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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In higher dimensional geometry resolutions are not unique. In dimension two there are still minimal resolutions, but embedded resolutions are already not unique. Threefold theory suggests to look at embedded canonical models, for which the ambient space has also at most canonical singularities. In this paper toric methods are used to construct such models for singularities, defined by functions which are nondegenerate for their Newton diagrams.
The paper first describes the Newton blow-up, the modification defined by the Newton diagram, in terms of an equivariant normalised blow-up, in arbitrary dimensions. The main results concern the surface case; let \(f\to k\) be nondegenerate for its Newton diagram, put \(S=\{f=0\}\), let \(\hat V\to V\) be the Newton blow-up, and let \(\hat S\) be the strict transform of \(S\). The authors prove that \(\hat S\) has only \(A_ k\)- singularities in smooth points of Sing\(\hat V\), and they describe in detail how the surface \(\hat S\) intersects the exceptional set (transversally, which has to be explained in the presence of singular points). This embedded canonical model \(\hat S\subset\hat V\) of \(S\subset V\) is minimal for toric morphisms.
As example the authors work out the case of \(E_ 6\), which has five terminal embedded resolutions, related by flops, but the toric canonical model is unique. \(E_ 6\); models for singularities; embedded canonical model; toric morphisms; flops Gonzalez-Sprinberg (G.) and Lejeune-Jalabert (M.).- Modèles canoniques plongés. I. Kodai Math. J., 14(2) p. 194-209 (1991). Zbl0772.14008 MR1123416 Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry Modèles canoniques plongés. I. (Embedded canonical models. I) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0672.00003.]
We will interpret the ``rank 4 quadrics problem'' in terms of the deformation theory of singularities. This problem was solved by \textit{M. Green} [Invent. Math. 75, 85-104 (1984; Zbl 0542.14018)], over the complex numbers. We will use the deformation theory of Kodaira-Spencer, Grothendieck, and Schlessinger to solve the problem, first over the complex numbers and then also in characteristic p. The precise statement is given as follows:
Theorem. Let C be a nonhyperelliptic curve of genus \(g\geq 5\) over an algebraically closed field of characteristic \(\neq 2.\) Let \(\Theta\) be a theta divisor in the jacobian of C, and let \(I_ 2(C)\) denote the vector space of quadratic polynomials vanishing on the canonical model of C in \({\mathbb{P}}^{g-1}\). Then the tangent cones to \(\Theta\) at rank 4 double points span \(I_ 2(C)\). deformation of singularities; rank 4 quadrics problem; nonhyperelliptic curve; theta divisor in the jacobian R. Smith and R. Varley , Deformations of singular points on theta divisors , Theta Functions- Bowdoin 1987, Proceedings Symp. Pure Math. vol. 49, Part I, A.M.S., 1989, 571-579. Local deformation theory, Artin approximation, etc., Theta functions and abelian varieties, Jacobians, Prym varieties Deformations of singular points on theta divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field, \(G\) be a finite small subgroup of \(\text{GL}( 2,k)\). Suppose that the characteristic of \(k\) is prime to \(|G|\). By assumption the natural \(G\)-action on \(\mathbb A^2_k\) has no fixed points except the origin. It is known that the \(G\)-Hilbert scheme \(G\)-Hilb\((\mathbb A^2_k)\) is the minimal resolution of the quotient \(\mathbb A^2_k/G\).
The main result of paper under review finds the generating sheaf of the ideal defining the universal
\(G\)-cluster over \(G\)-Hilb\((\mathbb A^2_k)\). More concretely, for a geometric point \(y\) of the exceptional set \(E\) of the resolution let \(\text{Gen}( I_y)\) be the minimal
\(k [G]\)-module generating the ideal \(I_y \subset \mathcal O_{\mathbb A^2_k}\) corresponding to \(y\). Then it is proven in this paper that the union of all \(\text{Gen}( I_y)\) over \(E\) is an \(\mathcal O_F\)-module \(G\)-isomorphic to
\[\left(\bigoplus_{\rho\neq \rho_0} E(\rho)(-1) \otimes_k \rho\right) \oplus \mathcal O_F(-F)\otimes_k \rho_0,\]
where \(\rho\) ranges over the set of all non-trivial special irreducible representations of \(G\), \( E (\rho ) \) is the irreducible component of \(E\) associated to \(\rho\) (under special McKay correspondence), and \(F\) is the fundamental divisor of \(E = F_{\text{red}}\).
The proof of this result is by means of the derived categorical methods. McKay correspondence; minimal resolution; generating sheaf McKay correspondence, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Extended McKay correspondence for 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 In the theory of singularities of complex spaces the two-dimensional case has received much attention, partly because the resolution of isolated surface singularities is a powerful tool to study them. In this paper the author studies an analogue of resolution for non-isolated surface singularities. A non-isolated singularity has a transverse singularity, whose type is well-defined on each irreducible component of the (one-dimensional) singular locus. An improvement is a modification, which preserves the transverse type, but otherwise reduces the singularities: a modification \(\pi:Y\to X\) is an improvement if (i) \(\pi\) is an isomorphism in codimension one on \(X\), (ii) \(S=\text{Sing}(Y)\) is the strict transform of \(\text{Sing}(X)\), (iii) \(S\), the normalization \(\tilde Y\) of \(Y\) and the inverse image \(\tilde S\) of \(S\) on \(\tilde Y\) are smooth, (iv) Y is Cohen-Macaulay. Improvements for surfaces with transverse \(A_1\) were first constructed by Shepherd-Barron in his study of degenerate cusps [\textit{N. I. Shapherd-Barron}, Prog. Math. 29, 33-84 (1983; Zbl 0506.14028)]. In this paper the author constructs improvements for arbitrary non- isolated surface singularities. He then studies the singularities, which can occur on improvements. In the transverse \(A_1\) case only ordinary double curves and pinch-points can appear, but the general case is hopeless. For the restricted class of surfaces with transverse simple \((A,D,E)\) singularities the author gives a finite list, and proves that improvements with only such singularities exist, for hypersurfaces even obtainable by embedded improvement. Examples show that without this restriction these results do not hold. resolution for non-isolated surface singularities; modification Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Improvements of non-isolated 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 G be a simple complex Lie group of type A, D, E. The author gives a construction of the hypersurface singularity of the same type in terms of conjugacy classes of G which is different from the one found by Grothendieck-Brieskorn [see \textit{F. Brieskorn}, ``Singular elements of semisimple algebraic groups'' in Actes Congr. Internat. Math. 1970, vol. 2, 279-284 (1971; Zbl 0223.22012)]. Namely, let X be the closed orbit of G under the adjoint representation in the projective space attached to the Lie algebra \({\mathfrak g}\) of G. Consider a regular nilpotent element \(y\in {\mathfrak g}\) and let H(y) be the hyperplane orthogonal to y with respect to the Killing form. Then \(X\cap H(y)\) has exactly one singularity which is simple of the desired type. Its dimension is quite high, for example equal to 57 in the case of \(E_ 8\). By varying y in \({\mathfrak g}\) one can realize a versal deformation of this singularity.
These results are generalized to algebraic groups over fields of any characteristic and applied to get interesting deformations in characteristic two and three. conjugacy classes of complex Lie group; hypersurface singularity; versal deformation Knop, F.: Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. Invent. Math. \textbf{90}, 579-604 (1987) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Simple groups, General properties and structure of complex Lie groups, Linear algebraic groups over arbitrary fields, Deformations of singularities Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. (A new connection between simple groups and simple singularities) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an interesting paper with many connections to other work on algebraic curves and group theory. The reviewer feels that the author's fine introduction cannot be improved upon.
The theory of algebraic curves associated with subgroups of finite index in the modular group \(\Gamma\) is highly developed for such subgroups of \(\Gamma\) as may be defined by means of congruences in the ring \({\mathbb{Z}}\) of rational integers. The situation in the case of non-congruence subgroups of \(\Gamma\), on the other hand, is drastically different. It reduces to a few isolated examples, two of which may be found in the author's paper in J. Reine Angew. Math. 268/269, 348-359 (1974; Zbl 0292.10021). Related research by \textit{A. O. L. Atkin} and \textit{H. P. F. Swinnerton-Dyer} began in Proc. Symp. Pure Math. 19, 1-25 (1971; Zbl 0235.10015).
Observing that Macbeath's curve [cf. \textit{A. M. Macbeath}, Proc. Lond. Math. Soc., III. Ser. 15, 527-542 (1965; Zbl 0146.427)] affords another pertinent example made us look at it more closely. Its automorphism group is isomorphic with the simple group PSL(2,8) of order 504. Accordingly, the associated subgroup \(\Delta\) of \(\Gamma\) is a maximal normal subgroup of index 504. We shall prove, in passing, that there are exactly two such subgroups in \(\Gamma\), neither of them a congruence group.
The view above renders Macbeath's curve as a covering of the projective line with Galois group G isomorphic to PSL(2,8). Corresponding to any of its Sylow 7-subgroups and its normalizer in G we find two intermediate curves \textbf{B} and \textbf{A}, respectively elliptic and rational, of which the former covers the latter 2-fold and with 4 branch points. In this classical situation the 4 points of \textbf{A} under the branch points may, moreover, be made explicit through Macbeath's model. Their cross ratio, or Legendre's modulus \(\lambda\), and in turn the absolute invariant J of \textbf{B} could then be calculated.
There is, however, more to gain with less effort. We find the said 4 points on \textbf{A}, after a Möbius transformation, to satisfy an algebraic 4-th-degree equation \(P(X)=0\) with integral rational coefficients. Thus \(Y^{2}=P(X)\) describes \textbf{B} as a curve over \({\mathbb{Q}}\). Its Weierstraß normal form yields the invariants \(g_ 2=196\), \(g_ 3=-196\), and \(J=(14/13)^ 2\). The Mordell-Weil rank of \textbf{B}(\({\mathbb{Q}})\) may then be seen to be greater than 1, by reduction modulo 29.
The genus \(g=7\) of Macbeath's curve and the order \(h=504\) of its automorphism group are related by \(h=84(g-1)\). G is then called a Hurwitz group. Many instances of such groups were recently constructed by \textit{J. M. Cohen} [Math. Proc. Camb. Philos. Soc. 86, 395-400 (1979; Zbl 0419.20034) and in ''The geometric Vein'', Coxeter Festschrift, 511-518 (1982; Zbl 0496.20033)], as abelian extensions of PSL(2,7). We should like to point out that all such extensions occur in the author's paper in Math. Ann. 174, 79-99 (1967; Zbl 0157.036); they correspond to the ideal I of algebraic integers in \({\mathbb{Q}}(\sqrt{-7})\), and their orders are \(h=168(N(I))^ 3\). This also accounts for the orders of some groups in a paper by \textit{A. Sinkov} [Ann. Math., II. Ser. 38, 577-584 (1937; Zbl 0017.05702)]. automorphisms group of Macbeath curve; algebraic curves associated with subgroups of finite index in the; modular group; non-congruence subgroups; covering of the projective line; Hurwitz group; algebraic curves associated with subgroups of finite index in the modular group Klaus Wohlfahrt, Macbeath's Curve and the Modular Group, Glasg. Math. J.27 (1985), p. 239-247 - ISSN : 2118-8572 (online) 1246-7405 (print) - Société Arithmétique de Bordeaux Coverings of curves, fundamental group, Arithmetic ground fields for curves, Structure of modular groups and generalizations; arithmetic groups, Special algebraic curves and curves of low genus, Singularities of curves, local rings, Unimodular groups, congruence subgroups (group-theoretic aspects), Subgroup theorems; subgroup growth, Finite automorphism groups of algebraic, geometric, or combinatorial structures Macbeath's curve and the modular 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 A particular case in the superstring theory where a finite group \(G\) acts upon the target Calabi-Yau manifold \(M\) in the theory seems to attract both physicists' and mathematician's attention. Define the ``orbifold Euler characteristic'': \(\chi (M,G)= {1\over |G |} \sum_{gh=hg} \chi (M^{\langle g, h\rangle})\), where the summation runs over all the pairs \(g,h\) of commuting elements of \(G\), and \(M^{\langle g,h \rangle}\) denotes the subset of \(M\) of all the points fixed by both of \(g\) and \(h\).
Vafa's formula-conjecture. If a complex manifold \(M\) has trivial canonical bundle and if \(M/G\) has a (nonsingular) resolution of singularities \(\widetilde {M/G}\) with trivial canonical bundle, then we have \(\chi (\widetilde {M/G} ) = \chi (M,G)\).
In the special case where \(M= \mathbb{A}^n\) an \(n\)-dimensional affine space, \(\chi (M,G)\) turns out to be the number of conjugacy classes, or equivalently the number of equivalence classes of irreducible \(G\)-modules. If \(n=2\), then the formula is therefore a corollary to the classical McKay correspondence.
Let \(G\) be a finite subgroup of \(SL(2, \mathbb{C})\) and \(\text{Irr} (G)\) the set of all equivalence classes of nontrivial irreducible \(G\)-modules. Let \(X=X_G: =\text{Hilb}^G (\mathbb{A}^2)\), \(S=S_G: =\mathbb{A}^2/G\), \({\mathfrak m}\) (resp. \({\mathfrak m}_S)\) the maximal ideal of \(X\) (resp. \(S)\) at the origin and \({\mathfrak n}: ={\mathfrak m}_S {\mathcal O}_{\mathbb{A}^2}\). Let \(\pi: X\to S\) be the natural morphism and \(E\) the exceptional set of \(\pi\). Let \(\text{Irr} (E)\) be the set of irreducible components of \(E\). Any \(I\in X\) contained in \(E\) is a \(G\)-invariant ideal of \({\mathcal O}_{\mathbb{A}^2}\) which contains \({\mathfrak n}\).
Definition: \(V(I): =I/({\mathfrak m} I+{\mathfrak n})\).
For any \(\rho\), \(\rho'\), and \(\rho''\in \text{Irr} (G)\) define \(E(\rho): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho)\}\)
\(P(\rho, \rho'): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho')\}\)
\(Q(\rho, \rho', \rho''): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho') \oplus V(\rho'')\}\).
Main theorem: (1) The map \(\rho \mapsto E(\rho)\) is a bijective correspondence between \(\text{Irr} (G)\) and \(\text{Irr} (E)\).
(2) \(E(\rho)\) is a smooth rational curve for any \(\rho\in \text{Irr} (G)\).
(3) \(P(\rho, \rho)= Q(\rho, \rho',\rho'') = \emptyset\) for any \(\rho,\rho', \rho''\in \text{Irr} (G)\). Hilbert schemes; orbifold Euler characteristics; irreducible components of exceptional set; superstring theory; McKay correspondence Ito, Y., Nakamura, I.: McKay correspondence and Hilbert schemes. Proc. Japan Acad. Ser. A Math. Sci., 72, 135--138 (1996) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) McKay correspondence and Hilbert schemes | 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 article is best viewed within the framework of mirror symmetry, a profound duality linking the symplectic geometry of an algebraic variety \(X\) (known as the \(A\)-side) to the complex algebraic geometry of its mirror algebraic variety \(Y\) (the \(B\)-side). Although there are various expressions of this duality, for the purposes of the present article, we focus on the \(A\)-side and on symplectic Gromov-Witten invariants of \(X\), which are counts of pseudo-holomorphic maps in \(X\).
More precisely, the author considers log Calabi-Yau surface pairs \((X,D)\), where \(X\) is a del Pezzo surface of degree \(\geq3\) and \(D\) is a smooth anticanonical divisor. The prototypical example for this consists of the pair of complex projective plane and elliptic curve. In this setting and for a curve class \(\beta\), the symplectic invariants \(N_\beta(X,D)\) of interest are the genus 0 class \(\beta\) log Gromov-Witten invariants of maximal tangency of \((X,D)\). These can be thought of as counts of maps \(\mathbb{A}^1\to X \setminus D\).
The main result of the present article is a correspondence theorem between the \(N_\beta(X,D)\) and tropical counts in the integral affine manifold with singularities (tropical manifold) \(B\) associated to a toric degeneration of \((X,D)\). This is where mirror symmetry and the mirror constructions of the Gross-Siebert programme come in.
A toric degeneration of \((X,D)\) is a degeneration of \((X,D)\) with special fibre the union of toric pieces. Given this datum, \(B\) is the dual intersection complex of the special fibre. The Gross-Siebert mirror construction then is an algorithm that produces a wall-and-chamber structure on \(B\). The author of the present paper considers \emph{vertical} Gross-Siebert walls and proves that their wall-crossing functions are given by generating functions of a slightly refined version of the \(N_\beta(X,D)\).
This result makes the link between deformation-theoretic mirror construction and enumerative geometry. This work extends [\textit{M. Gross} et al., Duke Math. J. 153, No. 2, 297--362 (2010; Zbl 1205.14069)] on the case of maximal boundary to the case of smooth boundary. This involves global computations based on infinitely many local computations. This work is then subsequently used in [\textit{P. Bousseau}, J. Algebr. Geom. 31, No. 4, 593--686 (2022; Zbl 1502.14137)] and [\textit{P. Bousseau}, ``A proof of N.Takahashi's conjecture for \((\mathbb{P}^2,E)\) and a refined sheaves/Gromov-Witten correspondence'' Duke Math. J. (to appear)] to prove a major conjecture of [\textit{N. Takahashi}, Commun. Math. Phys. 220, No. 2, 293--299 (2001; Zbl 1066.14048)]. tropical correspondence; log Gromov-Witten invariants; mirror symmetry; Gross-Siebert program Enumerative problems (combinatorial problems) in algebraic geometry, Mirror symmetry (algebro-geometric aspects), Logarithmic algebraic geometry, log schemes, Applications of tropical geometry Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The mathematical content of this paper is a computation of the fundamental group of
\(\mathbb{P}^2 \backslash\) (critical locus of the family \(W)\), \(W\) the family of quintic threefolds over \(\mathbb{P}^2\):
\[
c(y^5_1 + \cdots + y^5_5) - 5y^2_4 y^2_5 (ay_4 + by_5) = 0
\]
\((a,b, c\) homogeneous coordinates on \(\mathbb{P}^2\), \(y_i\) homogeneous coordinates on \(\mathbb{P}^5)\) and its monodromy representation on a certain three-dimensional space of periods (of a holomorphic 3-form). The result is to identify the fundamental group \(\Gamma\) with the braid group \(B_5 \), this is obtained by an explicit computation of the critical locus and its singularities, thus the generators correspond to the set of critical points of a Lefschetz pencil, and the relations are found by van Kampen's theorem [\textit{E. R. van Kampen}, Am. J. Math. 55, 255-260 (1933; Zbl 0006.41502)].
The monodromy representation is computed by an suitable explicit description of the Gauß-Manin connection. It leads to a projective \(U(1,2)\)-representation of \(\Gamma\). The introduction contains furthermore comments about the relation of this result to supersymmetric conformal field theory. Calabi-Yau spaces; monodromy; fundamental group; quintic threefolds; braid group; Gauß-Manin connection; supersymmetric conformal field theory Structure of families (Picard-Lefschetz, monodromy, etc.), Homotopy theory and fundamental groups in algebraic geometry, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Target space duality of Calabi-Yau spaces with two moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It was observed by \textit{V. I. Arnol'd} [Russ. Math. Surv. 30, No.5, 1-75 (1975); translation from Usp. Mat. Nauk 30, No.5(185), 3-65 (1975; Zbl 0338.58004)] that for each exceptional unimodular hypersurface singularity (X,0) there is a unique exceptional unimodular hypersurface singularity (Y,0) with the following property: The Dolgachev numbers (these are data given by the resolution graph) of (X,0) coincide with the Gabrielov numbers (these are invariants of the system of vanishing cycles in the homology of the Milnor fibre) of (Y,0), and conversely. The correspondence (X,0)\(\leftrightarrow (Y,0)\) is called the ''strange duality'' among the exceptional unimodular hypersurface singularities. The authors define an extension of this duality to the class of all Kodaira singularites (i.e. singularities for which the exceptional set in the minimal resolution \(\tilde X\to X\) of the singularity is isomorphic to a minimal exceptional fibre in an elliptic pencil - but with different neighbourhood in \(\tilde X)\). They discuss the relation between resolution graphs, Dynkin diagrams and Milnor lattices and other invariants of dual Kodaira singularities. Most of these invariants are also computed explicitly for all Kodaira singularities. deformation; Dolgachev numbers; Gabrielov numbers; strange duality; exceptional unimodular hypersurface singularities; Kodaira singularites; resolution graphs; Dynkin diagrams; Milnor lattices Ebeling W., Wall C.T.C.: Kodaira singularities and an extension of Arnold's strange duality. Compositio Math. 56, 3--77 (1985) Singularities of surfaces or higher-dimensional varieties, Deformations of singularities, Complex singularities, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Kodaira singularities and an extension of Arnold's strange 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 This thesis determines the singularities of the coarse moduli space \(\overline S_g\) over \(\mathbb{C}\) of spin curves. The precise description of the singularities yields the result that pluricanonical forms on the smooth locus of \(\overline S_g\) lift holomorphically to a desingularisation. The moduli space \(\overline S_g\) constructed by \textit{M. Cornalba} [Moduli of curves and theta-characteristics. Teaneck, NJ: World Scientific Publishing Co. 560--589 (1989; Zbl 0800.14011)] compactifies the coarse moduli space \(S_9\) of smooth spin curves. These are pairs \((C,L)\) of a smooth curve of (arithmetic) genus \(g\geq 2\) and a theta characteristic \(L\) on \(C\), i.e. a line bundle \(L\) on \(C\) such that \(L^{\otimes 2}\) is isomorphic to the canonical bundle \(\omega_C\). This compactification is compatible with the Deligne-Mumford compactification \(\overline M_g\) of the coarse moduli space \(M_g\) of smooth curves of genus \(g\) via stable curves [\textit{P. Deligne} and \textit{D. Mumford}, Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803) and Matematika, Moskva 16, No. 3, 13--53 (1972; Zbl 0233.14008)]. In particular there exists a natural morphism \(\pi:\overline S_g\to\overline M_g\) which sends the moduli point of a spin curve to the moduli point of the underlying curve. \(\pi\) is a finite map of degree \(2^{2g}\).
This thesis focuses on the local (analytic) structure of the moduli space \(\overline S_g\). As in the case of \(\overline M_g\) an analytic neighbourhood of the moduli point of a spin curve in \(\overline S_g\) is isomorphic to the quotient \(V/G\) of a \(3g-3\)-dimensional vector space \(V\) with respect to a finite group \(G\). This group is essentially the automorphism group of the spin curve under consideration. A careful analysis of the occurring quotients gives a description of the locus of canonical singularities of \(\overline S_g\) with the help of the Reid-Tai criterion. This locus has codimension 2 in \(\overline S_g\). Moreover, the smooth locus \(\overline S^{\text{reg}}_g\subset\overline S_g\) is determined. The morphism \(\pi\) plays an important role in these calculations, since it establishes a connection between the well-understood singularities of \(\overline M_g\) by \textit{J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016)] and those of \(\overline S_g\). In order to understand this connection the ramification of the finite map \(\pi\) is described.
These local results are used to prove that all pluricanonical forms on \(\overline S^{\text{reg}}_g\), i.e. sections in \(\Gamma(\overline S^{\text{reg}}_g,{\mathcal O}_{\overline S_g}(kK_{\overline S_g}))\), extend holomorphically to a desingularisation \(\widetilde S_g\) of \(\overline S_g\). An important ingredient is the analogous result for \(\overline M_g\) by Harris and Mumford [loc. cit.]. moduli space; spin curve; singularities; pluricanonical forms Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, moduli of curves (algebraic) Moduli of spin curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present a method based on mutations of helices which leads to the construction (in the large-volume limit) of exceptional coherent sheaves associated with the (\(\sum_a l_a=0\)) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi-Yau hypersurface. The method is based on two conjectures which lead to the analog, in the general case, of the Beilinson quiver for \(P^n\). We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space. Gepner models; gauged linear sigma model; Kähler moduli space Govindarajan S., Jayaraman T.: D-branes, exceptional sheaves and quivers on Calabi-Yau manifolds: From Mukai to McKay. Nucl. Phys. B 600, 457--486 (2001) String and superstring theories in gravitational theory, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics D-branes, exceptional sheaves and quivers on Calabi-Yau manifolds: from Mukai to McKay | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Dimer models are introduced by string theorists to study four-dimensional \(N=1\) superconformal field theories. See, e.g., a review by \textit{K. D. Kennaway} [Int. J. Mod. Phys. A 22, No. 18, 2977-3038 (2007; Zbl 1141.81328)] and references therein for a physical background. A dimer model is a bipartite graph on a real two-torus which encodes the information of a quiver with relations. A typical example of such a quiver is the McKay quiver determined by a finite Abelian subgroup \(G\) of \(\text{SL}(3,\mathbb{C})\) [see \textit{M. Reid}, ``McKay correspondence'', \url{arXiv:alg-geom/9702016v3}, \textit{K. Ueda} and \textit{M. Yamazaki}, Commun. Math. Phys. 301, No. 3, 723-747 (2011; Zbl 1211.81090)]. In this case, the moduli space of representations of the McKay quiver (for the dimension vector \((1,1,\dots,1)\)) coincides with the moduli space of \(G\)-constellations considered by \textit{A. Craw} and \textit{A. Ishii} [Duke Math. J. 124, No. 2, 259-307 (2004; Zbl 1082.14009)]. For a generic choice of a stability parameter \(\theta\), the moduli space of \(G\)-constellations is a crepant resolution of the quotient singularity \(\mathbb{C}^3/G\) and the derived category of coherent sheaves on the moduli space is equivalent to the derived category of finitely generated modules over the path algebra of the McKay quiver. It is expected that these kinds of statements can be generalized to the case of dimer models that are `consistent' in the physics context, which should be called `brane tilings'. In this note, we discuss a slightly weaker notion of non-degenerate dimer models, which is strong enough to ensure that the moduli space is a crepant resolution of the three-dimensional toric singularity determined by the Newton polygon of the characteristic polynomial (see Theorem 6.4). We expect that one has to impose further conditions to prove the derived equivalence.
For the proof, we use a generalization of the description of a torus-fixed point on the moduli space in terms of a choice of a covering by hexagons of the fundamental region of a real 2-torus due to \textit{I. Nakamura} [J. Algebr. Geom. 10, No. 4, 757-779 (2001; Zbl 1104.14003)]. Many of the arguments are similar to those of \textit{A. Ishii} [in Clay Mathematics Proceedings 3, 227-237 (2005; Zbl 1156.14308)]. There is also a physics paper by \textit{S. Franco} and \textit{D. Vegh} [Moduli spaces of gauge theories from dimer models: proof of the correspondence, \url{arXiv:hep-th/0601063v2}] which deals with the relation between brane tilings and moduli spaces. dimer models; superconformal field theories; bipartite graphs; quivers with relations; McKay quivers; moduli spaces; representations of quivers; crepant resolutions; quotient singularities A. Ishii and K. Ueda, \textit{On moduli spaces of quiver representations associated with dimer models}, arXiv:0710.1898. Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory On moduli spaces of quiver representations associated with dimer models. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a local hypersurface ring with isolated singularity, and let \(M\) and \(N\) be \(A\)-modules. \textit{Hochster's theta invariant}, introduced in [\textit{M. Hochster}, Lect. Notes Math. 862, 93--106 (1981; Zbl 0472.13005)], is given by
\[
\theta(M,N) = l(\mathrm{Tor}^A_{2i}(M,N)) - l(\mathrm{Tor}^A_{2i+1}(M,N))
\]
for \(i \gg 0\), where \(l(-)\) denotes length as an \(A\)-module. Some conjectures involving the vanishing of the theta invariant have inspired a good deal of work in recent years. For instance, a conjecture of \textit{H. Dao} in [Trans. Am. Math. Soc. 365, No. 6, 2803--2821 (2013; Zbl 1285.13018)] states that \(\theta(M,N)=0\) whenever \(\dim(A)\) is even. A brief history of this conjecture and several others involving the \(\theta\) invariant is provided in a preliminary section of the article under review. The authors also pose some new conjectures related to the \(\theta\) invariant in this section. \vskip\baselineskip The authors go on to establish several new results involving the \(\theta\) invariant. Their first main result is that, if \(\mathrm{Spec}(A)\) admits a resolution of singularities, \(\theta(M, -)\) is the zero map whenever \(M\) is \textit{numerically equivalent to 0}: that is, whenever \(\sum_i (-1)^i l(H_i(F_\bullet \otimes_A M)) = 0\) for all bounded complexes \(F_\bullet\) of finite free \(A\)-modules with finite length homologies. The authors show that, in their setting, numerical equivalence implies \textit{algebraic equivalence} as defined in [\textit{O. Celikbas} and \textit{M. E. Walker}, Math. Ann. 353, No. 2, 359--372 (2012; Zbl 1252.13009)], and hence they recover a result of Celikbas-Walker stating that \(\theta\) factors through \(G_0(A)\) modulo algebraic equivalence. \vskip\baselineskip The authors also show that \(\theta\) is positive semi-definite when \(\dim(A) = 3\), among other results on \(\theta\) in small dimensions. In addition, they show that if \(\dim(A) = 3\) and \(A\) admits a disingularization, the class group of \(A\) is finitely generated and torsion free. local hypersurface; isolated singularity; Hochster's theta invariant; intersection multiplicity; divisor class group; Grothendieck group; numerical equivalences Dao, H; Kurano, K, Hochster's theta pairing and numerical equivalence, J. K-Theory, 14, 495-525, (2014) Homological functors on modules of commutative rings (Tor, Ext, etc.), Grothendieck groups, \(K\)-theory and commutative rings, Homological conjectures (intersection theorems) in commutative ring theory, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry Hochster's theta pairing and numerical equivalence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article, the authors define and investigate the \textit{triangulated category of relative singularities} associated to a closed subscheme \(Z\) of a separated Noetherian scheme \(X\) with enough vector bundles, where \(\mathcal{O}_Z\) has finite flat dimension as an \(\mathcal{O}_X\)-module. It is given by the quotient of \(\text{D}^b(Z)\) by the thick subcategory generated by the image of the derived inverse image functor \(\mathbb{L}i^*: \text{D}^b(X) \to \text{D}^b(Z)\), and it is denoted by \(\text{D}^b_{\mathrm{Sing}}(Z/X)\). When \(X\) is regular, \(\text{D}^b_{\mathrm{Sing}}(Z/X)\) is precisely the singularity category of \(Z\) as defined by \textit{R.-O. Buchweitz} in [``Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings'', unpublished manuscript (1987)].
In Section 1 of the article, the authors establish some general results regarding \textit{derived categories of the second kind} (cf. [\textit{L. Positselski}, Mem. Am. Math. Soc. 996, i-iii, 133 p. (2011; Zbl 1275.18002)]) associated to curved dg-modules over curved dg-rings. These results are applied later in the article to \textit{matrix factorizations}, which may be considered as curved dg-modules over a certain curved dg algebra.
In Section 2, the authors prove what they refer to as their main result. Let \(X\) be as above, let \(\mathcal{L}\) be a line bundle on \(X\), and let \(w \in \mathcal{L}(X)\) be a section. Let \(X_0 \subseteq X\) denote the closed subscheme given by the zero locus of \(w\). Assume the morphism of sheaves \(w: \mathcal{O}_X \to \mathcal{L}\) is injective. Let \((X, \mathcal{L}, w)-\text{coh}\) denote the category of \textit{coherent matrix factorizations} of \(w\); that is, the pair of \(\mathcal{O}_X\)-modules underlying the matrix factorization is allowed to be a pair of coherent modules, rather than locally free. The authors construct an equivalence
\[
\text{D}^{\text{abs}} ((X, \mathcal{L}, w)-\text{coh})) \to \text{D}^b_{\mathrm{Sing}}(X_0/X),
\]
where \(\text{D}^{\text{abs}}(-)\) denotes a certain derived category of the second kind. When X is regular, this theorem recovers a well-known theorem of \textit{D. Orlov} (Theorem 3.5 of [Math. Ann. 353, No. 1, 95--108 (2012; Zbl 1243.81178)]).
The authors also establish, in this section, what they refer to as covariant and contravariant Serre-Grothendieck duality theorems for matrix factorizations. In Section 3, the authors give some general results on ma- trix factorizations with a support condition, and also pushforwards and pullbacks of matrix factorizations. Hochschild (co)homology of dg categories of matrix factorizations is discussed in an appendix. matrix factorizations; relative singularities of cartier divisors; triangulated categories of singularities; derived categories of the second kind; coderived categories; direct and inverse images; covariant Serre-Grothendieck; localization theory A. I. Efimov & L. Positselski, ``Coherent analogues of matrix factorizations and relative singularity categories'', Algebra Number Theory9 (2015) no. 5, p. 1159-1292 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories and commutative rings, Representation theory of associative rings and algebras Coherent analogues of matrix factorizations and relative singularity categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a complex reductive affine algebraic group and \(\Gamma\) a finitely generated group. The set of group homomorphisms \(\mathrm{Hom}(\Gamma, G)\) is an affine algebraic set on which \(G\) acts rationally by conjugation. The (affine) geometric invariant theoretic (GIT) quotient \[\mathfrak{X}(\Gamma,G):=\mathrm{Hom}(\Gamma,G)/\!\!/G\] is called the \textit{\(G\)-character variety of \(\Gamma\)}.
In [\textit{S. Lawton}, J. Algebra 313, No. 2, 782--801 (2007; Zbl 1119.13004)] the reviewer explicitly describes \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) where \(F_2\) is a free group of rank 2. In particular, the reviewer shows that \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) is a normal hypersurface in \(\mathbb{C}^9\) that branch double covers \(\mathbb{C}^8\), and the branching locus properly contains the singular locus (shown to be exactly the reducible homomorphisms). This theorem is proved using non-commutative algebra, in particular, the theory of polynomial identity algebras (PI Theory). Later, in [\textit{C. Florentino} and \textit{S. Lawton}, Math. Ann. 345, No. 2, 453--489 (2009; Zbl 1200.14093)] the reviewer geometrically describes the branching locus as exactly the symmetric homomorphisms (Proposition 6.11, ibid) and uses this as a step in proving that \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) strong deformation retracts onto a topological 8-sphere despite the fact that \(\mathfrak{X}(F_2,\mathrm{SL}(3,\mathbb{C}))\) is not a (topological) manifold (in the analytic topology).
There has been much work since then on \(\mathrm{SL}(3,\mathbb{C})\)-character varieties. In particular, the reviewer has described global coordinates [\textit{S. Lawton}, J. Algebra 320, No. 10, 3773--3810 (2008; Zbl 1157.14030)], local coordinates [\textit{S. Lawton}, J. Algebra 324, No. 6, 1383--1391 (2010; Zbl 1209.14036)], Poisson structures [\textit{S. Lawton}, Trans. Am. Math. Soc. 361, No. 5, 2397--2429 (2009; Zbl 1171.53052)], and \(E\)-polynomials [\textit{S. Lawton} and \textit{V. Muñoz}, Pac. J. Math. 282, No. 1, 173--202 (2016; Zbl 1335.14003)] on \(\mathfrak{X}(F_r,\mathrm{SL}(3,\mathbb{C}))\) for arbitrary rank free groups \(F_r\). There are many other examples by many other authors too.
For \(\mathrm{SL}(2,\mathbb{C})\)-character varieties of \(\Gamma\) a complete algebraic picture is given in [\textit{C. Ashley} et al., Geom. Dedicata 192, 1--19 (2018; Zbl 1390.14187)]. In other words, an explicit presentation of the (reduced) coordinate ring of \(\mathfrak{X}(\Gamma, \mathrm{SL}(2,\mathbb{C}))\) is given for any finitely presented group \(\Gamma\). This is only possible since the free group case is solved for all ranks for \(\mathrm{SL}(2,\mathbb{C})\). The only case where generators and relations (of the coordinate ring of the character variety) are known for free groups and \(\mathrm{SL}(3,\mathbb{C})\) is the reviewer's theorem for \(F_2\) referenced above. So one naturally wonders if the 2-generator group problem can likewise be solved for \(\mathfrak{X}(\Gamma, \mathrm{SL}(3,\mathbb{C}))\); that is, can one describe the coordinate ring of \(\mathfrak{X}(\Gamma, \mathrm{SL}(3,\mathbb{C}))\) for any \(\Gamma\) finitely presentable with 2-generators.
Using the results and methods of the reviewer mentioned above, in particular the explicit description of \(\mathfrak{X}(F_2, \mathrm{SL}(3,\mathbb{C}))\) and PI theory, the author of the paper under review presents a method for determining the irreducible locus of \(\mathfrak{X}(\Gamma, \mathrm{SL}(3,\mathbb{C}))\) for any 2-generated finitely presented group \(\Gamma\) and demonstrates this method with some examples (various link complements). character variety; irreducible representation; two-generator group; double twist link; symmetric slice; asymmetric slice Character varieties, Invariants of 3-manifolds (including skein modules, character varieties) \(\mathrm{SL}(3,\mathbb{C})\)-character varieties of two-generator 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 a finite subgroup \(G\) of \(\mathrm{GL}(2,\mathbb{C})\), we consider the moduli space \(\mathscr{M}_\theta\) of \(G\)-constellations. It depends on the stability parameter \(\theta\) and if \(\theta\) is generic it is a resolution of singularities of \(\mathbb{C}^2/G\). In this paper, we show that a resolution \(Y\) of \(\mathbb{C}^2/G\) is isomorphic to \(\mathscr{M}_\theta\) for some generic \(\theta\) if and only if \(Y\) is dominated by the maximal resolution under the assumption that \(G\) is abelian or small. \(G\)-constellation; maximal resolution; quotient singularity Algebraic moduli problems, moduli of vector bundles, McKay correspondence, Singularities of surfaces or higher-dimensional varieties \(G\)-constellations and the maximal resolution of a quotient surface singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \({\mathbb{P}}^ n\) is the simplest example of a homogeneous variety with ample anticanonical class (other being quotients of a simple complex Lie group modulo a parabolic subgroup). On such a variety algebraic vector bundles can be described more or less explicitly in terms of derived categories. It had been noted that, moreover, on these varieties there exist sets of exceptional bundles (with no higher Exts between elements of the set) that generate the K-functor (any complex of bundles is quasiisomorphic to a complex of direct sums of exceptional bundles). Furthermore, on \({\mathbb{P}}^ 2\) and a 2-dimensional quadric any exceptional set can be obtained from any other one by a sequence of perestroikas (modular operations).
In the paper under review the opposite case is considered, that of varieties (more exactly, surfaces) with \(p_ g>0\). The paper consists of 3 parts.
In part I, a symplectic structure on the ``moduli space'' of vector bundles on the surface is defined. Since the moduli space itself consists of a monstrously huge number of undescribable connected components, there is given an existence theorem rather than explicit description. (However, an elucidation of some of these connected components is obtained as a byproduct.) This result is a generalization of a similar one for K3- surfaces due to Mukai. - If we turn to classification of coherent sheaves rather then vector bundles, there are examples of explicitly describable moduli spaces, e.g. of sheaves of ideals of 0-cycles. These moduli are birationally equivalent to symmetric powers of the surface itself.
In part II it is shown that in certain cases perestroikas (more specifically, ``universal extensions'' and ``universal divisions'') do not change the birational type of the components of the moduli space. This being the case for the sheaves of ideals of 0-cycles together with the fact that certain perestroikas of them are sheaves of sections of vector bundles determines the birational type of some components of the moduli space of vector bundles. - Unlike for bundles on homogeneous spaces, perestroikas of some exceptional bundles on a surface with \(p_ g>0\) do not give all other exceptional bundles.
In part III the author proves even more: the number of the corresponding equivalence classes is infinite. moduli space of vector bundles; modular operations; classification of coherent sheaves; universal extensions; universal divisions Tyurin A.N., Symplectic structures on the moduli spaces of vector bundles on algebraic surfaces with p g \> 0, Math. USSR-Izv., 1989, 33(1), 139--177 Moduli, classification: analytic theory; relations with modular forms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli, classification: algebraic theory Symplectic structures on the varieties of moduli of vector bundles on algebraic surfaces with \(p_ g>0\) | 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 The present article is expository in nature. In \S 1 we give a description of Grassmannians \(G(p,q)\), explaining the universal bundles, the tensor product decomposition and the structure of the Fano variety of lines (degree-1 rational curves) on them. The flatness of the corresponding \(G\)-structures will be verified by using the usual covering of \(G(p,q)\) by coordinate charts of affine spaces. In \S 2 we give a geometric proof of Ochiai's Theorem characterizing irreducible Hermitian symmetric spaces \(S\) of the compact type and of rank \(\geq 2\) in terms of flat \(G\)-structures. We do this by showing that any local holomorphic map on \(S\) transforming cones of minimal rational curves to each other induces a meromorphic map on the homogeneous Fano varieties of lines on them, and derive from there that such maps are restrictions of biholomorphisms of \(S\). In \S 3 we consider the question of flatness (also called integrability) of \(G\)-structures. We explain how the obstruction to the existence of \(k\)-th order flattening coordinates can be measured by holomorphic ``curvature tensors''. In \S 4 we discuss distributions spanned by cones of minimal rational tangents, yielding an obstruction to the linear degeneration of such cones on a Fano manifold \(X\) with \(b_2(X)=1\).
In \S 5 we state our result in [J. Reine Angew. Math. 490, 55-64 (1997; Zbl 0882.22007)] characterizing irreducible Hermitian symmetric spaces of the compact type among uniruled (in particular Fano) manifolds in terms of \(G\)-structures, sketching a proof with details provided for the case of \(G(p,q)\). In \S 6 we state our result in [Invent. Math. 131, 393-418 (1998; Zbl 0902.32014)] on the rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation and give an outline of the proof. Wherever possible, the special case of \(G(p,q)\) is discussed to illustrate methods and results that are applicable in the general theory.
For the first three sections in view of the lack of a comprehensive reference we provide full details as far as possible, especially in the geometric proof of Ochiai's theorem as presented in \S 2. The last three sections are essentially a reorganized summary of the two articles cited above. For this reason the presentation is more sketchy, with an aim at providing an overview of the perspective adopted in the cited articles. flat \(G\)-structures; Grassmannians; Fano variety; Fano manifold; irreducible Hermitian symmetric spaces; compact type; Kähler deformation Differential geometry of homogeneous manifolds, Fano varieties, General properties and structure of complex Lie groups, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Other complex differential geometry, Rigidity results, \(G\)-structures Characterization and deformation-rigidity of compact irreducible Hermitian symmetric spaces of rank \(\geq 2\) among Fano manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study constructive embedded resolutions of irreducible quasi-ordinary singularities in \(\mathbb{C}^3\). A surface germ \((V,p)\subset (\mathbb{C}^3,0)\) is a quasi-ordinary singularity if it admits a finite projection \(\pi:(V,p) \to(\mathbb{C}^2,0)\) such that the discriminant locus (i.e., the plane curve over which \(\pi\) ramifies) has only normal crossings. If \(f\) is such a singularity and is irreducible, it admits a parametrization (analogous to the Puiseux series of an irreducible algebroid plane curve) from which certain pairs of numbers, called the characteristic pairs, can be extracted. They are important in the study of the germ. For instance, explicit resolutions of such a singularity have been studied by \textit{J. Lipman} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 2, 161-172 (1983; Zbl 0521.14014)]. In this process, which does not lead to embedded resolutions, an important role is played by the characteristic pairs. Recall that, informally, ``embedded resolution'' means a process where along which the singular variety \(V\) one transforms the ambient space where it is defined, so that eventually the strict transform of \(V\) is non-singular and its union with the exceptional divisor is locally defined by simple, ``nice'' equations. Recently, several (closely related) methods to constructively (or canonically) obtain embedded resolutions of general singularities have been proposed. That is, procedures which involve a finite sequence of blowing-ups and which tell us, at each stage of the resolution process, how to choose the center of the transformation.
More precisely, in this note the authors explicitly study what results from the application of the general canonical process devised by \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207-302 (1997; Zbl 0896.14006)] to an irreducible quasi-ordinary singularity. It turns out that the process depends on the (suitably normalized) characteristic pairs of the singularity only. The description of the algorithm given in the paper is very explicit. The authors apply it to a non-trivial example, and they affirm that the method can be actually implemented by a computer.
The authors plan to apply similar techniques to the problem of simultaneous desingularization of a family of quasi-ordinary singularities in a future work. canonical resolution; quasi-ordinary singularities; characteristic pairs; embedded resolutions C. Ban and L. J. McEwan, Canonical resolution of a quasi-ordinary surface singularity , Canad. J. Math. 52 (2000), 1149--1163. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex surface and hypersurface singularities Canonical resolution of a quasi-ordinary surface singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the introduction to the article the author writes: ``This account shall serve as a quick guide to the historical development, the main contributors and the basic notions in the context of resolution of singularities. For detailed information we give references to the literature for each item. The presentation, which is necessarily subjective, does not claim completeness or utmost rigor. It shall merely help the reader to access the field and to find further sources.''
The article is very compressed and appears to be more a dictionary than an account of the history of the resolution of singularities. It is divided into several short sections: Main achievements; Some research problems; Contributors; Dictionary; Surveys; Miscellaneous; Selection of references before 1930; Selection of references after 1930.
Wisely the author preempts much of the criticism that can be raised to his selection of material with remarks as ``We list a selections of \dots'', ``The following list is far from complete \dots'', ``For technical and human reasons not all relevant contributions could be included.'' It is clear that there is no hope of writing a complete overview of a vast area like resolutions of singularities. On the other hand it would have been nice to get a suggestion of how the selection of the material was made. Is it just a result of ``human limitations'' or is it a conscious choice? It would also be of use to have some comments on related topics, like the resolutions of special varieties, that may not be considered, by the experts, as belonging to the field proper. The author has succeeded in writing a short and concise guide to students, and experts from other parts of algebraic geometry. Resolutions of singularities is a vast area of mathematics that has contributed to the development of several parts of algebraic geometry and algebra. The present article will make it considerably easier to access the area, and is a real service to the mathematical community. Among the many useful items of the article is a list of problems that, although quite specialized, will inspire younger mathematicians to work in the area. resolution of singularities; curves; surfaces; threefolds Hauser [Hauser 00] H., Resolution of Singularities (2000) History of algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), History of mathematics in the 19th century, History of mathematics in the 20th century, Singularities in algebraic geometry, Global theory of complex singularities; cohomological properties, Modifications; resolution of singularities (complex-analytic aspects), Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties Resolution of singularities 1860--1999 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a revised version of the main body of the author's recent Ph.D. thesis written at the University of Tokyo, Japan, under the supervision of H. Matumoto. Its main objective is to describe the category of bi-equivariant vector bundles on a bi-equivariant smooth (partial) compactification of a connected, reductive algebraic group \(G\) with normal crossing boundary divisors. Roughly speaking, the author's results generalize the description of the category of equivariant vector bundles on toric varieties established by \textit{A. A. Klyachko} [Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010)] fifteen years ago. As an application of his general method, the author proves a splitting property of equivariant vector bundles of low rank on the so-called ``wonderful compactification'' of an adjoint simple group in the sense of \textit{C. De Concini} and \textit{C. Procesi} [in: Invariant Theory, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. More precisely, let \(G\) be a connected, reductive algebraic group over an algebraically closed field \(k\) of characteristic zero. Let \(X\) be a \((G\times G)\)-equivariant smooth (partial) compactification of \(G\) with normal crossing boundary divisors, as it exists, for instance, in the cases of smooth toric varieties and of adjoint semi-simple groups à la De Concini-Procesi.
Then the author succeeds in constructing an equivalence between the category \(EV(X)\) of \((G\times G)\)-equivariant vector bundles on \(X\) and a category \({\mathcal C}(N)\) of certain asymptotic filtration data on \(X\), whose involved notions and technical subtleties occupy the better part of the paper. The crucial ingredient for establishing this categorical equivalence is the author's novel transversality condition for the respective asymptotic filtration data on the compactifying space \(X\) which finally enables the author to generalize related previous constructions by A. A. Klyachko, T. Uzawa, M. Brion, R. Steinberg, E. Strickland, and others in a fundamental way. The consequences and applications of the author's equivalent description of the category \(EV(X)\) discussed in the final section of the paper, include a detailed comparison with Klyachko's characterizing category, a splitting principle for vector bundles on the ``wonderful compactification'' of De Concini-Procesi, and a solution to a problem raised by B. Kostant in 2002. In fact, B. Kostant posed the question of the existence of a canonical extension of an equivariant vector bundle on a (complexified) symmetric space to its ``wonderful compactification''. After suitably translating this problem into the conceptual setting of the present paper, the author obtains a precise answer to Kostant's (unpublished) question.
No doubt, this paper represents an important contribution to, and a major step forward in the theory of group completions. The exposition is utmost thorough, detailed, rigorous, systematic, and skillfully arranged, thereby making it easily readable. compactifications; symmetric spaces; toric varieties; tensor categories; Tannaka categories S. Kato, Equivariant vector bundles on group completions, J. Reine Angew. Math. 581 (2005) 71 -- 116. Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Abstract manifolds and fiber bundles (category-theoretic aspects), Monoidal categories (= multiplicative categories) [See also 19D23], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representation theory for linear algebraic groups Equivariant vector bundles on group completions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Der Quotient \(V/G\), \(V\) komplexer Vektorraum, \(G\) endliche Gruppe, ist im allgemeinen eine singuläre algebraische Varietät und man sucht nach guten Auflösungen.
Falls \(V=\mathbb C^2\) und \(G \subset \text{SL}(2,\mathbb C)\) die standard-symplektische Form auf \(V\) erhält, existiert eine solche Auflösung \(X \to V/G\), wobei insbesondere \(X\) ein triviales kanonisches Bündel besitzt. Man nennt dies eine crepante Auflösung, beschrieben durch die McKay-Korrespondenz.
In der vorliegenden Arbeit wird bei der Verallgemeinerung auf höhere Dimensionen ein symplektischer Vektorraum \(V=V_0 \oplus V_0^{\star}\) betrachtet und als notwendige Bedingung für eine crepante Auflösung gezeigt, dass die Aktion von \(G\) auf \(V_0\) durch komplexe Spiegelungen erzeugt wird. Der Beweis erfolgt durch Reduktion auf den klassischen Fall des Quotienten \(X_0=V_0^{\star}/G\), der unter obiger Voraussetzung für \(G\) dann glatt ist.
Eine strengere Version des Hauptergebnisses wurde unabhängig von \textit{M. Verbitsky} [Asian J. Math. 4, 553--563 (2000; Zbl 1018.32028)] bewiesen. symplectic quotient singularity D. Kaledin, On crepant resolutions of symplectic quotient singularities, Sel. Math. New Ser., 9 (2003), 529--555. Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) On crepant resolutions of symplectic quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author constructs explicitly some resolutions of singularities in many relevant examples that appear often in moduli problems. The base is an algebraically closed field of characteristic \(p\geq 2\), which is then generalized to \(\text{Spec}({\mathbb{Z}}_p )\). The main new idea is to use compactification of symmetric spaces and line bundles on flag varieties. In the first part of the paper, which has an independent interest in itself, it is shown that the compactification of symmetric spaces \(X=G/H\) given by \textit{C. De Concini} and \textit{C. Procesi} [Invariant theory, Proc. 1st 1982 Sess. C. I. M. E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] works in any characteristic. More precisely \(G\) is an adjoint semisimple Chevalley group and \(H\) is the invariant group of an involution of \(G\). This part relies on many other results in the literature, but it is almost self-contained and clearly written.
If \(\lambda\) is a dominant weight, the compactification \(\overline{X}_{\lambda}\) is defined as the closure of \(X\) in a representation space \({\mathbb{P}} (V({\lambda}^*))\). There is a smooth \(\overline{X}\) which dominates the other \(\overline{X}_{\lambda}\). The author shows that there is a Frobenius splitting of the above compactification (i.e. a section of the injection \({\mathcal O}\to \text{Frob}_* ({\mathcal O})\) ). The splitting induces compatible splittings on all strata. The cohomology of line bundles on \(\overline X\) can be computed in many cases by using this splitting. It follows by these cohomological computations and by a criterion of \textit{G. Kempf} [``Toroidal embeddings. I'', Lect. Notes Math. 339, 41-52 (1973; Zbl 0271.14017)] that \(\overline {X}_{\lambda}^{norm}\) has rational singularities, improving previous results of several authors that showed that these singularities are Cohen-Macaulay. Then the author checks the projective normality of \(\overline{X}_{\lambda}\) in many cases, e.g. for \(X\) equal to \(G\times G\) quotiented by the diagonal. This machinery is applied to several examples of singularities. Additional informations about the strata at infinity are also obtained. A basic type of singularity studied in the examples is that of two \(n\times n\) matrices \(B\) and \(C\) satisfying \(B\cdot C=C\cdot B=p\text{ Id}\). These singularities appear in the author's paper [\textit{G. Faltings}, Math. Ann. 304, 489-515 (1996; Zbl 0847.14018)] about moduli-stacks for bundles on semistable curves. Some analogous examples for the other classical groups are also considered. resolutions of singularities; characteristic \(p\); compactification of symmetric spaces; moduli space; rational singularities; moduli-stacks for bundles on semistable curves Faltings, G.: Explicit resolution of local singularities of moduli-spaces. J. reine angew. Math. 483, 183-196 (1997) Global theory and resolution of singularities (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Singularities of curves, local rings Explicit resolution of local singularities of moduli-spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The fixed point set under a natural torus action on projectivized moduli spaces of simple representations of quivers is described. As an application, the Euler characteristic of these moduli is computed.
The main theme of the present paper is to apply the localization principle to projectivizations of quiver moduli parametrizing simple representations of quivers, to be introduced and discussed in section 2 (the projectivization being necessary in order to obtain torus fixed points). Based on an analysis of the fixed points in section 3 (which is complicated by the fact that the moduli spaces in case do not arise as quotients by a free group action), the first main result, Theorem 2.7, gives a description of a set of torus fixed points as a disjoint union of moduli spaces of the same type, that is, of projectived moduli of simple representations for other quivers. In fact, the quivers appearing in this description are `almost universal Abelian coverings' of the original quiver. Therefore, in the course of the proof, some principles of covering theory of finite-dimensional algebras have to be studied in the context of simple quiver representations in section 4.
In [\textit{M. Reineke}, Int. Math. Res. Not. 2006, No. 17, Article ID 70456 (2006; Zbl 1113.14018)], it is proven that the number of rational points of models of quiver moduli over finite fields behaves polynomially in the cardinality \(q\) of the field. Starting from computer experiments, a formula for the value of the first derivative of these counting polynomials at \(q=1\) was conjectured (the value at \(q=1\) of the polynomial itself is almost always zero, corresponding to the fact that the natural torus action on the unprojectivized moduli space does not admit fixed points except in trivial cases). Using some arithmetic geometry, this number can be reinterpreted as the Euler characteristic in cohomology with compact support of the projectivized quiver moduli.
In this form, the conjecture is proven in section 5. Namely, Theorem 2.6 identifies the Euler characteristic in question as the number of cyclic equivalence classes of certain primitive cycles in the given quiver. In fact, the localization techniques of this paper were developed for the purpose of computing this number. This result is proved via iterated localization: by building up a recursion over all quivers and all dimension vectors, one arrives at a finite number of iterated torus fixed points after finitely many applications of Theorem 2.7.
The description of the fixed point set provides an interesting interaction between algebraic geometry and representation theory of quivers: the detection of torus fixed points naturally leads to the study of covering techniques for simple representations. After such methods are developed, they can be successfully applied to the geometry of quiver moduli. The description also motivates consideration of a very special class of combinatorially defined simple representations of quivers, called string representations (Definition 6.1), which are in some sense `responsible' for aspects of the global topology of the moduli. fixed point sets; torus actions; projectivized moduli spaces; simple representations of quivers; Euler characteristic; Abelian coverings Markus Reineke, Localization in quiver moduli, J. Reine Angew. Math. 631 (2009), 59 -- 83. Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Group actions on varieties or schemes (quotients), Rational points Localization in quiver moduli. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is concerned with the equivalence of two ostensibly different properties of an algebraic stack $X$. \par On the one hand, $X$ is said to have the \textit{resolution property} if every finite type, quasi-coherent sheaf on $X$ is a quotient of a vector bundle. This definition is natural from the point of view of $K$-theory, insofar as it gives a natural sufficient criterion for the natural map $K_*^{\text{naive}}(X) \to K_*(X)$ (from the ``naive'' $K$-groups, defined in terms of vector bundles, to the honest $K$-groups, defined in terms of perfect complexes) to be an isomorphism. \par On the other hand, $X$ is \textit{basic} if it is a stack quotient of the form $[U/\mathrm{GL}_n]$ for some quasi-affine scheme $U$ acted on by $\mathrm{GL}_n$ (over $\text{Spec} \mathbb{Z}$) for some $n \geq 0$. \par The main result of this paper is that for $X$ a quasi-compact and quasi-separated algebraic stack over $\text{Spec} \mathbb{Z}$, $X$ is basic if and only if the closed points of $X$ have affine stabilizer groups and $X$ has the resolution property. This result builds on work of previous authors: \textit{R. W. Thomason} [Adv. Math. 65, 16--34 (1987; Zbl 0624.14025)] proved that a Noetherian basic stack has the resolution property (and it is easy to see that closed points on a basic stack have affine stabilizer groups), and \textit{B. Totaro} [J. Reine Angew. Math. 577, 1--22 (2004; Zbl 1077.14004)] proved the converse when $X$ is normal and Noetherian. The author's new contribution is thus to remove the normal and noetherian hypotheses on $X$ in the statement of the equivalence. An important input in the proofs is the recent work on noetherian approximation of algebraic stacks of \textit{D. Rydh} [J. Algebra 422, 105--147 (2015; Zbl 1308.14006)]. \par When $\mathrm{GL}_n$ acts on $U$, the tautological $\mathrm{GL}_n$-torsor $\pi: U \to [U/\mathrm{GL}_n]$ endows the target with a rank $n$ vector bundle $\mathcal{V}$, and $U$ and $\pi$ may then be recovered as the frame bundle of $\mathcal{V}$. Thus for a stack $X$, in the context of present paper, it is natural to ask for a characterization of the vector bundles $\mathcal{V}$ on $X$ such that the associated frame bundle has quasi-affine total space $U$. The author supplies a beautiful answer: when the closed points on $X$ have affine stabilizer groups, the frame bundle of $\mathcal{V}$ has quasi-affine total space if and only if $\mathcal{V}$ is a \textit{tensor generator} for the category of quasi-coherent sheaves on $X$ in a certain sense (which generalizes the usual notion of tensor generator in Tannaka theory for linear algebraic groups over a field). resolution property; locally free sheaf; vector bundle; generating family; tensor generator; algebraic space; algebraic stack; quotient stack; AF-scheme Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Tensor generators on schemes and stacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 earlier work [Proc. Lond. Math. Soc. (3) 113, No. 2, 185--212 (2016; Zbl 1375.13033)], the same three authors explained how the cluster algebra structure on the homogeneous coordinate ring of a Grassmannian, as described by \textit{J. Scott} [Proc. Lond. Math. Soc., III. Ser. 92, No. 2, 345--380 (2006; Zbl 1088.22009)], is additively categorified by the category of maximal Cohen-Macaulay modules over a certain Gorenstein order.
Among other properties reflecting the combinatorics of the Grassmannian cluster algebra, this category has `rank \(1\)' indecomposable objects in bijection with Plücker coordinates, which are Ext-orthogonal if and only if the corresponding Plücker labels are non-crossing. This leads to a bijection between the cluster-tilting objects of the category (or at least those mutation equivalent to one for which all indecomposable summands have rank \(1\)) and the seeds in the Grassmannian cluster algebra. This bijection is compatible with mutation, and the quiver of a seed can be computed as the Gabriel quiver of the endomorphism algebra of the corresponding cluster-tilting object.
The Grassmannian coordinate ring admits a natural quantisation, and \textit{J. E. Grabowski} and \textit{S. Launois} [Proc. Lond. Math. Soc. (3) 109, No. 3, 697--732 (2014; Zbl 1315.13036)] show that the cluster algebra structure can be quantised compatibly. In the present paper, the authors show that additional quantum information -- precisely, the powers of \(q\) appearing in quasi-commutation relations among quantum cluster variables -- may be computed from the same category of Cohen-Macaulay modules which categorifies the unquantised cluster structure. Thus this same category, with no modification such as passing to graded modules, categorifies the quantum Grassmannian cluster algebra.
For rank \(1\) modules, the \(\kappa\)-invariant, which the authors introduce to compute the quasi-commutation power of the corresponding Plücker coordinates, may also be expressed as an invariant MaxDiag of a pair of Young diagrams, as in work of \textit{K. Rietsch} and \textit{L. Williams} [Duke Math. J. 168, No. 18, 3437--3527 (2019; Zbl 1439.14142)].
We note that while this paper yields a categorical interpretation of the quasi-commutation data (or \(L\)-matrix) for each seed in the quantum Grassmannian cluster algebra, it does not provide a quantum cluster character, computing expressions for quantum cluster variables as quantum Laurent polynomials in a chosen initial seed. This is well-known to be a difficult problem, especially for cluster algebras without acyclic seeds, as is the case for most Grassmannians, and for now remains open. quantum Grassmannian; quantum cluster algebra; categorification Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Cohen-Macaulay modules in associative algebras, Quantum groups (quantized function algebras) and their representations Categorification and the quantum Grassmannian | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well-known that the quotient of \(\mathbb{C}^2\) by a finite group \(G\subset \mathrm{SL}(2, \mathbb{C})\) is a hypersurface in \(\mathbb{C}^3\) defined by a single equation \(F=0\). The hypersurface has a unique singularity at the origin, which is of type \(A, D, E\). We say the polynomial \(F\) is of type \(A, D, E\) if the hypersurface singularity is of type \(A, D, E\). The polynomial \(F\) induces a map \(\mathbb{C}^3 \to \mathbb{C}\), whose only singular fiber the hypersurface defined by \(F=0\).
The author of the paper under review studies the rationality question of the generic fiber (over the function field \(\mathbb{C}(t)\)) and the automorphism group of the hypersurface.
Theorem 1. The generic fiber is rational if and only if the polynomial \(F\) is of type \(A_n\).
The proof goes by finding a suitable compactification of the generic fiber and then using some rationality criterion of minimal del Pezzo surface and conic bundles over a perfect field.
The author goes further by considering the minimal field extension of \(\mathbb{C}(t)\) to make the \(D, E\) case rational.
The automorphism group of \(A_n\) type hypersurfaces is quite large, as it contains the group
\[
\{(x, y, z) \mapsto (x+yP(y), y, z+\frac{(x+yP(y))^n-x^n}{y})| P \in \mathbb{C}[y]\}.
\]
As a contrary, the author shows the following:
Theorem 3. The followings are equivalent:
1. Each automorphism of \(F=0\) extends to \(\mathbb{C}^3\).
2. The group of automorphisms has finite dimension.
3. The polynomial \(F\) is of type \(D\) or \(E\).
The author also determines the automorphism groups. Klein surfaces; ADE singularities; rationality; fibrations Rationality questions in algebraic geometry, Reflection and Coxeter groups (group-theoretic aspects), Lie algebras of linear algebraic groups, Deformations of singularities, Rational and birational maps, Group actions on affine varieties Non-rationality of some fibrations associated to Klein 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 finite dimensional associative \(k\)-algebra \(A\) (the base field \(k\) is assumed to be algebraically closed and have characteristic zero in the paper) is \textit{tilted} if it is the endomorphism algebra of a multiplicity-free tilting module over a connected path algebra of an acyclic quiver. The present paper characterizes tameness of a tilted algebra in terms of the invariant theory related to \(\text{mod}(A,d)\), the affine varieties of \(d\)-dimensional \(A\)-modules for various dimension vectors \(d\).
In particular, the following are equivalent for a tilted algebra \(A\): (1) \(A\) is tame; (2) for each generic root \(d\) and each irreducible indecomposable component \(C\) of \(\text{mod}(A,d)\), the field \(k(C)^{\text{GL}(d)}\) of rational invariants on \(C\) is isomorphic to \(k\) or \(k(x)\) (the field of fractions of the univariate polynomial ring); (3) for each generic root \(d\) and each irreducible indecomposable component \(C\) of \(\text{mod}(A,d)\), the moduli space \(M(C)_\theta^{ss}\) of \(\theta\)-semistable \(A\)-modules is either a point or the projective line, where \(\theta\) is an integral weight for which \(C\) contains a \(\theta\)-stable point; (4) \(M(C)_\theta^{ss}\) is smooth for all \(d\), \(C\) and \(\theta\) as in (3). -- Along the way some useful technical reductions are proved about the behaviour of these moduli spaces with respect to tilting functors and the \(\theta\)-stable decomposition. exceptional sequences; moduli spaces; bound quivers; rational invariants; tame algebras; wild algebras; tilted algebras; tilting modules; connected path algebras Chindris, C.: On the invariant theory for tame tilted algebras. Algebra Number Theory \textbf{7}(1), 193-214 (2013)\textbf{(MR 3037894)} Representations of associative Artinian rings, Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Geometric invariant theory, Trace rings and invariant theory (associative rings and algebras) On the invariant theory for tame tilted 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 A theory of the resolution of singularities should not only prove that every space \(Z\) can be resolved by a modification \(Z' \to Z\), but the resolution should be
\begin{itemize}
\item canonical in the sense that the theory distinguishes one resolution \(Z_{res} \to Z\);
\item constructive in the sense that there is an explicit algorithmic procedure to obtain the distinguished resolution \(Z_{res} \to Z\);
\item functorial in the sense that for every smooth morphism \(Y \to Z\), the resolution \(Y_{res} \to Y\) is the pull-back (in the appropriate category) of \(Z_{res} \to Z\).
\end{itemize}
The article is the first in an announced series which establishes such a resolution for morphisms \(Z \to B\) of fine saturated logarithmic Deligne-Mumford stacks. The article itself deals only with the case \(B = \mathrm{Spec}(k)\) the trivial log point, i.e., with the absolute case.
The starting point is resolution of logarithmic schemes. The authors ask for an extended functoriality principle, i.e., the resolution should not only be functorial for classically smooth maps \(Y \to Z\) but also for log smooth maps \(Y \to Z\). This makes it necessary to resolve via blowups of Kummer centers, i.e., ideals in the Kummer étale topology of \(Z\). Such a blowup is not necessarily a logarithmic scheme but a logarithmic Deligne-Mumford stack; thus the natural setup for the resolution are (fine saturated) logarithmic Deligne-Mumford stacks. A log smooth Deligne-Mumford stack is called a toroidal orbifold.
The algorithm comes in two versions, an embedded one -- the principalization of ideals on a toroidal orbifold -- and a non-embedded one -- the resolution of singularities.
The algorithm does not specialize to a classical one for trivial logarithmic structures. When it starts with a variety \(Z\) with trivial logarithmic structure, then \(Z_{res}\), in general, neither has trivial logarithmic structure nor is a scheme but a honest toroidal orbifold.
The algorithm is less complicated than algorithms for resolving singularities of classical schemes since it does not need to take separate care about the exceptional divisor, but it is nonetheless intricate. It employs a logarithmic version of the induction over hypersurfaces of maximal contact to reduce the log order of marked ideals. Additionally, in several so-called cleaning processes it must be ensured that the ideal to resolve has a nice interplay with the logarithmic structure.
After this article was finished, Ming Hao Quek -- a student of one of the authors -- has found a simpler, more canonical, and presumably faster algorithm to resolve logarithmic singularities in the same setup, see [\textit{M.~H.~Quek}, ``Logarithmic resolution via weighted toroidal blow-ups'', Preprint, \url{arXiv:2005.05939}]. This is achieved by allowing more general centers which are no longer ideals in the Kummer étale topology; it is a logarithmic variant of the \emph{dream algorithm} of [\textit{D.~Abramovich} et al., ``Functorial embedded resolution via weighted blowings up'', Preprint, \url{arXiv:1906.07106}]. resolution of singularities; logarithmic geometry; algebraic stacks Global theory and resolution of singularities (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), Logarithmic algebraic geometry, log schemes Principalization of ideals on toroidal orbifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 lecture note examines the 6 types of ``quadrilateral'' singularities \(J_{3,0}\), \(Z_{1,0}\), \(Q_{2,0}\), \(W_{1,0}\), \(S_{1,0}\), \(U_{1,0}\): if \(X\) is a class of them, \(PC(X)\) denotes the set of Dynkin graphs \(G\) with components of type \(A\), \(D\), \(E\) having the following property: There is a fibre \(Y\) in the versal deformation of a singularity in \(X\), such that \(Y\) has only rational double points, and \(G\) is given as a (disjoint) union of their Dynkin graphs. Quadrilateral singularities are of modality 2, and \(PC(X)\) is studied for the case, if \(X\) is one of the relevant normal forms [cf. \textit{V. I. Arnold}, Invent. Math. 35, 87- 109 (1976; Zbl 0336.57022)].
Due to Looijenga, \(PC(X)\) can be studied using the lattice embedding of the associated root-lattice into the even unimodular lattice with signature (19,3). Using Nikulin's results for such embeddings, this gives a possibility to determine whether or not \(G\) belongs to \(PC(X)\). The book under review gives a systematic treatment of all cases; technical tools are the root systems \(A,\dots,F\) as well as the nonreduced root systems \(BC\) arising in the relevant constructions. Using Dynkin graphs (in a terminology slightly different from the standard one), the description of \(G \in PC(X)\) (in the cases of \(X = J_{3,0}\), \(Z_{1,0}\), \(Q_{2,0})\) is given by the following theorem: \(G\) belongs to \(PC(X)\) iff it is in a list of exceptions or can be obtained by applying elementary or ``tie transformations'' (in the sense, studied by the author in a previous paper) twice to some of a certain list of essential basic Dynkin graphs and if \(G\) contains no short root. This theorem can be interpreted in the language of elliptic \(K3\)-surfaces: It describes the possible combinations of singular fibres on \(K3\)-surfaces with a singular fibre of type \(I^*_ 0\) (in Kodaira's notation). -- The remaining cases of \(X = W_{1,0}\), \(S_{1,0}\), \(U_{1,0}\) require further efforts. Appearing graphs can be characterized by additional rules (presence of ``obstruction components'' and/or ``dual elementary transformations'').
The book starts with an introduction to quadrilateral singularities: In chapter 1 Looijenga's results (which are basic to reduce the problem to lattice embeddings) are reviewed.
After an introduction to lattices, another section is devoted to a theory of root systems, adapted to the situation considered later. Further, the technical tools for manipulating graphs are introduced, followed by a section, where conditions are given for a Dynkin graph \(G\) to be in \(PC(X)\). The chapter concludes explaining Coxeter-Vinberg graphs associated with hyperbolic spaces.
Chapter 2 deals with the first three types of quadrilateral singularities, as indicated above, whereas chapter 3 and 4 are devoted to the study of the cases \(X = W_{1,0}\), \(S_{1,0}\), \(U_{1,0}\) respectively.
In an appendix, similar questions for plane sextic curves are considered: This is a revised version of the authors' earlier paper [in Singularities, Proc. IMA Particip. Inst. Conf., Iowa City 1986, Contemp. Math. 90, 295-316 (1989; Zbl 0698.14023)], where the above methods are applied to study conditions for a Dynkin graph to correspond to a configuration of \(A,D,E\)-singularities on a sextic curve.
This book gives insight to deep properties of deformations of a class of bimodal singularities. The author points out essential ideas stemming from a discussion at Oberseminar Brieskorn (University of Bonn), especially from \textit{F. J. Bilitewski}, who considered several of the cases treated here, already. The methods employed may be useful to study other types of singularities as well. union of Dynkin graphs; quadrilateral singularities; tie transformations; singular fibres on \(K3\)-surfaces; versal deformation of a singularity; modality; root systems; Dynkin graphs; bimodal singularities T. Urabe, ''Dynkin graphs and triangle singularities,'' In:Proc. of Workshop on Topology and Geometry, Hanoi, Vietnam, March (1993). Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Global theory of complex singularities; cohomological properties, Singularities in algebraic geometry Dynkin graphs and quadrilateral 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 It is well known that the Thetanullwerte \(\vartheta_ m\) give rise to a holomorphic immersion \(\theta(q)\) of \(\Gamma_ g (q^ 2, 2q^ 2)\setminus \mathbb{H}_ g\) into \(\mathbb{P}_{N(q)}\mathbb{C}\), \(N(q):= q^{2g}-1\), defined by \(\tau\mapsto (\vartheta_ m (\tau))_{m\in \Sigma_ q}\) where the theta-characters \(m\) range over the kernel \(\Sigma_ q\) of the multiplication by \(q\) in \((\mathbb{Q}/\mathbb{Z})^{2g}\). Clearly, \(\mathbb{H}_ g\) is the Siegel upper half plane of genus \(g\) and \(\Gamma_ g (\ell, 2\ell)\) the subgroup of the congruence subgroup \(\Gamma_ g(\ell)\) of the Siegel modular group \(\Gamma_ g:= \text{Sp}_ g (\mathbb{Z})\) of those elements \({{a\;b} \choose {c\;d}}\) such that \(\text{diag}^ t ab\equiv \text{diag}^ t cd\equiv (2\ell)\). In a preprint by the same author it is shown, that the mappings \(\theta^ 2(2)\) resp. \(\theta^ 4(2)\) defined by \(\tau\mapsto (\vartheta^ 2_ m (\tau))_{m\in \Sigma_ 2}\) resp. \((\vartheta^ 4_ m (\tau))_{m\in \Sigma_ 2}\) also induce an injection of the quotients \(\Gamma_ g(2,4) \setminus \mathbb{H}_ g\) resp. \(\Gamma_ g(2) \setminus \mathbb{H}_ g\) into \(\mathbb{P}_{N(2)} \mathbb{C}\). Now, in the present paper these results are generalized in several directions.
First for arbitrary \(q\), \(\theta^ 2(q)\) gives rise to an injection of \(\Gamma_ g (q^ 2/ 2,q^ 2) \setminus \mathbb{H}_ g\) into \(\mathbb{P}_{N(q)} \mathbb{C}\) whereas \(\theta^ 4(q)\) induces an injection of \(\Gamma_ g (q^ 2/2) \setminus \mathbb{H}_ g\) resp. \(\Gamma_ g (q^ 2/4, q^ 2/2) \setminus \mathbb{H}_ g\) into the projective space according \(q\equiv 2\) resp. \(q\equiv 0(4)\). Similar results are obtained for the analogous map \(\theta^{8k} (q)\), \(k\) of the form \(2^ p u\) with \(u\) odd (theorem 5).
All these maps may be extended holomorphically to the respective Satake compactifications but only in the case \(q=2\) will the extensions be injective, too. Moreover, in the special case \(q\equiv 0(4)\) the maps \(\theta^ 2(q)\), \(\theta^ 4(q)\), \(\theta^{8k}(q)\) are immersions of the respective (smooth) modular varieties into projective space (theorem 6).
The proof of these results stems from a generalization of Igusa's fundamental lemma, i.e. a characterization of the graded ring of respective modular forms as a suitable integral closure (theorem 4) from which it follows immediately that all the mappings are at least generically injective. The full injectivity is achieved by the same argument as in the case \(q=2\). Theta-functions; Siegel modular forms; graded ring of modular forms; Thetanullwerte; holomorphic immersion; Siegel modular group; Satake compactifications; projective space; injectivity Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Theta functions and abelian varieties Maps defined by Thetanullwerte | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f : ({\mathbb C}^n,0) \rightarrow ({\mathbb C}^p,0)\) be a \(\mathcal K\)-finite map germ. Based on the approach developed in a previous paper [\textit{J. J. Nuño Ballesteros} and \textit{M. J. Saia}, Glasg. Math. J. 40, No. 1, 21-32 (1998; Zbl 0919.58010)], the authors investigate the problem of computing the number of stable isolated Boardman singularities of type \(\Sigma^{i_1, \ldots, i_k},\) that may occur in a generic deformation of \(f.\) The case \(n = p = 2\) has been studied by T. Fukuda and G. Ishikawa, J. Rieger, T. Gaffney and D. Mond [see, e.g., \textit{T. Fukuda} and \textit{G. Ishikawa}, Tokyo J. Math. 10, 375-384 (1987; Zbl 0652.58012)] and the case \(n =2\), \(p =3\) has been considered in [\textit{D. Mond}, Lect. Notes Math. 1462, 221-234 (1991; Zbl 0745.32020)]. In the former case, the number of cusps which are in fact singularities of type \(\Sigma^{1,1,0}\) is counted, while in the latter the number of cross cusps or, equivalently, of singularities with Boardman symbol \(\Sigma^{1,0}\) is calculated.
In the paper under review, the authors prove that the number in question is equal to the multiplicity of the Jacobian ideal for the corresponding Boardman symbol if either the map \(f\) has rank \(n-i_1,\) where \(n\) is equal to the codimension of the Boardman manifold \(\Sigma^{i_1, \ldots, i_k}\) in the jet space \(J^k(n,p),\) or \(f\) is a singularity of type \(\Sigma^{i_1,i_2}\) and \(p=6.\) In conclusion, some examples are considered. Consequently, the authors express a hope that they have described all possible situations when this number is equal to the multiplicity of the Jacobian ideal. isolated singularities; stable perturbations; Boardman strata; Boardman symbol; Jacobian ideal; determinantal ideals; multiplicity; Minor number; the number of cusps; cross cusps T Fukui, J J Nuño Ballesteros, M J Saia, On the number of singularities in generic deformations of map germs, J. London Math. Soc. \((2)\) 58 (1998) 141 Classification; finite determinacy of map germs, Deformation of singularities, Deformations of complex singularities; vanishing cycles, Complete intersections, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Multiplicity theory and related topics, Implicit function theorems, Jacobians, transformations with several variables On the number of singularities in generic deformations of map germs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors give a survey about the development and the results concerning Grothendieck's conjectures relating simple singularities of surfaces and the geometry of finite dimensional complex Lie algebras. Let \(\mathfrak{g}\) be a simple complex Lie algebra of type \(A_n, D_n, E_6, E_7, E_8\) and \(G\) the corresponding simply connected simple Lie group. Let \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\) and \(\mathfrak{B}\) a Borel subalgebra of \(\mathfrak{g}\). Let \(B\) be the corresponding Borel subgroup of \(G\). There is a canonical map \(\gamma: \mathfrak{g}\to \mathfrak{h}/W\), \(W\) the Weyl group, defined by \(\gamma(x)=(\gamma_1(x), \ldots, \gamma_r(x))\) where \(\gamma_1, \ldots,\gamma_r\) are the homogeneous \(G\)--invariant polynomials generating \(\mathbb{C}[\mathfrak{g}]^G\). Grothendiek conjectured that \(\gamma\) has a simultaneous resolution. He also conjectured that for a subregular nilpotent element \(y\) and a transversal slice \(X\) at \(y\) to the orbit \(Gy\) the germ of the surface \(X\cap \gamma^{-1}(\gamma(0))\) at \(y\) is a simple surface singularity of the same type as the Lie algebra \(\mathfrak{g}\). simple singularities; resolution of singularities; Lie algebras; subregular nilpotent elements Lê, D. T.; Tosun, M., Simple singularities and simple Lie algebras, \textit{TWMS J. Pure Appl. Math.}, 2, 1, 97-111, (2011) Singularities in algebraic geometry, Complex surface and hypersurface singularities, Coadjoint orbits; nilpotent varieties, Deformation of singularities Simple singularities and simple Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0619.00007.]
Theorem [cf. \textit{H. Knörrer}, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033) and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and the author, ibid. 165-182 (1987; Zbl 0617.14034)]: A hypersurface singularity is simple iff it has finite CM-representation type, i.e. iff there are only finitely many isomorphism classes of indecomposable maximal Cohen- Macaulay modules.
I include a complete proof of this result in this survey article. My emphasis lies on the construction of maximal Cohen-Macaulay modules via matrix factorizations (section 2), from which we deduce Knörrer's periodicity result (section 3). In section 4 we construct for every nonsimple hypersurface singularity infinitely many indecomposable maximal Cohen-Macaulay modules.
Theorem [cf. the author's joint paper with Buchweitz and Greuel cited above]: A hypersurface has countable CM-representation type iff it is isomorphic to \(A_{\infty}\) or \(D_{\infty}.\)
In section 6 we describe all indecomposable maximal Cohen-Macaulay modules on \(A_ n\) and \(D_ n\) singularities for \(n=1,2,...,\infty\) and their Auslander-Reiten-sequences. - A complete classification of complex analytic singularities with finite CM-representation type is known only for dimension \(\leq 2\). A list of the known examples is contained in (7.1). In (7.2) I give a few more examples of singularities which have countable representation type. finite CM-representation; maximal Cohen-Macaulay modules; hypersurface singularity; Auslander-Reiten-sequences Schreyer, Frank-Olaf: Finite and countable CM-representation type. Lecture notes in math. 1273, 9-34 (1987) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Finite and countable CM-representation type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This monograph deals with the important invariants of an isolated complete intersection singularity, namely the Milnor lattice H, the vanishing cycles and the monodromy group. The author also introduces relative invariants, i.e., the relative homology group \(\hat H,\) the set of thimbles and the relative monodromy. In order to calculate these invariants, an appropriate notion of a Dynkin diagram for an isolated complete intersection singularity is given and the author generalizes the methodes due to Gabrielov and Lazzari in order to compute these Dynkin diagrams. By using these methods, the Dynkin diagrams for large classes of two codimensional isolated complete intersection singularities are computed in the third chapter. This chapter contains also supplementary results about the extension of the strange duality.
Based on the computations of Chapter 3, Chapter 4 deals with the main results. First the isolated intersection singularities with a definite, parabolic or hyperbolic intersection form are classified. Next it is shown that the monodromy groups and the vanishing cycles of all even- dimensional isolated complete intersection singularities except the hyperbolic singularities can be described in a purely arithmetic way. The monograph ends with Chapter 5 containing purely algebraic results about lattices which are necessary for the proofs of the main results. invariants; isolated complete intersection singularity; Milnor lattice; vanishing cycles; monodromy group; Dynkin diagram W. Ebeling, ''The monodromy groups of isolated singularities of complete intersections,'' Lecture Notes in Mathematics, 1293, Springer-Verlag, Berlin, 1987 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Local complex singularities, Complete intersections, Singularities in algebraic geometry The monodromy groups of isolated singularities of complete intersections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A germ of irreducible normal complex analytic space is called Hirzebruch-Jung singularity if it is the normalization of an \(n\)-dimensional irreducible quasi-ordinary germ (i.e., there exists a finite morphism to a smooth space of the same dimension). Let \(W\) be a lattice and \(\sigma\) a strictly convex finite rational polyhedral cone in \(W_\mathbb R=W\otimes \mathbb R\), \(M\) the dual lattice and \(\check{\sigma}\) the dual cone,
\[
Z(W, \sigma)=\operatorname{Spec} \mathbb C\;[\check{\sigma}\cap M].
\]
\((W, \sigma)\) is called a maximal simplicial pair if \(\sigma\) and \(W_\mathbb R\) have the same dimension and \(\sigma\) is simplicial. Hirzebruch-Jung singularities are analytically isomorphic with the germ \((Z(W,\sigma), 0)\) at the \(0\)-dimensional orbit of an affine toric variety defined by a maximal simplicial pair \((W, \sigma)\). It is proven that the analytical type of a Hirzebruch-Jung singularity \((Z(W,\sigma), 0)\) determines the pair \((W, \sigma)\) up to isomorphism. A normalization algorithm for quasi-ordinary hypersurface singularities is given. Hirzebruch-Jung singularities; quasi-ordinary singularities; toric singularities --------, On higher dimensional Hirzebruch-Jung singularities , Invariants of analytic local rings, Toric varieties, Newton polyhedra, Okounkov bodies On higher dimensional Hirzebruch-Jung 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 In this paper, the authors determine all finite groups which occur as finite subgroups of the automorphism groups of three-dimensional complex tori. Their strategy for carrying out this program is as follows:
As any finite subgroup \(G\subset\Aut(X)\), for a complex torus \(X\), is a subgroup of a finite subgroup that is maximal in the isogeny class of the torus \(X\), the problem is basically reduced to finding all those ``maximal'' finite automorphism subgroups in the case of \(\dim X=3\). This, in turn, amounts to determining all finite subgroups of maximal orders of the semi-simple \(\mathbb{Q} \)-algebra \(\text{End}_\mathbb{Q}(X)/\text{rad}(X)\) of the 3-dimensional torus \(X\).
Now, by applying the general version of Poincaré's reducibility theorem for complex tori, the authors show that there are exactly five isogeny types for 3-dimensional complex tori, for which the program of determining all finite subgroups of maximal order in the semi-simplification \(\text{End}_\mathbb{Q} (X)/\text{rad}(X)\) can be carried out very explicitly and completely. This is quite a subtle and tricky matter, ending up in a deep and beautiful classification result which can be regarded as a first step towards a generalization (to higher dimensions) of the classification of hyperelliptic algebraic surfaces à la \textit{F. Enriques} and \textit{F. Severi} [Acta Math. 32, 283--392 (1909; JFM 40.0684.01)] and \textit{G. Bagnera} and \textit{M. de Franchis} [Atti IV Cong. Int. Mat., Roma 1908, No. 2, 242--248 and 249--256 (1909; JFM 40.0686.01 and JFM 40.0686.02)]. finite subgroups of the automorphism groups; three-dimensional complex tori; isogeny types Birkenhake, C., González, V., Lange, H.: Automorphism groups of 3-dimensional complex tori. J. Reine Angew. Math. 508, 99--125 (1999) Group actions on varieties or schemes (quotients), Isogeny, Toric varieties, Newton polyhedra, Okounkov bodies, \(3\)-folds, Automorphism groups of \(\mathbb{C}^n\) and affine manifolds, Analytic theory of abelian varieties; abelian integrals and differentials, Compact complex \(3\)-folds Automorphism groups of 3-dimensional complex tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The basic object in this work is a differential graded (dg) category \(\mathcal A\) over a base-field \(k\). This is a category enriched over complexes of \(k\)-vector spaces, and the category of dg algebras over k embeds into the dg category of dg categories over \(k\) by giving a dg algebra \(A\) the structure of a dg category with a single object. Also, the category of perfect complexes (i.e. complexes quasi-isomorphic to a locally free complex) \(\text{perf}(X)\) on a quasi-compact, quasi-separated \(k\)-scheme \(X\) has a canonical dg enhancement \(\text{perf}_{\text{dg}}(X)\).
In this article, a dg functor \(F:\mathcal A\rightarrow\mathcal B\) is a Morita equivalence if it induces an equivalence of categories \(\mathcal D(\mathcal B)\overset{\simeq}{0\rightarrow}\mathcal D(\mathcal A)\). The category of (essentially small) dg categories is denoted \(\text{dgcat}(k)\), and \(\text{Hmo}(k)\) denotes its localization in the class of Morita equivalences. A functor \(E:\text{dgcat}(k)\rightarrow\mathcal T\) where \(\mathcal T\) is a triangulated category is called a localizing invariant if it inverts the Morita equivalences, and sends short exact sequences of dg categories to distinguished triangles in a way functorial for strict morphisms of exact sequences. The functor \(E\) is called \(\mathbb A^1\) homotopy invariant if it inverts the canonical dg functors \(\mathcal A\rightarrow\mathcal A[t].\) The authors list the main examples of such functors:
In homotopy \(K\)-theory, when \(\text{Ho(Spt)}\) is the homotopy category of spectra, Weibel's homotopy \(K\)-theory gives rise to a functor \(KH:\text{dgcat}(k)\rightarrow\mathbf{Ho(Spt)}\) which is both a localizing invariant and an \(\mathbb A^1\)-homotopy invariant. When applied to \(A\) and \(\text{perf} _{\text{dg}}(X)\), the functor computes the homotopy \(K\)-theory of \(A\), respectively \(X\).
In nonconnective algebraic \(K\)-theory with coefficients, for a prime \(l\), when \(l\nmid\text{char}(k)\), mod-\(l^\nu\) nonconnective algebraic \(K\)-theory gives rise to a functor \(\mathbb K(-;\mathbb Z/l^\nu):\text{dgcat}(k)\rightarrow\text{Ho(Spt)}\) which is a localizing invariant and an \(\mathbb A^1\)-homotopy invariant. When \(l\mid\text{char}(k)\), consider the functor \(\mathbb K(-)\otimes\mathbb Z[1/l]\). Applied to \(A\) and \(\text{perf} _{\text{dg}}(X)\), these functors computes the nonconnective algebraic \(K\)-theory with coefficients of \(A\) and \(X\).
In Étale \(K\)-theory, let \(l\) be an odd prime. Dwyer-Friedlander's étale \(K\)-theory induces a functor \(K^{\text{ét}}(-;\mathbb Z/l^\nu):\text{dgcat}(k)\rightarrow\text{Ho(Spt)}\), localizing and \(\mathbb A^1\) invariant. When \(l\nmid\text{char}(k)\) and \(X\) is regular and of finite type over \(\mathbb Z[1/l]\), \(K^{\text{ét}}(\text{perf}_{\text{dg}}(X);\mathbb Z/l^\nu)\) gives the étale \(K\)-theory of \(X\).
In periodic cyclic homology, for a field \(k\) of characteristic zero and \(\mathcal D^\pm(k)\) the derived category of \(\mathbb Z/2\)-graded \(k\)-vector spaces, the periodic cyclic homology gives a functor \(HP:\text{dgcat}(k)\rightarrow\mathcal D^\pm(k)\), localizing and \(\mathbb A^1\) invariant. Applied to \(A\), resp. \(\text{perf}_{\text{dg}}(X)\) it computes the periodic cyclic homology of \(A\), resp. \(X\). When \(X\) is smooth, the classical Hochschild-Kostant-Rosenberg theorem gives the identifications with de Rham homology: \(HP^+(X)\simeq\bigoplus_{n\text{ even}}H^n_{dR}(X),\;HP^-(X)\simeq\bigoplus_{n\text{ odd}}H^n_{dR}(X).\)
In Noncommutative motives, let \(\text{Mot}(k)\) be the closed symmetric monoidal triangulated category of such. It has a symmetric monoidal functor \(U:\text{dgcat}(k)\rightarrow\text{Mot}(k)\) that is localizing and \(\mathbb A^1\) invariant.
With the given examples above as a basis, the main result of the article is stated (nearly) verbatim:
Theorem 1.9 (Gysin triangle). Let \(X\) be a smooth \(k\)-scheme, \(i:Z\hookrightarrow X\) a smooth closed subscheme, and \(j:U\hookrightarrow X\) the open complement of \(Z\). For every functor \(E:\text{dgcat}(k)\rightarrow\mathcal T\) which is localizing and \(\mathbb A^1\) invariant, we have an induced triangle \(E(Z)\overset{E(i_\ast)}{\rightarrow} E(X)\overset{E(j^\ast)}{\rightarrow} E(U)\overset{\partial}{\rightarrow}\Sigma E(Z),\) where \(i_\ast\), resp. \(j^\ast\), stands for the push-forward, resp. pull-back, dg functor.
This theorem generalizes naturally. One can replace the schemes \(X,\;Z,\;U\) by algebraic spaces, and the categories \(\text{perf}_{\text{dg}}(Z)\), \(\text{perf}_{\text{dg}}(X)\), \(\text{perf}_{\text{dg}}(U)\) can be replaced by their tensor product with a dg category \(\mathcal A\). In particular, it is proved that when \(\mathcal A=\text{perf}_{\text{dg}}(Y)\) for a quasi-compact, quasi-separated scheme \(Y\), this corresponds to replacing \(X,Y,U\) by their tensor product over \(k\).
To prove the theorem, one has to study \(\text{perf}_{\text{dg}}(X)_Z\), the full subcategory of the perfect complexes of \(\mathcal O_X\)-modules supported on \(Z\). This involves a study of this category in terms of a formal dg \(k\)-algebra in the affine case, and then a Zariski Descent argument, linking the study to derived geometry in some sense. The proof thus explains the applications to particular cases in a very visible way:
The fundamental theorem (in homotopy theory) is illustrated when \(X=\mathbb A^1=\text{Spec}(k[t])\), \(Z=\text{Spec}(k[t]/(t))\), \(U=\text{Spec}(k[t,t^{-1}])\). Then the general Gysin triangle reduces to \(E(k)\overset{E(i_\ast)}{\rightarrow} E(k)\overset{E(j^\ast)}{\rightarrow} E(k[t,t^{-1}])\overset{\partial}{\rightarrow}\Sigma E(k).\) It follows that \(E(i_\ast)=0\) so that \(E(k[t,t^{-1}])\simeq E(k)\oplus\Sigma E(k)\). Generalizing to a dg category \(\mathcal A\), \(E(k[t,t^{-1}])\simeq E(\mathcal A)\oplus\Sigma E(\mathcal A)\). Then letting \(E=KH\), resp. \(E=HP\) this gives the fundamental theorems in homotopy \(K\)-theory, resp. periodic cyclic homology.
Also, Quillen's localization theorem is particular case of the general theorem: The homotopy \(K\)-theory agrees with Quillen's algebraic \(K\)-theory on smooth schemes, so when \(E=KH\) the general Gysin triangle reduces to Quillen's localization theorem \(K(Z)\overset{K(i_\ast)}{\rightarrow} K(X)\overset{K(j^\ast)}{\rightarrow} K(U)\overset{\partial}{\rightarrow}\Sigma K(Z)\). This proof differs from Quillen's proof based on the dévissage theorem for abelian categories, and the generalization of theorem 1,9 here applies in addition to algebraic spaces.
The maps \(i:Z\hookrightarrow X\) and \(j:U\hookrightarrow X\) give homomorphisms on de Rham cohomology \(H^n_{dR}(i_\ast):H^n_{dR}(Z)\rightarrow H^{n+2c}_{dR}(X)\) and \(H^n_{dR}(j^\ast):H^n_{dR}(X)\rightarrow H^n_{dR}(U)\), \(c=\text{codim}(i)\). When \(E=HP\) the long exact sequence associated to the general Gysin triangle reduces to an explicit six-term exact sequence which turns out to be the 2-periodization of the Gysin long exact sequence on de Rham cohomology constructed by Hartshorne.
The general theorem also leads to a noncommutative motivic Gysin triangle: When \(E=U\) the general Gysin triangle reduces to the \textit{noncommutative motivic Gysin triangle} \(U(Z)\overset{U(i_\ast)}{\rightarrow} U(X)\overset{U(j^\ast)}{\rightarrow} U(U)\overset{\partial}{\rightarrow}\Sigma U(Z)\). This is strongly (and explicitly remarked by the authors) related to the motivic Gysin triangles constructed by Morel and Voevodsky.
The work in this article is a highly nontrivial, and very roughly it can be said to use general representation theoretic methods to prove results in derived geometry. This gives more natural, and formal, proofs for known results which because of this generalizes to new applications. differential graded (dg) category; perfect complex; dg enhancement; localizing invariant; \(\mathbb A^1\) homotopy invariant; noncommutative algebraic \(K\)-theory; periodic cyclic homology; noncommutative graded motives; Gysin triangle; the fundamental theorem in homotopy theory; periodic cyclic homology; Quillen's algebraic \(K\)-theory; Quillen's localization theorem; noncommutative motivic Gysin triangle; motivic Gysin triangles ] \bysame, The Gysin triangle via localization and \(\bbA^{1}\)-homotopy invariance, Transactions of the American Mathematical Society 370 (2018), no. 1, 421-446. Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Motivic cohomology; motivic homotopy theory, Enriched categories (over closed or monoidal categories), \(K\)-theory and homology; cyclic homology and cohomology The Gysin triangle via localization and \(A^1\)-homotopy invariance | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \((X,x)\) be a normal, isolated 3-dimensional singularity. By Mori there exists a minimal resolution \(\sigma:Y\to X\) of \((X,x)\), that is, \(\sigma\) is a partial resolution, \(Y\) having only terminal singularities and the canonical divisor \(K_ Y\) being nef with respect to \(\sigma\). \((X,x)\) is called simple \(K3\) if it is quasi-Gorenstein and if the exceptional set of \(Y\) consists of a single normal \(K3\) surface \(D\). Let \(\{y_ i\}\) denote the set of (terminal) singularities of \(Y\) along \(D\) and \(r_ i\) the index of the terminal singularity \(y_ i\). Now assume that \((X,x)\) is a simple \(K3\) isolated hypersurface singularity defined by a weighted homogeneous polynomial \(f\) for positive weights \(p,q,r,s\) of the variables where \(\text{gcd}(p,q,r,s)=1\). Then the author proves the following formula describing the distribution of the terminal singularities on the minimal resolution of \((X,x)\):
\[
24-\Sigma(r_ i- 1/r_ i)=(p+q+r+s)(pq+qr+rs+sp)/pqrs.
\]
The proof goes by equating the (arithmetic) Poincaré series of the graded ring \(\mathbb{C}[x,y,z,u,v]/(f)\) of \((X,x)\), the grading being given by the weights, and the Poincaré series \(\sum_{m\geq 0}c_ m(X,x)t^ m\), where \(c_ m(X,x)=\dim_ \mathbb{C}\Gamma(Y,{\mathcal O})/\Gamma(Y,{\mathcal O}(-(m+1)D))\). The latter one can be computed by a Riemann-Roch type theorem for normal isolated singularities by the author and Kawamata-Viehweg vanishing. quasi-homogeneous sipmle \(K3\) singularity; distribution of the terminal singularities on the minimal resolution of isolated 3-dimensional singularity K. WATANABE, Distribution formula for terminal singularities on the minimal resolution of quasi-homogeneous simple ^3-singularity, Thoku Math. J. 43 (1991) 275-288. Singularities in algebraic geometry, \(3\)-folds, Local cohomology and commutative rings, \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties Distribution formula for terminal singularities on the minimal resolution of a quasi-homogeneous simple \(K3\) singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this paper is to study properties of the algebras of
planar quasi-invariants. These algebras are Cohen-Macaulay and
Gorenstein in codimension one. Using the technique of matrix
problems, the authors classify all Cohen-Macaulay modules of rank
one over them and determine their Picard groups. In terms of this
classification, they describe the spectral modules of the planar
rational Calogero-Moser systems. Finally, the authors elaborate
the theory of the algebraic inverse scattering method, computing a
new unexpected explicit example of a deformed Calogero-Moser
system. This paper is organized as follows. Section 1, is an
introduction to the subject and a description of the results
obtained. Section 2, deals with ring-theoretic properties of the
algebra of surface quasi-invariant. Section 3, deals with rank one
Cohen-Macaulay modules over the algebra of planar
quasi-invariants. In this section, the authors classify all
Cohen-Macaualy \(A\)-modules of rank one, specifying those of them,
which are locally free in codimension one. Next, they give an
explicit description of a canonical module of \(A\) (algebra of
planar quasi-invariant) and describe the Picard group \(Pic(A)\)
viewed as a subgroup of the group \(\mathbf{CM}^{lf}_1(A)\) (the set
of the isomorphism classes of Cohen-Macaualy \(A\)-modules of rank
one, which are locally free in codimension one). Section 4, deals
with spectral module of a rational Calogero-Moser system of
dihedral type. In this section, the authors discuss a link between
results on Cohen-Macaulay modules over an algebra planar
quasi-invariants with the theory Calogero-Moser systems. Section
5, discusses the elements of the higher-dimensional Sato theory.
Section 6, concerns the algebraic inverse scattering method in
dimension two. In this section, the authors discuss some examples
of the theory developed in the previous section. The paper is
supported with an appendix concerning the compactified Picard
variety of an affine cuspidal curve. Cohen-Macaulay modules; algebra of planar quasi-invariants; Calogero-Moser systems Singularities in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Pseudodifferential operators and other generalizations of partial differential operators, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser 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 Complex simple Lie algebras with simply laced root systems are classified by Dynkin diagrams of type \(A_{n}\), \(D_{n}\), \(E_6\), \(E_7\), and \(E_8\). Also the dual graphs of the minimal resolution of Kleinian singularities are precisely the same aforementioned Dynkin diagrams. In this work, we recall the basic definitions and some results of the theory of complex Lie algebras and of Kleinian singularities, in order to present a relation between finite dimensional complex simple Lie algebras and the Kleinian singularities, given by a theorem by Brieskorn. We also present the extension of Brieskorn's theorem to the simple elliptic singularity \(\tilde{D}_5\). Simple, semisimple, reductive (super)algebras, Singularities in algebraic geometry, Lie algebras of Lie groups, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to algebraic geometry Finite dimensional Lie algebras in 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 \(G\) be a finite subgroup of SL\((2,C)\) of type A or D. The paper under review studies the geometry and topology of the Hilbert scheme of points on \(C^2/G\), in two different flavors. First, the classical coarse Hilbert scheme Hilb\((C^2/G)\) of the singular quotient space; second, the orbifold Hilbert scheme Hilb\(([C^2/G])\), the moduli space of \(G\)-invariant finite colength subschemes of \(C^2\). The first one decomposes into components according to the number of points, and the second one decomposes according to finite dimensional representations of \(G\). Using these decompositions one defines the generating series of Euler characteristics in the two cases. The coarse generating series hence is a series in one variable, and the orbifold gerenating series is a series in \(n+1\) variables corresponding to the irreducible representations of \(G\).
The goal of the paper is to give explicit combinatorial formulas for these two generating series. The first main result is Theorem 1.4 which proves a decomposition of the orbifold Hilbert scheme to affine spaces (locally closed strata), parametrized by combinatorial objects called Young walls. Using some combinatorics and representation theory the authors show how this theorem implies Nakajima's formula for the Euler characteristic generating series of the orbifold Hilbert scheme. The second main result of the paper, Theorem 1.7, proves that the coarse generating series is a particular specialization of the orbifold one, essentially at roots of unities. Although this result was known in type A (by Dijkgraaf-Sulkowski, Toda) the direct combinatorial proof of this paper is new, as well as the result in type D. The authors formulate an analogous conjecture in type E. Hilbert scheme; singularities; Euler characteristic; generating series; Young wall Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial aspects of representation theory Euler characteristics of Hilbert schemes of points on simple 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 problem of resolving the singularities of an algebraic variety by a sequence of birational transformations has a long history. A ground- breaking step forward was made in 1940, when O. Zariski developed an ingenious method for uniformizing hypersurface singularities (in characteristic zero) by a process of successive substitutions of variables in polynomial or power series [cf. \textit{O. Zariski}, Ann. Math., II. Ser. 41, 852-896 (1940; Zbl 0025.21601)]. Then, in 1964, H. Hironaka gave an affirmative answer to the whole problem of resolving singularities in characteristic zero, essentially by generalizing Zariski's approach to a general process of successive ``permissible'' blow-up transformations, expressible in the full scheme-theoretic framework [cf. \textit{H. Hironaka}, Ann. Math., II. Ser. 79, 109-326 (1964; Zbl 0122.386)]. After that celebrated paper of Hironaka's, many attempts have been undertaken to analyse the constructiveness of his process of desingularization in various concrete situations. Among them are the papers of \textit{S. S. Abhyankar} [cf. ``Weighted expansions for canonical desingularization'', Lect. Notes Math. 910 (1982; Zbl 0479.14009)], \textit{O. E. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], \textit{E. Bierstone} and \textit{P. Milman} [J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007)] and others.
In the present paper, the author provides another approach to the presentation and uniformization of hypersurface singularities. Generalizing Zariski's method and systematizing Hironaka's ``quasi- canonical resolution procedure'' for hypersurface singularities with a normal crossing factor, he constructs a numerical sequence for any hypersurface singularity, which classifies the singularity completely and, moreover, describes a permissible resolution procedure in a very concrete and effective way. As the author points out, his systematized approach has the advantage of being applicable to the study of hypersurface singularities in positive characteristic, too [cf. the author, Publ. Res. Inst. Math. Sci. 23, No. 6, 965-973 (1987; Zbl 0657.14002)]. resolving the singularities; uniformization of hypersurface singularities [M]Moh, T. T., Canonical uniformization of hypersurface singularities of characteristic zero.Camm. Algebra 20 (1992), 3207--3251. Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Quasi-canonical uniformization of hypersurface singularities of characteristic zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an almost simple complex Lie group. The paper under review is devoted to the study of certain principal \(G\)-bundles on a curve \(C\) of genus \(g \geq 2\). Let \(\mathcal{M}_G\) be the moduli space of such semi-stable vector bundles. The Picard group of each component \(\mathcal{M}^{\bullet}_G\) of \(\mathcal{M}_G\) is infinite cyclic and let \(\mathcal{L}^{\bullet}\) be the generator. Mainly, the author considers the cases \(G=\text{ SO}_r\) and \(\text{ Sp}_{2r}\) and the rational map \(\varphi_G:\mathcal{M}_G \dashrightarrow | \mathcal{L}^{\bullet}| ^*\) given by global sections of \(\mathcal{L}^{\bullet}\). Let \(J^{g-1}\) be the component of the Picard variety of \(C\) parametrizing line bundles of degree \(g-1\). The first part of the paper is devoted to exploring the relation between \(\varphi_{\text{ SO}_r}\) and degree \(r\) theta functions on \(J^{g-1}\). Let \(\mathcal{M}_{\text{ SO}_r}^{\pm}\) be the two components of \(\mathcal{M}_{\text{ SO}_r}\) and \(| r\Theta| ^{\pm}\) the subspaces of respectively even and odd theta functions. We recall the well-known theta-map \(\theta: {M}_{\text{ SO}_r} \dashrightarrow | r\Theta| \) that associates to an orthogonal bundle \((E,q)\) the divisor
\[
\Theta_E:= \{L \in J^{g-1}| H^0(C,E\otimes L)\neq 0\}\in | r\Theta| .
\]
The main result of this part of the paper is the following: there are canonical isomorphisms
\[
| \mathcal{L}^{\pm}_{\text{ SO}_r}| ^* \overset{\sim}{\rightarrow} | r\Theta| ^{\pm}
\]
which identify \(\varphi_{\text{ SO}_r}^{\pm}: \mathcal{M}_{\text{ SO}_r}^{\pm} \dashrightarrow | \mathcal{L}^{\pm}_{\text{ SO}_r}| ^*\) with the map \(\theta^{\pm}: \mathcal{M}_{\text{ SO}_r}^{\pm} \dashrightarrow | r\Theta| ^{\pm}\) induced by \(\theta\). This is equivalent to showing that the pull-back map
\[
\theta^*: H^0(J^{g-1}, \mathcal{O}(r\Theta))^* \rightarrow H^0(\mathcal{M}_{\text{ SO}_r},\mathcal{L}_{\text{ SO}_r})
\]
is an isomorphism. The injectivity is proven by restricting to a subvariety of \(\mathcal{M}_{\text{ SO}_r}\) and the surjectivity via the Verlinde formula. In the second section the author considers the same question for the symplectic group \(\text{ Sp}_{2r}\). The main difference is that in this case the theta-map does not involve \(J^{g-1}\) but the moduli space \(\mathcal{N}\) of semistable rank 2 vector bundles on \(C\) with canonical determinant. Let \(\mathcal{L_N}\) be the determinant bundle on \(\mathcal{N}\); then for a general \((E,\sigma)\in \mathcal{M}_{\text{ Sp}_{2r}}\) the reduced subvariety
\[
\Delta_E=\{F\in \mathcal{N}| H^0(E\otimes F)\neq 0\}
\]
is a divisor on \(\mathcal{N}\) that belongs to the linear system \(| \mathcal{L_N}^r| \). This defines a map \(\mathcal{M}_{\text{ Sp}_{2r}} \dashrightarrow | \mathcal{L_N}^r| \). The author conjectures (and can show in some cases) that this map should coincide, up to a canonical isomorphism, with \(\varphi_{\text{ Sp}_{2r}}\). vector bundles on curves; determinant bundle; moduli spaces Arnaud Beauville, Orthogonal bundles on curves and theta functions, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1405 -- 1418 (English, with English and French summaries). Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Orthogonal bundles on curves and theta functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider versal deformation of a reflexive module over a rational surface singularity. Especially, for a rational double point, we determine the closure relation of the stratification of the mini-versal deformation space of a reflexive module with respect to isomorphism classes. We also determine the singularity of the closure of a minimal stratum. As a reusult, we obtain singular spaces together with their desingularizations which are similar to quiver varieties of Dynkin type. For singularities of type \(A\), we obtain the orbit stratifications and desingularizations of the nilpotent varieties and their subvarieties. The existence of an algebraic versal deformation of an isolated singularity was proved by \textit{R. Elkik} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 6, 553-603 (1973; Zbl 0327.14001)] and the same arguments can be applied to our case. In section 2, we define two kinds of deformation functors \(\text{Def}_E\) and \(\text{Def}_E'\) of a module \(E\) over a local ring \({\mathcal O}\) and give a proof of existence of a versal deformation for each of them by following \textit{M. Artin} [Invent. Math. 27, 165-189 (1974; Zbl 0317.14001)] and \textit{R. Elkik} (loc. cit). We will see in section 4 that the miniversal deformation spaces for the two functors have the same reduced parts (but different scheme structures in general) in the case where \({\mathcal O}\) is a rational surface singularity. versal deformation; rational surface singularity; desingularizations; quiver varieties; Dynkin type Ishii, A., Versal deformation of reflexive modules over rational double points, Math. Ann., 317, 2, 239-262, (2000), MR 1764236 Singularities of surfaces or higher-dimensional varieties, Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Versal deformation of reflexive modules over rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities 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 It is of interest to describe not only the topology of the Milnor fibre of a singularity \((X,x)\), but also the constellations of singularities which can appear on singular fibres in a deformation of \((X,x)\).
In this paper the author gives a simple algorithm for determining which constellations of simple singularities can appear on the fibres of a deformation of a triangle singularity. The 14 triangle singularities are a class of unimodal hypersurface singularity; under the name ``exceptional unimodal singularities'' they are the subject of the strange duality reported on by \textit{W. Ebeling} in his paper in this volume [ibid., 55-77 (1999; Zbl 0958.14021)]. The name ``triangle singularities'' arose because they can be realised as quotients of \(\mathbb{C}^2\) by certain finite groups of automorphism, constructed from triangles in the hyperbolic plane. The algorithm is in terms of the combinatorics of Dynkin diagrams (weighted graphs which encode information about the intersection form on the Milnor fibre). deformation of a triangle singularity; unimodal hypersurface singularity; Dynkin diagrams Tohsuke Urabe, Dynkin graphs, Gabrièlov graphs and triangle singularities, Singularity theory (Liverpool, 1996) London Math. Soc. Lecture Note Ser., vol. 263, Cambridge Univ. Press, Cambridge, 1999, pp. xvii -- xviii, 163 -- 174. Deformations of complex singularities; vanishing cycles, Deformations of singularities Dynkin graphs, Gabriélov graphs and triangle 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 A well known theorem due to \textit{D. O. Orlov} [J. Math. Sci., New York 84, 1361--1381 (1997; Zbl 0938.14019)] asserts that any equivalence \(F\) of derived categories of sheaves on smooth projective varieties \(X\) and \(Y\) is representable, i.e. there exists an object \(e\) of \(D(X \times Y)\) such that \(F\) is the integral functor \(\Phi^e\) associated to \(e\). In the paper under review the author extends this result to derived categories of sheaves on smooth stacks. More precisely he proves that, given stacks \(\mathcal{X}\) and \(\mathcal{Y}\) associated to normal projective varieties with quotient singularities \(X\) and \(Y\), any exact fully faithful functor \(F\) admitting a left adjoint is representable by a unique object (up to isomorphism).
The first step is the construction of a (possibly infinite) Beilinson-type resolution of the diagonal over \(X \times X\). It is pointed out here that this construction agrees with \textit{A. Canonaco}'s resolution for weighted projective spaces [J. Algebra 225, 28--46 (2000; Zbl 0963.14007)]. After proving some boundedness results, the author uses this resolution to define the representing object \(e\). The isomorphism between the integral functor \(\Phi^e\) and the original functor \(F\) is first constructed on a spanning class of locally free sheaves.
Some applications are also considered. The first one is a comparison of numerical invariants for varieties with quotient singularities having equivalent derived categories. The second one is an extension to orbifolds of \textit{A. Bondal} and \textit{D. Orlov}'s reconstruction theorem for varieties with ample (anti)canonical bundle [Compos. Math. 125, 327--344 (2001; Zbl 0994.18007)]. Fourier-Mukai transform; integral functors; representable functors; sheaves on orbifolds; weighted projective spaces Kawamata, Y., \textit{equivalences of derived categories of sheaves on smooth stacks}, Amer. J. Math., 126, 1057-1083, (2004) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Generalizations (algebraic spaces, stacks), Derived categories, triangulated categories Equivalences of derived categories of sheaves on smooth stacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \documentclass[a4paper,11pt]{article} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage[english]{babel} \usepackage{amssymb} \usepackage{amsfonts, amsmath}
\date{} \begin{document}
In this paper the author deals with some problems related to the construction of the moduli space of Higgs bundles using analytic methods. Recall that a Higgs bundle on a closed Riemann surface \(X\) (with genus \(g\geq2\)) was introduced by Hitchin in the seminal paper [Proc. London Math. Soc. (3) 55 (1987), no. 1, 59-126]. It is a pair \((\mathcal{E},\Phi)\) consisting of a holomorphic bundle \(\mathcal{E}\longrightarrow X\) and a holomorphic section \(\Phi\in H^0(\mbox{End }\mathcal{E}\otimes\mathcal{K}_X)\), where \(\mathcal{K}_X\) is the canonical bundle of \(X\). It is well known that the Kuranishi slice method can be used to construct the moduli space of semistable Higgs bundles on a closed Riemann surface as a complex space [J. Fixed Point Theory Appl. 11 (2012), no. 1, 1-41, Adv. Courses Math. CRM Barcelona, 165-219, Birkhäuser/Springer, Cham (2016)]. The aim of this paper is to give a proof in detail of this result. The author also gives a direct proof that the moduli space is locally modeled on an affine GIT (or geometric invariant theory) quotient of a quadratic cone by a complex reductive group. The paper is organized as follows. Section 1 is an introduction to the subject. In Section 2, after reviewing the deformation complex for Higgs bundles, the author introduces another useful Fredholm complex that will be used later. Section 3 is devoted to Kuranishi local models. A crucial ingredient in the Kuranishi slice method is the Kuranishi maps. Here the author deals with some questions concerning Kuranishi maps, perturbed Kuranishi maps and open embeddings into the moduli space. Section 4 deals with gluing local models. Section 5 deals with singularities in Kuranishi spaces. In this section, the author shows that Kuranishi spaces have only cone singularities. Section 6 is devoted to comparison with the algebraic construction.
\end{document} Higgs bundles; moduli space Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli Construction of the moduli space of Higgs bundles using analytic methods | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((M,J,\omega)\) be a closed connected Kähler manifold. The complex structure \(J\) is called Campana-simple if the union of all complex, proper subvarieties of \(M\) has measure zero in \(M\). (The only other case is when this union is all \(M\).) Even-dimensional tori equipped with Kähler symplectic forms and closed hyper-Kähler manifolds of maximal holonomy are examples of Campana-simple manifolds. Furthermore \(J\) is said to be approximated by Campana-simple complex structures if \(J\) can be approximated by a smooth family \(\{J_t\}_{t\in B^{2n}\subset\mathbb{C}^n}\) of complex structures with \(J_0=J\) and if there is some sequence \({t_i}\subset B^{2n}\) with limit 0 such that every \(\{J_{t_i}\}\) is Campana-simple.
The main result of the present article asserts that if \(J\) can be approximated by Campana-simple complex structures then packing \(M\) by ellipsoids is unobstructed, i.e., any finite collection of pairwise disjoint closed ellipsoids in the standard symplectic \(\mathbb{R}^{2n}\) of total volume less than the symplectic volume of \(M\) can be symplectically embedded into \(M\). The proof of this flexibility result follows the sphere-packing results of \textit{D. McDuff} and \textit{L. Polterovich} [Invent. Math. 115, No. 3, 405--429 (1994; Zbl 0833.53028)] which study the structure of the symplectic cone in the cohomology of a blow-up of \(M\). The authors here consider instead the Kähler resolution (which is a smooth manifold) of the weighted blow-ups of \(M\) (which produce orbifolds). This result is a generalization of [\textit{J. Latschev} et al., Geom. Topol. 17, No. 5, 2813--2853 (2013; Zbl 1277.57024)]. ellipsoid packing; symplectic embedding Global theory of symplectic and contact manifolds, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, \(K3\) surfaces and Enriques surfaces Unobstructed symplectic packing by ellipsoids for tori and Hyperkähler manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0527.00002.]
The paper deals with some properties of the 3-dimensional singularities with equations arising from Arnold's list, considered over an algebraically closed field of characteristic \(\neq 2,3\). These singularities are absolutely isolated, i.e. they have resolutions obtained by successively blowing up points, the ''canonical'' resolutions. These were considered first by P. J. Giblin in the topological context. - The article under review gives the algebraic description of the canonical resolution together with the intersections. Fundamental cycles (i.e. minimal negative embeddings of the exceptional loci) are computed. A method of calculating some cohomology groups on the nonreduced exceptional divisors is developed and applied to the normal bundles and the structural sheaves, thus getting vanishing theorems, applied to study the local moduli of the exceptional loci (embedded into the canonical resolution). For the \(A_ n\)-resolutions, the moduli space turns out to be smooth, and the dimension of its tangent space is the number of irreducible components isomorphic to the ruled surface \(F_ 2\). deformation; exceptional divisor; \(D_ n\); \(E_ n\); 3-dimensional singularities; vanishing theorems; local moduli of the exceptional loci; canonical resolution; \(A_ n\) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Some properties of the canonical resolutions of the 3-dimensional singularities \(A_ n\), \(D_ n\), \(E_ n\) over a field of characteristic \(\neq 2\) | 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 \textit{A. Tsuchiya, K. Ueno} and \textit{Y. Yamada} [Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)]\ have given a mathematical formulation of the Wess-Zumino-Novikov-Witten model associated to a finite-dimensional simple Lie algebra \(\mathfrak g\). The correlation functions (\(N\)-point functions) are defined as flat sections of the vector bundle of conformal blocks over the moduli space (or the configuration space) of \(N\)-pointed curves of genus \(g\). This bundle is constructed via representations of the affine Lie algebra \(\widehat{\mathfrak g}\). Roughly speaking, the vector bundle of conformal blocks is the bundle of linear forms on the representation bundle which are annihilated by a certain geometrical subalgebra.
For \(g=0\), the Riemann sphere with \(N\) marked points, these linear forms are already determined if one knows them on the ``zero degree part''. It consists of the representations of the finite-dimensional Lie algebra \(\mathfrak g\). The flat section are solutions of the Knizhnik-Zamolodchikov (KZ) equations. In this article the author generalizes this to the genus 1 case. He characterises the \(N\)-point function as vector valued functions satisfying a system of differential equations. This system contains the generalization of the KZ equation by Bernard and Felder, the KZB equation. The article concentrates on \(\mathfrak g=\mathfrak s\mathfrak l(2,\mathbb C)\).
First the author recalls the \(g=0\) situation. In particular, the relation of the \(N\)-point functions defined via conformal blocks with the traces of certain vertex operator algebras (see also Tsuchiya and Kanie) is explained.
Next the \(g=1\) case, the elliptic curve case, is examined. Here it is not true anymore that we know the linear form if we know its restriction to the zero degree part. The essential idea to overcome this difficulty, which goes back to Bernard, Eguchi, Ooguri and more recently \textit{G. Felder} and \textit{C. Wieczerkowski} [Commun. Math. Phys. 176, 133-161 (1996)] is to twist the space of conformal blocks by introducing an additional free twisting parameter. For generic values of the twisting parameter a certain restriction result is valid and a system of differential equations (containing the KZB equation) can be given which is fulfilled by the \(N\)-point functions. The author gives a complete set of differential equations (consisting of 4 different types) which determines the solution space completely (Theorem 3.3.4). By a sewing procedure the elements of the solution space can be described equivalently also as traces of vertex operators (Theorem 3.4.3). Explicit formulas are given for the 1-point case. Heat kernel type equations and theta functions appear. conformal blocks; highest weight representations; WZNW models; Knizhnik-Zamolodchikov-Bernard equation; Sugawara construction; vertex operator algebras Suzuki, T., Differential equations associated to the SU(2) WZNW model on elliptic curves, Publ. Res. Inst. Math. Sci. Kyoto, 32, 207, (1996) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Vertex operators; vertex operator algebras and related structures, Lie algebras of vector fields and related (super) algebras, Families, moduli of curves (algebraic), Elliptic curves Differential equations associated to the \(SU(2)\) WZNW model on elliptic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it.
In one sense, the question can be answered almost completely. Roughly, the space of all possible linearizations is divided into finitely many polyhedral chambers within which the quotient is constant and when a wall between two chambers is crossed, the quotient undergoes a birational transformation which, under mild conditions, is a flip in the sense of Mori. Moreover, there are sheaves of ideals on the two quotients whose blow-ups are both isomorphic to a component of the fibred product of the two quotients over the quotient on the wall. Thus the two quotients are related by a blow-up followed by a blow-down.
The ideal sheaves cannot always be described very explicitly, but there is not much more to say in complete generality. To obtain more concrete results, we require smoothness, and certain conditions on the stabilizers which, though fairly strong, still include many interesting examples. The heart of the paper is devoted to describing the birational transformations between quotients as explicitly as possible under these hypotheses. In the best case the blow-ups turn out to be just the ordinary blow-ups of certain explicit smooth subvarieties, which themselves have the structure of projective bundles.
The last three sections of the paper put this theory into practice, using it to study moduli spaces of points on the line, parabolic bundles on curves, and Bradlow pairs. An important theme is that the structure of each individual quotient is illuminated by understanding the structure of the whole family. So even if there is one especially natural linearization, the problem is still interesting. Indeed, even if the linearization is unique, useful results can be produced by enlarging the variety on which the group acts, so as to create more linearizations. I believe that this problem is essentially elementary in nature, and I have striven to solve it using a minimum of technical machinery. For example, stability and semistability are distinguished as little as possible. Moreover, transcendental methods, choosing a maximal torus, and invoking the numerical criterion are completely avoided. The only technical tool relied on heavily is the marvelous Luna slice theorem. This theorem is used, for example, to give a new, easy proof of the Bialynicki-Birula decomposition theorem. geometric invariant theory; linearization of the group action; flip; blow-up; blow-down; birational transformations between quotients; Luna slice theorem Thaddeus, Michael. \(Geometric invariant theory and flips\). J. Amer. Math. Soc. 9 (1996), no. 3, 691-723. Group actions on varieties or schemes (quotients), Geometric invariant theory, Birational geometry Geometric invariant theory and flips | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a simple algebraic group \(L\) over a characteristic zero algebraically closed field and let \(P\) be a parabolic subgroup with aura, i.e., with Abelian unipotent radical. Let \(l=\text{Lie }L\), and let \(G=l(0)\). Here the representation of \(G\) on \(l(1)\) is examined. The author gives a unified construction of a \(G\)-equivalent resolution of singularities for both the closure of an orbit of \(G\) in \(l(1)\) as well as for the closure of the conormal bundle of the orbit.
It is shown that \(L\cdot l(1)\cap(l(1)\oplus l(-1))=\bigcup_i{\mathfrak E}_i\), where \({\mathfrak E}_i\) is the closure of the orbit \({\mathcal O}_i\). Also, \(\bigcup_i{\mathfrak E}_i=\{(x,y)\mid x\in l(1),\;y\in l(-1),\;[x,y]=0\}\). The \(G\)-orbit structure of this variety is related to the double coset space \(G\backslash L/P\).
If \(U\) is the maximal unipotent subgroup of \(G\), then \(k[l(1)]\) is a free \(k[l(1)]^U\)-module, which here is equivalent to the quotient map \(\pi_{l(1)}\colon l(1)\to l(1)/ /U\) being equidimensional. To prove this, the paper provides a sufficient condition for the quotient map \(\pi_X\colon X\to X/ /U\) to be flat, where \(X\) is an affine \(G\)-variety. As a result, the irreducible representations of simple algebraic groups with aura are classified. parabolic subgroups; simple algebraic groups; Abelian unipotent radicals; resolutions of singularities; orbit structure; irreducible representations M. Brion, \textit{Invariants et covariants des groupes algébriques réductifs}, in: \textit{Théorie des Invariants et Géometrie des Variétés Quotients}, Travaux en Cours, t. 61, Paris, Hermann, 2000, pp. 83-168. Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory Parabolic subgroups with Abelian unipotent radical as a testing site for invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((J, \Theta)\) be a \(g\)-dimensional principally polarized abelian variety and \(A\) the affine ring of \(J-\Theta\), \(J = {\mathbb{C}}^{g}/\Gamma\) - the quotient of the \(g\)-dimensional vector space by some lattice and \(\Theta\) as the zero locus of a theta function \(\theta(z)\) with \(z = (z_{1},\dots, z_{g})\) being linear coordinates of \({\mathbb{C}}^{g}\). In particular, that Jacobian varieties of hyperelliptic curves of genus \(g > 2\) and non-hyperelliptic curves of genus \(g > 4\) are excluded. Analytically \(A\) is isomorphic to the ring of meromorphic functions on \(J\) which have poles only on \(\Theta\). This means that \(A\) become a module over of ring of differential operators \(D = {\mathbb{C}}[\partial_{1},\dots, \partial_{g}]\), where \(\partial_{i} = \frac{\partial}{\partial_{i}}\). It is a very curious problem to determine generators and relations of the \(D\)-module \(A\). The aim of this paper is to study these problems for \((J, \Theta)\) with \(\Theta\) being non-singular. principally polarized abelian variety; theta function; Jacobian varieties of hyperelliptic curves and non-hyperelliptic curves; module over of ring of differential operators Cho K., Nakayashiki A., Differential structure of Abelian functions, Internat. J. Math., 2008, 19(2), 145--171 Theta functions and abelian varieties, Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems Differential structure of abelian functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The vanishing lattice of an isolated surface singularity \(f:\mathbb{C}^3 \to \mathbb{C}\) can be characterised by a Coxeter-Dynkin diagram. If \(f\) deforms to \(f_t\) then the vanishing lattices of the singular points of \(f_t\) embed in the vanishing lattice of \(f\). It is the author's programme to classify possible combinations of rational double points in the fibre \(f_t\) solely in terms of operations on Coxeter-Dynkin diagrams. In this survey he describes and gives examples of two such operations, called elementary transformation and tie transformation.
The author states his theorem that a collection of rational double points is adjacent to a \(Q_{10}\), \(Z_{11}\) or \(E_{12}\) singularity if and only their Coxeter-Dynkin diagrams can be obtained by applying two operations as above on, respectively, the \(E_6\), \(E_7\) or \(E_8\) diagram.
The author notes that the same principles of operating on graphs dominate how singularities appear on global algebraic varieties. The examples he gives in the local and global case have in common a relation to Del Pezzo surfaces or the \(K3\) lattice. Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry Dynkin graphs and the singularity theory, local and global | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Modules over a bound quiver algebra \(A\) are parameterized by affine algebraic varieties \(\mathrm{mod}(A,d)\) endowed with an action of a product \(\mathrm{GL}(d)\) of general linear groups such that the orbits are in bijection with the isomorphism classes of \(A\)-modules of dimension vector \(d\). The present paper fits into the study of the following question: what are the implications of the tameness of the representation type of \(A\) on the structure of various \(\mathrm{GL}(d)\)-quotient varieties of \(\mathrm{mod}(A,d)\) constructed via invariant theory?
The paper deals with the case when \(A\) is a triangular gentle algebra (so is known to be tame, and the indecomposable \(A\)-modules are classified, but it exhibits so-called non-polynomial growth). The main result is that the field of rational invariants \(k(C)^{\mathrm{GL}(d)}\) is a purely transcendental extension of \(k\) (an algebraically closed base field of characteristic zero) whose transcendence degree equals the sum of the multiplicities of the indecomposable irreducible regular components occurring in the generic decomposition of \(C\). If \(C\) is itself an irreducible regular component and \(\theta\) is a weight such that \(C\) contains a \(\theta\)-semistable point, then the moduli space of \(\theta\)-semistable points in \(C\) is a product of projective spaces. Moreover, an explicit transcendence basis of \(k(C)^{\mathrm{GL}(d)}\) is given in terms of generalized Schofield's semi-invariants associated to generic modules. -- We note that related results on tame quasitilted algebras were obtained by \textit{G. Bobiński} [Algebra Number Theory 8, No. 6, 1521-1538 (2014; Zbl 1319.16014)]. gentle algebras; bound quiver algebras; module varieties; moduli spaces of modules; rank sequences; rational invariants; up and down graphs; tame representation type; Schofield semi-invariants Carroll, Andrew T.; Chindris, Calin, On the invariant theory for acyclic gentle algebras, Trans. Amer. Math. Soc., 367, 5, 3481-3508, (2015) Representations of associative Artinian rings, Representation type (finite, tame, wild, etc.) of associative algebras, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Representations of quivers and partially ordered sets On the invariant theory for acyclic gentle algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a survey article about the study of the links of some complex hypersurface singularities in \(\mathbb{C}^3\). We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group \(\mathrm{SU}(2)\), \(\mathrm{Nil}^3\) or \(\mathrm{Sol}^3\), respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of \(\mathrm{S}^5\). Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in \(\mathrm{S}^5\) as codimension two contact submanifolds. links of complex hypersurface singularities, contact structures Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Symplectic and contact topology in high or arbitrary dimension On the links of simple singularities, simple elliptic singularities and cusp 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 Homological mirror symmetry for Brieskorn-Pham singularities is obtained by showing that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial is equivalent to the triangular category of singularities associated to the same polynomial together with a grading by an abelian group of rank one (Theorem 1.3., cf. [\textit{K. Ueda}, ``Homological mirror symmetry and simple elliptic singularities'', \url{arXiv:math/0604361}; \textit{M. Kontsevich}, in: Proceedings of the international congress of mathematicians, ICM '94, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021); \textit{L. Katzarkov}, in: Real and complex singularities. Proceedings of the 1st Australian-Japanese workshop, University of Sydney, Australia, 2005. Hackensack, NJ: World Scientific. 176--206 (2007; Zbl 1183.14056)]).
A Brieskorn-Pham singularity is a hypersurface singularity defined by a Brieskorn-Pham polynomial
\[
f_p= x^{p_1}_1+\cdots+ x^{p_n}_n,\quad p= (p_1,\dots, p_n).
\]
The Milnor lattice of a Brieskorn-Pham singularity is the tensor product of the Milnor lattices for \(f_{p_i}= x^{p_i}\). The Milnor lattice for \(f_p= x^p\) is a free abelian group generated by \(C_i\), \(i= 1,\dots,p- 1\) with the intersection form
\[
(C_i,C_j)= \begin{cases} 2\quad &i= j,\\ -1\quad &|i-j|= 1,\\ 0\quad &\text{otherwise},\end{cases}
\]
(cf. [\textit{M. Sebastiani} and \textit{R. Thom}, Invent. Math. 13, 90--96 (1971; Zbl 0233.32025)]). Categorification of this description is the main result of this paper. For this purpose, the differential graded category \({\mathfrak A}_p\), whose objects are \((C_1,\dots, C_p)\) and morphisms are
\[
\text{hom}(C_i, C_j)= \begin{cases} \mathbb{C}\cdot\text{id}_{C_i}\quad &\text{if }i= j,\\ \mathbb{C}[-1]\quad &\text{if }i=j-1,\\ 0\quad &\text{otherwise},\end{cases}
\]
is introduced. Then after a precise study of distinguished basis of vanishing cycles of the Lefschetz fibration \(W_p: \mathbb {C}^n\to\mathbb C\), with
\[
W_p(x_1,\dots, x_n)= f_p(x_1,\dots, x_n)+ (\text{lower order terms}),
\]
a quasi equivalence of \(A_\infty\)-categories
\[
{\mathfrak F}{\mathfrak u}{\mathfrak k}W_ p\cong{\mathfrak A}_{p_1-1}\otimes\cdots \otimes{\mathfrak A}_{p_n-1},
\]
where \({\mathfrak F}{\mathfrak u}{\mathfrak k}W_p\) is the Fukaya category of \(W_p:\mathbb C^n\to\mathbb C\) is proved by induction on \(n\) adopting the symplectic Picard-Lefschetz theory of [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001), Theorem 1.1]. Proof is given in \S2 and \S3.
In \S4, along the lines of [Ueda, loc. cit.], Theorem 5, triangulated category of a Brieskorn-Pham singularity \(D^{\text{gr}}_{\text{Sg}}(A_p)\) [\textit{D. O. Orlov}, Proc. Steklov Inst. Math. 246, 227--248 (2004); translation from Tr. Mat. Inst. Steklova 246, 240--262 (2004; Zbl 1101.81093)] is shown to be equivalent to the bounded derived category \(D^b({\mathfrak A}_{p_1-1}\otimes\cdots\otimes{\mathfrak A}_{p_n-1})\) (Theorem 1.2). Precisely, the authors use the stabilized derived category [\textit{R. O. Buchweitz},``Maximal Cohen-Macaulay modules and the Tate-cohomology over Gorenstein rings'' (1986), \url{https://tspace.library.utoronto.ca/handle/1807/16682}], which is the same as Orlov's triangular category. By Theorems 1.1 and 1.2, we have the equivalence
\[
D^b{\mathfrak F}{\mathfrak u}{\mathfrak k}W_p\cong D^{\text{gr}}_{\text{Sg}}(A_p),
\]
that is, we obtain homological mirror symmetry for
Brieskorn-Pham singularities (Theorem 1.3; for \(n=2\), this theorem is proved in [Ueda, loc. cit.]).
The authors also remark, adopting the Calabi-Yau/Landau-Ginzburg correspondence [\textit{D. Orlov}, Prog. Math. 270, 503--531 (2009; Zbl 1200.18007)], that if \({1\over p_1}+\cdots+{1\over p_n}= 1\), then
\[
D^{\text{gr}}_{\text{Sg}}(A_p)\cong D^b\text{coh\,}Y_p,
\]
where \(D^b\text{coh\,}Y_p\) is the derived category of coherent sheaves on a stack \(Y_p\) (Theorem 1.4). homological mirror symmetry; Brieskorn-Pham singularity; triangular category; derived category; coherent sheaves Futaki, M; Ueda, K, Homological mirror symmetry for Brieskorn-pham singularities, Sel. Math. (N.S.), 17, 435-452, (2011) Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Cohen-Macaulay modules in associative algebras, Mirror symmetry (algebro-geometric aspects) Homological mirror symmetry for Brieskorn-Pham 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 In the present doctoral dissertation, the author develops a theory of generalized tori from a purely algebraic viewpoint. Starting from the classical and well-established theory of complex tori, she replaces the standard chain of field extensions \(\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}\) by an arbitrary chain of field extensions \(K\subset F\subset E\) satisfying \([E:F]= d\in\mathbb{N}\), defines (generalized) tori as pairs \((V, L)\) consisting of a finite-dimensional \(E\)-vector space \(V\) and a lattice \(L\) (i.e., a \(K\)-vector space \(L\subset V\) with \(F\otimes_K L= v\) and \(\dim_KL= d\cdot\dim_EV\)) in \(V\), and studies these objects under various algebraic and categorical aspects. With respect to the obvious definition of morphisms, these (generalized) tori form an abelian category, and it is shown that many classical constructions for complex tori (e.g., period matrices, complex multiplication, etc.) can be generalized to this more general abelian category. Also, the duality functor for complex tori can be generalized in a suitable manner, within this algebraic framework, making the category of (generalized) tori into a Hermitian category with sesquilinear forms on its objects.
The main task of the thesis under review is to investigate all these generalized constructions more thoroughly, to develop a coherent theory as far as possible, and to study important special cases in greater detail. The outcome is a comparatively voluminous treatise of remarkable comprehensiveness, in which many interesting new results are presented.
The entire work is divided into five chapters, each of which is subdivided into several sections. Chapter 1 provides the necessary categorical preliminaries, with special emphasis put on constructions in Hermitian categories (duality, quadratic forms, Witt groups, extensions of objects, and representation types). Chapter 2 introduces the category of (generalized) tori, including representations of homomorphisms of tori, extensions of tori, representation types, and period matrices. Chapter 3 deals particularly with endomorphisms of tori, culminating in the classification of both simple tori without CM (the so-called RM-tori) and full CM-tori. Chapter 4 is devoted to tori in the special case of \([E: F]=[F: K]= 2\), which is closest to the classical case of complex tori. The structure of the corresponding tori is completely described, and the full classification of tori of dimension one or two, including CM-tori and RM-tori, is certainly one of the highlights of the entire treatise. In Chapter 5, the author turns to duality of (generalized) tori and the allied quadratic theory. This includes the definition of a suitable duality functor on the category of tori, the description of a dual torus by a period matrix, the generalization of the Riemann period relations to generalized tori, and the construction of sesquilinear forms on one-dimensional tori. The latter leads to an explicit description of self-dual one-dimensional tori.
In the sequel, the author investigates the operation of duality on the groups \(\text{Ext}(T^{\text{dual}},T)\) for tori \(T\) extensions of tori, Grothendieck groups and Witt groups, the behavior of forms under field extensions, the representation type of the abelian category of (generalized) tori, and the differences between self-dual CM-tori and self-dual RM-tori. No doubt, this concluding chapter is the most important part of the whole work, as it is the most advanced one, containing a wealth of new and enlightening results. In two appendices, at the end of the thesis, the author explains some auxiliary constructions used in the course of the main text.
Altogether, this thesis, thematically inspired by Prof. W. Scharlau (University of Münster, Germany), represents a highly useful contribution to the algebraic theory of tori, in general, and it stands out by its high degree of completeness, clarity, rigor, and comprehensiveness. Numerous remarks and hints, in the course of the text, point to further problems of research in this area, thereby enforcing both the value and the utmost cultured style of presentation of this fine piece of work. field extensions; lattices; complex multiplication; duality theory; Witt groups Algebraic theory of abelian varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Complex multiplication and abelian varieties, Abelian varieties of dimension \(> 1\), Complex multiplication and moduli of abelian varieties, Witt vectors and related rings A generalization of complex tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This book originates from a series of lectures and seminar talks for graduate students at various universities in Japan. It is a research monograph describing research by the author on finite branched coverings of projective complex manifolds in connection with the theory of algebraic functions of several complex variables. The author presents a theory generalizing earlier work in one complex variable, in particular work of \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)].
The book has three chapters: 1. ``Branched coverings of complex manifolds'', 2. ``Fields of algebraic functions'', 3. ``Weil-Tōyama theory''.
The main result in chapter 1 is a theorem providing a suitable sufficient condition for the existence of a finite Galois covering of a projective manifold, branched over a given divisor. In the case of compact Riemann surfaces, this originates in a problem posed by Fenchel. In addition, chapter 1 contains many well chosen examples of finite branched coverings. -- Chapter 2 studies finite abelian coverings using the theory of currents developed by \textit{G. de Rham} and \textit{K. Kodaira} in ``Harmonic integrals'', Lecture Notes Inst. Adv. Study (Princeton 1950). There are results providing necessary and sufficient conditions for the existence of a finite abelian covering onto a projective complex manifold branched over a given divisor. Furthermore, the set of all isomorphism classes of finite abelian branched coverings of a projective complex manifold is described using the notion of rational divisor classes. The results can be considered as higher dimensional generalizations of results of Iwasawa from 1952. -- Chapter 3 studies similar questions for finite Galois coverings, but according to the author the results obtained are not completely satisfactory. The book should be a good source for inspiration to further work. finite branched coverings of projective complex manifolds; algebraic functions; finite Galois coverings M. NAMBA, Branched coverings - algebraic functions, Pitman Research Notes in Mathematics Series 161. Zbl0706.14017 MR933557 Coverings of curves, fundamental group, Coverings in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Low-dimensional topology of special (e.g., branched) coverings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Compact Riemann surfaces and uniformization Branched coverings and algebraic functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a summary of the author's forthcoming paper ``Deformation of reflexive modules over rational double points''. Let \(\text{Def}_E\) be the semi-universal deformation space of a reflexive module \(E\) over a rational double point. \(\text{Def}_E\) has a stratification whose strata correspond to some well described isomorphism classes of reflexive modules, the stratification graph being given for some examples. A versal family of a reflexive module over an \(A_n\)-singularity is given using Faltings' construction. The author believes that the reduced part of \(\text{Def}_E\) is isomorphic with the quiver variety of a Dynkin diagram [see \textit{H. Nakajima}, Duke Math. 76, No. 2, 365-416 (1994; Zbl 0826.17026)]. rational double points; semi-universal deformation; reflexive modules; stratification graph; quiver variety Deformations of singularities, Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Semi-universal family of reflexive modules over a rational double point 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 The authors start with a connected complex semi-simple Lie group \(G\) and consider the moduli stack \({\mathcal M}\) of algebraic principal \(G\)-bundle on a smooth complex projective curve \(X\) of genus at least two. The main result is a description of the vector space of jets of regular functions at every point of \({\mathcal M}\). This description is not in terms of the moduli stack but involves certain sheaves on a canonical resolution of cartesian powers of \(X\) such that the preimage of the diagonal divisor has normal crossings.
The first part of the paper is devoted to the combinatorial description of this resolution. The authors use results from and their methods are related to the papers of \textit{H. Esnault}, \textit{V. V. Shehtman}, and \textit{E. Viehweg}, ``Cohomology of local systems on the complement of hyperplanes'' (preprint 1991) and \textit{N. V. Shekhtman} and \textit{A. N. Varchenko} [Invent. Math. 106, No. 1, 139-194 (1991; Zbl 0754.17024)].
The paper under review contains almost no proves. Details and applications are announced to appear elsewhere. moduli stack of algebraic principal bundles on a smooth complex projective curve; normal crossings; N. V. Shekhtman; A. N. Varchenko A. Beilinson, V. Ginzburg, Infinitesimal structure on moduli space of \(G\)-bundles, Intl. Math. Res. Notices 4 (1992), 63-74. Complex-analytic moduli problems, Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Semisimple Lie groups and their representations, Fine and coarse moduli spaces, Divisors, linear systems, invertible sheaves, Formal neighborhoods in algebraic geometry Infinitesimal structure of moduli spaces of \(G\)-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 (X,x) be a germ of an analytic quotient surface singularity. Let \(\pi: \tilde X\to X\) be the minimal desingularization of X with exceptional system \(\{E_ i\}_{1\leq i\leq r}\) and fundamental cycle \(Z=\sum ^{r}_{i=1}r_ iE_ i.\quad For\) each reflexive module M on X the sheaf \~M\(=\pi ^ *M/torsion\quad is\) locally free on \~X and the first Chern class is represented by a divisor which is transversal to the exceptional set E of \(\pi\). - The subject of this article is a generalization of the theorem of Artin-Verdier and the multiplication formula of Esnault-Knörrer on the McKay correspondence for rational double points to the case of an arbitrary quotient surface singularity.
Main result: (i) For \(E_ i\) there is exactly one isomorphism class of an indecomposable reflexive module \(M_ i\) on (X,x) with \(c_ 1(\tilde M_ i).E_ j=\delta _{ij}\), \(1\leq i,j\leq r\), and \(R^ 1\pi _ *(\tilde M_ i^{\vee})=0\) for the dual sheaf \~M\({}_ i^{\vee}\) of \~M\({}_ i\). The rank of \(M_ i\) is \(r_ i\). - \((ii)\quad If\) \(0\to \tau (M)\to N_ M\to M\to 0\) is an almost split exact sequence then \(c_ 1(\tilde N_ M)=c_ 1(\tau (M)^{\sim})+c_ 1(\tilde M)\) if \(M\neq M_ 1,...,M_ r\), and \(c_ 1(\tilde N_ M)=c_ 1(\tau (M)^{\sim})+c_ 1(\tilde M)+E_ i\) if \(M=M_ i\). The fundamental sequence \(0\to \omega _ X\to N_{{\mathcal O}_ X}\to {\mathcal O}_ X\to {\mathbb{C}}\to 0\) induces \(c_ 1(\tilde N_{{\mathcal O}_ X})=c_ 1(\omega _{\tilde X})-Z\). first Chern class; McKay correspondence for rational double points; quotient surface singularity; reflexive module Jürgen Wunram, Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), no. 4, 583 -- 598. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Characteristic classes and numbers in differential topology Reflexive modules on 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 Given a simple, simply connected complex Lie group \(G\), the Verlinde formula is a combinatorial function \(V^ G: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\) associated with \(G\). The expressions \(V^ G (k,g)\) were first introduced by E. Verlinde in the context of conformal quantum field theory [cf. \textit{E. Verlinde}, Nucl. Physics B, Field Theory and Statistical Systems 300, No. 3, 360-376 (1988)]. Their significance in algebraic geometry stems from the (originally conjectural) fact that they are related to the Hilbert functions of moduli spaces of semi-stable vector bundles over compact Riemann surfaces of genus \(g\). The corresponding relation between those Hilbert functions of moduli spaces and the ``Verlinde numbers'' \(V^ G (k,g)\) used to be called the ``Verlinde conjecture'' for the respective moduli spaces, and its verification, mainly in the cases of \(G= \text{SU} (n)\) and \(G= \text{SL} (n)\), has been a central topic of research in the last five years.
More precisely, the Verlinde conjecture, in most general setting, can be roughly stated as follows. Let \(C\) be a compact Riemann surface of genus \(g\), and let \({\mathcal M}_ C^ G\) be the (factually existing) moduli space of principal G-bundles over \(C\). Then there is an ample line bundle \({\mathcal L}\) over \({\mathcal M}^ G_ C\), a so-called generalized theta bundle, such that \(\dim_ \mathbb{C} H^ 0 ({\mathcal M}_ C^ G, {\mathcal L}^{\otimes k})= V^ G (k_ h, g)\) for any \(k\in \mathbb{Z}\), where \(h\) denotes the dual Coxeter number of the group \(G\).
In this form, and for \(G= \text{SL} (n)\), the Verlinde conjecture has recently been verified by \textit{G. Faltings} [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]; \textit{A. Beauville} and \textit{Y. Laszlo} [Commun. Math. Phys. 164, No. 2, 385-419 (1994; Zbl 0815.14015)]; \textit{S. Kumar}, \textit{M. S. Narasimhan} and \textit{A. Ramanathan} [Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)]; \textit{A. Bertram} and the author [Topology 32, No. 3, 599-609 (1993; Zbl 0798.14004)], and others in less general cases. -- In the present paper under review, the author discusses the origin and properties of the Verlinde formulas \(V^ G (k,g)\) and, in addition, their connection with the famous Witten conjectures in the intersection theory of moduli spaces of algebraic curves. After a brief overview of the structure of topological field theories, fusion algebras and Verlinde's formal calculus, the explicit behavior of \(V^ G (k,g)\) as a function of \(k\) is studied, again in the special case of \(\text{SL} (n)\). The main result is a residue formula for \(V^{\text{SL} (n)} (k,g)\) yielding the deep fact that these numbers are integer-valued polynomials in \(k\). The author then shows how this residue formula for \(V^{\text{SL} (n)} (k,g)\) can be related, via the Grothendieck- Hirzebruch-Riemann-Roch theorem, to Witten's conjectures on the intersection numbers of moduli spaces of curves [cf. \textit{E. Witten}, J. Geom. Phys. 9, No. 4, 303-368 (1992; Zbl 0768.53042)]. Assuming the validity of Witten's formulas, a quick proof of the Verlinde conjecture (as stated above) is given.
This very instructive approach puts the Verlinde formulas into a wider context, and provides some more evidence of Witten's conjectures from this now well-established complex of results. vector bundles; conformal quantum field theory; Verlinde formula; Hilbert functions of moduli spaces of semi-stable vector bundles; compact Riemann surface; generalized theta bundle; Witten conjecture; intersection theory of moduli spaces of algebraic curves; topological field theories; fusion algebras Szenes, A.: The combinatorics of the Verlinde formulas In: Vector Bundles in Algebraic Geometry, Hitchin, N.J., et al., (eds.), Cambridge University Press, 1995 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Theta functions and abelian varieties, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The combinatorics of the Verlinde formulas | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Lie algebra \({\mathfrak g}={\mathfrak{sl}}(z, \mathbb C)\oplus {\mathfrak {sl}}(z, \mathbb C)\) is used to construct the semi-universal deformation of simple elliptic singularities of type \(\widetilde{D}_5\). The nilpotent variety of the Lie algebra above is \(\mathcal N=\{\binom{a\;b}{c-a}\;| \;a^2+bc=0\}\times \{\binom{d\;e}{f-d}\;| \;d^2+ef=0\}\). The surface singularity \((\mathcal N\cap S, 0)\) is simple elliptic of type \(\widetilde{D}_5\) for a generic slice \(S\) at \(0\). If the slice is defined by \(c=d+e\) and \(f=a+b\) the construction is as follows. Let \(\mathfrak{h}=\{\binom{a\;0}{0-a}\}\oplus\{\binom{d\;0}{0-d}\}\) and consider the adjoint quotient \(\chi:\mathfrak{g}\to\mathfrak{h}/W\;,\;\chi(\binom{a\;b}{c-a}, \binom{d\;e}{f-e})=(-a^2-bc, -d^2-ef)\). \(W\cong\mathbb Z/2\mathbb Z \oplus\mathbb Z/2\mathbb Z\) is the Weyl group of \(\mathfrak{g}\). For \((\alpha, \beta)\in \mathbb C^2\) let \(f_{(\alpha,\beta)}(\binom{a\;b}{c-a},\;\binom{d\;e}{f-e}) = (-a^2-bc-\alpha e, -d^2-ef-\beta b)\) and the slice \(S_{(\gamma, \delta, \varepsilon)}\) be defined by \(c=d+e+\gamma, f=a+b+\delta e+\varepsilon\). Let \(S:=\mathbb C^2\times \mathbb C^3\times \mathfrak{h}/W\) and \(X=\{(A, \alpha, \beta, \gamma, \delta, \varepsilon, \lambda, \mu)\in \mathfrak{g}\times S\;| \;f_{(\alpha,\beta)}(A)=(\lambda, \mu), A\in S_{(\gamma, \delta, \varepsilon)}\}\). Let \(o:=(0,0,0,0,0,0,0)\in S\) and \(q:=(0,o)\in X\) then the morphism \((X,q)\to (S,o)\) is a semi-universal deformation of the simple elliptic singularity defined by the slice \(c=d+e, f=a+b\). simple elliptic singularity; semi-universal deformation Deformations of singularities, Deformations of complex singularities; vanishing cycles Semi-universal deformation spaces of some simple elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this survey the author gives an overview and an explanation of the techniques used in [\textit{P. Cascini} and \textit{V. Lazić}, Duke Math. J. 161, No. 12, 2415--2467 (2012; Zbl 1261.14007)] and [\textit{V. Lazić}, ``Adjoint rings are finitely generated'', \url{arXiv:0905.2707}, 2009] for proving the finite generation of the canonical ring for varieties of general type with at most Kawamata log terminal singularities. The proofs mostly rely on induction on dimension and Kawamata-Viehweg vanishing theorem, all connected with the convex geometry of the Néron-Severi vector space. The work aims to explain the difference in the approach with respect to the previous one in [\textit{C. Birkar} et al., J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)] , trying to lighten the topic, making it less technical and more accessible. Various applications related to classical birational geometry are given. Minimal model Program; Kawamata-Viehweg Vanishing Theorem; finite generation; Canonical Ring; Mori Dream Spaces Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)), Special varieties Around and beyond the canonical class | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(G\) is a finite group of birational isomorphisms of the smooth projective variety \(Y\), then a resolution of the indeterminacy of the pair \((Y,G)\) is any birational map \(\phi : X \rightarrow Y\) with \(X\) a smooth projective variety, such that for any \(\tau \in G\) the birational map \(\phi^{-1} \tau \phi : X \rightarrow X\) is biregular, i.e. an automorphism of \(X\). Such resolutions always exist (see section 1), and the problem is to study them more closely in some particular occurences.
In this paper is studied in some detail the case when \(\dim \;Y = 2\), i.e. when \(Y\) is a smooth projective surface. In section 2 is established a correspondence between birational morphisms of smooth surfaces and finite closed subsets of algebraic valuations. These results are used in section 3 to prove the existence of a minimal resolution of a pair \((Y,G)\) in the case when \(\dim Y = 2\).
In the last section 4 is introduced a birational invariant of each subgroup of prime order of the group \(\text{Bir}(Y)\) of birational isomorphisms of a projective surface \(Y\). This invariant is used to determine when two subgroups of prime order of \(\text{Bir}({\mathbb P}^2)\) are conjugate. As mentioned in the paper, a close study of resolutions of indeterminacy of pairs in the case \(\dim\;Y = 3\) has been undertaken by \textit{I. A. Cheltsov} [Math. Notes 76, No.~2, 264--275 (2004; Zbl 1059.14019)]. birational isomorphism; resolution of indeterminacy de Fernex (T.) and Ein (L.).- Resolution of indeterminacy of pairs. Algebraic geometry, p. 165-177, de Gruyter, Berlin (2002). Zbl1098.14008 MR1954063 Rational and birational maps, Birational automorphisms, Cremona group and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) Resolution of indeterminacy of pairs. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma\) be a torsion-free subgroup of finite index in the elliptic modular group \(\Gamma (1):=SL_ 2({\mathbb{Z}})\) and E(\(\Gamma\)) the associated elliptic surface, which is fibered over the Riemann surface \(X(\Gamma):=\overline{\Gamma \setminus H}\). For any \(\Gamma \triangleleft \Delta \subset \Gamma (1)\) the quotient group \({\mathcal G}=\Delta /\Gamma\) acts on E(\(\Gamma\)), in particular there is an intrinsic minimal (geometric) model S(\(\Gamma\)) of the quotient space \(E(\Gamma)/<-1>\) which is by a result of \textit{D. Burns} [in Algebraic geometry, Proc. Conf., Ann. Arbor/Mich. 1981, Lect. Notes Math. 1008, 1-29 (1983; Zbl 0543.14022)] nothing but the tangential ruled surface of any projective embedding of X(\(\Gamma\)). Therefore constructing E(\(\Gamma\)) from X(\(\Gamma\)) and \({\mathcal G}\) amounts to study the branch locus of E(\(\Gamma\))\(\to S(\Gamma)\), which was done by Burns (loc. cit.):
The image of the branch curve in S(\(\Gamma\)) is the unique \({\mathcal G}\)- invariant divisor in its linear equivalence class. Adapting the methods developed there this result is extended in a two-fold way under a mild condition on \(\Gamma \triangleleft \Delta\). The divisor class D of the nontrivial component of the branch locus is easily calculated and characterized by an isomorphism between \(Gr(H^ 0(S,{\mathcal O}_ S(D))^{{\mathcal G}}\) and \(H^ 0(X(\Delta),\Xi_{\Delta})\), where \(\Xi_{\Delta}\) is a rank-4 vector bundle on the Riemann surface X(\(\Delta\)). Moreover for normal \(\Delta\) the dimension of \(H^ 0(X(\Delta),\Xi_{\Delta})^{\Gamma (1)/\Delta}\) equals 1. elliptic modular surface; quotient space; branch curve; divisor class Elliptic surfaces, elliptic or Calabi-Yau fibrations, Divisors, linear systems, invertible sheaves, Modular and Shimura varieties, Group actions on varieties or schemes (quotients), Unimodular groups, congruence subgroups (group-theoretic aspects) On the divisor of involutions in an elliptic modular 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 The author provides and discusses an analogue of a flasque resolution of a torus for a connected reductive linear algebraic group.
Namely, by a flasque resolution of a group \(G\) one calls a central extension of algebraic groups \(1 \to S \to H\to G\to 1\), where \(H\) is a quasi-trivial group and \(S\) is a flasque torus.
Here quasi-trivial means an extension of a quasi-trivial torus by a simply connected semisimple group.
The paper is organized as follows:
In sections 0-4 the author recalls basic properties of linear algebraic groups, establishes the existence and properties of such flasque and coflasque resolutions. In section 5 he discusses relations between resolutions and universal torsors of smooth compactifications of linear groups.
In section 6 the author using the language of resolutions introduces and studies an algebraic fundamental group \(\pi_1(G)\) of a connected linear group \(G\).
In section 7 he provides two formulas for the Brauer group of a smooth compactification of \(G\): one in terms of flasque resolutions of \(G\) and another in terms of the algebraic fundamental group of \(G\).
In the last two sections he applies flasque resolutions to obtain information on \(R\)-equivalence classes of \(G(k)\). flasque resolution; linear algebraic group; universal torsor; algebraic fundamental group Colliot-Thélène, J.-L., Resolutions flasques des groupes linéaires connexes, J. Reine Angew. Math., 618, 77-133, (2008) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Galois cohomology of linear algebraic groups Flasque resolutions of linear connected 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 \(w : \mathbb C^N \to \mathbb C\) be a quasi-homogeneous polynomial whose total degree \(d\) is equal to the sum of the weights of each variable. Let \(G \le SL_N (\mathbb C)\) be a diagonal subgroup of automorphisms of \(w\). In the article [\textit{Y.-P. Lee} et al., Ann. Sci. Éc. Norm. Supér. (4) 49, No. 6, 1403--1443 (2016; Zbl 1360.14133)] the commutativity of a diagram called LG/CY square is proven. The top row vertices of this square are \(GWT_0([\mathbb C^N /G])\), the genus zero Gromov-Witten theory of \([\mathbb C^N /G]\), and \(GWT_0(\mathrm{tot}(\mathcal O_{\mathbb P(G)}(-d)))\), the genus zero Gromov-Witten theory of a partial crepant resolution of \([\mathbb C^N /G]\). The bottom row vertices are \(FJRW_0(w, G)\), the genus zero FJRW theory of the Landau-Ginzburg model given by the pair \((w, G)\), and \(GWT_0(\mathcal Z)\), the genus zero Gromov-Witten theory of a hypersurface \(\mathcal Z\) defined as the vanishing locus of \(w\) in an appropriate finite quotient of weighted projective space \(\mathbb P(G)\). The arrows in the square are the crepant transformation conjecture, quantum Serre duality, the LG/CY correspondence, and the local GW/FJRW correspondence.
The goal of the paper under review is to relate each of the above correspondences to an integral transform between appropriate derived categories, i.e. to lift the LG/CY square to the derived category to obtain a cube of relations. It is known that the crepant transformation conjecture (the top horizontal arrow of the LG/CY square) is compatible with a natural Fourier-Mukai transform. A similar result is known for the bottom horizontal arrow at least when \(G\) is cyclic.
The paper under review shows that there are natural derived functors corresponding to both of the vertical arrows of LG/CY square as well, after restricting to subcategories of \(D([\mathbb C^N /G])\) and \(D(\mathrm{tot}(\mathcal O(-d)))\) with proper support. This requires a reformulation of both the local GW/FJRW correspondence and quantum Serre duality in terms of narrow quantum \(D\)-modules, which turns out to be a more natural way of describing these correspondences.
Even though the corresponding square of derived functors does not commute in general the paper shows that the induced maps on \(K\)-theory commute. Gromov-Witten theory; FJRW theory; crepant resolution conjecture; LG/CY correspondence; wall crossing; Fourier-Mukai Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Integral transforms and quantum correspondences | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 reprove the following theorem. This theorem is a weak version of the equivariant case of Hironaka's famous theorem on resolution of singularities. It was announced by Hironaka, but a complete proof wasn't easily accessible for a long time.
Theorem. Let \(X\) be a projective variety of finite type over \(k\), and let \(Z\subset X\) be a proper closed subset. Let \(G\subset \Aut_k(Z \subset X)\) be a finite group. Then there is a \(G\)-equivariant modification \(r:X_1\to X\) such that \(X_1\) is a nonsingular projective variety, and \(r^{-1} (Z_{\text{red}})\) is a \(G\)-strict divisor with normal crossings.
The proof takes a completely different approach. It uses two ingredients: First, we assume that we know the existence of resolution of singularities without group actions; second, we use equivariant toroidal resolution of singularities. The authors devote section 2 to the proof of the following theorem.
Theorem. Let \({\mathcal U}\subset X\) be a strict toroidal embedding, and \(G\subset \Aut (U\subset X)\) be a finite group acting toroidally. Then there is a \(G\)-equivariant toroidal ideal sheaf \({\mathcal I}\) such that the normalized blow-up of \(X\) along \({\mathcal I}\) is a nonsingular \(G\)-strict toroidal embedding. equivariant modification; resolution of singularities; toroidal embedding D. Abramovich and J. Wang, Equivariant resolution of singularities in characteristic \(0\) , Math. Res. Lett. 4 (1997), 427-433. Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Equivariant resolution of singularities in characteristic 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 The paper under review studies the smoothness and singularities of two important toroidal compactifications of \(\mathcal A_g\), the moduli of principally polarized abelian varieties of dimension \(g\). A toroidal compactification of \(\mathcal A_g\) is a compactification constructed from an admissible rational polyhedral decomposition of \(\mathcal C_g\), the cone of positive-definite real quadratic forms over \(\mathbb R^g\). More precisely, the support of the decomposition is the rational closure \(\overline{\mathcal C}_g^{\mathrm{rc}}\) consisting of semi-positive-definite real quadratic forms with null spaces defined over \(\mathbb Q\). The two concerned compactifications are \(\mathcal A_g^{\mathrm{Perf}}\), the compactification associated to the perferct cone decomposition (also called the first Voronoi decomposition), and \(\mathcal A_g^{\mathrm{Vor}}\), the compactification associated to the second Voronoi decomposition. The main results are:
(Corollary 1.2) \(\mathcal A_g^{\mathrm{Perf}}\) is smooth for \(g\leqslant 3\) and the codimension of the singular locus and the non-simplicial locus are both \(10\) for \(g\geqslant 4\).
(Theorem 1.4) \(\mathcal A_g^{\mathrm{Vor}}\) is smooth for \(g\leqslant 4\) and the codimension of the non-simplicial locus is \(3\) for \(g\geqslant 5\).
Here what we mean by singularities are those ``essential'' singularities that can not be resolved simply by passing to the level covers. In other words, the singularities are the same with the toric singularities of the affine toric varieties for each rational polyhedral cone \(\sigma\subset \overline{\mathcal C}_g^{\mathrm{rc}}\) in the decompositions, and are independent on the actions of the arithmetic groups on \(\overline{\mathcal C}_g^{\mathrm{rc}}\). If we identify a quadratic form over \(\mathbb R^g\) with its associated symmetric matrix, and regard \(\mathcal C_g\) as a cone in the vector space of symmetric matrices \(\mathrm{Sym}^2(\mathbb R^g)\), then being smooth is equivalent to the corresponding monoid \(\sigma\cap \mathrm{Sym}^2(\mathbb Z^g)\) being generated by a subset of some \(\mathbb Z\)-basis of \(\mathrm{Sym}^2(\mathbb Z^g)\) (Such cones will be called \textit{basic}.), and being simplicial (meaning the toric variety has abelian finite quotient singularities) is equivalent to the primitive vectors of the rays of \(\sigma\) being linearly independent over \(\mathbb R\) (Such cones will be called \textit{simplicial}.). The main results thus follow from the following statements about the cone decompositions,
(Theorem 1.1) Every cone of dimension at most \(9\) in the perfect cone decomposition is basic. Moreover, with the exception of the cone of the root lattice lattice \(D_4\), every cone in the perfect cone decomposition of dimension at most \(10\) is simplicial.
(Theorem 1.4) For \(g\leqslant 4\), every cone in the second Voronoi decomposition is basic. For \(g\geqslant 5\), there are non-simplicial cones of dimension \(3\) in the second Voronoi decomposition.
Most of the paper is on the proof of Theorem 1.1. The proof can be understood without any background in the toroidal compactifications or the moduli of abelian varieties. The basic idea is to classify the cones of low dimensions (\(\leqslant 9\)) and check whether they are basic or simplicial by computers. However, when the dimension of the cone is \(10\), and the quadratic forms are over \(\mathbb R^g\) for \(g\geqslant 8\), a complete classification is not readily available. Then it is proved directly that the cones in those cases are always simplicial (Lemma 3.1). The proof is thus a combination of mathematical reasoning and computer-aided computations. Furthermore, a complete list of all \(9\)-dimensional perfect cones, which should be interesting on its own, is obtained during the proof and is provided in a separate electronic form. abelian varieties; Siegel modular varieties; toroidal compactifications; Voronoi reduction theories; the perfect cone decomposition M. Dutour Sikirić, K. Hulek and A. Schürmann, Smoothness and singularities of the perfect form compactification of {{\mathcal{A}}_{g}}, Algebraic Geom. 2 (2015), no. 5, 642-653. Algebraic moduli of abelian varieties, classification, Stacks and moduli problems, Quadratic forms (reduction theory, extreme forms, etc.), Toric varieties, Newton polyhedra, Okounkov bodies, Modular and Shimura varieties Smoothness and singularities of the perfect form and the second Voronoi compactification of \(\mathcal A_g\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f: X\to \Delta\) be a proper projective morphism with \(X\) smooth and \(\Delta\) an open disk. Let \(D\) be a divisor on \(X\) and set \(U=X\setminus D\). Assume that \(X_0\cup D\) is a divisor with simple normal crossings. This paper centers around the question whether the number \(\nu_{f_U,\lambda}^j\) of Jordan blocks of largest possible size (\(j+1\) if \(0\leq j \leq n\)) for the eigenvalue \(\lambda\) of the monodromy on \(H^j(U_t,\mathbb C)\) can be computed as \(\dim H^j C^\bullet_{f,\lambda}\), where \(C^\bullet_{f,\lambda}\) is a complex determined by the combinatorics of the intersection lattice consisting of the irreducible components of \(X_0\) and their intersections. For \(\lambda=1\) this is true by the theory of limit mixed Hodge structures. An explicit example at the end of the paper shows that this is in general not the case, even when \(f\) is obtained by a desingularisation of a good compactification of an isolated hypersurface singularity. The \(\nu_{f_U,\lambda}^j\) depend on the positions of the singular points in the embedded resolution. Therefore there are no simple combinatorial formulas. The authors construct a complex \(B^\bullet_{f,\lambda}\) with \(B^j_{f,\lambda}\) a direct factor of \(C^j_{f,\lambda}\), such that \(\nu_{f_U,\lambda}^j=\dim H^j B^\bullet_{f,\lambda}\). The problem is the global triviality of certain local systems. For good compactifications of isolated singularities \(\nu_{f_U,\lambda}^n\) can be computed from the Euler characteristic of the complex \(B^\bullet_{f,\lambda}\). The authors give also sufficient conditions for equality of \(B^j_{f,\lambda}\) and \(C^j_{f,\lambda}\). These are in particular satisfied for superisolated surface singularities.
There is a partial generalisation of the formulas for the case of singular total spaces. local monodromy; limit mixed Hodge structure; nearby cycles Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects) Number of Jordan blocks of the maximal size for local monodromies | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(G_ i\), \(i=1,2\) be complex analytic groups and let \(R(G_ i)\) denote their complex Hopf algebras of representative functions (an analytic function on \(G\) is representative if its \(G\) translates span a finite dimensional complex vector space). Suppose that \(R(G_ 1)\) and \(R(G_ 2)\) are isomorphic as complex Hopf algebras. The author's main theorem establishes that then there must be a complex algebraic group \(F\) and a torus \(Y\) in the radical of \(F\) which contains no non-trivial central element of \(F\) and analytic embeddings \(\rho_ i: G_ i\to F\) with Zariski dense images such that \(F\) is the semi-direct product \(F=Y\cdot \rho_ i(G_ i)\) and such that if \(M\supseteq Y\) is any maximal reductive subgroup then \(M\cap\rho_ 1(G_ 1)=M\cap \rho_ 2(G_ 2)\). Conversely, if such an \(F\) exists then the Hopf algebras \(R(G_ 1)\) and \(R(G_ 2)\) are isomorphic.
A triple \((F,Y,\rho_ i)\) as above is called a normal reduced split hull for \(F_ i\). Split hulls \((F,Y,\sigma)\) and \((H,Z,\tau)\) for the analytic group \(G\) are said to be equivalent if there is an algebraic group isomorphism \(\theta: F\to H\) such that \(\theta(Y)=Z\) and \(\tau\theta=\sigma\). Recall that a nucleus of \(G\) is a simple connected solvable normal subgroup with linearly reductive quotient, and that the intersection of the kernels of all the semi-simple representations of \(G\) is the representation radical of \(G\). The author shows that if the representation radical of \(G\) is a nucleus then \(G\) is the unique reduced split hull of \(G\). If not, then there are equivalent normal reduced split hulls unless the connected component of the center of any maximal reductive subgroup of \(G\) is also central in \(G\) itself. complex analytic groups; complex Hopf algebras of representative functions; complex algebraic group; analytic embeddings; semi-direct product; maximal reductive subgroup; normal reduced split hull; semi- simple representations General properties and structure of complex Lie groups, Affine algebraic groups, hyperalgebra constructions, Complex Lie groups, group actions on complex spaces, Other representations of locally compact groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Complex analytic 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 main result of this article is a new proof of Looijenga's conjecture on cusp singularities. A cusp singularity \((\overline{V},p)\) is a germ of surface singularity such that the exceptional divisor \(D\) of the minimal resolution \(\pi: V\to \overline{V}\) is a reduced cycle \(\pi^{-1}(p)= D=D_1+\cdots+D_n\in |-K_V|\) of smooth rational curves \(D_i\) meeting transversely. It is known that each cusp surface singularity \((\overline{V},p)\) has its \textit{dual} cusp singularity \((\overline{V}',p')\). One of the constructions relating \((\overline{V},p)\) and \((\overline{V}',p')\) is via \textit{hyperbolic Inoue surface}, and it is recalled in the paper. \textit{E. Looijenga} [Ann. Math. (2) 114, 267--322 (1981; Zbl 0509.14035)] proved that whenever a cusp singularity \((\overline{V},p)\) is smoothable, the exceptional divisor \(D\) of the minimal resolution of its dual cusp singularity \((\overline{V}',p')\) (\(D'\) is then called dual to \(D\)) is an anticanonical divisor of some smooth rational surface \(Y'\). Looijenga also conjectured the converse, i.e., that existence of such an \textit{anticanonical pair} \((Y',D')\) implies the smoothability of the cusp singularity with cycle \(D\). The first proof of Looijenga's conjecture was obtained by \textit{M. Gross} et al. [Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)], as an application of mirror symmetry. In the present paper, P. Engel gives an alternative proof of Looijenga's conjecture.
The new proof does not make any explicit use of mirror symmetry. Instead, it relies mainly on Friedman-Miranda's criterion of smoothability of cusps with cycle \(D\), see [\textit{R. Friedman} and \textit{R. Miranda} [Math. Ann. 263, 185--212 (1983; Zbl 0488.14006)], and on the theory of integral-affine structures on real surfaces. The work of Friedman and Miranda is quoted in Gross, Hacking, and Keel's paper, but seems not to be used. The scheme of the proof is, roughly, as follows. First, an anticanonical pair \((Y',D')\) is represented as a sequence of blow-ups and smoothings of a toric surface \((\overline{Y}',\overline{D}')\). The moment polygon \(\overline{S}'\) of \((\overline{Y}',\overline{D}')\) is endowed with an appropriate integral-affine structure and then it admits a sequence of surgeries parallel to that of \((\overline{Y}',\overline{D}')\). In this way, a new integral-affine real surface \(S\) is produced. After completing \(S\) to a topological triangulated sphere \(\hat{S}\) and passing to a certain refinement \(\hat{S}[k]\), it is shown that to each vertex \(v_i\), \(i>0\), of the triangulation one can associate an anticanonical pair \((V_i,D_i)\), and there is also a special vertex \(v_0\) to which one associates the hyperbolic Inoue pair \((V_0,D)\), where \(D\) is the cycle dual to \(D'\). All the surfaces \(V_i\), \(i\geq 0\), glue well and form a reducible surface \(\mathcal{X}_0\) whose dual complex is \(\hat{S}[k]\) and which is smoothable by Friedman-Miranda's criterion.
The proofs of Looijenga's conjecture by Gross, Hacking, and Keel and by Engel make an impression of being of the same spirit. At the same time, both are highly non-trivial and a direct connection between them is hard to specify. Some comments on this are given by P. Engel in the introduction to his paper. cusp surface singularity; anticanonical pair; hyperbolic Inoue surface; integral-affine structure; smoothability Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Looijenga's conjecture via integral-affine geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Q\) be a finite acyclic quiver. Denote by \(\tilde{Q}\) its framed quiver, that is, for each vertex of \(Q\), we add a new vertex and an arrow pointing from the old vertex to the new vertex. The singular Nakajiama category \(\mathcal{S}\) is a full subcategory of a certain mesh category of the repetition quiver \(\mathbb{Z}\tilde{Q}\). More generally, some quotient category of \(\mathcal{S}\) is studied. By \textit{B. Leclerc} and \textit{P.-G. Plamondon} [Publ. Res. Inst. Math. Sci. 49, No. 3, 531--561 (2013; Zbl 1285.14050)], the representations of \(\mathcal{S}\) are related to the points on graded affine quiver varieties.
The main result of this paper constructs a \(\delta\)-functor \(\Phi\) from the module category of \(\mathcal{S}\) to the bounded derived category of representations of \(Q\). The constructed functor is called the stratifying functor. Indeed, for two \(\mathcal{S}\)-modules, they are in the same stratum of the quiver variety if and only if their images in the derived category are isomorphic. In this Dynkin case, the functor \(\Phi\) is constructed using the singularity category of \(\mathcal{S}\), where \(\mathcal{S}\) is \(1\)-Gorenstein of infinite global dimension. In the non-Dynkin case, the category \(\mathcal{S}\) has global dimension one. quiver variety; Gorenstein; singularity category; stratifying functor Keller, B., Scherotzke, S.: Graded quiver varieties and derived categories. J. reine. angrew. Math. 10.1515/crelle-2013-0124 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets, Derived categories and associative algebras, Derived categories, triangulated categories Graded quiver varieties and derived categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Tannakian duality describes the problem of recovering a group or a group scheme from its category of representations. In this context, the notion of a Tannakian category was introduced by \textit{N. Saavedra Rivano} in [Categories tannakiennes. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0241.14008)], and later by \textit{P. Deligne} in [Prog. Math. 87, 111--195 (1990; Zbl 0727.14010)] who also corrected some inaccuracies in Saavedra's work. A Tannakian category over a field \(K\) consists of a \(K\)-linear rigid symmetric monoidal tensor category \(\mathcal{T}\) together with a faithful exact additive tensor functor \(\omega:\mathcal{T}\to \mathrm{Mod}_X\) (called \textit{fibre functor}) to the category \(\mathrm{Mod}(X)\) of \(\mathcal{O}_X\)-modules for some \(K\)-scheme \(X\) such that the endomorphism ring of the unit object \(1_{\mathcal{T}}\in \mathcal{T}\) is \(\mathrm{End}(1_{\mathcal{T}})\cong K\). In the neutral case, i.e. when \(X=\mathrm{Spec}(K)\), one obtains an affine group scheme \(G\) over \(K\) from \((\mathcal{T},\omega)\) as the group of tensor automorphisms of \(\omega\), and the Tannakian category is equivalent to the category \(\mathrm{Rep}_K(G)\) of finite dimensional \(K\)-representations of this group \(G\) via the functor \(\omega\). On the other hand, for every affine group scheme \(G\) over \(K\), the category \(\mathrm{Rep}_K(G)\) is a neutral Tannakian category with fibre functor being the forgetful functor, and the Tannakian group scheme obtained from this category is the group scheme \(G\) again.
In this article, the authors generalise this Tannakian duality to a duality for flat affine group schemes \(G\) over a Dedekind ring \(R\).
One such generalisation has already been considered by Saavedra in [loc. cit.] which is recalled by the authors in the first section. This approach defines the notion of a (neutral) Tannakian category \((\mathcal{C},\omega)\) over a Dedekind ring \(R\) which as above turns out to be equivalent to the category \(\mathrm{Rep}_R(G)\) of representations of \(G=\mathrm{Aut}^\otimes(\omega)\) on finitely generated \(R\)-modules, and \(G\) is a flat affine group scheme. In the core of this approach is a ``subcategory of definition'' \(\mathcal{C}^0\) of \(\mathcal{C}\) which corresponds to the full subcategory \(\mathrm{Rep}_R^0(G)\) of representations on finitely generated projective \(R\)-modules.
The author's approach focuses on the subcategory \(\mathrm{Rep}_R^0(G)\). They introduce the notion of a ``neutral Tannakian lattice'' over \(R\) (a certain \(R\)-linear category \(\mathcal{T}\) with fibre functor \(\omega:\mathcal{T}\to \mathrm{Mod}^0(R)\); cf. Definition 2.2.2 ibid.). The main theorem in this context is Theorem 2.3.2: For a neutral Tannakian lattice \((\mathcal{T},\omega)\) over a Dedekind ring \(R\), the group scheme \(G=\mathrm{Aut}^\otimes(\omega)\) of tensor automorphisms of \(\omega\) is faithfully flat over \(R\) and \(\omega\) induces an equivalence between \(\mathcal{T}\) and the category \(\mathrm{Rep}_R^0(G)\).
One should be aware that the definition they use is different from the definition of a Tannakian lattice over valuation rings introduced by \textit{T. Wedhorn} in [J. Algebra 282, No. 2, 575--609 (2004; Zbl 1088.18008)].
Since affine group schemes are in one-to-one correspondence with commutative Hopf algebras, and a representation of an affine group scheme is just a comodule of the corresponding Hopf algebra, large parts of the paper are formulated in terms of the latter.
In the later parts of the article, the authors also discuss homomorphisms of flat coalgebras via the Tannakian duality as well as torsors of an affine flat group scheme corresponding to different fibre functors \(\eta:\mathcal{T}\to \mathrm{Mod}(S)\) (\(S\) being an \(R\)-algebra). This is a direct generalisation of Theorem 3.2 in [\textit{P. Deligne} and \textit{J. S. Milne}, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)]. Tannakian category; Dedekind rings; affine flat group schemes; coalgebras Affine algebraic groups, hyperalgebra constructions, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Tannakian duality over Dedekind rings and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is an important contribution to the the broader program of Kontsevich's Homological Mirror Symmetry conjecture. It relates the Fukaya category \( F(\Sigma_g)\) of a closed surface \(\Sigma_g\) of genus \(g\geq2\) and the Fukaya category \( F(Q_0^{2g}\cap Q_1^{2g})\) of the complete intersection \(Q_0 \cap Q_1\) of two smooth quadric hypersurfaces in \(\mathbb P^{2g+1}\). The main result of the paper, stated in Theorem 1.1, is the following \(\mathbb C\)-linear equivalence of \(\mathbb Z_2\)-graded split-closed triangulated categories:
\[
D^{\pi} F(\Sigma_g) \simeq D^{\pi} F(Q_0^{2g} \cap Q_1^{2g};0),
\]
where \(D^{\pi} C\) denotes the cohomological category \(H(Tw^{\pi} C)\) underlying the split-closure of the category of twisted complexes of an \(A_{\infty}\)-category \( C\) and \( F(\bullet;0)\) denotes the summand corresponding to the \(0\)-eigenvalue of quantum cup-product by the first Chern class. The proof constructs equivalences with quasi-isomorphic images
\[
D^{\pi} F(\Sigma_g) \;\hookrightarrow \;D^{\pi} F(Z) \;\hookleftarrow \;D^{\pi} F(Q_0\cap Q_1;0),
\]
where the relative quadric \(Z\) is given by blowing up \(\mathbb P^{2g+1}\) along \(Q_0\cap Q_1\).
The author also gives nice applications of the main result to representation varieties and instanton Floer homology, making the paper of wide interest to experts on geometric topology and mathematical physics. The paper is well organized: Section 1 is a very readable summary of the motivation and background of the paper, as well as an outline of the proof. Section 2 contains the applications to topology, assuming the main theorem. Section 3 is an exposition of Fukaya category. Finally the main theorem is proved in Sections 4 and 5.
The paper could be read together with [\textit{P. Seidel}, J. Algebr. Geom. 20, No. 4, 727--769 (2011; Zbl 1226.14028)]. 23 I. Smith, 'Floer cohomology and pencils of quadrics', \textit{Invent. Math.}189 (2012) 149-250. Mirror symmetry (algebro-geometric aspects), Symplectic aspects of Floer homology and cohomology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Floer cohomology and pencils of quadrics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a continuation of the author's expository notes [Sûgaku 38, No. 2, 97--115 (1986; Zbl 0622.20040)]. In the first part, he interprets the relations between his theory of systems of general weights and the corresponding transformation groups and singularities of surfaces in \(\mathbb C^3\). In the present notes, he introduces the generalized root systems and shows that the root lattice can be imbedded in \(K3\)-lattice as a sublattice and discusses the invariants of the Weyl groups of root systems.
To a regular system of weights with minimal exponent \(\varepsilon >0\), there corresponds a finite root system by McKay correspondence (cf. first part). Here, for any regular system of weights, he defines a ``generalized'' root system, where the root lattice \(H\) is the homology group \(H_2(X_1,\mathbb Z)\) of the Milnor fiber \(X_1\) corresponding to the system of weights, and the set \(R\) of the roots is the subset of vanishing cycles of \(H\) which is a natural generalization of finite root systems for the case \(\varepsilon >0\). He poses a number of reasonable axioms for ``generalized'' root systems which in fact is a generalization of finite or infinite root systems, and discusses Dynkin diagrams, Coxeter transformations which are related to the work of Gabrielov, Arnold, Brieskorn etc.
To analyze the root lattice (in particular for the case \(\varepsilon <0)\), he discusses the imbedding of the root lattice into the homology group of \(K3\)-surface (\(K3\)-lattice) as a sublattice, namely he deals with the compactification of the Milnor fiber \(X_1\) for each case \(\varepsilon >0\), \(\varepsilon =0\), \(\varepsilon =-1\) or \(\epsilon\leq -1\) and \(\varepsilon\) is the only negative exponent. Let H be the root lattice corresponding to a regular system of weights, the Weyl group acts on the \({\mathbb{C}}\)-vector space \(Hom_{{\mathbb{Z}}}(H,{\mathbb{C}})\). He introduces and expounds his work on the ring \(S^+\) of W-invariant functions on a certain domain in \(Hom_{\mathbb Z}(H,\mathbb C)\), e.g. Chevalley type theorem for \(S^+\) and flat structure of \(S^+\) for the case \(\varepsilon >0\) and \(\varepsilon =0\).
Finally, the author proposes and discusses some open problems related to the present theme [cf. Publ. Res. Inst. Math. Sci. 21, 75--179 (1985; Zbl 0573.17012); Adv. Stud. Pure Math. 8, 479--526 (1986; Zbl 0626.14028)]. Milnor lattice; generalized root systems; root lattice; imbedded in K3- lattice; invariants of the Weyl groups of root systems; regular system of weights; homology group of K3-surface; Chevalley type theorem Simple, semisimple, reductive (super)algebras, Special surfaces, Singularities of surfaces or higher-dimensional varieties, Compact complex surfaces, Infinite-dimensional Lie (super)algebras The theory of systems of general weights and related topics. II: Its effects on the theory of singularities, general Weyl groups and their invariants, etc | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 goals of this article are as follows: (1) To determine the irreducible components of the affine varieties \({\mathbf {Rep}}_{\mathbf {d}}(\Lambda )\) parametrizing the representations with dimension vector \(\mathbf {d}\), where \({\Lambda }\) traces a major class of finite dimensional algebras; (2) To generically describe the representations encoded by the components. The target class consists of those truncated path algebras \({\Lambda }\) over an algebraically closed field \(K\) which are based on a quiver \(Q\) without oriented cycles. The main result characterizes the irreducible components of \({\mathbf {Rep}}_{\mathbf {d}}(\Lambda )\) in representation-theoretic terms and provides a means of listing them from quiver and Loewy length of \({\Lambda }\). Combined with existing theory, this classification moreover yields an array of generic features of the modules parametrized by the components, such as generic minimal projective presentations, generic sub- and quotient modules, etc. Our second principal result pins down the generic socle series of the modules in the components; it does so for more general \({\Lambda }\), in fact. The information on truncated path algebras of acyclic quivers supplements the theory available in the special case where \({\Lambda }= KQ\), filling in generic data on the \(\mathbf {d}\)-dimensional representations of \(Q\) with any fixed Loewy length. representations of finite dimensional algebras; quivers with relations; parametrizing varieties; irreducible components; generic properties of representations B. Huisgen-Zimmermann, I. Shipman, Irreducible components of varieties of representations. The acyclic case, preprint. Representations of associative Artinian rings, Representations of quivers and partially ordered sets, Special varieties Irreducible components of varieties of representations: the acyclic case | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In his famous paper, Chevalley has defined analogues over arbitrary fields of the complex simple Lie groups. Further, Chevalley and Demazure have constructed a certain group scheme G over \({\mathbb{Z}}\) associated to every reductive group over \({\mathbb{C}}\) which is characterized by a few simple properties, making G(R) for any commutative ring R the natural analogue of the group over R. The group schemes are classified by data \(D=(I,\Lambda,(\alpha_ i)_{i\in I}\), \((h_ i)_{i\in I})\) consisting of a finite set I, a finitely generated free abelian group \(\Lambda\), and two maps \(i\mapsto \alpha_ i\) and \(i\mapsto h_ i\) of I in \(\Lambda\) and in its dual \(\Lambda^{\vee}\), respectively, these data being subject only to the condition that the matrix \(A=(A_{ij})=(<\alpha_ j\), \(h_ i>)\) be a Cartan matrix. If we assume for A the only conditions \(A_{ii}=2\), \(A_{ij}\leq 0\) if \(i\neq j\) and \(A_{ij}=0 \Leftrightarrow A_{ij}=0\), then A is called a generalized Cartan matrix.
In this paper, to any data D with generalized Cartan matrix A, the author associates a group functor \(G_ D\) on the category of commutative rings with 1. Further, the author states a few axioms which should hold for any reasonable extension of the Chevalley-Demazure group schemes to the Kac- Moody situation and shows that any functor satisfying these axioms coincides with \(G_ D\) over fields.
The author has treated already these problems [cf. Annuaire College de France 82, 91-105 (1981/82)], but the present method is simpler and more direct and it is based on a presentation ``à la Steinberg'' of the Kac- Moody groups. An important feature of the situation is that it leads to groups endowed with two distinct BN-pairs having the same group N (double BN-pairs), and the new structure turns out to be much richer than that consisting of a single BN-pair.
These axiomatic characterization of Kac-Moody groups provides an easy answer to certain recognition problems. For instance, Chevalley groups of classical types are isomorphic with corresponding classical groups, that has been proved in various ways, and it also can be applied to Kac-Moody groups of affine types which also have ``classical interpretation''. Further, the axiomatic set up should also apply to suitably defined twisted Kac-Moody groups. Steinberg presentation; reductive group; group schemes; generalized Cartan matrix; BN-pairs; Kac-Moody groups; Chevalley groups Mühlherr, B., Petersson, H.P., Weiss, R.M.: Descent in Buildings. Annals of Mathematics Studies, vol. 190. Princeton University Press, Princeton (2015) Infinite-dimensional Lie groups and their Lie algebras: general properties, Group schemes, Generators, relations, and presentations of groups, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Linear algebraic groups over local fields and their integers, Linear algebraic groups over arbitrary fields Uniqueness and presentation of Kac-Moody groups over 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 Let \(k\) be algebraically closed field, and let \(Q=(Q_0,Q_1,s,e) \) be a Dynkin quiver of type \(\mathbb A_n\) for \( n\geq 1\). Let \({\mathbf d}\in\mathbb N^{Q_0}\) be a dimension vector. It was previously known [\textit{S. Abeasis, A. Del Fra, H. Kraft}, Math. Ann. 256, 401--418 (1981; Zbl 0477.14027)] that, in the case where \(Q\) is an equioriented Dynkin quiver and \(k\) is of characteristic zero, the orbit closures in \(\text{rep}_Q({\mathbf d})\) are normal and Cohen-Macauley varieties and have rational singularities. Here the same result is proved when \(Q\) has arbitrary orientation.
To accomplish this, the authors prove the following. Let \(Q'\) and \(Q\) be Dynkin quivers of type \(\mathbb A\), and let \(A=kQ'\) (resp. \(B=kQ)\) be the path algebras of quivers \(Q'\) (resp. \(Q\)). If we assume there is a full embedding of translation quivers \(F: \Gamma_B\to\Gamma_A\), then there exists a functor \({\mathcal F}: \text{mod }B\to\text{mod }A\) which is hom-controlled. This short paper is dedicated to proving this proposition. The desired result above follows from a paper by \textit{G. Zwara} [Proc. Lond. Math. Soc. (3) 84, No. 3, 539--558 (2002; Zbl 1054.16009)]. Dynkin quivers; orbit closures; Cohen-Macaulay varieties; rational singularities; path algebras; translation quivers Bobiński, Grzegorz; Zwara, Grzegorz, Normality of orbit closures for Dynkin quivers of type \(\mathbb{A}_n\), Manuscripta Math., 105, 1, 103-109, (2001) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Normality of orbit closures for Dynkin quivers of type \(\mathbb A_n\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((N,G)\), where \(N\unlhd G\leq \operatorname{SL}_n(\mathbb{C})\), be a pair of finite groups and \(V\) a finite-dimensional fundamental \(G\)-module. We study the \(G\)-invariants in the symmetric algebra \(S(V ) = \bigoplus_{k \geq 0}S^k(V\) ) by giving explicit formulas of the Poincaré series for the induced modules and the restriction modules. In particular, this provides a uniform formula of the Poincaré series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincaré series in terms of Tchebychev polynomials in the sense that one needs only the dimensions of the subgroups and their group-types to completely determine the Poincaré series. symmetric algebra; Poincaré series; McKay-Slodowy correspondence; invariants; quantum Cartan matrix McKay correspondence, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Group rings of finite groups and their modules (group-theoretic aspects) Poincaré series of relative symmetric invariants for \(\operatorname{SL}_n(\mathbb{C})\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main object of the paper under review is a central simple algebra defined over a function field \(F\) of transcendence degree one over a ground field \(k\) of characteristic zero. In other words, \(F=k(X)\) is the field of rational functions of a smooth geometrically connected projective curve \(X\) defined over \(k\). Roughly speaking, the authors' goal is to understand to what extent a given algebra is determined by its local invariants, or, in other words, to describe the reciprocity laws for local invariants.
To be more precise, one can write down a complex of abelian groups
\[
0\to\text{{Br }}X\to\text{{Br }}k(X)\overset{\text{{inv}}}{}\bigoplus_{x\in X^1}H^1(k(x),\mathbb Q/\mathbb Z)\overset{\Sigma }{} H^1(k,\mathbb Q/\mathbb Z),
\]
where \(\text{{inv}}\) is the direct sum of local invariants and \(\Sigma\) is the sum of corestriction maps. If \(X={\mathbb P}^1\), it is known (Faddeev-Grothendieck) that this complex is exact and, moreover, \(\Sigma\) is surjective (local invariants of a central simple algebra over \(k(t)\) satisfy the Faddeev reciprocity law, and, conversely, any system satisfying this reciprocity law comes from some central simple algebra). If \(X\) is a curve of positive genus, this is no longer true: in general, \(\text{{im}}(\text{{inv}})\neq\text{{ker}}(\Sigma)\). The authors study the quotient \(\Delta =\text{{ker}}(\Sigma)/\text{{im}}(\text{{inv}})\) which can be thought of as the obstruction group for realizing characters satisfying the reciprocity law as systems of local invariants of algebras over \(k(X)\). The authors focus on the case where \(k\) is a number field and \(X\) has a \(k\)-point. \textit{B.~Bekker} [Mat. Zametki 17, 419-422 (1975; Zbl 0321.14015)] proved that under these hypotheses \(\Delta\) is a 2-group (trivial if \(k\) is totally imaginary). The main result of the paper under review makes this more precise: \(\Delta\cong ({\mathbb Z}/2)^{r_1+\dots +r_s-s}\); here \(s\) is the number of real embeddings \(\rho _i\colon k\to\mathbb R\) and \(r_i\) is the number of components of \(X_i(\mathbb R)\) in the real topology where \(X_i=X\times _k\mathbb R\) with the product taken with respect to \(\rho _i\). Brauer group; field of rational functions; reciprocity law; projective curve Brauer groups of schemes, Arithmetic theory of algebraic function fields, Varieties over global fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Algebraic functions and function fields in algebraic geometry Reciprocity laws for simple algebras over function fields of number curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let $k$ be a perfect field of characteristic $p$. Let $W_n$ be the ring of length $n$ Witt vectors of $k$. For a smooth scheme $P$ over $W_n$, \textit{M. Emerton} and \textit{M. Kisin} [The Riemann-Hilbert correspondence for unit \(F\)-crystals. Paris: Société Mathématique de France (2004; Zbl 1056.14025)] established a ``Riemann-Hilbert correspondence'' for unit \(F\)-crystals on $P$. \par The ``topological'' side of the correspondence is the derived category of étale constructible complexes of $\mathbb{Z}/p^{n}$-modules of finite tor-dimension; the ``\(\mathcal{D}\)-module'' side is a certain triangulated category $D^b_{\mathrm{lfgu}}(P/W_n)^{\circ}$ whose objects are some special complexes of \(\mathcal{D}\)-modules with Frobenius actions. Among other technical assumptions, the Frobenius actions on cohomology sheaves of complexes in $D^b_{\mathrm{lfgu}}(P/W_n)^{\circ}$ are required to be isomorphisms. \par Parallel to the classical Riemann-Hilbert correspondence, the solution functor gives an anti-equivalence from the \(\mathcal{D}\)-module side to the topological side. The ``smooth'' version of this equivalence goes back to [\textit{N. M. Katz}, Lect. Notes Math. 350, 69--190 (1973; Zbl 0271.10033)]. \par In the paper under review, the author generalizes the result of Emerton and Kisin to any ``$W_n$-embeddable'' (e.g., quasi-projective) $k$-variety $X$. \par The definition of the ``\(\mathcal{D}\)-module side'' derived category is given in Definition~4.7. Let $X$ be a $k$-variety. Assume that there is a locally closed embedding $X \to P$, where $P$ is a smooth proper $W_n$-scheme. Then the category $D^b_{\mathrm{lfgu}}(X/W_n)^{\circ}$ is the full subcategory $D^b_{\mathrm{lfgu}}(P/W_n)^{\circ}$ consisting of complexes that are supported on $\overline{X}$, the closure of $X$ in $P$, but have zero support on $\overline{X}-X$. The author shows that this definition is independent of the choice of the embedding, and shows that this category is amenable to the six-functor-formalism. \par The generalization of Emerton-Kisin's theorem is Theorem~4.12. The idea is to reduce the Riemann-Hilbert correspondence for $X$ to Emerton-Kisin's Riemann-Hilbert correspondence for $P$, using the fact that the solution functor on $D^b_{\mathrm{lfgu}}(P/W_n)^{\circ}$ takes a complex supported on $X$ to an étale complex on $P$ supported on $X$. \par In section 5, the last section of the paper, the author assumes $W_n=W_1=k$, and discusses some topics on the various t-structures on the two triangulated categories considered before. For example, it is shown that the solution functor takes the standard t-structure on $D^b_{\mathrm{lfgu}}(X/k)^{\circ}$ to Gabber's perverse t-structure of the étale derived category. \(\mathcal{D}\)-modules; \(F\)-crystals; étale sheaves \(p\)-adic cohomology, crystalline cohomology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Riemann-Hilbert correspondence for unit \(F\)-crystals on embeddable algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(2\) and \(V\) a \(3\)-dimensional \(k\)-vector space. A linear map \(\varphi: V\to V\) is called a pseudoreflection if the set of fixed points is a hyperplane in \(V\). Let \(G\) be a finite subgroup of \(\mathrm{GL}(V)\) generated by pseudoreflections of order \(2^r, r\geq 1\). Assume that \(G\) has a two-dimensional invariant subspace \(W\) and the restriction of \(G\) to \(W\) is isomorphic to the group \( \mathrm{SL}_2(\mathbb{F}_2 n), n>1\), in its natural representation.
It is proved that the ring of invariants \(S(V^{\ast})^G\) is polynomial. As a corollary one obtains a generalization of a theorem of \textit{G. Kemper} and \textit{G. Malle} [Transform. Groups 2, No. 1, 57--89 (1997; Zbl 0899.13004)]:
If \(G\subseteq \mathrm{GL}(V)\) is any finite subgroup then \(S(V^{\ast})^G\) is polynomial iff \(G\) is generated by pseudoreflections and the pointwise stabilizer in \(G\) of any non-trivial subspace of \(V\) has a polynomial ring of invariants.
Another corollary is the following:
If \(V\) is a \(3\)-dimensional vector space over an arbitrary field \(k\) and \(G\) a finite subgroup of \(\mathrm{GL}(V)\) such that the variety \(V/G\) has an isolated singularity then \(V/G\) is isomorphic to one of the non-modular isolated quotient singularities from Vincent's classification. quotient singularities; ring of invariants; finite group Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients) Three-dimensional isolated quotient singularities in even characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The first part of this book concerns the question of classification of the automorphic representations of the projective symplectic group PGSp(2) of similitudes over a number field \(F\). The author reduces this question to that for the projective general linear group PGL(4) by means of the theory of liftings with respect to the dual group embedding of Sp(2,\(\mathbb C\)) in SL(4,\(\mathbb C\)). He introduces the notion of packets and quasi-packets of representations, admissible and automorphic, of PGSp(2). The lifting implies a rigidity theorem for packets and multiplicity one theorem for the discrete spectrum of PGSp(2). The classification uses the theory of endoscopy, and twisted endoscopy. This leads to a notion of stable and unstable packets of automorphic forms. The stable ones are those which do not come from a proper endoscopic group.
The second part of the book concerns the decomposition of the étale cohomology with compact supports of the Shimura variety associated with PGSp(2), over an algebraic closure \(\overline{F}\), with coefficients in a local system. This is a Hecke-Galois bi-module, and its decomposition into irreducibles associates to each geometric automorphic representation a Galois representation. They are related at almost all places as the Hecke eigenvalues are the Frobenius eigenvalues, up to a shift. In the stable case the author obtains Galois representations of dimension \(4^{[F:\mathbb Q]}\). In the unstable case the dimension is half that, since endoscopy shows up. The statement, and the definition of stability, is based on the classification and lifting results of the first, main, part. The description of the zeta function of the Shimura variety, also with coefficients in the local system, follows formally from the decomposition of the cohomology.
The third part of the book, which is written for non-experts in representation theory, consists of a brief introduction to the Principle of Functoriality in the theory of automorphic forms.
Parts one and two are examples of the general, mainly conjectural, theory described in this last part. Part three can be read independently of parts one and two. automorphic forms; Shimura varieties; group of similitudes; automorphic representations; projective symplectic group PGSp(2) of similitudes Flicker, Y. Z., Automorphic forms and Shimura varieties of \(\operatorname{PGSp}(2)\), (2005), World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ Representation-theoretic methods; automorphic representations over local and global fields, Arithmetic aspects of modular and Shimura varieties, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Modular and Shimura varieties, Research exposition (monographs, survey articles) pertaining to number theory Automorphic forms and Shimura varieties of \(\text{PGSp}(2)\). | 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 paper contains a description of all terminal singularities of three- folds [see \textit{M. Reid}, Journées géom. algebr., Angers/France 1979, 273-310 (1980; Zbl 0451.14014); Problem (0.9); see also Algebr. varieties and analytic varieties, Proc. Symp. Tokyo 1981, Adv. Stud. Pure Math. 1, 131-180 (1983; Zbl 0558.14028)]. It was proved by M. Reid (ibid.) that terminal singularities are cyclic quotients of either a smooth point or a cDV point (compound DuVal singularity). The case when this cover is smooth was described by \textit{V. I. Danilov} [Math. USSSR, Ivz. 21, 269- 280 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 971-982 (1982; Zbl 0536.14008)] and \textit{D. R. Morrison} and \textit{G. Stevens} [Proc. Am. Math. Soc. 90, 15-20 (1984; Zbl 0536.14003)].
The paper under review contains the case when the cover is a cDV point.
The cDV points are divided (naturally) into three different types, cA, cD and cE. (They are intimately related to the \(A_ n\), \(D_ n\) and E type singularities for surfaces.) Instead of repeating the results, let me just describe one cA which occurs. If the order m of the cyclic group is bigger than 4, then the cA point is isomorphic to \(xy+z^ 2+u^{im}=0\) where x,y,z,u are co-ordinates in 4-space. The group action is the following: for a suitable w, a primitive m-th root of unity, and \(\sigma\) a generator of the group,
\[
\sigma (x)=wx,\quad \sigma (y)=w^{- 1}y,\quad \sigma (z)=z,\quad \sigma (u)=w^ au
\]
where a and m are relatively prime. Since cDV points are all hypersurfaces, they are given by \(\phi =0\), \(\phi\) a function on 4-space. Since they are singular, the linear part \(\phi_ 1\) of \(\phi\) is zero. If rk \(\phi\) \({}_ 2\geq 2\), one gets a cA-point. If rk \(\phi\) \({}_ 2=1\) then the group has order 2 or 3 and \(\phi\) is described explicitly (up to isomorphism) in case \(\phi_ 3\) is not power of a linear form. These are cD-type singularities. Finally when \(\phi_ 3\) is also power of a linear form, the order of the group is 2 and a reasonable description of \(\phi\) is made. These are cE-type points. This paper, thus settles the initial but crucial problem of explicitly describing all possible terminal singularities of 3-folds. terminal singularities of three-folds; compound DuVal singularity Mori, Shigefumi, On \(3\)-dimensional terminal singularities, Nagoya Math. J., 98, 43-66, (1985) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds On 3-dimensional terminal 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 Denote by \(X\) a separated Noetherian scheme of finite Krull dimension which has enough locally free sheaves of finite rank (i.e., every coherent sheaf on \(X\) is a quotient of a locally free sheaf of finite rank). Assume that \(U\subseteq X\) is an open subscheme.
\textit{D. O. Orlov} introduced [Proc. Steklov Inst. Math. 246, 227--248 (2004); translation from Tr. Mat. Inst. Steklova 246, 240--262 (2004; Zbl 1101.81093)] an invariant, called the singularity category of \(X\) defined to be the Verdier quotient triangulated category: \(\mathbf D_{sg}(X)=\mathbf D^b(\text{coh} X)/\text{perf} X\). Orlov proved (ibid) that if the singular locus of \(X\) is \(\subseteq U\), then the triangle functor \(\overline j^*:\mathbf D_{sg}(X)\longrightarrow \mathbf D_{sg}(U)\) is an equivalence. Another result of \textit{D. Orlov} [Adv. Math. 226, No.~1, 206--217 (2011; Zbl 1216.18012)] proves that \(\mathbf D_{sg}(X)=\text{thick}\langle q(\text{coh}_\mathbb Z X)\rangle\), under the assumption that the singular locus of \(X\) is \(\subseteq \mathbb Z\) and \(q:\mathbf D^b(coh X)\longrightarrow \mathbf D_{sg}(X)\) is the quotient functor. The author in a sense unifies (and generalizes) these results of Orlov (in a spirit resembling a result of \textit{H. Krause} [Compos. Math. 141, No. 5, 1128--1162 (2005; Zbl 1090.18006)]).
This unifying result is that the triangle functor \(\overline j^*:\mathbf D_{sg}(X)\longrightarrow \mathbf D_{sg}(U)\) induces a triangle equivalence \(\mathbf D_{sg}(X)/\text{thick}\langle q(\text{coh}_\mathbb Z X)\rangle\cong \mathbf D_{sg}(U)\). This is proved through some lemmas and it was shown that Orlov's results are then corollaries. The author also proves a non-commutative version of his main result, for a left Noetherian ring \(R\); here \(X\) is replaced by \(R\) and \(U\) by \(eRe\), for an idempotent \(e\) that is subject to some reasonable conditions. singularity category; quotient functor; Schur functor; Verdier quotient; triangle functor; triangle equivalence; left Noetherian ring; idempotents; Orlov; Krause Chen, Xiao-Wu, Unifying two results of Orlov on singularity categories, Abh. Math. Semin. Univ. Hambg., 0025-5858, 80, 2, 207-212, (2010) Chain complexes (category-theoretic aspects), dg categories, Singularities in algebraic geometry, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Unifying two results of Orlov on singularity categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider the moduli space of tuples \((C,\eta,x,y)\) with \(C\) a smooth genus \(g\) curve, \(\eta\) an even spin structure on \(C\) (i.e., a square root of the canonical bundle with \(h^0(C,\eta)\) even), and \(x, y\) points on \(C\). In this space there is the \textit{theta-null divisor} defined by the condition that \(h^0(C,\eta)\) is positive, and away from the theta-null divisor one can define the \textit{universal Scorza correspondence} by the condition that \(h^0(C,\eta+x-y)\) is positive. The universal Scorza correspondence has codimension one, is symmetric under switching \(x\) and \(y\), and is disjoint from the diagonal \(x = y\). One can ask: how does the closure of the Scorza correspondence intersect the theta-null divisor?
Gavril Farkas has conjectured that for a generic curve \(C\) with a theta-null \(\eta\), the closure of the Scorza correspondence inside \(C \times C\) is given by the union of the diagonal with multiplicity two and the curve \(\{x,y \, :\, h^0(\eta - x - y) \neq 0 \}\). The paper under review proves a stronger statement when the genus is three: that this is true for \textit{any} genus three curve with a theta-null. (The genus three curves with a theta-null are precisely the hyperelliptic curves.)
The proof is very explicit and does not generalize to higher genus. They use the difference map \(C \times C \to J(C)\) in order to consider the image of the Scorza correspondence in of the universal abelian variety over \(A_3\). There they are able to write down equations for the Scorza correspondence using theta functions and the geometry of Van der Geer and Van Geemen's linear system \(\Gamma_{00}\). Scorza correspondence; spin curves; theta functions DOI: 10.1007/s00229-012-0564-z Families, moduli of curves (algebraic), Theta functions and curves; Schottky problem, Analytic theory of abelian varieties; abelian integrals and differentials, Special divisors on curves (gonality, Brill-Noether theory) The Scorza correspondence in genus 3 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study the cup product on the Hochschild cohomology of the stack quotient \([X/G]\) of a smooth quasi-projective variety \(X\) by a finite group \(G\). More specifically, they provide a \(G\)-equivariant sheaf of graded algebras on \(X\) whose \(G\)-invariant global sections recover the associated graded algebra of the Hochschild cohomology of \([X/G]\), under a natural filtration.
Using Hochschild cohomology, the autors provide a proof that Kontesvich's formality theorem does not hold for Deligne-Mumford stacks for the cup product.
In the case of a symplectic group action on a symplectic variety \(X\), the authors discuss relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, the authors employ compatible trivializations of the determinants of the normal bundles of the fixed loci in \(X\), which requires (for the cup product) a nontrivial normalization missing in previous literature.
The structure of the paper is: in Section \(2\) we find a discussion on some projective examples. Section \(3\) is dedicated to background material. In Section \(4\) the authors introduce the smash product and establish the necessary relationships between the local Hochschild cohomology \(HH_\star(X)\) and the cohomology of the stack quotient \(X\). Section \(5\) is devoted to some generic relations between normal bundles and tangent bundles for the fixed spaces, which are used in Sections \(6\) and \(7\) to give a geometric description the Hochschild cohomology \(HH_\star(X)\) as a sheaf of algebras on \(X\).
The authors discuss formality and Calabi-Yau orbifolds in Section \(8\) and in Section \(9\) they describe the Hochschild cohomology of the quotient orbifold of a symplectic variety by symplectic automorphisms. Finally, in Section \(10\) the authors establish some linear identifications between Hochschild cohomology and orbifold cohomology, and provide a number of conjectures regarding the relationship between the cup product on Hochschild cohomology and the orbifold product on orbifold cohomology. Hochschild cohomology; orbifolds; orbifold cohomology; formality Generalizations (algebraic spaces, stacks), Topology and geometry of orbifolds, Orbifold cohomology, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) The Hochschild cohomology ring of a global quotient orbifold | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is the authors' ongoing efforts to establish the \(K\)-equivalence conjecture [\textit{C.-L. Wang}, J. Algebr. Geom. 12, No. 2, 285--306 (2003; Zbl 1080.14510)] for general ordinary flops, which is an important case of the crepant transformation conjecture.
A flop is a birational surgery on a smooth algebraic variety which modifies a small part of the variety called the exceptional locus. The exceptional locus of an ordinary flop is in general a projectivised non-split vector bundles over an arbitrary base. For a simple flop where the base is a point, the \(K\)-equivalence conjecture has been proved in [\textit{Y.-P. Lee} et al., Ann. Math. (2) 172, No. 1, 243--290 (2010; Zbl 1272.14040)] in the genus zero case, and in [\textit{Y. Iwao} et al., J. Reine Angew. Math. 663, 67--90 (2012; Zbl 1260.14068)] for all genera.
The background and some recent development on the \(K\)-equivalence relation among birational manifolds has been surveyed in [\textit{C.-L. Wang}, in: Second international congress of Chinese mathematicians. Proceedings of the congress (ICCM2001), Taipei, Taiwan, December 17--22, 2001. Somerville: International Press. 199--216 (2004; Zbl 1328.14022)].
The current paper is a continuation of \textit{Y.-P. Lee} et al. [``Invariance of quantum rings under ordinary flops. I: Quantum corrections and reduction to local models'', Preprint, \url{arXiv:1109.5540}; ``Invariance of quantum rings under ordinary flops. II: A quantum Leray-Hirsch theorem'', Preprint, \url{arXiv:1311.5725}] where the authors proved that a general ordinary flop over an smooth base induces an isomorphism of big quantum rings in the genus zero case. In the first two papers, the authors showed that the general case can be reduced to the case of the standard local models for such flops; they then verified the case of split vector bundles for the local models. The major effort of the third paper is to further reduce the non-split case of local models to the split case.
To prove the reduction, the authors observed that after a sequence of blow-ups of the base of a given local model, the vector bundle can be deformed to a split one. Since Gromov-Witten invariants are deformation invariant, the challenge is to analyze the invariants under blow-ups of the base. To relate these invariants of the blow-ups, the authors employed similar ideas as in [\textit{D. Maulik} and \textit{R. Pandharipande}, Topology 45, No. 5, 887--918 (2006; Zbl 1112.14065)], and applied the degeneration formula for Gromov-Witten invariants [\textit{J. Li}, J. Differ. Geom. 60, No. 2, 199--293 (2002; Zbl 1063.14069)] and [\textit{A.-M. Li} and \textit{Y. Ruan}, Invent. Math. 145, No. 1, 151--218 (2001; Zbl 1062.53073)]. Lee, Yuan-Pin; Lin, Hui-Wen; Qu, Feng; Wang, Chin-Lung, Invariance of quantum rings under ordinary flops III: A quantum splitting principle, Camb. J. Math., 4, 3, 333-401, (2016) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Invariance of quantum rings under ordinary flops. III: A quantum splitting principle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Mirror symmetry was discovered in the late 1980's by physicists working with super-conformal field theories. At the ICM in Zürich in 1994, Kontsevich proposed that mirror symmetry can be described as an equivalence of triangulated categories between the derived Fukaya category \(D\text{ Fuk}(X)\) and the bounded derived category of coherent sheaves \(D^b(Y)\) on the mirror manifold \(Y\). Although \(X\) and \(Y\) are three dimensional Calabi-Yau varieties in the situation envisaged by physicists, it is also interesting to study homological mirror symmetry in lower dimension. The one dimensional case was first studied by \textit{A. Polishchuk} and \textit{E. Zaslow} [Adv. Theor. Math. Phys. 2, No.~2, 443--470 (1998; Zbl 0947.14017)]. In the paper under review, the author explicitly calculates the mirror map in the case of dimension one. This mirror map sends a complexified symplectic form on a two-torus to the complex moduli parameter of the corresponding mirror manifold, which is an elliptic curve. More explicitly, each \(\tau\) in the upper half-plane gives a complexified symplectic form \(\tau dx\wedge dy\) on the two-torus and the mirror map should send it to \(j(\tau)\), the absolute invariant of the elliptic curve given by the lattice \({\mathbb Z}\tau\oplus {\mathbb Z}\) in \({\mathbb C}\). The interesting point is that this mirror map, i.e. the \(j\)-function, can be calculated purely in terms of the structure of the Fukaya category, because the homogeneous coordinate ring \(\bigoplus_{k\geq0} H^0({\mathcal O}(k))\) of the mirror elliptic curve is isomorphic to \(\bigoplus_{k\geq0} \text{ Hom}((\psi({\mathcal O}),\psi({\mathcal O}(k)))\) via the equivalence \(\psi\) from the conjecture. Of course, it is crucial here that the derived Fukaya category is explicitly known in the one dimensional case. In order to actually carry out calculations, the objects \(\psi({\mathcal O}(k))\) of the Fukaya category have to be identified. This is done under the assumption that the line bundle \({\mathcal O}(1)\) is of degree three, so that the mirror elliptic curve appears as a plane cubic. To describe the \(\psi({\mathcal O}(k))\), an important ingredient is the general fact that the monodromy around the large complex structure limit point corresponds to tensor product by \({\mathcal O}(1)\) on the derived category. This monodromy is then seen to be equal to the cube of a minimal Dehn twist. This is a sufficient basis for the calculations which involve interesting relations among theta functions. theta function; modular function; symplectic two-torus; mirror map; Lagrangian sub-manifold Eric Zaslow, Seidel's mirror map for the torus, Adv. Theor. Math. Phys. 9 (2005), no. 6, 999 -- 1006. Calabi-Yau manifolds (algebro-geometric aspects), Symplectic aspects of Floer homology and cohomology Seidel's mirror map for the torus | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 finite-dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite-dimensional simple spherical representations of double affine Hecke algebras.
Double affine Hecke algebras (DAHA) were introduced by Cherednik about fifteen years ago to prove MacDonald conjectures. The understanding of their representation theory has progressed very much recently, in particular by the classification of the simple modules in the category \(\mathcal O\) by \textit{E. Vasserot} [in Duke Math. J. 126, No. 2, 251--323 (2005; Zbl 1114.20002)] (when the parameters are not roots of unity). The latter is very similar to Kazhdan-Lusztig classification of simple modules of affine Hecke algebras. One can show that any simple module in the category \(\mathcal O\) is the top of a module induced from an affine Hecke subalgebra (see Corollary A.3.6). However, the representation theory of DAHA has some specific features that have no analogues for affine Hecke algebras (e.g., it is very difficult to classify the finite-dimensional simple modules).
This can be approached in several ways. The DAHA, denoted by \(\mathbf H\), admits two remarkable degenerated forms. The first one, the degenerated DAHA, denoted by \(\mathbf H'\), is an analogue of the degenerate Hecke algebras introduced by Drinfeld and Lusztig. Its representation theory is more or less the same as that of \(\mathbf H\). The second one was introduced by \textit{P. Etingof} and \textit{V. Ginzburg} [Invent. Math. 147, No. 2, 243-348 (2002; Zbl 1061.16032)] and is called the rational DAHA (or rational Cherednik algebra). We denote it by \(\mathbf H''\).
In this article, we concentrate on the spherical finite-dimensional modules. The case of nonspherical modules can probably be done with similar techniques. We come back to this issue later. The article contains two main results.
First, we classify all spherical finite-dimensional simple \(\mathbf H''\)-modules in Theorem 2.8.1. Since the finite-dimensional simple \(\mathbf H''\)-modules belong to the category \(\mathcal O\), each of them is the top of a standard module. The spherical ones are the top of a polynomial representation (which is equal to a standard module induced from the trivial representation of the Weyl group). So they are labelled by the value of the parameter of \(\mathbf H''\), which is a rational number \(c=k/m\) with \((k,m)=1\) and \(m>0\). Surprisingly, the classification we get is extremely simple and nice. The spherical finite-dimensional simple modules correspond to the integers \(k,m\) such that \(k<0\) and \(m\) is an elliptic regular number (i.e., the integer \(m\) is the order of an elliptic element of the Weyl group which is regular in Springer's sense). In type \(E_8\), for instance, there are twelve elliptic regular numbers. The only known cases before were the case where \(m\) is the Coxeter number in arbitrary type and the dihedral types (in particular, all rank 2 types). Notice that in this article we assume that \(\mathbf H''\) is crystallographic with equal parameters. The proof is as follows. Any simple spherical finite-dimensional \(\mathbf H''\)-module \(M''\) also has the structure of a simple spherical \(\mathbf H\)-module, denoted by \(M\). The algebra \(\mathbf H''\) has two remarkable polynomial subalgebras (yielding, under induction, two representations) called the polynomial representations. A spherical finite-dimensional \(\mathbf H''\)-module is a quotient of both polynomial representations. Using this, one can identify \(M\) with the top of a standard \(\mathbf H\)-module with explicit Langlands parameters (see [Vasserot, loc. cit.] for the terminology). Using the Fourier-Sato transform of perverse sheaves shows that this explicit module is finite-dimensional precisely when \(m\) is elliptic regular.
In the second part of the article, we describe explicitly all the spherical Jordan-Hölder factors (modulo a technical hypothesis). This classification (contrarily to the first one) relies on a case-by-case computation. It is quite remarkable that it involves interesting combinatorial objects that already appear in local Langlands correspondence for \(p\)-adic groups. Affine Hecke algebras are related to unramified Langlands correspondence via Bernstein's functor. DAHAs seem to be related to the tamely ramified correspondence.
The first chapter contains standard facts on elliptic regular elements in Weyl groups, conjugacy classes of tori in \(G\), and homogeneous regular semisimple elements in the loop Lie algebra \(\mathfrak g=\text{Lie}(G_0)\otimes\mathbb C((\varepsilon))\). In particular, Corollary 1.3.3 gives a criterion for the existence of homogeneous elliptic regular semisimple elements in \(\mathfrak g\) which is important for the rest of the article.
In Sections 2.1, 2.2, and 2.3, we recall the definitions and the main properties of DAHAs, degenerate DAHAs, and rational DAHAs. We introduce the category \(\mathcal O\), the polynomial representation, and the spherical modules for each of these algebras. Propositions 2.1.7 and 2.2.4 are analogues of theorems of Lusztig on affine Hecke algebras which compare the categories \(\mathcal O\) of \(\mathbf H\) and \(\mathbf H'\). Proposition 2.3.1 compares the categories \(\mathcal O\) of \(\mathbf H'\) and \(\mathbf H''\).
In Section 2.4, we introduce the affine Springer fibers and the \(\widehat{\mathbf H}\)-action in their homology. The algebra \(\widehat{\mathbf H}\) is another version of \(\mathbf H\) (both algebras are isogenous); it is the one that appears in the geometric picture. The comparison of the modules of \(\mathbf H\) and \(\widehat{\mathbf H}\) is given in Corollary 2.5.8. The simple modules of \(\widehat{\mathbf H}\), \(\mathbf H\) are classified in Proposition 2.5.1 and Theorem 2.5.3, respectively. Section 2.6 contains generalities on the Fourier-Sato transform. In particular, it is related to the Iwahori-Matsumoto involution in Lemma 2.6.1. In Section 2.7, we give a geometric description of the polynomial representation (in Lemma 2.7.2), which yields a simple characterization of the simple finite-dimensional spherical \(\widehat{\mathbf H}\)-modules in Lemma 2.7.3. Section 2.8 contains the main result of the article (i.e., Theorem 2.8.1), which gives a complete list of all simple finite-dimensional spherical \(\mathbf H''\)-modules.
Section 3 contains a description of the simple finite-dimensional spherical modules that appear (with multiplicities) in the homology of affine Springer fibers. First,
Theorem 3.3.1 classifies the isomorphism classes of Jordan-Hölder composition factors of the homology of the affine Springer fiber of an elliptic semi-simple regular element of \(\mathfrak g\) in terms of a set \(\mathcal X_{c,1,RS}\) of local systems. Then Theorem 3.3.6 yields an explicit description of the set \(\mathcal X_{c,1,RS}\) under the technical hypothesis of Conjecture 3.3.3. The proof of Theorem 3.3.6 is based on an explicit description of the affine Springer map in the case we are interested in. It uses the technical results in Sections 3.1 and 3.2, which are proved via the evaluation map \(\mathbb C((\varepsilon))\to\mathbb C\), \(f(\varepsilon)\mapsto f(1)\). The assumption of Conjecture 3.3.3 is checked in a large number of cases in Section 4 (on a case-by-case analysis). double affine Hecke algebras; degenerate Hecke algebras; simple modules in category \(\mathcal O\); simple spherical representations; induced modules; rational Cherednik algebras; spherical finite-dimensional modules M. Varagnolo and E. Vasserot, Finite dimensional representations of DAHA and affine Springer fibers: the spherical case, Duke Math. J., 147 (2007), 439--540. Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds Finite-dimensional representations of DAHA and affine Springer fibers: the spherical case. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\lambda\) be a root of unity of odd order \(\ell\) and \(\mathfrak{u} = \mathfrak{u}_\lambda(\mathfrak{sl}_2)\) the small quantum group of order \(\ell^3\) (a slight variation of the usual one). The author introduces a chain of finite-dimensional complex algebras \((\mathcal D_{\lambda,N} (\mathfrak{sl}_2))_{N\in \mathbb N_0}\), with \(\mathcal D_{\lambda,N-1}(\mathfrak{sl}_2) \hookrightarrow \mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) a cleft extension of \(\mathfrak{u}\)-comodule algebras. Putting all of them together in \(\mathcal D_{\lambda} (\mathfrak{sl}_2) := \lim\limits_{\to} \mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) one gets a quantized version of the algebra of distributions of \(SL_2\) (in positive characteristic). The algebras \(\mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) have triangular decompositions, hence highest weight modules; the classification of the simple representations of \(\mathcal D_{\lambda,N} (\mathfrak{sl}_2)\) follows in a familiar way. Every simple \(\mathcal D_{\lambda,N} (\mathfrak{sl}_2)\)-module admits a tensor product decomposition, where the first factor is a simple \(\mathfrak{u}_\lambda(\mathfrak{sl}_2)\)-module and the second factor is a simple \(\mathcal D_{\lambda,N-1}(\mathfrak{sl}_2)\)-module; this factorization is meant to be a quantum version of the celebrated Steinberg decomposition theorem. The motivation behind these constructions and results is a new approach to a character formula of simple modules over a simple algebraic group proposed in [\textit{G. Lusztig}, Represent. Theory 19, 3--8 (2015; Zbl 1316.20049)], in turn stimulated by the counterexamples to a previous conjecture presented in [\textit{G. Williamson}, J. Am. Math. Soc. 30, No. 4, 1023--1046 (2017; Zbl 1380.20015)]. pointed Hopf algebras; Frobenius-Lusztig kernels; algebras of distributions Angiono, Iván Ezequiel, A quantum version of the algebra of distributions of {\({\mathrm SL}_2\)}, Publications of the Research Institute for Mathematical Sciences, 54, 1, 141-161, (2018) Affine algebraic groups, hyperalgebra constructions, Quantum groups (quantized enveloping algebras) and related deformations, Hopf algebras and their applications, Linear algebraic groups over arbitrary fields A quantum version of the algebra of distributions of \(\mathrm{SL}_2\) | 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 The very interesting paper under review can be seen as a finite type version of a celebrated result in representation theory, due to \textit{E. Date} et al. [J. Phys. Soc. Japan 50, 3806--3812 (1981; Zbl 0571.35099)]. The latter supplied a precise description of the ring \(\mathbf{C}[x_1,x_2,\ldots]\) in infinitely many indeterminates as a representation of the Lie algebra of the complex valued matrices of infinite size with finitely many non-zero diagonals.
The DJKM representation is based on the fact that the so-called \textit{Fermionic Fock Space} can be seen as the fundamental representation of the infinite Lie algebra \(gl_\infty(\mathbb{C})\). From a representation theoretical point of view, the \textit{Fermionic Fock space} can be roughly thought of an infinite wedge power of the vector space \(\mathbb{C}[X^{-1}, X]\) of Laurent polynomials in the indeterminate \(X\). The natural question the authors asked themselves is how the DJKM picture can be detected already for polynomial rings in finitely many indeterminates. To explain what and how they do let us walk a few steps backward, to render more precisely the feeling of their result.
To begin with, le \(K[X]\) be the ring of polynomials in one indeterminate over a field \(K\) of characteristic zero. Let \(B_r:=K[x_1,\ldots,x_r]\) be the \(K\)-algebra of polynomials in the indeterminates \(\mathbf{x}:=(x_1,\ldots,x_r)\). It is easy to convince oneself that \(B_r\) is isomorphic to the \(r\)-th exterior power \(\bigwedge^rK[X]\). Although a number of mathematicians like to explain this fact as a special case of some sophisticated \textit{Geometric Satake Correspondence}, the naive reason is that both spaces possess a basis parametrized by all the partitions of length at most \(r\). It is also easy to see that, as the \(r\)-th exterior power of \(K^n\) is a representation of the Lie algebra \(gl_n(K)\) of the \(K\)-valued \(n\times n\) square matrices, then \(\bigwedge^rK[X]\) turns into a representation of the Lie algebra \(gl_\infty(K)\) of the \(K\)-valued matrices \((a_{ij})_{i,j\geq 0}\) whose entries are all zero but finitely many. The isomorphism \(B_r\rightarrow \bigwedge^rK[X]\) then makes \(B_r\) itself into a representation of \(gl_\infty(K)\).
The main result of the paper under review, Theorem 4.11, consists in determining what the authors call the \textit{generating formal power series} \(\mathcal{E}(z,w^{-1}, t_1,\ldots, t_r)\), which describes \(B_r\) as a representation of the Lie algebra \(gl_\infty(K)\). This means the following. Recall that the elementary matrices \(E_{i,j}\) with all entries zero but \(1\) in position \((i,j)\) form a basis of \(gl_\infty(K)\) and that, interpreting \(B_r\) as the ring of symmetric polynomials in \(r\) indeterminates, it possesse a basis of Schur determinants \(\Delta_{\lambda}\) constructed out of the complete symmetric polynomials, parametrized by partitions of length at most \(r\). If \(\mathbf{t}_r:=(t_1,\ldots,t_r)\) is an \(r\)-tuple of formal variables, denote by \(s_{\lambda}(\mathbf{t}_r)\) the Schur polynomials in the indeterminates \(\mathbf{t}_r\), like at p. 40 of the book by \textit{I. G. Macdonald} [Symmetric functions and Hall polynomials. With contributions by A. V. Zelevinsky. Reprint of the 1998 2nd edition. Oxford: Oxford University Press (2015; Zbl 1332.05002)].
The determination of the generating formal power series of the aforementioned Theorem 4.11 heavily relies on the techniques introduced in the 2005 reviewer's paper [\textit{L. Gatto}, Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)] and substantially improves results by \textit{L. Gatto} and \textit{P. Salehyan} [Commun. Algebra 48, No. 1, 274--290 (2020; Zbl 1442.14156)].
How does this relate with DJKM work? This is widely discussed in the last section of the paper under review. In a nutshell, taking a suitable limit \(r\to\infty\), one obtains an isomorphism from the ring \(B:=B_\infty\) to the charge zero vector subspace \(\mathcal{F}_0\) of the \textit{fermionic Fock space} \(\mathcal{F}:=\bigwedge^{\infty/2}K[X^{-1}, X]\). There is a vector space isomorphism of \(\mathcal{F}_0\) with the ring \(B\) of polynomials in infinitely many indeterminates \((x_1,x_2,\ldots)\) (the \textit{bosonic Fock space}), because both spaces possess a basis parametrized by partitions. The literature (see, e.g., [\textit{V. G. Kac} et al., Highest weight representations of infinite dimensional Lie algebras. World Scientific (2013)]) often refers to this isomorphism as the \textit{boson-fermion correspondence.} As a consequence, \(B\) is made into a representation of the Lie algebra \(gl(\infty)\), induced by the natural one living on the infinite wedge power. Moreover the vertex operators occurring in the DJKM representation are nothing but an infinite dimensional version of what the authors call \textit{Schubert derivations}, borrowing the own reviewer terminology, which, as the name suggest, are devices useful to cope with Schubert Calculus. The paper concludes itself with an essential but comprehensive reference list. Hasse-Schmidt derivations; vertex operators on exterior algebras; representation of Lie algebras of matrices; bosonic and fermionic representations by Date-Jimbo-Kashiwara-Miwa; symmetric functions Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Vertex operators; vertex operator algebras and related structures, Exterior algebra, Grassmann algebras, Symmetric functions and generalizations On the vertex operator representation of Lie algebras of matrices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the present paper, the authors define the relation of mirror symmetry on the class of pairs \((A,\omega_A)\), where \(A\) is an abelian variety and \(\omega_A\) is an element of the complexified ample cone \(C_A \subset NS_A (\mathbb{C})\) of the abelian variety \(A\). This notion of mirror symmetry is purely algebraic, on the one hand, but it is compatible with the usual analytic notion of mirror symmetry for Calabi-Yau manifolds, on the other hand.
As this notion of mirror symmetry for the so-called algebraic pairs \((A, \omega_A)\) of structured abelian varieties is extremely subtle and conceptually involved, it takes the authors ten chapters to rigorously establish it, via various delicate and original constructions, and to analyze its properties as well as its significance in the general realm of mirror symmetry.
After a very careful and motivating introduction to the subject of study, the first eight chapters provide the methodical and technical base for the main goal of the paper. Finally, in the remaining two chapters, the authors define the notion of mirror symmetry for algebraic pairs of complex tori and abelian varieties, analyze its properties and discuss another variant of this construction.
Without any doubt, this paper provides very substantial ideas and constructions towards the just as deep as fascinating problem of mirror symmetry in geometry and physics. Although being technically highly involved and methodically rather extensive (and intensive), the exposition is throughout very clear, detailed, rigorous and well-structured. The numerous remarks and comments that point to related developments, in this context, testify to both the comprising reflection of the authors and the actual value of this work. Néron-Severi group; mirror symmetry; abelian variety; complex tori Golyshev V., Lunts V. and Orlov D. (2001). Mirror symmetry for abelian varieties. J. Algebraic Geom. 10: 433--496 Calabi-Yau manifolds (algebro-geometric aspects), Abelian varieties and schemes, Lie algebras of linear algebraic groups Mirror symmetry for abelian varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Kontsevich moduli stacks of stable maps arise as generalizations of the
classical Deligne-Mumford spaces of stable curves. Their intersection theory
has been intensively studied in the last decade in relation to enumerative
geometry and string theory.
Partial results are known about the cohomology or the Chow groups of the
Deligne-Mumford spaces in low codimension or low genus. Higher genera are
particularly difficult since nontautological classes do exist. Very little
is known even about the tautological rings. However, the genus zero case is
well understood. Keel proved that the cohomology is tautological, in fact,
generated by boundary classes of curves with fixed dual graph.
As the moduli spaces of stable curves are
examples of Kontsevich spaces, it was suggested that it may be useful
to push the investigation of the tautological rings in the context of Gromov-
Witten theory. Here, we study the generalization of \textit{S. Keel}'s theorem [Trans. Am. Math. Soc. 330, No. 2, 545--574 (1992; Zbl 0768.14002)] to the genus zero Kontsevich spaces of maps to flag varieties \(X\).
There are natural cohomology classes defined on the moduli spaces of stable
maps; their intersection numbers are the Gromov-Witten invariants of \(X\). We
show that these natural classes generate the rational cohomology. This implies
that the Gromov-Witten invariants essentially capture the entire intersection
theory of the Kontsevich moduli spaces.
Among the natural cohomology classes, we single out the boundary classes
of maps with fixed dual graph. We may impose additional constraints making
the marked points or nodes map to certain Schubert subvarieties of \(X\) and
requiring that the image of the map intersect various Schubert subvarieties.
There is a cleaner way of bookkeeping the geometric Schubert-type classes we
just described which leads to the definition of the tautological rings.
The section following this introduction contains
generalities about the spaces of stable maps. We collect there known results
and we fix the notation. We also discuss the Hodge theory needed for our
arguments. We will indicate the proof of Theorem 1 in the second section. The
last section presents a few conjectures. The appendix contains a discussion of
the higher genus tautological systems. Oprea, D.: The tautological rings of the moduli spaces of stable maps Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds The tautological rings of the moduli spaces of stable maps to flag varieties | 0 |
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