product of three cyclic algebras. -- In all this the ground field \(k\) is assumed to be algebraically closed, but an appendix discusses the general situation, when only partial (rather technical) results are obtained. tensor products of cyclic algebras; division algebras of prime index; division algebras over function fields; cubic divisors; central division algebras; ramification divisors; Brauer groups; exponents Michel Van den Bergh, Division algebras on \?² of odd index, ramified along a smooth elliptic curve are cyclic, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995) Sémin. Congr., vol. 2, Soc. Math. France, Paris, 1997, pp. 43 -- 53 (English, with English and French summaries). Finite-dimensional division rings, Arithmetic theory of algebraic function fields, Quaternion and other division algebras: arithmetic, zeta functions, Brauer groups of schemes Division algebras on \(\mathbb{P}^2\) of odd index, ramified along a smooth elliptic curve are cyclic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Lusztig conjectured that the almost characters of a finite reductive group are up to scalar the same as the characteristic functions of the rational character sheaves defined on the corresponding algebraic group.
The author now considers certain groups of type \(B_2\), \(G_2\) and \(F_4\). These groups come with an automorphism defining a twisted finite Chevalley group (a Suzuki group in case \(B_2\), a Ree group in cases \(G_2\), \(F_4\)). The conjecture is confirmed and tables are given.
There is also a version where the character sheaves live on a coset of the identity component in a disconnected reductive group. The conjecture is checked for a disconnected group of type \(B_2\). In case \(F_4\) the picture is not complete yet. Tables are given, but it still needs to be shown they match the values of almost characters. Lusztig algorithm; character sheaves; Ree groups; Suzuki groups; cyclic extensions; disconnected groups; almost characters; Lusztig conjecture; characteristic functions; irreducible characters; finite reductive groups Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Étale and other Grothendieck topologies and (co)homologies On Lusztig's conjecture for connected and disconnected exceptional 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 In this thoroughly written article the author gives details and full proofs of results already announced in [Proc. Japan Acad., Ser. A 66, 195-200 (1990; Zbl 0723.11020)] without any restrictions on the class number: there is a natural action \(f_ \bullet\) of the Hecke ring \(\text{HR} (\Gamma (N)_ K, G_ 2^ + ({\mathcal O}_ k))\) of a totally real algebraic number field \(K\) of degree \(g\) with ring of integers \({\mathcal O}_ K\) on the singular homology \(H_ \bullet ((\Gamma(N)_ K \setminus \mathbb{H}^ g)^ \sim, \mathbb{Z})\) of a smooth projective toroidal compactification of the Hilbert modular variety \(\Gamma(N)_ K \setminus \mathbb{H}^ g, \Gamma(N)_ K\) being the principal congruence subgroup of level \(N\geq 3\) as well as an \(\ell\)-adic version \(f^ \bullet\) for the \(\ell\)-adic cohomology of \((\Gamma(N)_ K \setminus \mathbb{H}^ g)^ \sim\) resp. \(D_ p(\Gamma (N)_ K)\) where the latter is a suitable proper smooth Hilbert modular variety (in the sense of Rapoport) over the field \(\overline {\mathbb{F}}_ p\), \(p\nmid N\) prime.
The main results are
(1) An estimation (based on the Weil conjecture) of the eigenvalues of the Hecke operators \(f^{(n)} (T_ p{\mathcal O}_ K)\)
\[
|\lambda_ p|\leq p^{n/2}+ p^{(2n-2)/2} \qquad \text{ for } 0\leq n\leq 2g \quad(\text{Theorem 8}).
\]
(2) The determination of the monic polynomial
\[
P_ n(X)= \text{det} (X^ 2- f^{(n)} (T_ p{\mathcal O}_ K)X+ (\sigma^ \sim_{pn}) p^ g)\in \mathbb{Z}[ X]
\]
where \(\sigma^ \sim_{pn}\) is the endomorphism of the \(\ell\)-adic cohomology induced by the element
\[
\sigma_ p\equiv \left( \begin{smallmatrix} p^{-1} &0\\ 0 &p\end{smallmatrix} \right)\bmod N
\]
of \(\text{SL} (2,\mathbb{Z})\), namely
\[
P_ n(X)= \text{det} (X- [\text{Frob} (p)]_ n) \text{ det} (X-[ \text{Frob} (p) ]_{2g-n}) \quad( \text{Theorem }10).
\]
Here \([\text{Frob} (p)]_ \bullet\) is the endomorphism of the \(\ell\)-adic cohomology induced by the \(\overline {\mathbb{F}}_ p\)-linear Frobenius endomorphism of \(D_ p (\Gamma (N)_ K)\).
(3) An expression of the local zeta function in terms of the above Hecke operators
\[
Z(D_ p (\Gamma(N)_ K, X)= \sqrt {\prod_{n=0}^{2g} \bigl[ \text{det} (1- f^{(n)} (T_ p {\mathcal O}_ K)X+ (\sigma^ \sim_{pn}) p^ g X^ 2) \bigr]^{(-1)^{n+1}}} \quad (\text{Theorem }11).
\]
The methods of proof heavily rely on the theory of toroidal compactifications, Deligne's work on the Weil conjecture as well as the author's work on the same problem in the Siegel case. Hecke operators on Hilbert varieties; estimates of eigenvalues; Hecke ring; Hilbert modular variety; \(\ell\)-adic cohomology; local zeta function; toroidal compactifications; Weil conjecture K. Hatada: On the local zeta functions of compactified Hilbert modular schemes and action of the Hecke rings. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci., 18, no. 2, 1-34 (1994). Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(p\)-adic theory, local fields, Hecke-Petersson operators, differential operators (several variables) On the local zeta functions of compactified Hilbert modular schemes and action of the Hecke 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 In this book the author discusses the Hilbert scheme \(X^{[n]}\) of points on a complex surface \(X\). This object is originally studied in algebraic geometry but, as it has been recently realized, it is related to many other branches of mathematics, such as singularities, symplectic geometry, representation theory, and even to theoretical physics. The book reflects this feature on Hilbert schemes and therefore the subjects are analyzed from various points of view. One sees that \(X^{[n]}\) inherits structures of \(X\), e.g., it is a nonsingular complex manifold, it has a holomorphic symplectic form if \(X\) has one, it has a hyper-Kähler metric if \(X= \mathbb{C}^2\), and so on. A new structure is revealed when one studies the homology group of \(X^{[n]}\). The generating function of Poincaré polynomials has a very nice expression. The direct sum \(\bigoplus_n H_* (X^{[n]})\) is a representation of the Heisenberg algebra.
The book, which is nicely written and well-organized, tries to tell the harmony between different fields rather than focusing attention on a particular one. The reader is assumed to have basic knowledge on algebraic geometry and homology groups of manifolds. Some chapters require more background, say spectral sequences, Riemannian geometry, Morse theory, intersection cohomology. symplectic structure; moment map; hyper-Kähler quotients; Dynkin diagrams; vertex algebra; symmetric products; Hilbert scheme of points; Poincaré polynomials; Heisenberg algebra; Morse theory; intersection cohomology H. Nakajima, \textit{Lectures on Hilbert schemes of points on surfaces}, \textit{University Lecture Series}\textbf{18}, American Mathematical Society, Providence RI U.S.A., (1999). Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Special Riemannian manifolds (Einstein, Sasakian, etc.) Lectures on Hilbert schemes of points on 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 This paper is about classification of certain rational cuspidal curves, meaning irreducible curves \(C\subset \mathbb{P}^2:= \mathbb{P}_{\mathbb{C}}^2\), whose only singularities are analytically irreducible. They have interesting properties, e.g., for such a curve \(C\) having at least three cusps, \(X= \mathbb{P}^2- C\) is an example of an affine surface of log-general type.
More precisely, let \(V\) be a minimal embedded resolution of singularities of \(C\subset \mathbb{P}^2\) and \(D\) the total transform of \(C\) to \(V\), so that \(D\) is a normal crossings divisor of \(V\), denote by \(\Theta\langle D\rangle\) the sheaf of germs of vector fields on \(V\) tangent to \(D\), \(\chi (\Theta \langle D\rangle)\) its Euler characteristic. A rational cuspidal curve \(C\) is of type \((d,m)\) if the degree of \(C\) is \(d\) and the maximum of the multiplicities of points of \(C\) is \(m\). In the present article the author proves that if \(d> 6\) and \(C\) is a rational cuspidal curve of type \((d,d-4)\) with \(\chi (\Theta \langle D\rangle)\leq 0\), then (for a suitable integer \(a\geq 2\)) the degree \(d\) of \(C\) is of the form \(3a+4\) and \(C\) has 3 cusps, where the structure of these singularities can be described explicitly in terms of \(a\) (i.e., the sequence of multiplicities when they are resolved by means of quadratic transformations). Moreover, \(\chi(\Theta \langle D\rangle)\) must be equal to zero. -- Conversely, if \(a\geq 1\), a cuspidal curve \(C_a\) of degree \(d= 3a+4\) having 3 singularities of the types found above and satisfying \(\chi (\Theta \langle D\rangle)= 0\) can be constructed, this \(C_a\) being unique up to projective equivalence.
The author uses an array of techniques from the (global and local) theory of plane curves as well as from the theory of algebraic surfaces. singularities; affine surface of log-general type; minimal embedded resolution; Euler characteristic; rational cuspidal curve; sequence of multiplicities T. Fenske, Rational cuspidal plane curves of type {(d,d-4)} with {\({\chi}\)(\({\Theta}\)_{V}\(\langle\) D\(\rangle\))\(\leq\) 0}, Manuscripta Math. 98 (1999), no. 4, 511-527. Singularities of curves, local rings, Plane and space curves, Families, moduli of curves (algebraic), Projective techniques in algebraic geometry, Special surfaces Rational cuspidal plane curves of type \((d,d-4)\) with \(\chi(\Theta_V \langle D\rangle)\leq 0\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper studies commutative dimension \(n\) polynomial formal groups of degree two over the valuation ring \(R\) of a local field \(K\), in particular, those which over \(K\) become isomorphic to \(\mathbb{G}_m^n\). These are used to construct Hopf orders over \(R\) in \(KC^n_p\): These Hopf orders are associated with certain \(n\times n\) lower triangular matrices, and may be described as iterated extensions of Tate-Oort Hopf algebras. The constructions yield Raynaud orders, that is, Hopf orders in \(KC_p^n\) which admit an action by \(\mathbb{F}_q\subset R/m\), the residue field of \(R\), when the matrices are made up of symmetric functions of roots of the minimal polynomial of a field generator of \(\mathbb{F}_q/ \mathbb{F}_p\). polynomial formal groups of degree two; valuation ring; Tate-Oort Hopf algebras; Raynaud orders; Hopf orders Childs, L. N.; Sauerberg, J.: Degree two formal groups and Hopf algebras. Mem. amer. Math. soc. 136, No. 651, 55-89 (1998) Formal groups, \(p\)-divisible groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group schemes, Class field theory; \(p\)-adic formal groups Degree two formal groups and Hopf algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The irreducible finite group \(G\), generated by reflections in the \(n\)-dimensional unitary space, acts on the polynomial ring in \(n\) variables over the field of complex numbers in a natural manner. It is well known that there exist \(n\) algebraically independent \(G\)-invariant homogeneous polynomials, called basic invariants, such that all \(G\)-invariant polynomials can be uniquely written as polynomials of the basic invariants. Given the group \(G\), there are infinitely many possible choices of basic invariants, but their degrees are well known and typical of the given group \(G\). It is possible to select, among the infinitely many basic invariants, some basic invariants, by requiring some supplementary conditions to be satisfied. So, it has been proved (L. Flatto, N. Nakashima, H. Terao, S. Tsujie) that it is possible to choose basic invariants in such a way that they satisfy a certain system of differential equations. Basic invariants of this kind are called canonical system of basic invariants. In this article we consider finite primitive groups \(G\), generated by reflections in four-dimensional unitary space. For all groups \(G\) are constructed in explicit form canonical systems of basic invariants. unitary space; reflection groups; algebra of invariants; basic invariant; canonical system of basic invariants Reflection groups, reflection geometries, Geometric invariant theory, Reflection and Coxeter groups (group-theoretic aspects) Canonical system of basic invariants for primitive reflection groups of four-dimensional unitary 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 This paper contains a new interpretation of an example of Zariski pairs introduced by the reviewer [\textit{E. Artal Bartolo}, J. Algebr. Geom. 3, No. 2, 223-247 (1994; Zbl 0823.14013)]. The example consists of curves with four irreducible components: a smooth cubic and three lines in general position which are tangent to inflection points of the cubic. In the reviewer's paper it was proven that the topological type of the embedding of these curves in the projective plane depends on the position of the inflection points on a line. The author proves this fact with the help of his theory about dihedral coverings. One main point of this paper is the new relationship of Zariski pairs and Mordell-Weil groups of elliptic \(K3\) surfaces. Zariski pairs; dihedral coverings; elliptic fibrations; topological type of the embedding; Mordell-Weil groups of elliptic \(K3\) surfaces Hiro-o Tokunaga, A remark on E. Artal-Bartolo's paper: ''On Zariski pairs'' [J. Algebraic Geom. 3 (1994), no. 2, 223 -- 247; MR1257321 (94m:14033)], Kodai Math. J. 19 (1996), no. 2, 207 -- 217. Coverings in algebraic geometry, Rational points, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Special algebraic curves and curves of low genus A remark on Artal's paper | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Alexandre Grothendieck, being undoubtedly one of the great men in mathematics after World War II, and being particularly famous for his revolutionary foundation of modern, scheme-theoretic algebraic geometry, had an impact on the development of contemporary mathematics, as a whole, that was probably matchless at his time. Apart from his numerous research papers, books (EGA), seminar notes (SGA), Bourbaki talks, and contributions to conference proceedings on many occassions, it is also the vast amount of sketches of his further-going ideas and his published programmatic treatises that still inspires generations of mathematical researchers in our days. Grothendieck's ingenious ideas, programs, outlines of what he thought that should propel pure mathematics in the next future, and his pioneering mathematical insight were much ahead of his time. Therefore it will take another number of generations to elaborate his programs in full depth, or at least to understand how far his mathematical thoughts went already back then. Alexandre Grothendieck quit his academic career in the mid 1980's, and ceased any publishing of scientific articles even ten years earlier, and that for reasons that are well-known to the mathematical (and political) community, but this does not mean at all that he had discontinued working mathematically. On the contrary, his ingenious mathematical mind has produced, in the meantime, a wealth of innovating and prospective ideas of extended programmatic character in algebraic and arithmetic geometry. His manuscripts from this period, written by hand or sometimes on a portable typewriter, amounted upto several hundred papers. More than ten years ago, Alexandre Grothendieck handed a great part of these mathematical documents over to Jean Malgoire (Université Montpellier II) and authorized him to possibly publish them on behalf of A. Grothendieck himself. -- Deciphering A. Grothendieck's (mostly hand-written) manuscripts, which must have been a Sisyphean-type of work, \textit{J. Malgoire} and his collaborators have prepared, during the period from 1990 to 1995, that part of Grothendieck's unpublished work which he had written between 1980 and 1981 under the title ``The long march through Galois theory''.
The brochure under review is the first part of this transcript. The text of this first volume encompasses the first thirty-seven sections of the entire text, while the remaining fourteen sections form the contents of the subsequent second volume. Altogether this is a treatise of about eight-hundred pages, the first twohundred-fifty of which are published in the present first volume. It would be an equally Sisyphean job to describe the contents of those first thirty-seven chapters, all the more so as A. Grothendieck has created a wealth of new concepts, notions, involved constructions, and terminological peculiarities in his notes. May it suffice here to say that the main theme of this first volume is, very roughly speaking, categorical-topological and cohomological abstract Galois theory and generalizes Galois-Teichmüller theory, with applications to the general moduli theory of algebraic and arithmetic curves. Already these few key words should make transparent that this part of A. Grothendieck's post-academic mathematical work is of greatest actuality and significance in current algebraic and arithmetical geometry, especially in the realm of moduli theory of curves and their recent applications to algebraic number theory and conformal field theories in mathematical physics.
I am not as insolent as to comment, or even make judgements on Alexandre Grothendieck's treatise on these subjects, and neither dare his trustees to do so, but I have no hesitation in saying that this work certainly gives propelling impulses to abstract algebraic geometry and its allied theories, just like Grothendieck's earlier pioneering work did. -- The publishers leave it to the readers to extract what they need, understand and find useful for their own work, and that should be a vast quantity and quality of inspiring new ideas, methods, and results.
The second part of this transcript, which is not included here, will present Grothendieck's unpublished work on group schemes, algebraic fundamental groups, and Fermats Last Theorem. Again, this reflects a high degree of actuality, and Grothendieck's far-sighted ingeniuity (back then in the early 1980's) just as well.
The publishers have rendered a great favor to the mathematical community by having transcribed A. Grothendieck's hand-written notes and made them available, in readable form, to a wider audience. It is hard to guess what an earlier publication of these notes would have brought about for the development of algebraic geometry, but finally they are there, thank Alexandre Grothendieck's generous decision to release them for publication and thank the publishers' marvellous job of deciphering and transcribing them. -- Whitout any doubt, and despite the fact that they are available only in their present form as the publication of some mathematics department, these notes ought to be in any modern library of algebraic geometry. cohomological Galois theory; topoi; sites; arithmetic curves; algebraic Teichmüller theory; modular groups; elliptic curves; moduli theory of curves; conformal field theories Grothendieck, Alexandre: La longue marche à travers la théorie de Galois tome 1, (1995) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Curves in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Topoi, Galois cohomology The long march through Galois theory. Vol. 1. Transcription of an unpublished manuscript. Edited and with a foreword by Jean Malgoire | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We introduce and study the local quiver as a tool to investigate the étale local structure of moduli spaces of \(\theta\)-semistable representations of quivers. As an application we determine the dimension vectors associated to irreducible representations of the torus knot groups \(G_{p,q}=\langle a,b\mid a^p=b^q\rangle\). local quivers; representations of quivers; dimension vectors; irreducible representations; torus knot groups Adriaenssens J., Le Bruyn L.: Local quivers and stable representations. Comm. Algebra 31(4), 1777--1797 (2003) Representations of quivers and partially ordered sets, Geometric invariant theory, Group actions on varieties or schemes (quotients), Trace rings and invariant theory (associative rings and algebras) Local quivers and stable representations. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this dissertation, the author studies invariants of Hilbert schemes of zero-dimensional subschemes of smooth varieties. He simplifies his arguments from Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007) to compute the Betti numbers of the Hilbert scheme \(\text{Hilb}^n (X)\) of zero-dimensional subschemes of length \(n\) of a smooth projective surface. The proof is based on an extension of a cell decomposition of the local Hilbert scheme presented by \textit{G. Ellingsrud} and \textit{S. A. Strømme} [in Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)], and on finding a good reduction mod \(p\) and using the Weil conjectures. The author proceeds to use similar methods to find the Betti numbers of \(\text{Hilb}^n (X)\) for Kummer varieties \(X\) of higher order, and for several kinds of Hilbert schemes of triangles.
The second part of the thesis deals with cases in which the Chow ring of Hilbert schemes can be computed. The author succeeds in the case of varieties of second and higher order data and applies his formulas to give enumerative results about contact varieties of projective varieties with linear spaces in \(\mathbb{P}^N\). He also describes the Chow ring of \(\text{Hilb}^3 (\mathbb{P} ({\mathcal E}),X)\), the relative Hilbert scheme of a projective bundle of a vector bundle \({\mathcal E}\) of rank 3 over a smooth variety \(X\), in analogy with the results of \textit{G. Elencwajg} and \textit{P. Le Barz} [Compos. Math. 71, No. 1, 85-119 (1989; Zbl 0705.14004)]. Also several varieties of triangles are treated. The results get rather messy and were determined using the aid of a computer.
The exposition is clear and on a very high level. The reader is assumed to have a thorough knowledge of a good portion of the work contained in the 108 references. invariants of Hilbert schemes of zero-dimensional subschemes; Betti numbers; Kummer varieties; Chow ring Göttsche, L.: Hilbert schemes of zero-dimensional subschemes of smooth varieties. Lect. Notes Math. vol. 1572, Berlin Heidelberg New York: Springer 1993 Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Algebraic cycles, Algebraic moduli problems, moduli of vector bundles Hilbert schemes of zero-dimensional subschemes of smooth 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 This survey is rich with ideas. The author expresses some regret that various questions that could have been included, could not be covered in the article. But, the ideas, motivations to questions, known results, conjectures and approaches towards them that are sketched here are already a veritable treasure for non-experts as well as useful to experts. The results and conjectures described in the article have been obtained in collaboration with Akshay Venkatesh.
Classically, only the free part of the homology groups of arithmetic groups was studied in detail. For instance, for congruence subgroups \(\Gamma_0(N)\) of \(\mathrm{SL}_2(R)\) where \(R\) is either \(\mathbb{Z}\) or the ring of integers of a real quadratic field, the free part \(H_i(\Gamma_0(N), \mathbb{Z})\) for \(i=1,2\) is intimately related to classical and Hilbert modular forms respectively. The torsion part appears to be little arithmetic interest. However, in marked contrast, when \(R\) is the ring of integers of an imaginary quadratic field, the free part is small but the torsion part is large and has deep arithmetic interest.
To give a few dramatic examples due to \textit{M. H. Şengün} [Int. J. Number Theory 8, No. 2, 311--320 (2012; Zbl 1243.30085)]:
When \(R = \mathbb{Z}[i]\), then
\[
\Gamma_0(41+56i)^{ab} \cong \mathbb{Z}/4078793513671 \mathbb{Z} \oplus \mathbb{Z}/292306033 \mathbb{Z} \oplus \cdots
\]
\[
\Gamma_0(118+175i)^{ab} \cong \mathbb{Z} \oplus T
\]
where \(T\) is a finite abelian group of order \(>10^{310}\).
As the author puts it, ``beyond the guilty pleasure of exhibiting gigantic randomly distributed primes, there are more serious reasons to study torsion in the homology''. Concerning congruence subgroups of \(\mathrm{SL}_2(R)\) for \(R\) ring of integers of an imaginary quadratic field, the following conjecture is stated:
\textbf{Conjecture.} Let \(M_n \rightarrow M_0\) (\(n \in N\)) be a sequence of congruence covers of a fixed congruence hyperbolic \(3\)-manifold \(M_0\) such that \(\mathrm{Vol}(M_n) \rightarrow \infty\). Then,
\[
\lim_{n \rightarrow \infty} \frac{\log|H_1(M_n, \mathbb{Z})_{\mathrm{tors}}|}{\mathrm{Vol}(M_n)} = \frac{1}{6 \pi}.
\]
A particular case of this conjecture for \(\mathrm{SL}_2(\mathbb{Z}[i])\) asserts more precisely that:
\[
\frac{\log |H_1(\Gamma_0(N),\mathbb{Z})_{\mathrm{tors}}|}{N^2} \rightarrow \frac{\lambda}{18 \pi}
\]
as \(N \rightarrow \infty\), where \(\lambda = L(\chi_{\mathbb{Q}(i)},2) = 1 - \frac{1}{9} + \frac{1}{25} - \cdots\)
The conjecture motivated the computations of Sengün mentioned above. The motivation for the conjecture itself comes from the study of Ray-Singer analytic torsion of a closed manifold, where a special case of a theorem of Cheeger and Müller implies an expression in terms of the Selberg zeta function that is analogous to expressing the central value of the $L$-function of an elliptic curve in terms of the rank and size of the Tate-Shafarevich group.
The author sketches an argument to explain that mod-\(p\) torsion classes in \(\Gamma_0(N)\) (very roughly) parametrize quadratic extensions of \(\mathbb{Q}(\sqrt{d})\) whose Galois group is a subgroup of \(\mathrm{GL}_2(\overline{\mathbb{F}_p})\). Conversely, certain conjectures that associate torsion classes to some field extensions are analyzed with a view to explaining the drastically different behaviour exhibited in the real quadratic and the imaginary quadratic cases. Motivated by this, the author considers posing a suitable conjecture for general arithmetic groups and states the following precise version for the case of subgroups of finite index in \(\mathrm{SL}_n(\mathbb{Z})\) for \(n \geq 3\).
\textbf{Conjecture.} Let \(\Gamma_i\) (\(i \geq 1\)) be a family of distinct subgroups of finite index in \(\mathrm{SL}_n(\mathbb{Z})\). Then, as \(i \rightarrow \infty\), the numbers \(\frac{\log|H_q(\Gamma_i, \mathbb{Z})_{tors}|}{[\mathrm{SL}_n(\mathbb{Z}):\Gamma_i]}\) tend to \(0\) unless we are in one of the following two situations:
\(n=3, q=2\) in which case the limit is \(\frac{\zeta(3)}{288 \pi^2}\); or
\(n=4=q\) i which case the limit is \(\frac{31 \sqrt{2} \zeta(3)}{259200 \pi^2}\).
The author also goes on to sketch a possible method of proving the conjectures. homology of arithmetic groups; torsion; mod-$p$ modular forms; $L$-functions; analytic torsion Cohomology of arithmetic groups, Modular and Shimura varieties Torsion homology growth in arithmetic 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 authors deal with the degrees of canonical maps of smooth varieties of general type. For curves it is classically known that the canonical map is either an embedding or a degree 2 map onto \(\mathbb P^1\), the latter happens precisely when the curve is hyperelliptic. For surfaces \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] showed that the degree of the canonical map is at most 36, and recently \textit{C. Rito} [``Surfaces with canonical map of maximum degree'', Preprint, \url{arXiv:1903.03017}] showed that this bound is sharp. In dimension three \textit{R. Du} and \textit{Y. Gao} [Geom. Dedicata 185, 123--130 (2016; Zbl 1391.14077)] showed that the degree of the canonical map is at most 360, but the highest value reached so far is 72, achieved by taking the product of a hyperelliptic curve and Rito's surface.
In the paper under review, the authors set a new record constructing a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96. The family is obtained by considering certain curves of genus five endowed with \((\mathbb Z_2)^4 \)-actions and by taking the quotient of the product of such three curves by a free \((\mathbb Z_2)^4 \)-action induced by the actions on the single curves. threefolds of general type; canonical map; finite group actions \(3\)-folds, Group actions on varieties or schemes (quotients) A family of threefolds of general type with canonical map of high 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 The authors develop a classification of simple parametrized curves over algebraically closed fields of characteristic \(p>0\); the case of simple multigerms of curves over the field of complex numbers was treated by \textit{P. A. Kolgushkin} and \textit{R. R. Sadykov} [Rev. Mat. Complut. 14, No. 2, 311--344 (2001; Zbl 1072.14501)]. In this paper, the authors treat the first part of this classification, of pairs of curves with regular first component.
The classification is based upon first finding a list of non-simple confining singularities, then finding a weak normal form using only left-equivalence, independent of the characteristic, and finally using right-equivalence for normal forms, characteristic dependent. \(\mathcal{A}\)-equivalence; characteristic \(p\); parameterized curves; simple singularities of multigerms Singularities of curves, local rings, Singularities in algebraic geometry, Plane and space curves Simple singularities of reducible curves in characteristic \(p > 0\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra using the equivariant cohomology of sheaves on \(\text{SL}_n\). This construction makes sense for all simple groups, so we obtain a generalization of the Jones-Ocneanu trace to Hecke algebras of other types. We give a geometric expansion of this trace in terms of the irreducible characters of the Hecke algebra and conclude that it agrees with a trace defined independently by Gomi.
Based on our proof, we also prove that certain simple perverse sheaves on a reductive algebraic group \(G\) are equivariantly formal for the conjugation action of a Borel \(B\), or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured.
This construction is also closely tied to knot homology. This interpretation of the Jones-Ocneanu trace is a more elementary manifestation of the geometric construction of HOMFLYPT homology given by the authors. irreducible characters of Hecke algebras; Jones-Ocneanu trace; equivariant cohomology of sheaves; perverse sheaves; reductive algebraic groups; Hochschild homology; Soergel bimodules Webster, B., Williamson, G.: The geometry of Markov traces. arXiv:0911.4494 Hecke algebras and their representations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Cohomology theory for linear algebraic groups The geometry of Markov traces. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 characterize Ulrich modules over cyclic quotient surface singularities using the notion of special Cohen-Macaulay modules. We also investigate the number of indecomposable Ulrich modules for a given cyclic quotient surface singularity, and show that the number of exceptional curves in the minimal resolution determines a boundary on the number of indecomposable Ulrich modules. Ulrich modules; special Cohen-Macaulay modules; McKay correspondence; cyclic quotient surface singularities Cohen-Macaulay modules, McKay correspondence, Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Ulrich modules over cyclic quotient surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper studies the normal quartic surfaces in \({\mathbb{P}}^ 3\) and the reduced sextic curves in \({\mathbb{P}}^ 2\) with a given singularity of one of the following type: a simple elliptic singularity \(\tilde E_ 8\), or a cusp singularity \(T_{2,3,7}\), or a unimodular exceptional singularity \(E_{12}\). The results obtained are of two types: (i) the description of the other singularities of the quartic surface or the sextic curve, or (ii) the existence of such surfaces or curves having prescribed singularities. The method used by the author relies heavily on Looijenga's paper [\textit{E. Looijenga}, Ann. Math., II. Ser. 114, 267-322 (1981; Zbl 0509.14035)] where a Torelli-type theorem for rational surfaces with effective anti-canonical divisors is established. Indeed, the author reduces himself to the study of double covers of \(P^ 2\) branched along a sextic curve, and proves that the surfaces in question are rational. Then he applies Looijenga' methods to construct the moduli space for them and examines closely the action of the Weyl group on the moduli space. Dynkin diagrams; normal quartic surfaces; sextic curves; prescribed singularities; action of the Weyl group on the moduli space Urabe, T.: On quartic surfaces and sextic curves with singularities of type \(\tilde E_8 \) ,T 2, 3, 7,E 12. Publ. RIMS, Kyoto Univ.20, 1185-1245 (1984) Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Special surfaces, Singularities in algebraic geometry On quartic surfaces and sextic curves with singularities of type \(\tilde E_ 8\), \(T_{2,3,7}\), \(E_{12}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the question of irreducibility of the Hilbert scheme of points \(\mathcal H\mathrm{ilb}_d\mathbb P^n\) and its Gorenstein locus. This locus is known to be reducible for \(d \geq 14\). For \(d \leq 11\) the irreducibility of this locus was proved in the series of papers [\textit{G. Casnati} and \textit{R. Notari}, J. Pure Appl. Algebra 213, No. 11, 2055--2074 (2009; Zbl 1169.14003); ibid. 215, No. 6, 1243--1254 (2011; Zbl 1215.14009); ibid. 218, No. 9, 1635--1651 (2014; Zbl 1287.13013)] and Iarrobino conjectured that the irreducibility holds for \(d \leq 13\). In this paper, we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function \((1,5,5,1)\) are smoothable, i.e. lie in the closure of the locus of smooth subschemes. This result completes the proof of irreducibility of the Gorenstein locus of \(\mathcal H\mathrm{ilb}_{12}\mathbb P^n\), see Theorem 2. smoothability; zero-dimensional schemes; Gorenstein algebras; Hilbert scheme J. Jelisiejew, \textit{Local finite-dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable}, J. Algebra App., 13 (2014), 1450056. Deformations and infinitesimal methods in commutative ring theory, Parametrization (Chow and Hilbert schemes), Commutative Artinian rings and modules, finite-dimensional algebras, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Local finite-dimensional Gorenstein \(k\)-algebras having Hilbert function (1,5,5,1) are smoothable | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well known that the automorphism group of a surface of general type is finite and bounded by a function of \(K^2\) [cf. \textit{A. Andreotti}, Rend. Mat. Appl., V. Ser. 9, 255-279 (1950; Zbl 0041.08502)]. Xiao obtained linear bounds for automorphism groups and for abelian automorphism groups [\textit{G. Xiao}, Invent. Math. 102, No. 3, 619-631 (1990; Zbl 0739.14024)] of a surface of general type. Then the upper bounds of various automorphism groups of genus 2 fibrations were obtained by \textit{Z. Chen} [Tôhoku Math. J., II. Ser. 46, No. 4, 499-521 (1994; Zbl 0837.14034)]. Based on these results, \textit{H. Xue} [Algebra Colloq. 4, No. 1, 65-78 (1997; Zbl 0899.14019] proved that, for a minimal complex surface \(S\) of general type with a pencil of genus 2, \(|\Aut(S)|\geq 576 K^2_S\). -- The main result is the following theorem:
Suppose \(S\) is a minimal surface of general type with a pencil of genus 2, which is not of type \(K^2_S=\chi ({\mathcal O}_S)=1\). Then any abelian automorphism group \(G\) of \(S\) satisfies \(| G|\leq 12.5 K^2_S+100\). abelian automorphism groups; surface of general type; genus 2 fibrations Surfaces of general type, Birational automorphisms, Cremona group and generalizations Bounds of abelian automorphism groups of surfaces with a pencil of genus 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a compact bordered Klein surface of (algebraic) genus \(p\geq 2.\) The reviewer showed that the order of an automorphism of X cannot be larger than \(2p+2\) if X is orientable and p is even; otherwise the order cannot be larger than 2p [Houston J. Math. 3, 395-405 (1977; Zbl 0379.14012)]. It was also established there that for each value of the genus \(p\geq 2,\) there is a unique topological type of orientable bordered surface with an orientation-preserving automorphism of maximum possible order. This topological type of surface has one boundary component if p is even and two boundary components if p is odd.
The paper under review is a thorough study of the topological types of bordered Klein surfaces with an automorphism of maximum possible order. Each remaining case is settled. The Klein surface X is represented as a quotient of the hyperbolic plane by a non-euclidean crystallographic (NEC) group, and NEC groups play a central role in the paper.
First assume that the bordered Klein surface X is orientable and f is an automorphism of X that reverses orientation. The author shows that if the \(genus\quad p\) is even and the other of f is \(2p+2\), then X has \(p+1\) boundary components. If p is odd, an improved upper bound for the order of f is obtained. In this case, \(o(f)\leq 2p-2\); again X has a unique topological type if the bound is attained. Finally, if X is non- orientable and has an automorphism of order 2p, then X has p boundary components. The author also obtains in each case (including the old one) the signature of the NEC group that produces the automorphism of maximum possible order.
The corrigendum contains two lines omitted from section 3. non-euclidean crystallographic group; compact bordered Klein surface; order of an automorphism; topological type; NEC groups Curves in algebraic geometry, Topological properties in algebraic geometry, Other geometric groups, including crystallographic groups Topological types of Klein surfaces with a maximum order automorphism. Corrigendum | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 organizing centre of an imperfect bifurcation problem \(F(u,\lambda,\alpha)=0\) is linked with a simple root of an auxiliary operator (the inflated mapping of F). Assuming that the particular singularity of the bifurcation equation has codimension \(\leq 2\), the classification of organizing centres is developed in terms of singularities of the germs of smooth mappings \(g: R_ m\times R_ 1\times R_ k\to R_ m\), by defining properly the inflated mapping. organizing centre of an imperfect bifurcation problem; simple root of an auxiliary operator; inflated mapping; singularities of the germs of smooth mappings Variational problems in abstract bifurcation theory in infinite-dimensional spaces, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Numerical solutions to equations with nonlinear operators, Singularities in algebraic geometry, Equations involving nonlinear operators (general) Inflated mappings for singularities of codimension \(\leq 2\) | 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 [For the entire collection see Zbl 0722.00009.]
As the author notes the paper under review reflects rather faithfully the lectures given by the author at the conference: ``Generators and relations in groups and geometries''. He introduces basic notions, formulates main theorems and considers some examples to provide an introduction to the topic of the title of the paper. The paper contains the following themes: affine algebraic varieties and their morphisms, fields of definition, linear algebraic groups and some examples of algebraic groups, subgroups, groups of rational points, algebraic groups over finite fields, Jordan decomposition in algebraic groups, conjugacy classes in groups of rational points, some general facts about conjugacy classes of semi-simple elements, the adjoint quotient map, simply connected groups, centralizers of semi-simple elements, properties of the adjoint quotient map, Borel groups, the simultaneous resolution. affine algebraic varieties; linear algebraic groups; groups of rational points; Jordan decomposition; conjugacy classes; semi-simple elements Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects) Geometric structure of conjugacy classes in 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 Let V be a finite dimensional representation of a complex reductive algebraic group G. Then (V,G) is called cofree, resp. coregular if the symmetric algebra S(V) is free over the algebra of invariants \(S(V)^ G\), resp. if \(S(V)^ G\) is a polynomial ring. The author classifies the pairs (\({\mathfrak g},K)\) which are cofree/coregular/ ``non-complicated'' in other ways. Here \({\mathfrak g}\) is the complexification of a real semisimple Lie algebra and K is a corresponding compact analytic subgroup of the adjoint group. cofree algebraic group; coregular algebraic group; finite dimensional representation; complex reductive algebraic group; symmetric algebra; algebra of invariants; real semisimple Lie algebra; compact analytic subgroup Nicolás Andruskiewitsch, On the complicatedness of the pair (\?,\?), Rev. Mat. Univ. Complut. Madrid 2 (1989), no. 1, 13 -- 28. Semisimple Lie groups and their representations, Geometric invariant theory, Simple, semisimple, reductive (super)algebras, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients) On the complicatedness of the pair \(({\mathfrak g},K)\) | 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 In [\textit{R. Bédard} and \textit{R. Schiffler}, Represent. Theory 7, 481--548 (2003; Zbl 1060.17001)] one gives the characterization of the orbits of representations of quivers of type \(A\) with Zariski closures which are rationally smooth. In the paper under review the author continues these investigations and studies the local rational smoothness of these closures. He obtains a description of the orbits with the property that the projectivization of their Zariski closures are rationally smooth.
The main idea of the method is to use that the change of basis between canonical, and PBW-basis of the positive part of the quantized enveloping algebra of type \(A_n\) has a geometric interpretation in terms of local intersection cohomology of some affine algebraic varieties, namely the Zariski closures of orbits of representations of a quiver of type \(A_n\). Then the author applies some methods already used in his paper with Bédard, and some results from there. Comparing the results of both papers, it turns out that the only non-smooth, projectively rationally smooth orbit closures are of type \(A_2\) and \(A_3\). projective representations of quivers; varieties of representations; rational smoothness; quantum groups; quantized enveloping algebra Quantum groups (quantized enveloping algebras) and related deformations, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Projective rational smoothness of varieties of representations for quivers of type \(A\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Motivated by appearances of Rogers' false theta functions in the representation theory of the singlet vertex operator algebra, for each finite-dimensional simple Lie algebra of ADE type, we introduce higher rank false theta functions as characters of atypical modules of certain \(W\)-algebras and compute asymptotics of irreducible characters which allows us to determine quantum dimensions of the corresponding modules. In the \(\mathrm{sl}_2\)-case, we recover many results from the authors' paper [Int. Math. Res. Not. 2015, No. 21, 11351--11387 (2015; Zbl 1359.17039)]. characters; false theta functions; Jacobi forms; modular forms; vertex algebras Bringmann, K.; Milas, A., W-algebras, higher rank false theta functions and quantum dimensions, Selecta math., 23, 2, 1249-1278, (April 2017) Vertex operators; vertex operator algebras and related structures, Relationship to Lie algebras and finite simple groups, Theta series; Weil representation; theta correspondences, Theta functions and curves; Schottky problem \(W\)-algebras, higher rank false theta functions, and quantum dimensions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a finite field of characteristic \(p>0\) and \(K\) be an algebraic function field of one variable over \(k\). Consider an elliptic curve \(E\) over \(K\) with \(j\)-invariant transcendental over \(\mathbb{F}_p\). In this paper the author proves that there exists a finite set of primes \(S\), with \(p\in S\), such that for every positive integer \(n\) not divisible by any of the primes of \(S\), the pure \(n\)-torsion points of the fibers of the Néron model of \(E\) describe after completion a smooth irreducible projective curve \(C_n\). The Selmer group \(S_n(E/K)\) embeds canonically into \(_n\text{Pic}^0 (C_n)\), and a necessary and sufficient condition is given for an element of \(_n\text{Pic}^0 (C_n)\) to belong to \(S_n(E/K)\). Moreover, the induced embedding \(E(K)/nE(K) \hookrightarrow {_n\text{Pic}^0} (C_n)\) is described explicitly. Shafarevich-Tate group; Picard groups; transcendental \(j\)-invariant; finite field; algebraic function field; elliptic curve; fibers of the Néron model; irreducible projective curve; Selmer group; embedding Elliptic curves over global fields, Global ground fields in algebraic geometry, Elliptic curves, Arithmetic ground fields for curves Selmer groups and Picard 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 [For the entire collection see Zbl 0517.00008.]
This is another version of an earlier paper of the author [Proc. Symp. Pure Math., Vol. 40, Part II, 479-484 (1983; Zbl 0524.57016)]. normal Gorenstein singularity; cobordism invariants; cobordism group of stably framed 3-manifolds; e-invariant; resolution of singularities; complex analytic surface; Milnor number Algebraic topology on manifolds and differential topology, Singularities of surfaces or higher-dimensional varieties, Complex singularities, Local complex singularities, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory, Stable homotopy of spheres Singularities of complex surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A Kähler group is the fundamental group of a compact Kähler manifold. The Lie algebra of the Mal'tsev completion of the fundamental group is quadratically presented if it is the quotient of the free Lie algebra on its abelianization by an ideal generated in degree two. It is known that the Lie algebras of such groups do not admit quadratic presentations. One purpose of this paper is to obtain an infinite family of examples of quadratically presented three-step nilpotent Lie algebras and to classify quadratically presented complex nilpotent Lie algebras with abelianization of dimension at most five. The second purpose of this article is to prove that if \(M\) is a compact Kähler manifold with nilpotent fundamental group which is not almost abelian and if \(\omega\) is a non-zero integral element of type (1,1) in its characteristic subspace \(\mathcal C\), then the rank of \(\omega\) is at least 8. Several applications of the above results are also given. Kähler group; fundamental group of a compact Kähler manifold; Mal'cev completion; quadratic presentations; three-step nilpotent Lie algebras; abelianization; nilpotent fundamental group; characteristic subspace James A. Carlson & Domingo Toledo, ``Quadratic presentations and nilpotent Kähler groups'', J. Geom. Anal.5 (1995) no. 3, p. 351-359, erratum in \(ibid.\)7 (1997), no. 3, p. 511-514 Algebraic topology on manifolds and differential topology, Global differential geometry of Hermitian and Kählerian manifolds, Transcendental methods, Hodge theory (algebro-geometric aspects), Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) Quadratic presentations and nilpotent Kähler groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A = {\mathbf C}[[x,y,z]]/(f, \partial f/\partial x, \partial f/\partial y, \partial f/\partial z)\) be the moduli local algebra of an isolated hypersurface singularity \((V,0) \subset ({\mathbf C}^3, 0)\) defined by the germ of an analytic function \(f=f(x,y,z).\) Due to \textit{J. Mather} and \textit{S.-T. Yau} [Invent. Math. 69, 243-251 (1982; Zbl 0499.32008)] it is a finite dimensional \({\mathbf C}\)-algebra which determines the analytic type of the singularity.
The authors study the case of \(\widetilde E_6\) singularities which are defined by the one parameter family of the germs of analytic functions \(f_t(x,y,z) = x^3 + y^3 + z^3 + txyz.\) The corresponding family of moduli algebras \(A_t = {\mathbf C}[[x,y,z]]/(3x^2+tyz, 3y^2+txz, 3z^2+txy)\) can be considered as a one parameter family of commutative Artinian algebras. The set of isomorphisms of such algebras \(A_t, t^3 \neq 0, 216, -27,\) is described. It consists in fact of 216 matrices in PGL\((3, \mathbf C).\) As a consequence the following result is obtained. Let \(k(t) = t^3(t^3-216)/(t^3+27)^3.\) Then two elements \(A_t\) and \(A_s\) of this one parameter family are isomorphic if and only if \(k(t) = k(s).\) Thus, \(k(t)\) is a modulus of the family.
The authors remark that they ``have enough evidence to show that Saito's computation of the \(j\)-invariant for \(\widetilde E_6\) is in error'' [cf. \textit{K. Saito}, Invent. Math. 23, 289-325 (1974; Zbl 0296.14019)]. simple elliptic singularities; moduli algebras; modulus; Artinian local algebras; \(j\)-invariant Chen, H., Seeley, C., Yau, S.S.-T.: Algebraic determination of isomorphism classes of the moduli algebras of \$\(\backslash\)tilde\{E\}\_\{6\}\$ singularities. Math. Ann. 318, 637--666 (2000) Equisingularity (topological and analytic), Infinitesimal methods in algebraic geometry, Commutative Artinian rings and modules, finite-dimensional algebras, Classification; finite determinacy of map germs Algebraic determination of isomorphism classes of the moduli algebras of \(\widehat E_6\) singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper extends the work of Dwyer and Friedlander about etale K-theory by defining a mod \(\ell^{\nu}\) topological K-homology theory for quasiprojective schemes X over k (k is a ''nice'' noetherian ring in which the prime \(\ell\) is invertible). There is a natural transformation \(\rho\) (X) from algebraic G-theory to (mod \(\ell^{\nu})\) topological K- homology theory. The Riemann-Roch theorem says that \(\rho\) (X) is natural for projective morphisms. Most part of the paper is concerned with the definition of this topological K-homology theory. After that, the deformation to the normal cone machinery is adapted to prove the theorem. The results are applied to compute the algebraic and topological K-groups of reductive group schemes, homogeneous spaces and complete rational surfaces. K-homology theory; quasiprojective schemes; Riemann-Roch theorem; normal cone; K-groups of reductive group schemes R. W. Thomason, Riemann-Roch for algebraic versus topological \?-theory, J. Pure Appl. Algebra 27 (1983), no. 1, 87 -- 109. Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry Riemann-Roch for algebraic versus topological K-theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0742.00080.]
From the introduction: The purpose of this article is to describe in modern terms some of the memoir ``Essai d'une théorie générale des formes algébriques'' by \textit{J. Deruyts} [Mém. Soc. R. Sci. Liège 17, 1-156 (1891; JFM 23.0110.01)]. In this remarkable work Deruyts anticipates by nearly a decade the main results of Schur's dissertation. His language is the language of invariant theory, and makes little use of matrices. But we now look back on Deruyt's work and find a wealth of methods which to our eyes are pure representation theory. Some of these methods are still unfamiliar today. One can speculate that the representation theory of linear groups could have taken a different course, or at least have matured more rapidly if Schur or Weyl had taken up the ideas which are lying just below the surface of Deruyt's memoir!\dots polynomial rings; covariants; polynomial representations; invariant theory; representation theory of linear groups Green, J. A.: Classical invariants and the general linear group. Progr. math. 95, 247-272 (1991) Representation theory for linear algebraic groups, Geometric invariant theory, Classical groups (algebro-geometric aspects) Classical invariants and the general linear 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 28 years after its first appearence, Irving Reiner's book ``Maximal orders'' has remained to be a standard reference to ``non-commutative arithmetics''. Apart from the classical applications already mentioned in the book, there are some areas of currently active research, like non-commutative algebraic geometry or non-Abelian Iwasawa theory, where the theory of maximal orders has become an important requisite.
Not only for maximal orders, but also for general orders over a Dedekind domain, Reiner's book provides an excellent introduction for students and serves as an indispensible reference for researchers. For a summary of contents we refer to the review Zbl 0305.16001 of the original. maximal orders; Dedekind domains; separable algebras; reduced norms; traces; localizations; completions; discrete valuation rings; Brauer groups; crossed products; simple algebras; hereditary orders I. Reiner, \textit{Maximal Orders}, London Mathematical Society Monographs. Vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Divisibility, noncommutative UFDs, Finite-dimensional division rings, Localization and associative Noetherian rings, Quaternion and other division algebras: arithmetic, zeta functions, Representations of orders, lattices, algebras over commutative rings, Algebras and orders, and their zeta functions, Brauer groups of schemes, Collected or selected works; reprintings or translations of classics 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 Let \(S\) be a smooth projective algebraic surface over an algebraically closed field \(\kappa\) and let \(H_d= \text{Hilb}_bS\) be the Hilbert scheme of 0-dimensional subschemes of length \(d\) in \(S\). Taking a point \(x\in S\), we shall study the set \(H_d[x]= \{\xi\in H_d |\text{Supp} \xi =x\}\). The set \(H_d[x]\) is obviously endowed with a variety structure, i.e., it is a reduced irreducible scheme of dimension \(d-1\) over \(\kappa\). Moreover, the set \(H_d[x]\) is a subscheme in \(H_d\). It is referred to as the punctual Hilbert scheme of the surface. For \(d=1\) and 2, its description is trivial: \(H_1[x]= \{\text{point}\}\) and \(H_2[x]= P(T_xS) \simeq \mathbb{P}^1\). Even for \(d=3\), the set \(H_d[x]\) acquires singularities: The set \(H_3[x]\) is a surface isomorphic to a cone over the space cubic curve in \(\mathbb{P}^3\). For higher dimensions \(d\), the singularities of the set \(H_d[x]\) are quite complicate. The description can, possibly, be made in terms of Iarrobino's stratification.
In this paper, we use a different approach based on the natural birational model \(X_d\) of the scheme \(H_d[x]\), which is obtained by ``lifting'' the scheme \(H_d[x]\) to the Hilbert scheme of complete flags \(\Gamma_{12 \dots d} =\{(\xi_1, \dots, \xi_d)\in \prod^d_{k=1} H_k |\xi_1 \subset\xi_2 \subset \cdots \subset \xi_d\}\). The model thus constructed is helpful in giving an algebraic-geometric description of the three-dimensional variety \(H_4[x]\) and its singularities. punctual Hilbert scheme of a surface; complete flags; singularities A. S. Tikhomirov, ''A smooth model of punctual Hilbert schemes of a surface,''Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],208, 318--334 (1995). Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties A smooth model for the punctual Hilbert scheme of a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.
For Part I, see [the first two authors, J. Eur. Math. Soc. (JEMS) 15, No. 6, 2353--2367 (2013; Zbl 1303.14019)]. compactified Jacobians; Hilbert scheme of points; reduced curves with locally planar singularities; perverse filtration; decomposition theorem; support theorem Jacobians, Prym varieties, Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) A support theorem for Hilbert schemes of planar curves. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group scheme over a field \(k\) of positive characteristic \(p\). In well-known work of \textit{E. Friedlander} and \textit{A. Suslin} [Invent. Math. 127, No. 2, 209-277 (1997; Zbl 0945.14028)], it was shown that the cohomology ring \(\text H^\bullet(G,k)\) is a finitely-generated Noetherian \(k\)-algebra. That argument was first reduced to the case that \(G\) is an infinitesimal group scheme. Any such \(G\) can be embedded in some \((GL_n)_r\), the \(r\)-th Frobenius kernel of the general linear group scheme \(GL_n\), and the key to the infinitesimal case was the construction of certain universal extension classes in \(\text{Ext}^{2p^{r-1}}_{GL_n}(k,\mathfrak{gl}_n^{(r)})\), where \(\mathfrak{gl}_n\) denotes the Lie algebra of \(GL_n\). That construction made use of strict polynomial functors.
In the work under review, the author gives a new construction of these universal extensions classes for \(GL_2\). The argument here does not make use of polynomial functors; something also done by \textit{W. van der Kallen} [in: Invariant theory in all characteristics. Proceedings of the workshop on invariant theory, Queen's University, Kingston, ON, Canada, April 8-19, 2002. Providence, RI: American Mathematical Society (AMS). CRM Proceedings \& Lecture Notes 35, 127-138 (2004; Zbl 1080.20039)]. Moreover, the author shows that the Ext-group \(\text{Ext}^{2p^{r-1}}_{GL_2}(k,\mathfrak{gl}_2^{(r)})\) is in fact one-dimensional. The author first shows the existence of such classes (and the dimensionality fact) over \(SL_2\) using a result of \textit{A. Parker} [Adv. Math. 209, No. 1, 381-405 (2007; Zbl 1112.20039)] on the nature of the Lyndon-Hochschild-Serre spectral sequence associated to a Frobenius kernel of \(SL_2\). The author then relates extensions over \(GL_2\) and \(SL_2\) to get the desired result. The finite-generation of \(\text H^\bullet(G,k)\) for an infinitesimal subgroup scheme \(G\) of \(GL_2\) then follows in a manner analogous to the argument given by Friedlander and Suslin. universal extension classes; general linear groups; special linear groups; infinitesimal group schemes; Frobenius kernels; cohomological finite generation Cohomology theory for linear algebraic groups, Cohomology of Lie (super)algebras, Group schemes Universal extension classes for \(GL_2\). | 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 In this very interesting and carefully written paper, the author discusses the problem of the effective termination of Kohn's algorithm for subelliptic multipliers for bounded smooth weakly pseudoconvex domains of finite type, introduced by \textit{J. J. Kohn} in [Acta Math. 142, 79--122 (1979; Zbl 0395.35069)]. The author gives here a complete proof of the effective termination of Kohn's algorithm for special domains, and gives directions on how the method is extended to the case of general bounded smooth weakly pseudoconvex domains of finite type. The termination of Kohn's algorithm in the real-analytic case was verified by \textit{K. Diederich} and \textit{J. E. Fornaess} [Ann. Math. (2) 107, 371--384 (1978; Zbl 0378.32014)] without effectiveness. In this paper, the author next formulates the Kohn algorithm geometrically in terms of the Frobenius theorem on integral submanifolds, and presents a proof of the real-analytic case of the ineffective termination of Kohn's algorightm from this geometric viewpoint. It is the author's hope that the geometric viewpoint developed in the paper will provide an easier and more transparent setting for further developments of the applications of algebraic-geometric techniques to general PDEs. subelliptic multipliers; Kohn's algorithm; complex Neumann problem; pseudoconvex domains; subelliptic estimates; finite type; multiplier ideal Siu, Y.T.: Effective termination of Kohn's algorithm for subelliptic multipliers. Pure Appl. Math. Q. \textbf{6}(4):1169-1241 (2010) \(\overline\partial\) and \(\overline\partial\)-Neumann operators, Multiplier ideals, Finite-type domains, Subelliptic equations Effective termination of Kohn's algorithm for subelliptic multipliers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Krichever correspondence between algebraic curves with additional structures and certain commutative algebras of differential operators, on which the Mulase-Shiota proof of the Novikov conjecture and solution of the problem of Riemann-Schottky are based, is refined and generalized to form an equivalence between a fibered category over the category of algebraic curves and some category of commutative algebras of matrix pseudo-differential operators, with the aim of a characterization of special (e.g. d-gonal) curves by differential equations for their theta functions. A generalization and reinterpretation of the generalized Jacobian and the singularization procedure of Rosenlicht-Serre is used to show that a deformation of the curve data is reflected by the matrix KP hierarchy on the operator side. Krichever correspondence; problem of Riemann-Schottky; matrix pseudo- differential operators; theta functions; generalized Jacobian; matrix KP hierarchy Theta functions and curves; Schottky problem, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Jacobians, Prym varieties, Pseudodifferential operators, General theory of partial differential operators, Partial differential equations of mathematical physics and other areas of application, Theta functions and abelian varieties Über den Kričever-Funktor in der Theorie der algebraischen Kurven. (On the Krichever functor in the theory of 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 A twisted Grassmann variety (a form of Grassmann variety), which is the variety representing the functor of right ideals of prescribed rank in a central simple algebra over a field, is represented by a linear section of a Grassmann variety (Theorem A). The Severi-Brauer schemes of some \(R\)-orders in the matrix ring \(M_4(R)\) of degree 4 over a regular local ring \(R\) are constructed (Theorem B). The variety of rank 4 right ideals of the associative \(k\)-algebra generated by \(x,y\) with the relations \(x^4=y^4=0\) and \(yx=\sqrt{-1}xy\) is described (Theorem C). orders; twisted Grassmann varieties; central simple algebras; Severi-Brauer schemes; regular local rings Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Grassmannians, Schubert varieties, flag manifolds, Brauer groups (algebraic aspects), Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) A realization of twisted Grassmann 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 In [\textit{E. Looijenga}, Invent. Math. 61, 1-32 (1980; Zbl 0436.17005)], a family \(A_ H\) of Abelian varieties, associated with an affine root system and parametrized by the complex upper half plane \(H\) was introduced and investigated. The natural action of the modular group \(SL_ 2(\mathbb{Z})\) lifts to that family, as well as to the ample linear bundle \({\mathcal L}_ H\) on \(A_ H\).
The purpose of the present paper is to study the invariant theory of \(A_ H\) and \({\mathcal L}_ H\) with respect to the action of the modular group (or a subgroup \(\Gamma\) of finite index) as well as of the Weyl group \(W\) of the corresponding root system. As a first step, the quotient by \(\Gamma\) is considered and the extension of the family \(A_ H/\Gamma\to H/\Gamma\) is obtained in a natural and explicit fashion, using a toroidal embedding technique of the author [Acta Math. 157, 159- 241 (1986; Zbl 0635.14015)]. Then the invariants are studied. They form a bi-graded algebra over the ring of modular forms, graded by weight (referring to the behaviour with regard to \(\Gamma\)) and index (referring to the appropriate power of \(\mathcal L\)), and are called Jacobi forms since in the special case of the root system of type \(A_ 1\) they reduce to (weak) Jacobi forms in the sense of \textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms (Birkhäuser, 1985; Zbl 0554.10018)]. The algebra of invariants is determined for all types of root systems excluding \(E_ 8\). It turns out to be a polynomial algebra over the ring of modular forms, with generators that do not depend on the particular choice of the group \(\Gamma\).
The result has an application in singularity theory to deformation of fat points in the plane with defining ideal \((x^ 2-y^ 3, y^ k)\) or \((x^ 2-y^ 3, xy^{k-1})\), \(k\geq 3\) (this will appear elsewhere). invariant theory; modular group; Weyl group; root system; toroidal embedding; Jacobi forms; polynomial algebra; deformation of fat points Wirthmüller, K., Root systems and Jacobi forms, Comp. Math., 82, 293, (1992) Simple, semisimple, reductive (super)algebras, Relationship to Lie algebras and finite simple groups, Jacobi forms, Singularities in algebraic geometry, Formal methods and deformations in algebraic geometry Root systems and Jacobi 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 This monograph (based on the author's Harvard dissertation) is concerned with the problem of compactification of the moduli space of principally polarised (\(g\)-dimensional) abelian varieties over \(\mathbb{Z}\). For elliptic curves \(E\), with \(j\) = the \(j\)-invariant of \(E\), the ''\(j\)-line'' \(\text{Spec}\mathbb{Z}[j]\) gives the moduli; the natural completion of the affine \(j\)-line \(\mathbb{A}^1\) is just \(\mathbb{P}^1\) and we have a 'canonical' solution for the compactification problem. When \(g>1\), the isomorphism classes of principally polarized abelian varieties correspond to points of the (coarse) moduli space \(A_ g\) of such abelian varieties; Mumford solved the classification problem, constructing a coarse moduli scheme \(A_ g\) over Spec \(\mathbb{Z}\), via his geometric invariant theory. Associated with the moduli scheme \(A_ g\), there exists a fundamental geometric problem underlying the need for the compactification: namely, for a given prime number \(p\),
\[
\begin{cases} \vtop{=.85 noindent is the geometric fibre \(A_ g \underset{\text{Spec}\mathbb{Z}}\times \text{Spec}\mathbb{F}_ p\) irreducible or equivalently, \smallskip\noindent is the moduli space of principally polarized abelian varieties irreducible, in characteristic \(p\)?} \end{cases} \tag{\(*\)}
\]
An affirmative answer to (\(*\)) for \(p>2\) is a consequence of the author's construction of toroidal completions of Siegel moduli schemes over \(\mathbb{Z}[\frac12]\) and extension (to positive characteristics) of a theorem of Tai on the ''projectivity of toroidal compactifications'' (substituting local holomorphic functions in Tai's proof with the algebraic machinery of theta functions). Let \(M(\mathbb{Z},k)\) denote the ring of Siegel modular forms of degree \(g\) and weight \(k\) with Fourier coefficients in \(\mathbb{Z}\), and \(R\) the graded ring \(\oplus_{k\geq 0} M(\mathbb{Z},k)\). For \(g=1\), it is well-known that \(R\) is finitely generated. The corresponding question for general \(g\) was raised by \textit{J. Iqusa} who also provided in a nice paper [Am. J. Math. 101, 149-183 (1979; Zbl 0415.14026)] an explicit set of generators for \(g=2\). The author's affirmative answer to (\(*\)) above (for \(p>2\)) implies that the graded ring of Siegel modular forms with Fourier coefficients in \(\mathbb{Z}[]\) is finitely generated over \(\mathbb{Z}[]\). --- A footnote on page xi refers to the question (\(*\)) of irreducibility for the case \(p=2\) having since been settled by Faltings.
Chapter I reviews the major results on Siegel moduli schemes used in subsequent chapters. The next chapter contains a treatment of semi-abelian varieties and with a vital definition of ''polarization'', the canonical construction of the quotient of a semi-abelian scheme by a discrete subgroup. These results are applied in chapter III to construct polarized semi-abelian schemes providing local coordinates, at the boundary, of toroidal completions of Siegel moduli schemes. Chapter IV contains the main result (theorem 4.2) of the author extending Tai's theorem to positive characteristics, and chapter V contains nice applications to Siegel modular forms.
There are three excellent appendices (dealing with theta functions); the results in the appendix on 2-adic theta functions with values in complete local fields \(k\) with a uniformization theorem for abelian varieties over k (with residue characteristic \(\neq2)\) are unpublished results due to Mumford. compactification of the moduli space of principally polarised; abelian varieties; coarse moduli scheme; toroidal completions of Siegel moduli schemes; semi-abelian varieties; 2-adic theta functions Chai, C.-L.: Compactification of Siegel moduli schemes. London Math. Soc. Lecture Note Series, vol. 107. Cambridge University Press (1985) Algebraic moduli problems, moduli of vector bundles, Algebraic moduli of abelian varieties, classification, Geometric invariant theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Structure of modular groups and generalizations; arithmetic groups, Families, moduli of curves (algebraic) Compactification of Siegel moduli 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 \(G\) be finite group scheme over an algebraically closed field \(k\) of characteristic \(p>0\). The group algebra \(kG\) of \(G\) is the dual of the (finite-dimensional) coordinate algebra \(k[G]\). A finitely generated \(kG\)-module \(M\) is said to be an endotrivial module if \(\Hom_k(M,M)\) is isomorphic as a \(kG\)-module to \(k\oplus P\) for a projective module \(P\). In other words, \(\Hom_k(M,M)\) is trivial in the stable module category. Endotrivial modules have been studied extensively for finite groups, and the goal of this paper is to begin a study in the more general setting of finite group schemes with special consideration given to the case of infinitesimal group schemes. Let \(T(G)\) denote the group of endotrivial modules (under appropriate equivalence).
The initial focus of the paper is on finite unipotent group schemes. For such a group \(U\), it is shown that there are a finite number of endotrivial modules of any given dimension. Further, the authors formulate a set of criteria under which \(T(U)\) is isomorphic to \(\mathbb Z\), that is, consists of syzygies of the trivial module.
For the remainder of the paper, the authors consider Frobenius kernels which form an important class of finite group schemes. From now on, let \(G\) be a semisimple simply connected algebraic group over \(k\) with Borel subgroup \(B\), and let \(U\) denote the unipotent radical of \(B\). Let \(G_r\), \(B_r\), \(U_r\), and \(P_r\) (for a parabolic subgroup \(P\subset G\)) denote the respective \(r\)-th Frobenius kernels. For \(U_1\), \(B_1\), and \(P_1\) (for a proper standard parabolic), the group of endotrivial modules is completely determined. In particular, except for some small rank cases, \(T(U_1)\) is shown to be isomorphic to \(\mathbb Z\).
\(T(G_1)\) is determined for \(G=\text{SL}_2\), but remains an open question in general. In particular, for \(\text{SL}_2\), it is shown that there are Weyl modules which are endotrivial over \(G_1\) but are not syzygies of the trivial module. Further, they make a precise determination of when simple, Weyl, and indecomposable tilting modules for \(G=\text{SL}_2\) are endotrivial over \(G_r\). On the other hand, for a group \(G\) of rank at least two (with some conditions on \(p\)) it is shown that there are no non-trivial simple, Weyl, or indecomposable tilting modules which are endotrivial upon restriction to \(G_r\).
Lastly, the authors note some connections with the Picard group of the projectivization of the cohomological spectrum and conclude with several interesting open questions. endotrivial modules; finite group schemes; Frobenius kernels; unipotent group schemes; semisimple simply connected algebraic groups; Picard groups; infinitesimal group schemes; indecomposable tilting modules Carlson, J.; Nakano, D.: Endotrivial modules for finite group schemes. J. reine angew. Math. 653, 149-178 (2011) Representation theory for linear algebraic groups, Group schemes, Modular Lie (super)algebras, Cohomology theory for linear algebraic groups Endotrivial modules for finite group schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is common to describe the deformation theory of algebraic objects by cohomological means; it is very interesting to note that this description is limited. Equivalence classes of infinitesimal deformations of a Lie algebra \({\mathfrak g}\) are in bijection to the Chevalley-Eilenberg cohomology space \(H^2({\mathfrak g},{\mathfrak g})\), second cohomology with adjoint coefficients. But Lie algebras with trivial \(H^2({\mathfrak g},{\mathfrak g})\) still may have non-trivial deformations. Examples for this phenomenon were given for the Lie algebra of meromorphic vector fields on the Riemann sphere which are holomorphic outside \(\{0,\infty\}\) in [\textit{A. Fialowski} and \textit{M. Schlichenmaier}, ``Global deformations of the Witt algebra of Krichever-Novikov type'', Commun. Contemp. Math. 5, No. 6, 921--945 (2003; Zbl 1052.17011)] and for current algebras of the form \({\mathbb C}[z,z^{-1}]\otimes {\mathfrak g}\) for a complex simple finite-dimensional Lie algebra \({\mathfrak g}\) in [\textit{A. Fialowski} and \textit{M. Schlichenmaier}, ``Global geometric deformations of current algebras as Krichever-Novikov type algebras'', Commun. Math. Phys. 260, No. 3, 579--612 (2005; Zbl 1136.17307)].
These examples illustrate the fact that the condition \(H^2({\mathfrak g},{\mathfrak g})=0\) implies infinitesimal and formal rigidity, but in general not rigidity (while for finite dimensional Lie algebra, one may conclude towards general rigidity).
The present article reviews several of these examples and discusses their geometric origin. Indeed, the deformations come from deforming the underlying algebraic curve with marked points. The point is that the Lie algebras of vector fields and the algebra of functions can be written down in terms of generators and relations for the explicit families of (marked) elliptic curves which degenerate to nodal or cuspidal cubics. An abstract way of capturing this transition from families of curves to deformations of Lie algebras of vector fields has been proposed in [\textit{F. Wagemann}, ``Deformations of Lie algebras of vector fields arising from families of schemes'', J. Geom. Phys. 58, No. 2, 165--178 (2008; Zbl 1175.17007)]. global deformations of Lie algebras; Lie algebras of meromorphic vector fields on marked Riemann surfaces; cuspidal cubic; nodal cubic; current Lie algebra; Krichever-Novikov Lie algebra Homological methods in Lie (super)algebras, Cohomology of Lie (super)algebras, Infinite-dimensional Lie (super)algebras, Lie algebras of vector fields and related (super) algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Virasoro and related algebras, Families, moduli of curves (algebraic), Elliptic curves Deformations of the Witt, Virasoro, and current algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a constructive bijective correspondence between (i) non-isotropic, linearly full harmonic maps \(\psi_0 : \mathbb{R}^2 \to \mathbb{C} \mathbb{P}^n\) of finite type, up to isometry, and (ii) triples \((X, \pi, {\mathcal L})\) consisting of a real, complete, connected algebraic curve \(X\), a rational function \(\pi\) on \(X\) and a line bundle \(\mathcal L\) over \(X\), all three of which are required to satisfy certain conditions.
Moreover, sufficient conditions are given for \(\psi_0\) to be doubly periodic. harmonic map; harmonic torus; maps of finite type; Jacobian; dressing orbit; polynomial Killing fields McIntosh I.: The construction of all non-isotropic harmonic tori in complex projective space. Int. J. Math. 6, 831--879 (1995) Harmonic maps, etc., Differential geometry of immersions (minimal, prescribed curvature, tight, etc.), Jacobians, Prym varieties A construction of all non-isotropic harmonic tori in complex projective 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 A Wahl singularity is a two-dimensional cyclic quotient singularity of type \(\frac{1}{p^2}(1,pq-1)\), where \(0 < q < p\) are coprime integers. As for any cyclic quotient singularity, there is an associated Hirzebruch-Jung continued fraction \[\frac{p^2}{pq-1} = b_1-\frac{1}{b_2-\frac{1}{\ddots -\frac{1}{b_{\ell}}}},\] where \(b_i \geq 2\) are integers. The index of the singularity is \(p\), and it satisfies \[p \leq F_{\ell}\] where \(F_i\) is the \(i\)-th Fibonacci number defined by \(F_{-1}=F_{0}=1\) and \(F_i=F_{i-1}+ F_{i-2}\). In this way, a bound for the length \(\ell\) gives a bound for the index of the Wahl singularity.
Let \(X\) be a normal complex projective surface with ample canonical divisor \(K_X\), and with only Wahl or Du Val singularities. Assume that the geometric genus \(p_g(X)\) is not zero. The authors prove that the length \(\ell\) for any Wahl singularity in \(X\) satisfies \[\ell \leq 4 K_X^2 +7.\] This is Theorem 1.1 in their paper, and it follows from a more general theorem, which is proved in the paper, on symplectic embeddings of rational homology balls \(B_{p,q}\) in symplectic \(4\)-manifolds \((X,\omega)\) with \(K_X=[\omega]\) and \(b^{+}(X)>0\).
Almost simultaneously, Rana and the reviewer proved in [\textit{J. Rana} and \textit{G. Urzúa}, Adv. Math. 345, 814--844 (2019; Zbl 1474.14057)] a slightly stronger and optimal result for non-rational surfaces \(X\) with one T-singularity, which uses only algebro-geometric methods, and it does not recovers the symplectic aspect of the general theorem of the paper under review.
It turns out that one can replace the hypothesis `` \(p_g>0\)'' by ``rational blow-up is not a rational surface'' in the theorems of this paper. This is done in [``Kodaira dimensions of almost complex manifolds II'', Preprint, \url{arXiv:2004.12825}], Theorem A.3 by \textit{H. Chen} and \textit{W. Zhang}.
Moreover, it is possible to improve the bound \(\ell \leq 4 K_X^2 +7\) to \(\ell \leq 4 K_X^2 +1\) by modifying Propositions 8.3 and 8.4 in Evans-Smith paper following what is done in Lemma 2.8 of Rana-Urzúa paper.
Therefore, with respect to a bound for \(\ell\) in the case of nonrational surfaces with one Wahl singularity, the results of Evans-Smith and Rana-Urzúa are essentially equivalent.
In relation to problems about optimal bounds for T-singularities in surfaces with ample canonical class, there are at least two interesting open questions left:
(1) The paper of Evans-Smith bounds the case of many Wahl singularities for each singularity, but it does not bound the length of all singularities at once, which may be better than one by one. It is open to achieve an optimal bound which involves all Wahl singularities in \(X\) and \(K_X^2\).
(2) The paper of Rana-Urzúa bounds the case of one T-singularity in a rational surface, but it involves a degree that could be artificially arbitrarily large. We know by Alexeev's boundedness that the rational surface's case is bounded after we fix \(K^2\), but it remains open to find an optimal bound. The bound \(4 K^2 +1\) discussed above is false for rational surfaces via examples, and so that result of Evans-Smith could not be extended to this case. Although, based again only on examples, an optimal bound could be close to \(4 K^2 +6\). Wahl singularities; surfaces of general type; rational homology balls; symplectic embeddings; Seiberg-Witten invariants Surfaces of general type, Global theory of symplectic and contact manifolds, Applications of global analysis to structures on manifolds Bounds on Wahl singularities from symplectic 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 Let the reductive group \(G\) act on the finitely generated commutative \(k\)-algebra \(A\). We ask if the finite generation property of the ring of invariants \(A^G\) extends to the full cohomology ring \(H^*(G,A)\). We confirm this for \(G=\text{SL}_2\) and also when the action on \(A\) is replaced by the ``contracted'' action on the Grosshans graded ring \(\text{gr\,}A\), provided the characteristic of \(k\) is large. reductive groups; finitely generated algebras; rings of invariants; cohomology rings; Grosshans graded rings W. van der Kallen, \textit{Cohomology with Grosshans graded coefficients}, in: \textit{Invariant Theory in All Characteristics}, H. E. A. E. Campbell, D. L. Wehlau eds., CRM Proceedings and Lecture Notes, Vol. 35 (2004), Amer. Math. Soc., Providence, RI, 2004, pp. 127-138. Cohomology theory for linear algebraic groups, Geometric invariant theory, Group actions on varieties or schemes (quotients) Cohomology with Grosshans graded coefficients. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an irreducible normal algebraic variety over \(\mathbb{C}\). The author introduces a notion of the local fundamental group of \(X\) at a point \(x\). His main concern in this paper is the following conjecture: Let \(X\) be affine and \(G\) be a reductive linear group acting algebraically on \(X\). If the local fundamental groups of \(X\) at all the points of \(X\) are finite, then the same is true for the (categorical) quotient variety \(X//G\), provided \(\dim X//G \geq 2\).
The aim of this paper is to prove this conjecture in the case when all the local rings of \(X\) have fully-torsion divisor class groups (in fact a more general result is proved). It is mentioned that \textit{Gurjar} obtained a proof of this conjecture as well but in the case when \(X\) is smooth. -- \textit{C. T. C. Wall's} conjecture follows from this result: If \(X = \mathbb{C}^ n\) and the action is linear then \(\dim \mathbb{C}^ 2//G = 2\) implies that \(\mathbb{C}^ 2//G\) is isomorphic to \(\mathbb{C}^ 2/ \Gamma\), where \(\Gamma\) is some finite group acting linearly on \(\mathbb{C}^ 2\). finite local fundamental groups; quotient variety; divisor class groups; action of linear group Kumar, S., Finiteness of local fundamental groups for quotients of affine varieties under reductive groups, Comment. Math. Helv., 68, 209-215, (1993) Homotopy theory and fundamental groups in algebraic geometry, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients) Finiteness of local fundamental groups for quotients of affine varieties under reductive groups | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.