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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 A sandwiched singularity is a local ring \(\mathcal O\) which is a birational extension of a two-dimensional regular local ring \(R\). Let us consider a surface \(S\) which has a point whose local ring is a sandwich singularity. The authors study in this paper the equisingularity classes of all birational projections of \(S\) to a plane. This problem was dealt with in \textit{M. Spivakovky}'s paper [Ann. Math. (2) 131, 411--491 (1990; Zbl 0719.14005)]. Spivakovsky divides this problem into two parts: discrete and continuous. In this paper the authors deal with the discrete part.
Any birational projection from a sandwiched singularity to a plane is obtained by the morphism of blowing up a complete \(\mathfrak m_{ O}\)-primary ideal \(J\) in the local ring of a regular point \(O\) on the plane. The goal of this paper is to give the equisingularity type of these ideals. More precisely: Let \(\mathcal O\) be a birational normal extension of a regular local ring \((R,\mathfrak m_O)\); the authors describe the equisingularity type of any complete \(\mathfrak m_O\)-primary ideal \(J\subset R\) such that its blow-up \(\text{Bl}_J(R)\) has some point \(Q\) whose local ring is analytically isomorphic to \(\mathcal O\). This is done by the Enriques diagram of the cluster of base points of any such ideal. Recall that an Enriques diagram is a tree together with a binary relation---proximity--- representing topological equivalence classes of clusters of points in the plane.
Therefore, in section 1 the authors give an overview concerning the language of infinitely near points, sandwiched surface singularities and graphs. A good source for this is the book of \textit{E. Casas-Alvero} [Singularities of Plane Curves. Cambridge University Press (2000; Zbl 0967.14018)]. In section two the authors introduce a technical device, namely the consept of contraction for a sandwiched singularity \(\mathcal O\). A contraction for a sandwiched singularity is the resolution graph \(\Gamma_{\mathcal O}\) of \(\mathcal O\), enriched by some proximity relations between their vertices. After having fixed a sandwich graph, the problem is to find the whole list of possibilities for such proximities. This is achieved in section 3. In section 4 the authors study the problem of describing the equisingularity classes of the ideals for a given sandwiched surface singularity, i.e., they describe all the possible Enriques diagram for~\(\mathcal O\). normal surface singularity; sandwiched singularity; complete ideal; Enriques diagram; equisingularity Alberich-Carramiñana, M.; Fernández-Sánchez, J.: Equisingularity classes of birational projections of normal singularities to a plane, Adv. math. 216, 753-770 (2007) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Regular local rings Equisingularity classes of birational projections of normal singularities to a plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\mathrm{SL}(3,\mathbb C)\), and let \(X=\mathbb C^3/G\) be the quotient variety with the nonsingular crepant resolution \(Y=G\)-Hilb\((\mathbb C^3)\). The famous theorem of Bridgeland-King-Reid proves an equivalence of derived categories \(D^b(\text{coh}([\mathbb C^3/G])) \cong D^b(\text{coh}(Y ))\) via a Fourier-MuKai transform. The left hand side of this equivalence is the bounded derived category of \(G\)-equivariant coherent sheaves on \(\mathbb C^3\) or equivalently the bounded derived category of the Deligne-Mumford stack \([\mathbb C^3/G]\). The right hand side of this equivalence is the bounded derived category of coherent sheaves on \(Y\).
The paper under review obtains a generalization of Bridgeland-King-Reid theorem in the case \(G\) is a finite subgroup of \(\mathrm{GL}(3, \mathbb C)\). Suppose that a \(\mathbb Q\)-divisor \(B\) on \(X\) is defined by \(\pi^*(K_X + B) = K_{\mathbb C^3}\), where \(\pi: \mathbb C^3\to X\) is the natural projection. Let \(V_j\subset \mathbb C^3 \quad (j = 1,\dots, m)\) be all the proper linear subspaces whose inertia subgroups \(I_j\) are non-trivial and not contained in \(\mathrm{SL}(3,\mathbb C)\), and let \(D_j <G\) be the subgroup keeping \(V_j\) invariant. The paper under review proves that there exist smooth affine varieties \(Z_i \quad (i = 1,\dots, l)\) and a projective birational morphism \(f:Y\to X\) from a normal variety \(Y\) with only terminal quotient singularities, which is called a maximal \(\mathbb Q\)-factorial terminalization for the pair \((X, B)\), such that there are fully faithful functors
\[
\phi_i : D^b(\text{coh}(Z_i)) \to D^b(\text{coh}([\mathbb C^3/G]))
\]
for \(i = 1,\dots, l\) and
\[
\phi : D^b(\text{coh}(\tilde Y )) \to D^b(\text{coh}([\mathbb C^3/G])),
\]
where \(\tilde Y\) is the smooth Deligne-Mumford stack associated to \(Y\). Moreover, these give a semi-orthogonal decomposition
\[
D^b(\text{coh}([\mathbb C^3/G]))\cong \langle \phi_1( D^b(\text{coh}(Z_1))),\dots, \phi_l ( D^b(\text{coh}(Z_l))),\phi( D^b(\text{coh}(Z))) \rangle.
\]
Finally, one of the following hold: {\parindent=6mm \begin{itemize}\item[(0)] \(\dim Z_i = 0\), and \(Z_i\cong V_j\) is the origin for some \(j\). \item[(1)] \(\dim Z_i = 1\), and \(Z_i\) is a smooth rational affine curve, and there is a finite morphism \(Z_i \to V_j/G_j\) for some \(j\), where \(G_j:=D_j/I_j\). \item[(2)] \(\dim Z_i = 2\), and \(Z_i \to V_j/G_j\) is the minimal resolution of singularities of the quotient surface for some \(j\).
\end{itemize}} The correspondence \(\{1, \dots, l\} \to \{1, \dots, m\}\) given by \(i \mapsto j\) from the last part of the result above is not necessarily injective nor surjective.
This result is an example of DK-hypothesis saying that equalities and inequalities of canonical divisors correspond to equivalences and semi-orthogonal decompositions of derived categories. McKay correspondence; derived category; minimal model program; toric variety Kawamata, Y., Derived mckay correspondence for \(\operatorname{GL}(3, \mathbb{C})\), in preprint McKay correspondence, Minimal model program (Mori theory, extremal rays), Toric varieties, Newton polyhedra, Okounkov bodies Derived McKay correspondence for \(\mathrm{GL}(3,\mathbb{C})\) | 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 Let \(R\) be an isolated hypersurface singularity. \textit{M. Hochster}'s theta invariant \(\theta(M,N)\) of any two finitely generated \(R\)-modules \(M\), \(N\) is defined to be \(\text{length}(\text{Tor}_{2i}^R(M,N))-\text{length}(\text{Tor}_{2i+1}^R(M,N))\) for \(i\gg 0\) [Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 93--106 (1981; Zbl 0472.13005)]. In a recent preprint Dao has conjectured that the pairing \(\theta\) vanishes if the dimension of \(R\) is even and \(R\) contains a field. The main result of the paper under review is a proof of this conjecture when moreover \(R\) is graded with its irrelevant maximal ideal giving the isolated singularity.
The paper's main technique is to factorize the \(\theta\)-pairing through étale or singular cohomology via the Chern character. This technique leads to interesting results also when \(\dim(R)\) is odd: if moreover the characteristic of \(R\) is zero the authors show that the pairing \((-1)^{\frac{n+1}{2}}\theta\) is positive semi-definite, they identify its kernel using the Hodge-Riemann bilinear relations and they finally show that an \(R\)-module \(M\) is Tor-rigid if \(\theta(M,M)=0\). In the final chapter the authors extend their results to hypersurfaces that are homogeneous with respect to a non-standard grading. Hochster's theta invariant; isolated hypersurface singularity; Hodge-Riemann bilinear relations; Tor-rigidity; Chern character; étale cohomology; singular cohomology; Hilbert series Moore, W. F.; Piepmeyer, G.; Spiroff, S.; Walker, M. E., \textit{hochster's theta invariant and the Hodge-Riemann bilinear relations}, Adv. Math., 226, 1692-1714, (2011) Homological functors on modules of commutative rings (Tor, Ext, etc.), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Riemann-Roch theorems, Chern characters Hochster's theta invariant and the Hodge-Riemann bilinear relations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a complex \(2\)-dimensional vector space and let \(\Gamma\) be a finite subgroup of \(\mathrm{SL}(V)\). Up to conjugacy, such subgroups are classified by the Dynkin diagrams of type \(A\), \(D\) and \(E\). Suppose \(\Delta\) is the Dynkin diagram of \(\Gamma\). The quotient \(V/\Gamma\) embeds as a hypersurface in \(\mathbb{A}^3\) and is a Kleinian singularity or rational double point of type \(\Delta\). Identifying \(V/\Gamma\) with the set of zeros of a weight homogeneous polynomial in \(\mathbb{C}^3\), it follows that there is a Possion bracket on the coordinate ring \(\mathbb{C}[V]^\Gamma\) and an associated Poisson structure on the polynomial ring \(\mathbb{C}[x,y,z]\).
In this paper, the author constructs all possible noncommutative deformations of \(V/\Gamma\) of type \(D\) in terms of generators and relations. He also proves that the moduli space of isomorphism classes of noncommutative deformations in type \(D_n\) is isomorphic to a vector space of dimension \(n\).
Earlier, \textit{P. Boddington} already constructed all possible noncommutative deformations of type \(D\) Kleinian singularities in his Ph.D. thesis [``No-cycle algebras and representation theory'', Ph.D. thesis, University of Warwick (2004)]. Boddington's parametrization of the noncommutative deformations is more closely related to the construction by \textit{W. Crawley-Boevey} and \textit{M. P. Holland}, [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)], than the construction in the article under review which uses methods similar to those of \textit{V. V. Bavula} and \textit{D. A. Jordan}, [Trans. Am. Math. Soc. 353, No. 2, 769-794 (2001; Zbl 0961.16016)]. Kleinian singularities of type \(D\); noncommutative deformations; simply laced Dynkin diagrams; coordinate rings; Poisson brackets; generators and relations; moduli spaces Levy, P., Isomorphism problems of noncommutative deformations of type \textit{D} Kleinian singularities, Trans. Amer. Math. Soc., 361, 5, 2351-2375, (2009) Deformations of associative rings, Rings arising from noncommutative algebraic geometry, Deformations of singularities, Noncommutative algebraic geometry Isomorphism problems of noncommutative deformations of type \(D\) Kleinian singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An usual way to study the complexity of a local noetherian ring \((A,m)\) is in terms of its associated category \(CM(A)\) of Maximal Cohen-Macaulay (MCM) finitely generated modules (recall that an \(A\)-module \(M\) is Maximal Cohen-Macaulay if \(\mathrm{depth}_A(M)=\dim(A)\)).
When the ring \(A\) is particularly well-behaved, MCM-modules correspond to familiar objects. For instance, when \(A\) is reduced of Krull dimension one, MCM A-modules are exactly the torsion free modules. Analogously, in the case of \(A\) being a normal surface singularity, MCM-modules correspond to the reflexive \(A\)-modules.
The study of MCM-modules rises from the theory of integral representations of finite groups and has revealed to own many connections with other fields, as it can be the theory of vector bundles on projective curves, McKay correspondence for the finite subgroups of \(\mathrm{SL}_2(\mathbb{C})\), etc.
A breakthrough in this area was the classification of hypersurface singularities \((R/f,m)\) which support only a finite number of non-isomorphic indecomposable MCM-modules: they correspond exactly to simple hypersurface singularities \(A_n, D_n,E_6, E_7\) and \(E_8\) [\textit{R. O. Buchweitz} et al., Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)]. Moreover, in the aforementioned paper two limiting non-isolated hypersurface singularities with a countable number of non-isomorphic MCM-modules were identified: \(A_{\infty}\) and \(D_{\infty}\).
\textit{F.-O. Schreyer} [Lect. Notes Math. 1273, 9--34 (1987; Zbl 0719.14024)] whether a non-isolated Cohen-Macaulay surface singularity \((A,m)\) of countable CM-type should be isomorphic to \(A\cong B^G\) which \(B\) either \(A_{\infty}\) or \(D_{\infty}\) and \(G\) a finite group of automorphisms of \(B\).
The monography under review deals with this set of problems. In particular, among the central results, it is exhibited in Theorem \(10.6\) a counterexample to Schreyer's conjecture. The authors also deal with rings of greater CM-complexity. For instance, it is shown that degenerate cusps, like for instance the ordinary triple point \(\mathbb{K}[[x,y,z]]/xyz\), as they were introduced by \textit{N. I. Shepherd-Barron} [Prog. Math. 29, 33--84 (1983; Zbl 0506.14028)], are of tame CM-type (see Definition \(8.13\) and Theorem \(8.15\)).
The technical core of this monography is the construction, associated to a non normal, reduced CM surface singularity \(A\), of the categories \(\mathrm{Tri}(A)\) (category of triples) and \(\mathrm{Rep}(\chi_A)\) (category of elements of a certain bimodule \(\chi_A\) and of two functors
\[
CM(A)\stackrel{F}{\rightarrow} \mathrm{Tri}(A)\stackrel{H}{\rightarrow} \mathrm{Rep}(\chi_A)
\]
such that F is an equivalence and H preserves indecomposibility and isomorphims (see Theorem 3.5 and Proposition 8.9). Therefore, the description of indecomposable MCM-modules is reduced to a problem of linear algebra (which the authors called ``a matrix problem''). Maximal Cohen-Macaulay modules; matrix factorizations; non-isolated surface singularities; degenerate cusps; tame matrix problems Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Cohen-Macaulay modules, Singularities of surfaces or higher-dimensional varieties Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 provides a new viewpoint on the remarkable relationship between rational double points, Dynkin diagrams and subgroups of SU(2) investigated in particular by \textit{E. Brieskorn} [Actes Congr. Int. Math. 1970, 2, 279-284 (1971; Zbl 0223.22012)]. The setting is that of the gradient flow of the function
\[
\phi (A_ 1,A_ 2,A_ 3)=\sum^{3}_{1}(A_ i,A_ i)+(A_ 1,[A_ 2,A_ 3])
\]
defined on \({\mathfrak g}\otimes R^ 3\). This can also be thought of as the Chern- Simons functional on a space of left-invariant G-connections on the 3- sphere, which reinterprets the space of trajectories as a space of instantons on \(S^ 4\). In this interpretation the space also has a natural hyperkähler structure, a fact which underlies many of the features in the paper, but which the author does not emphasize too strongly. The main analytical result is to parametrize the solutions of the ODE which describes the gradient flow of \(\phi\) by using the complex nilpotent orbits of the complex group \(G^ c\). This is carried out in a manner modelled on Donaldson's parametrization of monopoles [\textit{S. K. Donaldson}, Commun. Math. Phys. 96, 387-407 (1984; Zbl 0603.58042)].
There are two particularly attractive outcomes of this result. One is a new proof of Brieskorn's theorem that the nilpotent variety has a singularity along the subregular orbit of the type \(C^ 2/\Gamma\) where \(\Gamma\) is a finite subgroup of SU(2). The other is a demonstration of two copies of SO(3) in the group \(E_ 8\) which intersect in the icosahedral group. Nahm's equation; simple singularity; instantons; nilpotent variety P.B. Kronheimer, \textit{Instantons and the geometry of the nilpotent variety}, \textit{J. Diff. Geom.}\textbf{32} (1990) 473 [INSPIRE]. Moduli problems for differential geometric structures, Analysis on real and complex Lie groups, Singularities of surfaces or higher-dimensional varieties Instantons and the geometry of the nilpotent variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(N\) be an integer greater than 1 and let \(\Gamma(N)\) denote the subgroup of the modular group \(\Gamma(1) = \mathrm{SL}_2 (\mathbb Z)\) consisting of the matrices \(\gamma\equiv 1\pmod N\). \(\Gamma(1)\) acts on the upper half plane \(\mathfrak H\) and the modular function \(j\) defines a complex analytic isomorphism \(\Gamma(1)\backslash\mathfrak H \to \mathbb P^1(\mathbb C) - \infty\) whose one-point compactification is the Riemann surface \(\mathbb P^1(\mathbb C)\). The subgroup \(\Gamma(N)\) acts on \(\mathfrak H\) and there is a complex analytic isomorphism \(\Gamma(N)\backslash\mathfrak H \to Y(N)\) lifting that defined by \(j\), where \(Y(N)\) is an affine curve, whose compactification \(X(N)\) is obtained by adding the inverse image of \(\infty\) on the \(j\)-line: thus \(X(N) =Y(N)\cup X^\infty(N)\), where the points in \(X^\infty(N)\) are called cusps, after the shape of the fundamental domain for \(\Gamma(N)\). If \(\mathfrak H^* = \mathfrak H \cup \mathbb Q\cup \{\infty\}\), then \(X^\infty(N)\) is the set of equivalence classes of \(\mathbb Q\cup \{\infty\}\) with respect to the action of \(\Gamma(N)\).
The affine ring of regular functions on \(Y(N)\) over \(\mathbb C\) is the integral closure of \(\mathbb C[j]\) in the function field of \(X(N)\) over \(\mathbb C\) and one can also consider, for example, curves defined over \(\mathbb Q(\mu_N)\) (the cyclotomic field of \(N\)-th roots of unity), in which case one obtains the integral closure of \(\mathbb Q[j]\). In what follows we are concerned primarily with functions defined over \(\mathbb Q(\mu_N)\).
The cuspidal divisor class group \(\mathcal C(N)\) consists of the group of divisors of degree 0 whose support is the set \(X^\infty(N)\) of cusps, modulo the group of divisors of functions on \(X(N)\) having neither poles nor zeros outside the cusps; that is, the subgroup \(\mathrm{Pic}^\infty X(N)\) of \(\mathrm{Pic } X(N)\). The units consist of those functions having no zeros nor poles in the upper half plane.
The primary object of the theory presented in this book is the study of \(\mathcal C(N)\) as a module over a certain Cartan group \(C(N)\); namely the reduction \(\bmod N\) of a subgroup of \(\mathrm{GL}_2 (\mathbb Z_N)\), where \(\mathbb Z_N = \prod_{p\mid N} \mathbb Z_p\). The theory is analogous to the study of the ideal class group of \(\mathbb Q(\mu_N)\) as a module over the group ring \(\mathbb Z[G]\), \(G\approx (\mathbb Z/N\mathbb Z)^*\), in the cyclotomic case. Unlike our present knowledge of the cyclotomic case, \(\mathcal C(N)\) admits a complete description. In chapter 5, the authors show that the divisor class group generated by the cusps can be represented as a quotient of the group ring of the Cartan group \(C(N)\) by an analogue of the Stickelberger ideal. Full use is made of the characterization of the units given in chapters 2 and 3 and there are beautiful connections with algebraic geometry. The order of the cuspidal divisor class group is computed in a manner analogous to that used by Iwasawa in the cyclotomic case, the role of the Bernoulli numbers \(B_{1,\chi}\) in the latter case being played by the second Bernoulli numbers \(B_{2,\chi}\) in the case of \(\mathcal C(N)\). Chapter 5 closes with an analysis of the eigenspace decompositions on \(X(p)\), \(p\) a prime, and it turns out that they involve ordinary Bernoulli numbers and Gauß sums.
Chapter 6 deals with the cuspidal divisor class group for the modular curve \(X_1(N)\) obtained as before from the subgroup \(\Gamma_1(N)\) of \(\Gamma(1)\) of matrices \(\gamma\equiv \begin{pmatrix} 1 & b \\ 0 & 1\end{pmatrix}\pmod N\), where \(b\) is arbitrary.
Chapters 7 to 13 are more specialised. In Chapter 7 the authors study the modular units on Tate curves; that is on the elliptic curves \(Y^2 -XY = X^3 - h_2X - h_3\) where \(h_2 =5\,\sum_{n=1}^\infty q^n/(1-q^n)\), \(h_3 = \sum_{n=1}^\infty (5n^3+7n^5)\cdot q^n/12 \cdot (1-q^n)\).
Chapter 8 is concerned with applications to Diophantine equations and in particular to
\[ \frac{X_3-X_1}{X_2-X_1} + \frac{X_2-X_3}{X_2-X_1} = 1 \]
which is satisfied by the \(\lambda\)-function.
The remaining chapters contain an exposition of the theory of Robert's elliptic units in arbitrary class-fields with a unit index computation due to Kersey. The modular units are an example of a universal even distribution. That is, of a mapping \(\varphi\colon \mathbb Q/\mathbb Z\to A\) to an Abelian group such that for every \(N\) and some positive integer \(k\), \(N^k \sum_{j=1}^{N-1} \varphi(x+\tfrac{j}{N}) = \varphi(Nx)\). Distributions of that kind occur in a number of contexts in number theory and the book begins with an account of the basic theory. Not only is the geometric theory of the cuspidal divisor class group analogous to the theory of cyclotomic fields, but also the most exciting developments in recent years have their origins in work of \textit{A. Wiles} [Invent. Math. 58, 1--35 (1980; Zbl 0436.12004)], who first showed how the connection between the two theories can be made.
The present book is to be welcomed both as an exposition of a fascinating subject, much of which appeared first in a series of papers in the Math. Ann. and also as an introduction to the dramatic developments due to Mazur and Wiles [see \textit{S. Lang}, Bull. Am. Math. Soc., New Ser. 6, 253--316 (1982; Zbl 0482.12002)]. modular units; class groups; modular forms; cyclotomic fields; divisor class group; cusp; Cartan group; Stickelberger ideal; Bernoulli numbers; Tate curve Kubert D and Lang S 1981 Modular Units \textit{Grundlehren Wissenschaften} vol 244 (Berlin: Springer) Elliptic and modular units, Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Cyclotomic extensions, Global ground fields in algebraic geometry, Holomorphic modular forms of integral weight Modular units | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Zbl 0743.00050.]
This survey paper consists of three parts and an extensive bibliography. The first part is a general introduction to the theory of singularities of complex analytic spaces, which centers on the many characterisations of the \(ADE\)-singularities. The by now classical highlights are treated: the connection with Lie groups, quotient singularities and the platonic solids, the resolution graph and the Dynkin diagram, simpleness in the sense of Arnol'd. The author also covers the characterisation of the \(ADE\)-singularities by finite representation type. In many places references to newer and further developments are given.
The last two chapters introduce more advanced topics of current research in a careful discussion of problems and questions. The first one is the deformation space of rational surface singularities. The last chapter concerns moduli spaces for singularities and modules over local rings, a subject to which the author has contributed substantially. Whereas in the global case moduli spaces are constructed with Geometric Invariant Theory, and the groups appearing are reductive, one deals here with unipotent groups. simple singularities; deformation theory; bibliography; platonic solids; Dynkin diagram; moduli spaces Greuel, G.-M., Deformation und klassifikation von singularitäten und moduln, 177-238, (1992), Stuttgart Local complex singularities, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Deformation and classification of singularities and modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies singularities of \(\overline{\text{Bun}}_G\) for \(G=\mathrm{SL}_2\), where \(\overline{\text{Bun}}_G\) is a canonical compactification of the moduli stack of \(G\)-bundles on a smooth projective curve, discovered by Drinfeld but still unpublished. The compactification is relative to the diagonal morphism of \(\text{Bun}_G\), and relies on the Vinberg semigroup of \(G\). For technical reasons it is more convenient to work with \(\text{VinBun}_G\), the total space of the canonical \(\mathbb{G}_m\) bundle on \(\text{Bun}_G\), that has the same singularities and is a one-parameter degeneration of \(\text{Bun}_G\).
From the author's summary: ``We study the singularities of this degeneration via the weight-monodromy theory of its nearby cycles: We give an explicit description of the nearby cycles sheaf in terms of certain novel perverse sheaves which we call ``Picard-Lefschetz oscillators'' and which govern the singularities of the degeneration. We then use this description to determine its intersection cohomology sheaf and other invariants of its singularities. We also discuss the relationship of our results for \(G=\mathrm{SL}_2\) with the miraculous duality of \textit{V. Drinfeld} and \textit{D. Gaitsgory} [Camb. J. Math. 3, No. 1--2, 19--125 (2015; Zbl 1342.14041); Sel. Math., New Ser. 22, No. 4, 1881--1951 (2016; Zbl 1360.14060); Ann. Sci. Éc. Norm. Supér. (4) 50, No. 5, 1123--1162 (2017; Zbl 1423.11118)] in the geometric Langlands program, as well as two applications of our results to the classical theory: to \textit{V. Drinfeld} and \textit{J. Wang}'s [Sel. Math., New Ser. 22, No. 4, 1825--1880 (2016; Zbl 1393.11044)] ``strange'' invariant bilinear form on the space of automorphic forms; and to the categorification of the Bernstein asymptotics map studied by \textit{R. Bezrukavnikov} and \textit{D. Kazhdan} [Represent. Theory 19, 299--332 (2015; Zbl 1344.20064)] as well as by \textit{Y. Sakellaridis} and \textit{A. Venkatesh} [Periods and harmonic analysis on spherical varieties. Paris: Société Mathématique de France (SMF) (2017; Zbl 1479.22016)].'' geometric Langlands program; Langlands correspondence for function fields; moduli stack of \(G\)-bundles; Drinfeld-Lafforgue-Vinberg compactification; singularities of the degeneration; miraculous duality of Drinfeld and Gaitsgory; Drinfeld-Wang strange invariant bilinear form S. Schieder, Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for \(\text{SL}_{2}\), Duke Math. J. 167 (2018), 835--921. Geometric Langlands program (algebro-geometric aspects) Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for \(\mathrm{SL}_2\) | 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 Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence was proposed and understood in physical terms to correspond to the study of two (or more) different ``phases'' of certain two dimensional gauge theories [\textit{E. Witten}, Nucl. Phys., B 403, No. 1--2, 159--222 (1993; Zbl 0910.14020)]. Mathematically, the context for the LG/CY correspondence -- at least for CYs realized as complete intersections in weighted projective spaces -- is understood to be, respectively, FJRW theory [\textit{H. Fan} et al., Commun. Pure Appl. Math. 61, No. 6, 745--788 (2008; Zbl 1141.58012)] and genus zero Gromov-Witten (GW) theory. This paper generalizes the notion of FJRW theory for LG models corresponding to a collection of singularities (i.e. the threefolds are complete intersections in projective space), focusing on the only two non-quintic examples, \(X_{3, 3} \subset \mathbb{P}^5\) and \(X_{2, 2, 2, 2} \subset \mathbb{P}^7\).
The main result of the paper is to establish the genus zero LG/CY correspondence for these two examples. This is accomplished by constructing the I-function of the hybrid model, whose coefficients expanded in powers of \(H^{(i)}\) span the solution space of a Picard-Fuchs equation associated to the corresponding Calabi-Yau (after a simple identification). The author establishes that the hybrid I-function and the usual genus zero hybrid J-function are related by an explicit change of variables (the mirror map) given for these two examples in Theorem 1.1.1. The I- and J-functions of the CY side of the correspondence are similarly related to one another in [\textit{A. Givental}, Prog. Math. 160, 141--175 (1998; Zbl 0936.14031)]. The latter is the generating function of genus zero GW invariants for the CY, explicitly \(J_{GW}(\mathbf{t}, z) = z + \mathbf{t} + \sum_{n, \beta}{1 \over n!} \langle \mathbf{t}(\psi), \ldots, \mathbf{t}(\psi), {\varphi_a \over z - \psi} \rangle^{GW}_{0, n+1, \beta} \varphi^a\) with \(\mathbf{t}(z) = t_0 + t_1 z + \ldots \in H_{GW}[z]\) and \(\varphi_a\) running over a basis for \(H_{GW}\). The small J-function is the restriction of the J-function to the degree-two component; for the genus zero correspondence to hold, the small \(J_{\mathrm{hyb}}\) and \(J_{GW}\) functions must coincide.
The author explicitly verifies a degree-preserving isomorphism between the state spaces, \(H_{\mathrm{hyb}}\) and \(H_{GW} = H^*(X)\) for these examples. From this isomorphism between the state spaces in which the respective I-functions take values, and the content of Theorem 1.1.1, it then follows that \(I_{\mathrm{hyb}}\) and the analytic continuation of \(I_{GW}\) in a certain coordinate patch are given in terms of a basis of solutions to the same differential equation and are related by a linear isomorphism that performs the change of basis. This establishes the genus zero LG/CY correspondence for these examples (Corollary 1.1.2). algebraic geometry; mathematical physics; FJRW; Landau-Ginzburg; Calabi-Yau Clader, E., Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections \(X\)\_{}\{3,3\} and \(X\)\_{}\{2,2,2,2\}, arXiv: 1301.5530. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections \(X_{3,3}\) and \(X_{2,2,2,2}\) | 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 We resume the study initiated in our former work [Compos. Math. 154, No. 8, 1659--1697 (2018; Zbl 1403.14097)]. For a generic curve \(C\) in an ample linear system \(| \mathcal{L} |\) on a toric surface \(X\), a vanishing cycle of \(C\) is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of \(C\) to a nodal curve in \(| \mathcal{L} |\). The obstructions that prevent a simple closed curve in \(C\) from being a vanishing cycle are encoded by the adjoint line bundle \(K_X \otimes \mathcal{L} \). In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on \(C\) respectively as an hyperelliptic involution and as a spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group \(\text{MCG}(C)\). We show that the image of the monodromy is the subgroup of \(\text{MCG}(C)\) preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture 1 in [loc. cit.] aiming to describe all the vanishing cycles for any pair \((X, \mathcal{L})\). mapping class group; Torelli group; spin structures Toric varieties, Newton polyhedra, Okounkov bodies, Other groups related to topology or analysis, Deformations of complex singularities; vanishing cycles The vanishing cycles of curves in toric surfaces. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The quotient singularities \({\mathbb{C}}^ 2/G\) where G is a finite subgroup of GL(2,\({\mathbb{C}})\) have been classified by \textit{E. Brieskorn} and the reviewer [``Ebene algebraische Kurven'' (1981; Zbl 0508.14018)]. Using this classification and classical invariant theory the author studies the modularity and equivariant Milnor number of functions on such singularities. In particular he computes the equivariant Milnor number of a generic function on such a singularity. This gives information about he codimension of a generic function on \({\mathbb{C}}^ 2/G\) under the group for right - and contact equivalence and thus a criterion for the existence and a description of simple functions on \({\mathbb{C}}^ 2/G\). Using this technique the author derives a classification of zero-modal functions on each of these singularities. simple singularities; quotient singularities; modularity; equivariant Milnor number; zero-modal functions Wall C T C, Functions on quotient singularities,Philos. Trans. R. Soc. London 324 (1987) 1--45 Singularities in algebraic geometry, Complex singularities Functions on 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 describes three related techniques to work efficiently with power series in the context of (embedded) resolution of singularities, where the key problem is the construction of local invariants for singularities and their control under blowup and localization. The techniques are intended to provide a general framework for producing such invariants and observing their transformation rules.
1. One considers local flags in a regular ambient space, and local coordinates subordinate to the flag. The author constructs canonically new local flags on any blowup with centre transversal to the given flag. Choosing local coordinates subordinate to the flags before and after the blowup allows one to define significant invariants and to read off easily their change under the blowup.
2. The analog of the classical Gauss decomposition is proved for the group \(G\) of automorphisms of the formal (or convergent) power series in the variables \(x_1,\dots,x_n\) (i.e. local automorphisms of affine \(n\)-space): any \(\varphi \in G\) is a product \(\varphi =bus\), where \(b\) is an upper triangular automorphism, \(u\) is a lower unipotent de Joncquière automorphism and \(s\) is a permutation. As the \` Borel automorphisms\'\ \(b\) are just those which stabilize the flag with subordinate coordinates \(x_1,\dots,x_n\), one can restrict to automorphisms \(us\) when constructing subordinate coordinates out of ordinary ones. This is shown to be crucial for controlling a flag invariant under blowup or localization.
3. Initial ideals of an ideal of power series (with respect to monomial orders on \(\mathbb N^n\)) are a frequently used tool to investigate singularities, in particular the coordinate independent \textsl{generic initial ideal}. Here the author proposes to order the set of initial ideals with respect to all coordinate choices of a given ideal (by the ordering on monomial ideals through the lexicographic order taken on their minimal monomial generator systems). Then the minimal initial ideal is just the generic one, but the maximal initial ideal contains much more information on the singularity. Its existence is however not obvious and is shown in the paper. The Gauss decomposition of \(G\) is then used to determine coordinates realizing this maximal ideal.
The author summarizes the most important resolution invariants in the literature and indicates that they are special cases or variants of his flag invariants or minimal/maximal initial ideal. It is also interesting that the constructions in the paper are characteristic independent, thereby providing a potential contribution to the still unsolved characteristic \(p\) case. resolution of singularities; flag; Gauss decomposition of formal automorphisms; initial ideal Herwig Hauser, Three power series techniques, Proc. London Math. Soc. (3) 89 (2004), no. 1, 1 -- 24. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Modifications; resolution of singularities (complex-analytic aspects) Three power series techniques. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose that \(G\subset\mathrm{SO}(3)\) is a polyhedral group of type \(\mathbb Z/n\mathbb Z\), \(D_{2n}\) or tetrahedral. By the result of Bridgeland-King-Reid [\textit{T. Bridgeland} et al., J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] and also Gomi-Nakamura-Shinoda [\textit{Y. Gomi} et al., Asian J. Math. 4, No. 1, 51--70 (2000; Zbl 0981.14002)] it is known that \(G\)-\(\text{Hilb}(\mathbb C^3)\) is always a projective crepant resolution of \(\mathbb{C}^3/G\) and the fiber over \(0\) is 1-dimensional with irreducible rational components in bijection with nontrivial irreducible representations of \(G\). By the result of \textit{Y. Ito} and \textit{H. Nakajima} [Topology 39, No. 6, 1155--1191 (2000; Zbl 0995.14001)] \(G\)-\(\text{Hilb}(\mathbb C^3)\) can be realized as a moduli space of stable representations of the McKay quiver \((Q,W)\) with suitable relations (arising from the potential \(W\)). The space of generic stability conditions of the McKay quiver is the disjoint union of finitely many chambers. For a given chamber \(C\) let \(\mathcal M_C\) be the corresponding moduli space. Bridgeland-King-Reid [Zbl 0966.14028] more generally prove that each \(\mathcal M_C\) is in fact a projective crepant resolution of \(\mathbb{C}^3/G\). \textit{A. Craw} and \textit{A. Ishii} [Duke Math. J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)] conjectured that conversely every projective crepant resolution of \(\mathbb{C}^3/G\) is of the form \(\mathcal M_C\) for some chamber \(C\). This conjecture is proven when \(G\) is abelian.
The paper under review proves this conjecture for any \(G\) as above. It moreover proves that there exists a one-to-one correspondence between flops of \(G\)-\(\text{Hilb}(\mathbb C^3)\) and those mutations of \((Q,W)\) which do not mutate the trivial vertex. As a result to each \(\mathcal M_C\) above is associated an iterated quiver \(Q_C\). The paper under review proves that the dual graph of the fiber over \(0\) of the crepant resolution \(\mathcal M_C\) is the same as the graph of \(Q_C\) with the trivial vertex removed, and that the number of loops at a vertex of \(Q_C\) determines the normal bundle of the corresponding rational component of the fiber. crepant resolutions; polyhedral singularities; flops; mutations; moduli spaces of quiver representations Nolla de Celis, Á.; Sekiya, Y., Flops and mutations for crepant resolutions of polyhedral singularities, Asian J. Math., 21, 1, 1-45, (2017) McKay correspondence, Representations of quivers and partially ordered sets Flops and mutations for crepant resolutions of polyhedral singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For \(G\) a complex reductive subgroup with Borel subgroup \(B\) and an involutive automorphism \(\theta\) one can consider the closure of orbits in \(G/B\) under the action of \(G^{\theta}\), obtaining what are called symmetric varieties. Here, the author investigates these symmetric varieties in the case where \(G\) is a general linear group and \(\theta\) is conjugation by a diagonal matrix, determining when such a \(G^{\theta}\)-orbit has rationally smooth closure.
Specifically, let \(G=\text{GL}(n,\mathbb{C})\) and write \(n=p+q\). Let \(\theta\) be conjugation by a diagonal matrix on \(G\) with eigenvalues \(1\) and \(-1\) or multiplicity \(p\) and \(q\) respectively. Then \(G^{\theta}\) is naturally identified with \(\text{GL}(p,\mathbb{C})\times\text{GL}(q,\mathbb{C}),\) or alternatively is the complexification of the maximal compact subgroup \(U(p)\times U(q)\) of the real form \(U(p,q)\) of \(G\). Taking \(B\leq G\) to be the upper-triangular matrices one knows that the \(G^{\theta}\)-orbits in \(G/B\) are parameterized by sequences \(\gamma=(c_{1},\dots,c_{n}),\) where \(c_{i}\) are symbols which are \(+\) or \(-\) or a natural number, in such a way that every natural number occurs twice or not at all. Such sequences are called clans, and correspond to collections of complete flags in \(\mathbb{C}^{n}.\) We will denote the orbit corresponding to \(\gamma\) by \(\mathcal{O}_{\gamma}.\)
The main (and only)result can be easily stated in terms of clans. If \(\gamma\) includes one of the patterns \((1,+,-,1)\), \((1,-,+,1)\), \((1,2,1,2)\), \((1,+,2,2,1)\), \((1,-,2,2,1)\), \((1,2,2,+,1),\) or \((1,2,2,-1,1)\) then \(\mathcal{O}_{\gamma}\) does not have rationally smooth closure. If \(\gamma\) does not include any of the patterns above it is a derived functor orbit and hence its closure is smooth. representation theory; flag varieties; rational smoothness William M. McGovern, Closures of \?-orbits in the flag variety for \?(\?,\?), J. Algebra 322 (2009), no. 8, 2709 -- 2712. Classical groups (algebro-geometric aspects) Closures of \(K\)-orbits in the flag variety for \(U(p,q)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities To any finite group \(\Gamma\subset\text{Sp}(V)\) of automorphisms of a symplectic vector space \(V\) we associate a new multi-parameter deformation, \(H_\kappa\) of the algebra \(\mathbb{C}[V]\#\Gamma\), smash product of \(\Gamma\) with the polynomial algebra on \(V\). The parameter \(\kappa\) runs over points of \(\mathbb{P}^r\), where \(r=\) number of conjugacy classes of symplectic reflections in \(\Gamma\). The algebra \(H_\kappa\), called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity \(V/\Gamma\). This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If \(\Gamma\) is the Weyl group of a root system in a vector space \(\mathfrak h\) and \(V={\mathfrak h}\oplus{\mathfrak h}^*\), then the algebras \(H_\kappa\) are certain `rational' degenerations of the double affine Hecke algebra introduced earlier by Cherednik.
Let \(\Gamma=S_n\), the Weyl group of \({\mathfrak g}=\mathfrak{gl}_n\). We construct a 1-parameter deformation of the Harish-Chandra homomorphism from \({\mathcal D}({\mathfrak g})^{\mathfrak g}\), the algebra of invariant polynomial differential operators on \(\mathfrak{gl}_n\), to the algebra of \(S_n\)-invariant differential operators with rational coefficients on the space \(\mathbb{C}^n\) of diagonal matrices. The second order Laplacian on \(\mathfrak g\) goes, under the deformed homomorphism, to the Calogero-Moser differential operator on \(\mathbb{C}^n\), with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: \({\mathcal D}({\mathfrak g})^{\mathfrak g}\twoheadrightarrow\) spherical subalgebra in \(H_\kappa\), where \(H_\kappa\) is the symplectic reflection algebra associated to the group \(\Gamma=S_n\). This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of `quantum' Hamiltonian reduction.
In the `classical' limit \(\kappa\to\infty\), our construction gives an isomorphism between the spherical subalgebra in \(H_\infty\) and the coordinate ring of the Calogero-Moser space. We prove that all simple \(H_\infty\)-modules have dimension \(n!\), and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of \(S_n\). Moreover, we prove that the algebra \(H_\infty\) is isomorphic to the endomorphism algebra of that vector bundle. finite groups of automorphisms; symplectic vector spaces; multi-parameter deformations; symplectic reflection algebras; coordinate rings; McKay correspondence; Weyl groups; double affine Hecke algebras; Harish-Chandra homomorphisms; invariant polynomial differential operators; Calogero-Moser differential operators P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero--Moser space, and deformed Harish-Chandra homomorphism, \textit{Invent. Math.}, 147 (2002), no. 2, 243--348. Zbl 1061.16032 MR 1881922 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Group actions on varieties or schemes (quotients), Lie algebras of vector fields and related (super) algebras, Applications of Lie algebras and superalgebras to integrable systems, Hecke algebras and their representations, Rings arising from noncommutative algebraic geometry, Deformations of associative rings, Noncommutative algebraic geometry Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 complex algebraic stack \(X\) has \textit{nodal (codimension one) singularities}, if it is locally isomorphic in the f.p.p.f. topology to \(\{ xy=0\} \times \mathbb{A}^n\). In this case, the singular locus \(D\) is smooth, and \(X\) is said \textit{first-order smoothable}, if the line bundle \(\mathcal{E}xt^1(\omega_X,\mathcal{O}_X)\) on \(D\) is trivial. Let \(W_0 = X_1 \sqcup_D X_2\) be a nodal, first order smoothable proper Deligne-Mumford stack, which is the union of two smooth components \(X_1\) and \(X_2\) meeting transversally along \(D\). In this paper the authors give a new definition of Gromov-Witten invariants for stacks of the form as \(W_0 = X_1 \sqcup_D X_2\), and of relative Gromov-Witten invariants for smooth Deligne-Mumford stacks. In the case of relative Gromov-Witten invariants the target space is a pair \((X,D)\), where \(X\) is a proper Deligne-Mumford smooth stack with projective coarse moduli space and \(D\) is a smooth divisor. A degeneration formula is also proved, which expresses the Gromov-Witten invariants of \(W_0 = X_1 \sqcup_D X_2\) in terms of the relative invariants of the pairs \((X_1, D)\) and \((X_2,D)\).
The approach ``follows closely that of \textit{J. Li}'' [J. Differ. Geom. 57, No. 3, 509--578 (2001; Zbl 1076.14540); J. Differ. Geom. 60, No. 2, 199--293 (2002; Zbl 1063.14069)]. One of the differences is to replace predeformable maps (used in J. Li's approach) with transversal maps, where auxiliary orbifold structures along the nodes of both source curves and target spaces are introduced. This has the advantage of simplifying both, the definition of the obstruction theory on the moduli space of maps (that are used to define the Gromov-Witten invariants), and the proofs of the properties of the Gromov-Witten invariants. Abramovich, D., Fantechi, B.: Orbifold techniques in degeneration formulas, Preprint, Math. AG/1103.5132 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fine and coarse moduli spaces Orbifold thechniques in degeneration 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 Let \(Q\) be a quiver consisting of a single vertex along with two loops \(\alpha\), \(\beta.\) For \(k\) an algebraically closed field, let \(kQ\) be the quiver algebra for \(Q\). For a given dimension \(d\) one has the quiver variety rep\((Q,d) =k^{d^{2}}\times k^{d^{2}}\) (where each term on the right corresponds to a matrix representation of a linear operator on \(\alpha\) and \(\beta\) respectively) upon which \(\text{GL}(d)\) acts: the orbits of this action correspond to isomorphism classes of representations. For nonzero \(q\in k\) let \(I_{q}\) be the ideal generated by \(\{\alpha^{2},\beta^{2},\beta\alpha+q\alpha\beta\}\): one then can define the closed subset rep\((Q,I_{q},d)\) of quiver representations which are trivial on \(I_{q}.\)
The work being reviewed is a study of the irreducible components of rep\((Q,I_{q},d).\) Initially, these components are described in terms of orbit closures of modules related to the Kronecker quiver (two vertices, \(1,2\) and two arrows \(\alpha,\beta:1\rightarrow2\)), but this formulation is shown to be equivalent to a more direct description involving representations satisfying numerical criteria depending on the parity of \(d\). Along with this description, results are obtained concerning intersections of irreducible representations; in particular it is shown that a nontrivial intersection of irreducible components of rep\((Q,I_{q},d)\) is irreducible.
The class of examples presented here are noteworthy for two reasons. First, it provides an description of irreducible components in terms of equations for modules over an algebra which is neither representation finite nor hereditary. Second, applying the results to the case \(q=-1\) and \(d=4\) provides for a description of a famous example by Carlson -- see [\textit{C. Riedtmann}, Ann.\ Sci. Èc. Norm. Supér. (4) 19, No. 2, 275--301 (1986; Zbl 0603.16025)]. quiver representations; irreducible representations Riedtmann, C; Rutscho, M; Smalø, SO, Irreducible components of module varieties: an example, J. Algebra, 331, 130-144, (2011) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Irreducible components of module varieties: an example | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present work is inspired by that of \textit{D. J. Benson, J. F. Carlson}, and \textit{J. Rickard} [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)] in the representation theory of finite groups. In [J. Pure Appl. Algebra 173, No. 1, 59-86 (2002; Zbl 1006.20035)] the author constructed the support cone of a possibly infinite dimensional representation of a Frobenius kernel. Now she introduces support cones for representations of arbitrary infinitesimal group schemes over an algebraically closed field of characteristic \(p>0\). This is not done in terms of cohomology, but in terms of the \(1\)-parameter subgroups of \textit{A. Suslin, E. M. Friedlander}, and \textit{C. P. Bendel} [J. Am. Math. Soc. 10, No. 3, 693-728 (1997; Zbl 0960.14023)]. Thus the support cone of a module \(M\) over an infinitesimal group \(G\) of height \(r\) is a subset of the affine scheme \(V_r(G)\) representing the functor of \(1\)-parameter subgroups of height \(r\) of \(G\). Eventually the author reproduces all the desirable features, like Rickard idempotents and detection of projectivity. But the arguments are necessarily different from those in the finite group setting. support cones; Rickard idempotents; Frobenius kernels; stable categories; infinitesimal group schemes; 1-parameter subgroups Pevtsova, Julia, Support cones for infinitesimal group schemes.Hopf algebras, Lecture Notes in Pure and Appl. Math. 237, 203\textendash213 pp., (2004), Dekker, New York Representation theory for linear algebraic groups, Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters Support cones for infinitesimal group schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface \(\mathbb C^2/G\) with \(G<\mathrm{SL}(2,\mathbb C)\) a finite subgroup, we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type \(A\) singularities. We announce a proof of our conjecture for singularities of type \(D\). The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type \(A\) and type \(D\), respectively arbitrary, simple singularities, confirming predictions of \(S\)-duality. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Euler characteristics of Hilbert schemes of points on surfaces with 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 The isomorphism classes of representations of dimension vector \(\alpha\) of a quiver \(Q\) are in a one-to-one correspondence with the orbits of \(\text{GL}(\alpha)\) (a product of general linear groups) acting linearly on an affine space \(R(Q,\alpha)\). The Zariski closed orbits correspond to semi-simple representations. It is also well motivated to restrict the action to the commutator subgroup \(\text{SL}(\alpha)\) (a product of special linear groups) of \(\text{GL}(\alpha)\). The \(\text{SL}(\alpha)\)-invariant polynomial functions on \(R(Q,\alpha)\) are called semi-invariants.
The author applies Luna's slice theorem to study the geometry of this action. Given an action of a connected reductive group \(G\) on an irreducible affine variety \(X\), the author calls a point locally semi-simple if its \(G'\)-orbit is closed, where \(G'\) denotes the commutator subgroup of \(G\). Working over an algebraically closed field \(k\) of characteristic zero, he gives some equivalent characterizations of locally semi-simple points, refines the Luna stratification of the algebraic quotient \(X//G'\) by considering stabilizers in the overgroup \(G\), and proves a variant of the Luna-Richardson theorem that provides an isomorphism between \(k[X]^{G'}\) and \(k[X^H]^N\), where \(H\) is the generic stabilizer (in \(G\)) of a locally semi-simple point, and \(N\) is its normalizer in \(G'\). This is applied to the \(\text{GL}(\alpha)\)-action on \(R(Q,\alpha)\) and yields a decomposition of the algebra \(k[R(Q,\alpha)]^{\text{SL}(\alpha)}\) of semi-invariants in terms of spaces of relative invariants on representation spaces with smaller dimension vectors occuring as summands of the generic locally semi-simple decomposition of \(\alpha\).
As a corollary, the structure theorem of \textit{A. Skowroński} and \textit{J. Weyman} [Transform. Groups 5, No. 4, 361-402 (2000; Zbl 0986.16004)] on the algebras of semi-invariants of tame quivers is reproved with the aid of the semi-invariants constructed by \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 385-395 (1991; Zbl 0779.16005)] and the fact [from \textit{H. Derksen} and \textit{J. Weyman}, J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)] that these semi-invariants span the algebra of semi-invariants. algebras of semi-invariants; quiver representations; semi-simple orbits; slice representations; Schur roots; generic stabilizers; tame quivers D. A. Shmelkin, Locally semisimple representations of quivers, Transform. Groups 12 (2007), 153--173. Representations of quivers and partially ordered sets, Geometric invariant theory, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Representation theory for linear algebraic groups Locally semi-simple representations of quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(E\) be an elliptic curve and let \(R\) be a real three dimensional root system. Let \(W\) be the Weyl group associated to \(R\), and put \(W_+=W\cap\text{SL}_3(\mathbb{C}).\) Then \(W_+\) acts naturally on \(E\otimes Q(R)\) and the quotient \(E\otimes Q(R)/W_+\) is singular with two natural crepant resolutions. One is the result of a Jung process of desingularization of singularities, the other the equivariant Hilbert scheme.
The author compares these resolutions case by case by writing up explicit equations, and in the Hilbert scheme case, an explicit example is given and a McKay correspondence is achieved. Finally, this correspondence results in a family of vector bundles on \(E\) parameterized by the \(W_+\)-Hilbert scheme. This article gives an interesting comparison of the two types of crepant resolutions in the dimension three case. crepant resolution; singularities; equivariant Hilbert scheme Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Local deformation theory, Artin approximation, etc. Resolution of non-abelian three-dimensional 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 We prove an analogue of the McKay correspondence for Landau-Ginzburg models. Our proof is based on the ideas introduced by \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], which reformulate and generalize the McKay correspondence in the language of derived categories, along with the techniques introduced by \textit{J.-C. Chen} [J. Differ. Geom. 61, No. 2, 227--261 (2002; Zbl 1090.14003)]. Quintero-Vélez A., McKay Correspondence for Landau-Ginzburg models, Commun. Number Theory Phys., 2009, 3(1), 173--208 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and birational maps, Quantum field theory; related classical field theories, Derived categories, triangulated categories McKay correspondence for Landau-Ginzburg 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 (X,0) be a two-dimensional rational singularity, and \(\pi: ({\mathfrak X},0)\to (S,0)\) be the universal deformation of (X,0). If (X,0) is not a double point, the parameter space S has many components \(S_ 1,S_ 2,..\). in general. Among them a component \(S_ i\) is called a smoothing component if for some point \(s\in S_ i\) sufficiently near to 0 the fiber \({\mathfrak X}_ s=\pi^{-1}(s)\) over s is smooth. Let \(\{S_{\lambda}\}_{\lambda \in \Lambda}\) be the set of smoothing components. On the other hand, we consider partial resolutions of X. Let f: \(M\to X\) be the minimal resolution of X. Let \(E=f^{-1}(0)\) be the exceptional curve. Let \(\{D_{\gamma}\}_{\gamma \in \Gamma}\) be the set of connected curves contained in E such that the contraction of \(D_{\gamma}\) on M defines a rational quadruple point and, for the corresponding canonical cycle \(K_{\gamma}\) of \(D_{\gamma}\), \(2K_{\gamma}\) is a Cartier divisor. Then, for every \(S_{\lambda}\) there exists \(D_{\gamma}\) with the following relation (this is the main result of this article and solves a conjecture of \textit{Kollár}):
Let \(D'_{\gamma}\) be the union of components of E disjoint from \(D_{\gamma}\). Let \(\sigma_{\gamma}: M\to Y_{\gamma}\) be the contraction of the union \(D_{\gamma}\cup D'_{\gamma}\) to a normal space. \(Y_{\gamma}\to X\) is a partial resolution. There is a one- parameter smoothing \(\tau: {\mathfrak Y}\to T\) with \(\tau^{-1}(0)\cong Y_{\gamma}\) for \(0\in T\) such that some multiple of the canonical class of \({\mathfrak Y}\) is Cartier. We can contract \(\sigma_{\gamma}(E)\) in \({\mathfrak Y}\) and obtain a smoothing \({\bar\tau}: \bar {\mathfrak Y}\to T\) of X. The image of the induced morphism \(T\to S\) is contained in \(S_{\gamma}.\)
As the sub-main result the following characterization is given: A two- dimensional quadruple rational singularity (Y,0) has a one-parameter smoothing \(\tau: {\mathfrak Y}\to T\) with \(\tau^{-1}(0)\cong Y\) for \(0\in T\) such that some multiple of the canonical class of \({\mathfrak Y}\) is Cartier if and only if, for the canonical cycle K on the minimal resolution \(M\to Y\), 2K is Cartier. minimal resolution; two-dimensional quadruple rational singularity Stevens, Jan. Partial resolutions of rational quadruple points. Internat. J. Math. 2 (1991), 205--221.DOI: 10.1142/S0129167X91000144 Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Partial resolutions of rational quadruple 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 Let \(G\) be a finite subgroup of \(\mathrm{SL}(3,\mathbb{C})\) acting naturally on \(\mathbb{C}^3\), and let \(Y=G\text{-Hilb}( \mathbb{C}^3)\) be the moduli space of \(G\)-clusters introduced by Nakamura. It is proven in [\textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] that \(Y\) is irreducible and the natural Hilbert-Chow morphism \(f:Y\to \mathbb{C}^3/G\) is a crepant resolution. Furthermore, the derived category of coherent sheaves on \(Y\) is equivalent to the derived category of \(G\)-equivariant coherent sheaves on \(\mathbb{C}^3\) via a Fourier-Mukai transformation.
The question of whether other crepant resolutions of \(\mathbb{C}^3/G\) can be realized as moduli spaces was studied in [\textit{A. Craw} and \textit{A. Ishii}, Duke Math. J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)]. Craw and Ishii conjectured that any projective crepant resolution of \(\mathbb{C}^3/G\) can be identified with a certain moduli space of representations of the McKay quiver with relations, called the moduli space of \(G\)-constellations. Craw and Ishii proved their conjecture in case \(G\) is abelian.
Let \(N\leq G\) be a normal subgroup. \(G/N\) acts on \(N\text{-Hilb} (\mathbb{C}^3)\) the crepant resolution of \(\mathbb{C}^3/N\). It follows from the main result of Bridgeland-King-Reid that \(G/N\text{-Hilb}( N\text{-Hilb}(\mathbb{C}^3))\) is a crepant resolution \(\mathbb{C}^3/G\).
The main result of the paper under review proves that the crepant resolution \(G/N\text{-Hilb}( N\text{-Hilb}(\mathbb{C}^3))\) is isomorphic to a moduli space of \(G\)-constellations, establishing Craw-Ishii conjecture for such crepant resolutions. The crepant resolutions \(Y=G\text{-Hilb}( \mathbb{C}^3)\) and \(G/N\text{-Hilb}( N\text{-Hilb}(\mathbb{C}^3))\) are usually nonisomorphic. In the case \(G\) is abelian, the paper under review presents a complete list of the cases where two crepant resolutions are isomorphic. For non-Abelian subgroups, the paper under review shows that these two crepant resolutions are not isomorphic when \(G\) is a finite small subgroup of \(\mathrm{GL}(2,\mathbb{C})\subset \mathrm{SL}(3,\mathbb{C})\). \(G\)-clusters; \(G\)-constellations; crepant resolution; Hilbert scheme A. Ishii, Y. Ito and Á. Nolla de Celis, On \(G/N\)-Hilb of \(N\)-Hilb, Kyoto J. Math. 53 (2013), no. 1, 91-130. MR3049308 McKay correspondence, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets On \(G/N\)-Hilb of \(N\)-Hilb | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(C\) be a non-singular complex curve of genus \(g>1\). For a fixed line bundle \(\xi\) over \(C\) of given degree \(d\), denote by \({\mathcal M}(2,\xi)\) the (coarse) moduli space of isomorphism classes of rank-2 stable vector bundles over \(C\) with determinant line bundle isomorphic to \(\xi\). By a result by \textit{J.-M. Drezet} and \textit{M. S. Narasimhan} [Invent. Math. 97, No 1, 53-94 (1989; Zbl 0689.14012)], the Picard groups \(\text{Pic}{\mathcal M}(2,\xi))\) are freely generated by the line bundle \({\mathcal O_ M}(\Theta)\) associated with an ample divisor \(\Theta\) in \({\mathcal M}(2,\xi)\). This divisor \(\Theta\) is called the generalized theta divisor of \({\mathcal M}(2,\xi)\), and its global sections \(f\in H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))\) are called the generalized theta functions on \({\mathcal M}(2,\xi)\).
Recent developments in conformal quantum field theory gave rise to conjectures about the dimension of the spaces \(H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)\) of generalized theta functions of order \(k\) [cf. \textit{E. Verlinde} and \textit{H. Verlinde}, ``Conformal field theory and geometric quantization'', Prepr. PUPT-89/1149 (1989)].
In the present paper, the author provides a partial verification of these conjectures, i.e., of the so-called Verlinde formulae. More precisely, he proves that in the special case of \(k=1\) and \(d=\deg\xi\) an odd integer, the conjectured formula
\[
\dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))=2^{g-1}\centerdot (2^ g-1)
\]
indeed holds true. Moreover, using previous results of \textit{A. Beauville} [Bull. Soc. Math. Fr. 119, No. 3, 259-291 (1991)], the author succeeds in exhibiting an explicit base for the vector space \(H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))\). The method of proof, in the particular case under investigation, is based upon the handy description of the moduli space \(M(2,\xi)\) for hyperelliptic ground curves, which is due to \textit{U. V. Desale} and \textit{S. Ramanan} [Invent. Math. 38, 161-185 (1976; Zbl 0323.14012)], and on a comparison argument with respect to the number of odd theta characteristics on the base curve \(C\).
A general verification of the conjectured Verlinde formulae, i.e., for \(\dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)\) with \(k\) and \(d=\deg \xi\) arbitrarily chosen, has recently been obtained by \textit{A. Szenes} and \textit{A. Bertram} [cf. ``Hilbert polynomials of moduli spaces of rank-2 vector bundles'', I, II (Harvard-University, September 1991 and November 1991)]. An announcement and sketch of their general results was published by A. Szenes, after the appearance of the present article [cf. \textit{A. Szenes}, Int. Math. Res. Not. 1991, No. 7, 93-98 (1991)]. coarse moduli space of rank-2 stable vector bundles; generalized theta divisor; generalized theta functions; odd theta characteristics Y. Laszlo, La dimension de l'espace des sections du diviseur thêta généralisé , Bull. Soc. Math. France 119 (1991), no. 3, 293-306. Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Fine and coarse moduli spaces, Theta functions and abelian varieties, Families, moduli of curves (algebraic) Dimension of the space of sections of the generalized theta divisor. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 firstly gives an overview of results concerning strongly canonical expansions and canonical expansions of \(\omega\)-categorical structures. There are certain such structures which have no nontrivial strongly canonical expansions or no nontrivial canonical expansions, respectively.
Throughout the paper \(\infty = \aleph_ 0\) is assumed as the dimension of the considered vector spaces. A symplectic space \(\text{SPG} (\infty, q)\) is a projective space \(\text{PG} (\infty, q)\) corresponding to an infinite dimensional vector space \(V(\infty, q)\) over the field \(\text{GF}(q)\) which is equipped with the ternary relation of collinearity and the binary relation of orthogonality. By Witt's Lemma a symplectic space \(\text{SPG} (\infty, q)\) is \(\omega\)-categorical.
A subgroup \(G\) of the group \(\text{P} \Gamma \text{Sp}(\infty, q)\) of all projective transformations induced on \(\text{PG} (\infty, q)\) by the group of semilinear transformations which preserve the symplectic form \(\sigma\) is said to be Witt homogeneous if \(G\) and \(\text{P} \Gamma \text{Sp}(\infty, q)\) have the same orbits on the set of finite-dimensional subspaces of \(\text{SPG} (\infty, q)\). So we are able to formulate the main result of the paper:
Let \(G \leq \text{P} \Gamma \text{Sp} (\infty, q)\) be Witt homogeneous and let \(\Gamma\) be a non-degenerate \(2t\)-dimensional subspace of \(\text{SPG} (\infty, q)\). Then the group \(\text{PSp} (\infty, q)\) of projective transformations induced by the linear transformations on \(V(\infty,q)\) preserving \(\sigma\) is a subgroup of \(G_{\{T\}}/G_{(T)}\). Moreover, if \(\mathcal M\) is a canonical expansion of \(\text{SPG} (\infty, q)\) and \(G = \text{Aut} ({\mathcal M})\) then \(\text{PSp} (\infty, q) \subseteq \text{P} \Gamma \text{Sp} (\infty, q)\). Hence, \(\text{SPG} (\infty, q)\) has finitely many canonical expansions. group actions; algebraic closures; totally isotropic subspaces; radical actions; canonical expansions of \(\omega\)-categorical structures; symplectic spaces; projective spaces; projective transformations; group of semilinear transformations; symplectic forms; Witt homogeneous Linear algebraic groups over finite fields, Categoricity and completeness of theories, Linear transformations, semilinear transformations, Polar geometry, symplectic spaces, orthogonal spaces, Applications of logic to group theory, Geometry of classical groups, Classical groups (algebro-geometric aspects) Canonical expansions of countably categorical structures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the subsequent paper of H. Knörrer's paper with the same title [see the preceding review]. The main object is to prove the converse of Knörrer's result, namely that a non-simple hypersurface singularity R is of infinite Cohen-Macaulay-representation type, i.e. there are infinitely many isomorphism classes of indecomposable maximal Cohen- Macaulay modules (MCM) over R (see theorem A and B). Moreover the authors classify those (non-isolated) hypersurface singularities \((A_{\infty}\) and \(D_{\infty})\) which are of countable CM-representation type (see theorem B).
An application to vector bundles on projective hypersurfaces with ''no cohomology in the middle'' is given (see theorem C). non-simple hypersurface singularity; maximal Cohen-Macaulay modules; vector bundles on projective hypersurfaces R.O. Buchweitz, G.M. Greuel and F.O. Schreyer: ''Cohen-Macaulay modules on hypersurface singularities II'', Invent. Math., Vol. 88, (1987), pp. 165--182. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Cohen-Macaulay modules on hypersurface singularities. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Y\) denote an integral projective curve of arithmetic genus \(g\geq 2\) with at most nodes as singularities. Let \(U'\) (resp. \(U_\xi '\)) be the moduli space of semistable vector bundles of rank two and even degree (resp. with a fixed determinant \(\xi \)) on \(Y.\) \(U'\) is a quasi-projective variety. It has a compactification \(U\) (respectively \(U_\xi \)) which is the moduli space of semistable torsionfree sheaves of rank two and even degree (resp. the closure of \(U_\xi '\) in \(U\)). If \(Y\) is a non-singular hyperelliptic curve of genus \(g,\) then one can associate to \(Y\) a regular pencil of quadrics given by a pair of non-singular quadrics \(Q_1\) and \(Q_2\) in \(\mathbb P^{2g+1}.\)
Let \(R_4\) denote the variety of \(g\)-dimensional linear subspaces \(V\) of \(\mathbb P^{2g+1}\) which
(i) belong to a fixed system of maximum isotropic spaces for \(Q_1\), and
(ii) \(Q_2\) restricted to \(V\) has rank \(\leq 4.\)
The hyperelliptic involution \(i\) can be made to act on \(U_\xi \) in a natural way, let \(U_\xi /i\) denote the quotient of \(U_\xi \) by this action. When \(Y\) has a single node \(y_0\), the associated pencil of quadrics is a singular pencil and \(Q_1\cap Q_2\) has a unique singular point \(P.\) Let \((N_0)_4\) denote the subset of \(R_4\) consisting of subspaces \(V\) which contain \(P,\) \(R_r\), \(r=1,2,3\), denote the subsets of \(R_4\) consisting of subspaces \(V\) such that \(Q_2\mid V\) has rank \(\leq r\) and \(U_3\) denote the \(i\)-fixed sub-variety of \(U_\xi \) and \(U_2\) the sub-variety corresponding to non-stable sheaves.
Theorem. \(R_4\) is a normalisation of \(U_\xi /i.\) The morphism \(\psi :R_4\rightarrow U_\xi /i\) maps \(R_4-(N_0)_4\) (respectively \((N_0)_4)\) isomorphically onto \(U_\xi '/i\) (respectively \((U_\xi -U_\xi ')/i).\) Also \(\psi \) maps \(R_3\) to \(U_3\) and \(R_2\) to \(U_2.\) Moreover this theorem is generalised to the case when \(Y\) has \(m\) nodes as singularities. hyperelliptic curve with nodes; vector bundle on curve Bhosle, U.N.: Vector bundles of rank two and degree zero on a nodal curve, Proc. Barcelona--Catania Conferences. 1994--1995. Lecture notes in pure and applied Mathematics, vol. 200, pp. 271--281. Marcel-Dekker, New York (1998) Vector bundles on curves and their moduli, Singularities of curves, local rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Vector bundles of rank 2, degree 0 on a nodal hyperelliptic curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\pi :X\rightarrow \tilde X=\mathbb C^r/G\) be a crepant resolution, i.e. the canonical divisor \(K_{\tilde X}\) is trivial. The authors prove the existence of such resolution for abelian quotient singularities. The interest in these singularities is coming from the fact that they are good candidates for McKay correspondence in higher dimension. Also, when \(G\) is abelian, \(\mathbb C^r/G\) can be directly investigated by means of toric varieties. It is shown that the existence problem for abelian \(G\) is equivalent to the problem of finding junior simplices having basic triangulations. They give many properties of this simplex and a detailed study of Gorenstein abelian quotient singularity via this simplex. Moreover, an algorithm for treating the existence problem in the abelian case is given. The paper is self-contained and gives the complete view of the problem. D. I. Dais, M. Henk, G. M. Ziegler, On the existence of crepant resolutions of Gorenstein abelian quotient singularities in dimensions \geqslant4 Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) On the existence of crepant resolutions of Gorenstein abelian quotient singularities in dimension \(\geq 4\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C_g\) be a hyperelliptic curve of genus \(g\). The authors introduce the hyperelliptic \(\theta\) and \(\sigma\) function generalizing the corresponding objects from the classical theory of elliptic functions. For this purpose they consider the Abel map \(\phi_k\) from the \(k\)-th symmetric product \(\mathrm{Sym}^k(C_g)\) to the Jacobian \(J(C_g)\), and introduce the stratification \(\Theta^{[k]} := \phi_k(\mathrm{Sym}^k(C_g))\) of \(J(C_g)\). Let \(\kappa : \mathbb{C}^g \to J(C_g)\) be the natural map. The main equality obtained by the authors is
\[
\frac{\sigma_{\sharp^{m+n}}(u+v)\sigma_{\sharp^{m+n}}(u-v)}{\sigma_{\sharp^{m}}(u)^2\sigma_{\sharp^{n}}(v)^2} = (-1)^{\delta(g,n)} \prod^m_{i=1}\prod^n_{j=1}(x_i-x'_j),
\]
where \(m\) and \(n\) are positive integers such that \(m+n \leq g+1\),
\[
u = \sum^m_{i=1}\int^{(x_i,y_i)}_\infty (\omega_1, \ldots,\omega_g) \in \kappa^{-1}(\Theta^{[m]}), v = \sum^n_{j=1}\int^{(x'_j,y'_j)}_\infty (\omega_1, \ldots,\omega_g) \in \kappa^{-1}(\Theta^{[n]}),
\]
\(\delta(g,n) = \frac{1}{2}n(n-1)+gn\), and \(\sigma_{\sharp^l}\) denotes certain partial derivative of \(\sigma\) associated with multi-index \(\sharp^l\). This generalizes the well-known addition formula
\[
\frac{\sigma(u+v)\sigma(v-u)}{\sigma(u)^2\sigma(v)^2} = \wp(v) - \wp(u)
\]
for elliptic \(\sigma\) and \(\wp\)-functions.
The authors discuss also possible applications of their results. Schottky-Klein formulae; hyperelliptic sigma functions; Jacobian V. Enolskii, S. Matsutani, and Y. Ônishi, ''The Addition Law Attached to a Stratification of a Hyperelliptic Jacobian Variety,'' Tokyo J. Math. 31(1), 27--38 (2008); arXiv:math.AG/0508366. Algebraic functions and function fields in algebraic geometry, Subvarieties of abelian varieties, Special algebraic curves and curves of low genus, Special divisors on curves (gonality, Brill-Noether theory) The addition law attached to a stratification of a hyperelliptic Jacobian variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Motivated by Ruan's famous conjecture on the relation between Gromov-Witten theory and birational geometry, and the Gromov-Witten/Stable pair correspondence, the paper under review studies the transformation of the stable pair invariants (defined by Pandharipande-Thomas) under blow-ups along curves and points.
Let \(X\) be a nonsingular projective complex threefold and \(p:\tilde{X}\to X\) be the blow-up of \(X\) along a point \(P\in X\) or a nonsingular irreducible curve \(C \subset X\). Denote by \(E\) the exceptional divisor of the blow-up, by \(e\) the class of a line in the fiber of \(E\), and by \(p^{!}: H_2(X,\mathbb{Z})\to H_2(\tilde{X},\mathbb{Z})\) the natural injection via the pull-back of 2-cycles. Let \(Z(q)\) be the generating of the stable pair invariants, \(\beta \in H_2(X,\mathbb{Z})\), \(\gamma_i \in H^*(X,\mathbb{Q})\), \(d_i\in \mathbb{Z}_{\geq0}\), and \(k\in \mathbb{Z}_{>0}\). The main result of the paper under review can be summarized as follows:
In the case of blow-up at \(P\) (assume \(\deg(\gamma_i)>0\)):
\[
Z \left(\tilde{X};q \mid \prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta+ke}=0.
\]
\[
Z \left(X;q \mid \prod_{i=1}^m\tau_{d_i}(\gamma_i)\right)_{\beta}=Z \left(\tilde{X};q \mid \prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta}.
\]
\[
Z \left(X;q \mid \tau_0(pt)\prod_{i=1}^m\tau_{d_i}(\gamma_i)\right)_{\beta}=(1+q)^2 Z \left(\tilde{X};q \mid \prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta-e}.
\]
\[
Z \left(X;q \mid \tau_1(pt)\prod_{i=1}^m\tau_{d_i}(\gamma_i)\right)_{\beta}=\frac{1}{2}(1-q^2) Z \left(\tilde{X};q \mid \tau_0(-E^2)\prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta-e}.
\]
In the case of blow-up at \(C\) (assume \(\gamma_i\) has support away from \(C\)):
\[
Z \left(\tilde{X};q \mid \prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta+ke}=0.
\]
\[
Z \left(X;q \mid \prod_{i=1}^m\tau_{d_i}(\gamma_i)\right)_{\beta}=Z \left(\tilde{X};q \mid \prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta}.
\]
\[
Z \left(X;q \mid \tau_0(C)\prod_{i=1}^m\tau_{d_i}(\gamma_i)\right)_{\beta}=(1+q) Z \left(\tilde{X};q \mid \prod_{i=1}^m\tau_{d_i}(p^*\gamma_i)\right)_{p^!\beta-e}.
\]
The paper under review also studies the implicaitons of the formulas above to the BPS invariants of \(X\) and \(\tilde{X}\). The main technical tools in the proofs of the results above are the degeneration formula and the absolute/relative correspondence for the stable pair invariants. stable pair invariant; BPS state count; blow-up; degeneration formula; absolute/relative correspondence; GW/DT/P correspondence Ke, H-Z, Stable pair invariants under blow-ups, Math. Z, 282, 867-887, (2016) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Stable pair invariants under blow-ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers nine of the fourteen triangle singularities in Arnold's list [cf. \textit{V. T. Arnold}, Invent. Math. 35, 87-109 (1976; Zbl 0336.57022)]. Namely the singularities \(E_{12}\), \(E_{13}\), \(E_{14}\), \(Z_{11}\), \(Z_{12}\), \(Z_{13}\), \(Q_{10}\), \(Q_{11}\), \(Q_{12}\). He defines elementary and tie transformations on Dynkin graphs with several components. The aim of this paper is to give a description of Dynkin graphs \(\Gamma\) with several components such that there exists a small deformation fibre \(Y\) of a triangle singularity and satisfying two technical conditions, in terms of elementary and tie transformations of certain Dynkin graphs. The author recalls the definition of a root system of type \(BC\), \(G\) and also explains why this method does not apply to the study of the remaining triangle singularities, namely \(W_{12}\), \(W_{13}\), \(S_{11}\), \(S_{12}\) and \(U_{12}\). He announces that some results on these singularities will appear elsewhere. triangle singularities; Dynkin graphs; root system Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Local deformation theory, Artin approximation, etc. Dynkin 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 If G is a connected Lie group and \(\Gamma\) a discrete subgroup, the tangent bundle of G/\(\Gamma\) admits a natural trivialization. Hence, if G/\(\Gamma\) is compact and if one fixes an orientation of G/\(\Gamma\), the Pontryagin construction yields an element [G/\(\Gamma\) ] in the stable homotopy group of a sphere. In the present paper, the authors complete their project of computing explicitly all the classes [G/\(\Gamma\)) for 3- dimensional groups G. The relevant homotopy group is \(\pi^ S_ 3\cong {\mathbb{Z}}/24\) which is mapped injectively into \({\mathbb{Q}}/{\mathbb{Z}}\) by the e- invariant. The Lie groups which have to be considered are the universal coverings of \(SL_ 2({\mathbb{R}})\), SU(2), \({\mathbb{R}}^ 3\), the Heisenberg group H, the group \(E^+(2)\) of orientation-preserving affine motions of \({\mathbb{R}}^ 2\), and the inhomogeneous Lorentz group E(1,1). The computation of e[G/\(\Gamma\) ] is achieved by interpreting G/\(\Gamma\) in terms of singularity theory: For G locally isomorphic to \(SL_ 2({\mathbb{R}})\), SU(2), H, or E(1,1), the quotient G/\(\Gamma\) is known to be diffeomorphic to the link of a normal Gorenstein singularity, and for \(E^+(2)\) there is a similar description. The case of \(SL_ 2({\mathbb{R}})\) is the most complex one; the treatment of E(1,1) is based on Hirzebruch's work on Hilbert modular surfaces. Also included are the values of the Rochlin invariant and the signature defect. Lie group; Pontryagin construction; stable homotopy group; e-invariant; singularity theory; Rochlin invariant; signature Seade, J. A.; Steer, B. F.: Complex singularities and the framed cobordism class of compact quotients of 3-dimensional Lie groups by discrete subgroups. Comment. math. Helv. 65, 349-374 (1990) Differential topology, Stable homotopy of spheres, Singularities of surfaces or higher-dimensional varieties Complex singularities and the framed cobordism class of compact quotients of 3-dimensional Lie groups by discrete subgroups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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, in the reviewers opinion, a major contribution to singularity theory.
Based on the remarkable paper by \textit{V. V. Nikulin} [Math. USSR Izv. 14, 103-167 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111- 177 (1979; Zbl 0408.10011)] the authors attack the problem of characterizing the smoothing components and the intersection forms of the corresponding Milnor fiber, for isolated complex-analytic surface singularities.
Let (X,0) be any such singularity, i.e. a Stein space of dimension \(2\) with a smooth \((C^{\infty})\) boundary \(\partial X=L\), and a unique singular point 0. L is usually called the link of the singularity. Let \(\pi: \tilde X\to X\) be a good resolution, i.e. one with an exceptional curve \(E=\pi^{-1}(0)\) consisting of nonsingular irreducible components intersecting transversally. Consider a smoothing \(f: {\mathcal X}\to \Delta,\) where \(\Delta\) is an open disc in \({\mathbb{C}}\) containing 0. Here \({\mathcal X}\) is a Stein space with a partial \(C^{\infty}\)-boundary \(\partial {\mathcal X}\). There is an isomorphism of analytic spaces with boundary \(i: f^{- 1}(0)\overset \sim \rightarrow X\) and \(f| int({\mathcal X}-\{0\})\) and \(f| \partial {\mathcal X}\) are both submersions. The Milnor fiber M, i.e. the generic fiber of f, has a boundary \(\partial M\) which is diffeomorphic to L. Therefore the 3-dimensional manifold L bounds both of the two oriented 4- dimensional manifolds \(\tilde X\) and M.
The main result of this paper, (3.7), shows that the classical linking pairing of the link L, \(b: H_ 1(L)_ t\times H_ 1(L)_ t\to Q/Z\) where the subscript t denotes the torsion part, is in an obvious sense the link between the intersection pairings of \(H_ 2(\tilde X)\) and \(H_ 2(M)\). In fact, the complex structures of \(\tilde X\) and M induces an, up to homotopy, unique complex structure on the sum of the trivial bundle R and the tangent bundle \(\tau_ L\) of L.
Using the first Chern class of \(-\tau_ L\oplus R\) the authors construct a quadratic function \(q: H_ 1(L)_ t\to Q/Z\) such that the corresponding bilinear form \(b(x,y)=-(q(x+y)-q(x)-q(y))\) is the linking pairing. Now, let \(K_{\tilde X}:=-c_ 1(\tau_{\tilde X})\in H^ 2(\tilde X)=Hom(H_ 2(\tilde X),Z)\) then the bilinear form of the quadratic function \(Q_{\tilde X}: H_ 2(\tilde X)\to Z\) defined by \(Q_{\tilde X}(x)=(x\cdot x+K_{\tilde X}(x)),\) is the intersection form of \(H_ 2(\tilde X)\). Notice that since this form is negative definite, there is an element \(k\in H_ 2(\tilde X):=\{x\in H_ 2(\tilde X)\oplus Q| \forall y\in H_ 2(\tilde X), x\cdot y\in Z\}\) such that \(K_{\tilde X}(x)=k\cdot X\) for all \(x\in H_ 2(\tilde X).\)
To \(Q_{\tilde X}\) there is associated a discriminant quadratic function (DQF) \(q_{\tilde X}: H_ 2(\tilde X)/H_ 2(\tilde X)\to Q/Z\) and there is a natural isomorphism \(H_ 2(\tilde X)/H_ 2(\tilde X)\simeq H_ 1(L)_ t\) identifying \(q_{\tilde X}\) and q. Moreover, see (3.7) and (4.5), let \(K_ M\in Hom(H_ 2(M),Z)\) be the image of \(-c_ 1(\tau_ M)\), then the corresponding bilinear form of the quadratic function \(Q_ M: H_ 2(M)\to Z\) defined by \(Q_ M(x)=(x\cdot x+K_ M(x))\) is the intersection form of \(H_ 2(M)\), and the associated DQF is isomorphic to the quadratic function \(q_ I: I^{\perp}/I\to Q/Z\) induced by q, where \(I:=im(H_ 2(M,L)_ t\to H_ 1(L)_ t)\subseteq H_ 1(L)_ t\) is q-isotropic.
Specializing to Gorenstein singularities, the authors observe, see (4.8), that \(k\in H_ 2(\tilde X)\) and that q is a quadratic form. Moreover, \(K_ M=0\) implying that \(\bar H{}_ 2(M):=H_ 2(M)\) modulo torsion and the radical of the intersection form, is an even lattice with DQF canonically isomorphic to \(q_ I.\)
For Gorenstein singularities the formulas of \textit{J. H. M. Steenbrink} [in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 513-536 (1983; Zbl 0515.14003)] express the Sylvester invariants \((\mu_ 0,\mu_+,\mu_-)\) of the intersection form of \(H_ 2(M)\) as linear functions in the Betti number \(b_ 1(\tilde X)\), the genus p(X) and \(k\cdot k.\)
Consider now a smoothing component of the base space of a versal deformation. Any two smoothings contained in the same component will determine the same q-isotropic subgroup \(I\subseteq H_ 1(L)_ t\) and the same quadratic function \(q_ I\). Therefore \((\mu_ 0,\mu_+,\mu_-)\), I and \(q_ I\) are invariants of the smoothing component. The authors refine these invariants to what they call a smoothing datum.
The last part of the paper is concerned with the problem of characterizing the smoothing components, together with the intersection forms of the corresponding Milnor fibers, studying the subset of ''permissible'', see {\S} 6, smoothing data, and applying the classification of Nikulin, loc. cit. The discussion is confined to the minimally elliptic singularities, i.e. Gorenstein and \(p(X)=1\). For simple elliptic and triangle singularities this analyses gives a rather complete answer to the problems posed. smoothing components; intersection forms; Milnor fiber; isolated complex- analytic surface singularities; link of the singularity; discriminant quadratic function Looijenga (E.) & Wahl (J.).- Quadratic functions and smoothing surface singularities. Topology 25, p. 261-291 (1986). Zbl0615.32014 MR842425 Deformations of complex singularities; vanishing cycles, Local complex singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Complex singularities Quadratic functions and smoothing 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 During the past decade, two tendencies in the study of conformal quantum field theories have become evident and significant: the extended use of methods of algebraic geometry as a basic mathematical framework, on the one hand, and the necessity of also including non-commutative algebro- geometric objects as ingredients for appropriate models, on the other hand. With a view to these developments, and the fact that the theory of abelian varieties and their theta functions has proved to be of particular importance for constructing two-dimensional conformal field theories, it is highly desirable to establish a non-commutative analogue of abelian varieties (over arbitrary complete normed fields) and their theta functions, together with a suitable concept of quantization (quantum deformations).
In the present paper, the author proposes a possible approach to this highly difficult problem. In the first section he defines the category of so-called (non-commutative) quantum tori. The ``analytic'' functions on these objects appear as certain infinite linear combinations of formal exponents. Among them are the so-called quantized theta functions, defined with respect to certain morphisms in the category of quantum tori, which replace the classical operations on usual abelian varieties. These quantized theta functions are discussed in section 2. In particular, the basic functional equation for them is derived, and the linear spaces of quantized theta functions of given ``formal theta type'' are investigated. This allows to generalize the concept of polarized (classical) abelian varieties to quantum tori.
The third section is devoted to a special case: quantum tori whose alternating pairing takes values that are roots of unity in the ground field. This case is particularly related to Brauer theory and classical commutative geometry. As the author points out, this ``root-of-unity case'' may lead to interesting applications in number theory. Generally, the author's approach to quantum tori and their (quantized) theta functions is very much motivated by the \(p\)-adic theory of abelian varieties and D. Mumford's results [cf. \textit{D. Mumford}, ``On the equations defining abelian varieties'', I, II, III, Invent. Math. 1, 287- 354 (1966), 3, 75-135 and 215-244 (1967; Zbl 0219.14024)] on abelian varieties over arbitrary ground fields. It turns out that the author's quantized theta functions share many properties with the classical theta functions, but there is still one major difficulty: quantized theta functions cannot be multiplied by each other, since their defining exponential factors do not commute. Anyway, the class of quantized theta functions constitutes a further enrichment of the tremendously growing family of quantized special functions in physics, and the author's basic constructions seem to be the promising beginning of a quantized theory of non-commutative tori, all the more as already a large part of the theory of classical Lie groups and algebraic groups has been extended, in an analogous manner, to non-commutative geometry, in particular by \textit{A. Connes, V. G. Drinfeld, L. D. Faddeev, N. Y. Reshetikhin, L. A. Takhtajan}, and others. quantum deformations; conformal quantum field theories; theta functions; non-commutative analogue of abelian varieties; quantization; quantum tori; quantized theta functions Manin, Y. I., Quantized theta-functions, \textit{Progress of theoretical physics. Supplement}, 102, 219-228, (1991) Noncommutative algebraic geometry, Theta functions and abelian varieties, Quantum groups (quantized enveloping algebras) and related deformations, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Quantized 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 In this interesting paper the authors study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities.
The main result is the following:
Theorem. For every finitely presented group \(G\) there is an isolated, \(3\)-dimensional, complex singularity (\(0\in X_{G}\)) with link \(L_{G}\) such that \(\pi _{1}(L_{G})\cong G\).
The authors consider that this result is a strong exception to the principle formulated by \textit{M. Goresky} and \textit{R. MacPherson} [Stratified Morse theory. Berlin etc.: Springer-Verlag (1988; Zbl 0639.14012)]:
``Philosophically, any statement about the projective variety or its embedding really comes from a statement about the singularity at the point of the cone. Theorems about projective varieties should be consequences of more general theorems about singularities which are no longer required to be conical. ''
In order to prove the above theorem, the authors show that for every finitely presented group \(G\) there is a complex projective surface \(S\) with simple normal crossing singularities only, so that the fundamental groups of \(S\) is isomorphic to \(G\).
At the end of the paper the authors prove that a finitely presented group \(G\) is \(\mathbb{Q}\)-superperfect (has vanishing rational homology in dimensions \(1\) and \(2\)) if and only if \(G\) is isomorphic to the fundamental group of the link of a rational \(6\)-dimensional complex singularity. foundamental group; isolated singularity; normal crossing singularities; germs of singularities; \(R\)-superperfect group. Kapovich M. and Kollár J., Fundamental groups of links of isolated singularities, J. Amer. Math. Soc. 27 (2014), no. 4, 929-952. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Homotopy theory and fundamental groups in algebraic geometry, Generators, relations, and presentations of groups, Global differential geometry of Hermitian and Kählerian manifolds Fundamental groups of links of 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 \(d\) be a positive integer, \(k\) be a field of characteristic zero containing all \(d\)-th roots of unity, and let \(G\) be a finite subgroup of order \(d\) in \(\text{SL}(k,n)\) acting on the affine space \(\mathbb A^n_k\). Consider a resolution \(Y\to X\) of singularities on the quotient \(X=\mathbb A_k^n/G\) and assume that \(Y\) is crepant, i.e. \(K_Y=0\). The McKay correspondence is a connection between irreducible representations of the group \(G\) and cohomology of \(Y\). In one form it says that the Euler number of \(Y\) is equal to the number of conjugacy classes in \(G\) [\textit{M. Reid}, in: Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276, 53--72 (2002; Zbl 0996.14006)], and it was proved by \textit{V. Batyrev} [J. Eur. Math. Soc. 1, No. 1, 5--33 (1999; Zbl 0943.14004)]. The present paper is devoted to a proof of the corresponding statement on the motivic level by means of motivic integration.
To be slightly more precise, let \(\mathcal M\) be the Grothendieck group of algebraic varieties over \(k\) with relation \([X]=[Z]+[X-Z]\) for Zariski closed \(Z\) in \(X\) and the product induced by products of varieties. Following the notation of the paper, let \(\mathcal M_{\text{loc}}\) be the localization \(\mathcal M[\mathbb L^{-1}]\), where \(\mathbb L=[\mathbb A^1]\). Let also \(F^m\mathcal M_{\text{loc}}\) be a subgroup generated by \([X]\mathbb L^{-i}\) with \(\dim (X)\leq i-m\). The filtration \(F^m\) gives the completion \(\hat {\mathcal M}\) of \(\mathcal M_{\text{loc}}\). At last, we add the relation \([V/G]=[V]\) for each \(k\)-vector space with a linear action of a finite group \(G\) getting the corresponding quotient ring \(\hat \mathcal M_{\slash }\).
The above rings are closely connected with the category of Chow motives \(\text{CHM}_k\) over \(k\) with coefficients in \(\mathbb Q\). Namely, there exists a function \(\chi _c\) from the set of varieties over \(k\) to \(K_0(\text{CHM}_k)\) satisfying the nice properties listed on page 283 of the paper. The analogous filtration on \(K_0(\text{CHM}_k)\) gives rise to the completion \(\hat K_0(\text{CHM}_k)\). The map \(\chi _c\) induces ring homomorphisms \(\chi _c:\mathcal M\to K_0(\text{CHM}_k)\) and \(\hat \chi _c:\hat \mathcal M\to \hat K_0(\text{CHM}_k)\), which can be factored through \(\mathcal M_{\slash }\) and \(\hat \mathcal M_{\slash }\) respectively.
Given a variety \(X\) over \(k\) let \(\mathcal L(X)\) be the scheme of germs of arcs on \(X\). For any field extension \(K/k\) one has a natural bijection \(\mathcal L(X)(K)\cong \text{Hom}_k(K[[t]],X)\) where \(K[[t]]\) is the ring of formal power series with coefficients in \(K\). If \(B^t\) is a set of \(k[t]\)-semi-algebraic subsets in \(\mathcal L(X)\), then there is a nice measure \(\mu :B^t\to \hat \mathcal M\), called a motivic measure on \(\mathcal L(X)\). Assume that \(X\) is an irreducible normal variety of dimension \(n\), which is Gorenstein with at most canonical singularities at each point. Some appropriate notion of integration with respect to \(\mu \) with values in the ring \(\hat \mathcal M\) gives rise to the notion of motivic Gorenstein measure
\[
\mu^{\text{Gor}}(A)=\int _A{\mathbb L}^{-\text{ord}}_{t\omega _X}{\text{ d}}\mu
\]
of each subset \(A\in B^t\). Here \(\text{ord}_t\omega _X\) is the order of a global section \(\omega _X\) of \(\Omega _X^n\otimes k(X)\) generating \(\Omega _X^n\) at each smooth point of \(X\) (use that \(X\) is Gorenstein with good singularities).
Now we are returning to the quotient \(X=\mathbb A^n_k/G\), where \(G\) is a finite subgroup of \(\text{SL}(k,n)\). Let \(\mathcal L(X)_0\) be the set of arcs whose origins are in the image of the point \(0\) in the quotient \(X\). The main result of the paper expresses the Gorenstein motivic measure \(\mu^{\text{Gor}}(\mathcal L(X)_0)\) in terms of weights \(w(\gamma )\) of conjugacy classes \(\gamma \) in \(G\). Namely, the equality
\[
\mu ^{\text{Gor }}(\mathcal L(X)_0)= \sum _{\gamma \in \text{Conj}(G)}\mathbb L^{-w(\gamma )}
\]
holds in \(\hat \mathcal M_{\slash }\), where \(\text{Conj}(G)\) is the set of conjugacy classes of \(G\). As a corollary, if \(h:Y\to X\) is a crepant resolution, then
\[
[h^{-1}(0)]=\sum _{\gamma \in \text{Conj}(G)}\mathbb L^{n-w(\gamma )}
\]
in the ring \(\hat \mathcal M_{\slash }\). This is already a motivic expression of the McKay correspondence. If \(k=\mathbb C\) and we pass to the Hodge realization, we get the result proved by Batyrev and conjectured by Reid. germs of arcs; motivic measure; irreducible representation; crepant resolution Denef, Jan; Loeser, François, Motivic integration, quotient singularities and the mckay correspondence, Compos. Math., 131, 3, 267-290, (2002) Arcs and motivic integration, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Schemes and morphisms, Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients) Motivic integration, quotient singularities and the McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Seit vor nun fast hundert Jahren Hilbert im Zusammenhang mit dem 12. Hilbertschen Problem das nähere Studium der Hilbertschen Modulgruppen \(\Gamma\) total-reeller algebraischer Zahlkörper \(K\) und ihrer Modulfunktionen als direkte Analoga zur klassischen Modulgruppe anregte, hat sich, ausgehend von den Arbeiten von Blumenthal und Hecke, eine umfangreiche Theorie entwickelt. Es gab jedoch keine einführende Darstellung, so daß jeder Interessierte von Anfang an auf die ständig wachsende Flut von Originalarbeiten angewiesen war. Anfang der siebziger Jahre gelang es F. Hirzebruch, die Singularitäten des kompaktifizierten Quotientenraumes von \(\Gamma\) im Fall \([K:\mathbb Q]=2\) aufzulösen, wodurch eine singularitätenfreie projektive algebraische Fläche entstand. Damit war die unmittelbare Verbindung zur algebraischen Geometrie hergestellt, für die diese Flächen unter anderem eine Menge interessanten Beispiele und Sonderfälle lieferten.
Der Verf. des vorliegenden Buches schrieb 1981 zusammen mit \textit{F. Hirzebruch} eine erste Einfüh\-rung in diesen Teil der Theorie der Hilbertschen Modulfunktionen (d.h., \(\Gamma\) für \([K:\mathbb Q]=2\), zugehörige Quotientenflächen und ihre algebraischen Untervarietäten) unter dem Titel ``Lectures on Hilbert modular surfaces'' [Séminaire de Mathématiques Supérieures, Séminaire Scientifique OTAN (NATO Advanced Study Institute), Dep. Math. Stat., Univ. Montréal 77 (1981; Zbl 0483.14009)]. Die neue Darstellung entspricht im Aufbau bis auf einige Umstellungen im wesentlichen dem vom Verf. stammenden zweiten Teil der ursprünglichen Einführung einschließlich des Stoffes des Kapitels über Hilbertsche Modulflächen des ersten, damals von Hirzebruch verfaßten Teiles [für den Inhalt sei auf die ausführliche Besprechung von \textit{D. Zagier} im Zbl 0483.14009 verwiesen], ist aber im einzelnen bedeutend ausführlicher. Am Anfang steht jetzt ein zusätzliches Kapitel mit einer kurzen Beschreibung der Grundlagen (``Spitzen, Fundamentalbereich, elliptische Fixpunkte, Modulformen'') und am Schluß ein Kapitel über die Tate-Vermutungen für Hilbertsche Modulflächen, in dem über die neuen Ergebnisse von \textit{G. Harder}, \textit{R. P. Langlands} und \textit{M. Rapoport} [J. Reine Angew. Math. 366, 53--120 (1986; Zbl 0575.14004)] und \textit{C. Klingenberg} [Invent. Math. 89, 291--318 (1987; Zbl 0601.14007)] berichtet wird. Hervorzuheben ist das ebenfalls neue Kapitel VIII mit einer Zusammenstellung vieler interessanter Beispiele.
Die übrigen Kapitel (II.\ Auflösung der Spitzensingularitäten, III.\ Lokale Invarianten, IV.\ Globale Invarianten, V.\ Modulkurven auf Modulflächen, VI.\ Kohomologie, VII.\ Klassifikation der Hilbertschen Modulflächen, IX.\ Humbertsche Flächen, X.\ Moduln abelscher Flächen mit reeller Multipikation') bringen jeweils eine gut lesbare Einführung und verleihen durch eine Übersicht über weitere Ergebnisse mit ausführlichen Literaturhinweisen dem Buch den Charakter eines Handbuchs. Angefügt ist eine Tabelle numerischer Invarianten für die Hilbertschen Modulgruppen reell-quadratischer Zahlkörper mit den Diskriminanten \(D<500\).
Das Buch dürfte für längere Zeit das Standardwerk über dieses Gebiet sein. classification of Hilbert modular surfaces; Hilbert-Blumenthal surfaces; resolution of cusp singularities; Humbert surfaces; moduli Van Der Geer, G., \textit{Hilbert modular surfaces}, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in mathematics and related areas (3)], Vol. 16, (1988), Springer, Berlin Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic ground fields for surfaces or higher-dimensional varieties, Modular and Shimura varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to number theory, Global theory and resolution of singularities (algebro-geometric aspects) Hilbert modular surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the first part of a series of papers devoted to the resolution of singularities (and containing joint work with K. Matsuki, as mentioned in 0.6). The goal is presenting a program towards the construction of a resolution algorithm which works for an algebraic variety over a perfect field in positive characteristic. The author's intention is however, to develop a program working in full generality, i.e. including characteristic 0 as well.
The intended parts of the complete work are:
I. Foundation; the language of the idealistic filtration
II. Basic invariants associated to the idealistic filtration and their properties
III. Transformations and modifications of the idealistic filtration
IV. Algorithm in the framework of the idealistic filtration
Part I (under review) establishes the notion and fundamental properties of the \textit{idealistic filtration}, which is considered the main language of this program.
Chapter 0 starts with a crash course on the existing algorithms in characteristic 0 and introduces the author's program as a ``new approach to overcome the main source of troubles in the language of the \textit{idealistic filtration}, which is a refined extension of such classical notions as the idealistic exponent by Hironaka, the presentation by Bierstone-Milman, the basic object by Villamayor, and the marked ideal by Wlodarczyk.''
Section 0.2.3 introduces the idealistic filtration and mentions some of its distinguished features as there are: leading generator systems as substitutes for hypersurfaces of maximal contact, construction of the strand of invariants through enlargements of an idealistic filtration, saturation and a ``new nonsingularity principle''.
In 0.3 (``Algorithm constructed according to the program'') the author refers to the forthcomimg part IV of the paper as far as termination of the algorithm (in case of positive characteristic) is concerned; he mentions that this question is not yet settled.
In 0.5 a brief account of (mainly references to) the history of the problem is given, as well as hints to recent announcements and approaches.
The remaining about 70 pages of the paper (Part I) contain essentially the ``local'' ingredients of the program. Below follows (a part of) the author's outline, taken from 0.8:
``In Chapter 1, we recall some basic facts on the differential operators, especially those in positive characteristic. Both in the description of the preliminaries and in Chapter 1, our purpose is not exhaustively cover all the material, bu only to minimally summarize what is needed to present our program and to fix our notation.''
``Chapter 2 is devoted to establishing the notion of an idealistic filtration and its fundamental properties. The most important ingredient of Chapter 2 is the analysis of the \(\mathcal D\)-saturation and the \(\mathcal R\)-saturation and that of their interaction. In our algorithm, given an idealistic filtration, we always look for its bi-saturation, called the \(\mathcal B\)-saturation, which is both \(\mathcal D\)-saturated and \(\mathcal R\)-saturated and which is minimal among such containing the original idealistic filtration. The existence of the \(\mathcal B\)-saturation is theoretically clear. However, we do not know a priori whether we can reach the \(\mathcal B\)-saturation by a repetition of \(\mathcal D\)-saturations and \(\mathcal R\)-saturations starting from the given idealistic filtration, even after infinitely many times. The main result here is that the \(\mathcal B\)-saturation is actually realized if we take the \(\mathcal D\)-saturation and then the \(\mathcal R\)-saturation of the given one, each just once in this order. In our algorithm, we do not deal with an arbitrary idealistic filtration, but only with those which are generated by finitely many elements with rational levels. We say they are of r.f.g. type (short for `rationally and finitely generated'). It is then a natural and crucial question if the propery of being of r.f.g. type is stable under \(\mathcal D\)-saturation and \(\mathcal R\)-saturation.''
``In Chapter 3, through the analysis of the leading terms of an idealistic filtration (which is \(\mathcal D\)-saturated), we define the notion of a leading generator system, which \dots plays the role of a collective substitute for the notion of a hypersurface of maximal contact.
Chapter 4 is the culmination of part I, establishing the new nonsingularity principle of the center for an idealistic filtration which is \(\mathcal B\)-saturated. Its proof is given via three somewhat technical but important lemmas, which we will use again later in the series of papers.'' resolution of singulairties; positive characteristic Kawanoue, H.: Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 819-909. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0653.00009.]
The paper is devoted to the study and explicit description of the \(T^ 1_ X\) of a 2-dimensional quotient singularity \(X=(X,0)=({\mathbb{C}}^ 2,0)/G\) with \(G\subset GL(2,{\mathbb{C}})\) finite. One wants to determine \(T^ 1_ X\) only in terms of certain concrete G-modules and their invariants. It turned out that it is sometimes easier to deal with \((T^ 1_ X)^*\) instead of \(T^ 1_ X\). Some calculations are done in the case of rational double points and the situation is stated in the ``exceptional'' cases. deformations of singularities; quotient singularity; exceptional Behnke, K. , Kahn, C. , Riemenschneider, O. , Infinitesimal deformations of quotient surface singularities , in ' Singularities,' Banach Center Publications, Polish Scientific Publishers, Warzaw, 1989, pp. 31-66. Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry Infinitesimal deformations of quotient surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors investigate a procedure to resolve singularities that has reasonable ``functorial'' properties. The main result is expressed in terms of ``marked ideals'', that is a 5-tuple \(\mathcal I = (M,N,E,I,d)\) where \(N\) is a subvariety of \(M\), both smooth over a field of characteristic zero, \(I\) is a coherent sheaf of ideals of \(N\), \(E\) is a normal crossings divisor of \(M\), transversal to \(N\) and \(d\) is a positive integer. The singular locus, or cosupport, of \(\mathcal I\) is the set of points \(x\) of \(N\) such that the order of the stalk \(I_x\) is \(\geq d\). One may define the transformation of such a marked ideal with center a suitable smooth subvariety of \(N\) (the \textit{admissible} transformations), the result is a new marked ideal.
The objective is to obtain, by means of a finite sequence of admissible transformations, a marked ideal whose cosupport is empty. Such a sequence is called a resolution sequence. If this can be done in a reasonable constructive (or algorithmic) way other more classical desingularization theorems follow rather easily. (This is explained in the present article). The authors define a notion of equivalence of marked ideals as follows. A test transformation is either an admissible one, or one determined by the blowing-up of a center which is the intersection of two components of \(E\), or one induced by a projection \(M \times {\mathbb{A}}^1 \to M\). Two marked ideals \(\mathcal I = (M,N,E,I,d)\) and \(\mathcal J = (M,N',E,J,d')\) are equivalent if they have the same sequences of of test transformations. Then they prove that there is way to associate to each marked ideal a resolution sequence such that if \(\mathcal I\) and \(\mathcal J\) are equivalent, then the resolution sequence associated to \(\mathcal I\) is obtained by using the same centers as were needed for the sequence of \(\mathcal J\). Moreover, this procedure is compatible with smooth morphisms \(M' \to M\). This is what is meant by functoriality of the process.
The procedure is a variant of that introduced by the authors in their fundamental paper \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)] incorporating some ideas from \textit{J. Włodarczyck}'s article [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)].
The exploitation of the explicit requirements on functoriality simplifies the verification of the fact that certain constructions are independent of the elections made, which is a hard problem in this type of desingularization work. The authors use the functorial character of the algorithm to show that it coincides with others recently introduced by J. Wlodarczyk and J. Kollár. resolution; blowing-up; marked ideal; admissible transformation; derivative ideals; test transformation Bierstone, E., Milman, P.: Functoriality in resolution of singularities. Publ. R.I.M.S. Kyoto Univ.~\textbf{44}, 609-639 (2008) Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Birational geometry, Modifications; resolution of singularities (complex-analytic aspects), Equisingularity (topological and analytic) Functoriality in resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a complex classical simple Lie algebra \(\mathfrak{g}\) and a nilpotent orbit \(O\) of \(\mathfrak{g}\), the fundamental group \(\pi_1(O)\) is finite. Taking the universal covering \(\pi^0: X^0 \to O\), \(\pi^0\) extends to a finite cover \(\pi: X \to \bar{O}\), where \(\bar{O}\) is the closure of \(O\) in \(\mathfrak{g}\). By using the Kirillov-Kostant form \(\omega_{KK}\) on \(O\), the normal affine variety \(X\) becomes a conical symplectic variety. Let \(\omega := (\pi^0)^*\omega_{KK}\), a \(G\)-invariant symplectic \(2\)-form on \(X^0\).
The author studies the birational geometry for the resolutions of \((X, \omega)\). A crepant projective resolution \(f: Y \to X\) of \(X\) is a projective birational morphism \(f\) from a nonsingular variety \(Y\) to \(X\) such that \(K_Y = f^*K_X\). In general, \(X\) does not have a crepant projective resolution. However, \(X\) always has a nice crepant projective partial resolution \(f: Y \to X\) known as \(\mathbb{Q}\)-factorial terminalization. A \(\mathbb{Q}\)-factorial terminalization \(f\) is a projective birational morphism from a normal variety \(Y\) to \(X\) such that \(Y\) has only \(\mathbb{Q}\)-factorial terminal singularities and \(K_Y = f^*K_X\).
The author provides an explicit construction of a \(\mathbb{Q}\)-factorial terminalization of \(X\) when \(O\) is a nilpotent orbit of a classical simple Lie algebra \(\mathfrak{g}\) and \(X^0\) is the universal covering of \(O\). That is, let \(Q \subset G\) be a parabolic subgroup of \(G\) and let \(Q = U L\) be a Levi decomposition of \(Q\) by the unipotent radical \(U\) and a Levi subgroup \(L\). Its Lie algebra \(\mathfrak{q} = \text{Lie}(Q) = \mathfrak{n} \oplus \mathfrak{l}\) decomposes as a direct sum of \(\mathfrak{n} := \text{Lie}(U)\) and \(\mathfrak{l} := \text{Lie}(L)\). Let \(O'\) be a nilpotent orbit of \(\mathfrak{l}\). Then there is a unique nilpotent orbit \(O\) of \(\mathfrak{g}\) such that \(O\) meets \(\mathfrak{n} + O'\) in a Zariski open subset of \(\mathfrak{n} + O'\). In such a case, \(O\) is induced from \(O'\) and write \(O = \mathrm{Ind}^{\mathfrak g}_{\mathfrak l}(O')\). There is a generically finite map \(\mu: G\times^Q (\mathfrak{n}+ \bar{O}') \rightarrow \bar{O}\), where \([g,z] \mapsto \text{Ad}_g(z)\), which the author calls a generalized Springer map. Let \((X')^0 \to O'\) be an etale covering and let \(X' \to \bar{O}'\) be the associated finite cover. Then the author considers the space \(\mathfrak{n} + X'\) which is a product of an affine space \(\mathfrak{n}\) and the affine variety \(X'\). There is a finite cover \(\mathfrak{n} + X' \to \mathfrak{n} + \bar{O}'\). If the \(Q\)-action on \(\mathfrak{n} + \bar{O}'\) lifts to a Q-action on \(\mathfrak{n} + X'\), then one can make \(G \times^Q(\mathfrak{n} + X')\) and get a certain commutative diagram involving the Stein factorization. For an arbitrary nilpotent orbit \(O\) of a classical Lie algebra \(\mathfrak{g}\), the author gives an explicit algorithm for finding \(Q\), \(O'\) and \(X'\) such that \(O = \mathrm{Ind}^{\mathfrak g}_{\mathfrak l}(O')\), \(X'\) has only \(\mathbb{Q}\)-factorial terminal singularities, and the Q-action on \(\mathfrak{n} + \bar{O}'\) lifts to a Q-action on \(\mathfrak{n} + X'\) and the finite covering \(Z \to \bar{O}\) in the commutative diagram coincides with the finite covering \(\pi: X \to \bar{O}\) associated with the universal covering \(X^0\) of \(O\). nilpotent orbits; birational geometry; conical symplectic variety; \(\mathbb Q\)-factorial terminalization; \(\mathbb Q\)-factorial terminal singularities Global theory and resolution of singularities (algebro-geometric aspects), Coadjoint orbits; nilpotent varieties Birational geometry for the covering of a nilpotent orbit closure | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is a survey about generalized theta functions as sections of ample line bundles on moduli spaces [see also \textit{G. Faltings}, J. Algebr. Geom. 18, No. 2, 309--369 (2009; Zbl 1161.14025)]. It has seven sections:
\textit{1. Introduction}. Here the frame of the paper is introduced: one considers moduli of \(G\)-bundles over a curve, where \(G\) is a semisimple simply connected algebraic group and the main results are described: the Picard group of \({\mathcal M}_G\) is infinite cyclic, (the sections of these vector bundles are generalized theta functions) and the dimension of the vector space of sections is given (via Verlinde formula).
\textit{2. Moduli spaces.} Here the construction of the stack \(\mathcal{M}_{r,d}\) of vector bundles of rank \(r\) and degree \(d\) over a curve is sketched. Then one describes in few words the changes if \(GL_r\) is replaced with a group as above.
\textit{3. The double quotient.} Here the following setting is considered: take \(C\) a smooth projective curve over an algebraic closed field \(k\), \(x\in C\) a point and \(C^0=C\setminus \{x\}\). Then the moduli space is a double quotient:
\[
\mathcal{M}_G(k)=G(C^0)\setminus G(k((t))) /G(k[[t]]) .
\]
One studies first the right quotient, which turns out to be an affine Grassmanian \({\mathbb D}_G\). Using \(\text{Pic}({\mathbb D}_G)\) one reduces the study of \(\text{Pic}({\mathcal M}_G)\) to \(G(C^0)\)-equivariant line bundles on \({\mathbb D}_G\).
\textit{4. Line bundles on \({\mathcal M}_G\).} Two important examples corresponding to \(G=SL_r\) and \(G=Spin(r)\) (spingroup of \(SO(r)\)) are sketched.
\textit{5. Construction of a line bundle} of invariant \(1\).
\textit{6. The Verlinde formula.} Here the use of Verlinde formula to compute, in characteristic zero, \(\text{dim }\Gamma ({\mathcal M}_G,{\mathcal L}_c)\) for a line bundle of invariant \(c\) is explained in a simplified way. Some comments about ``das große offene Problem'' of describing geometrically these sections are given.
\textit{7. The Hitchin fibration.} Here the use of the Hitchin fibration to obtain results in positive characteristic is described, in a simplified version. vector bundles; moduli spaces; theta functions; algebraic group Vector bundles on curves and their moduli, Theta functions and curves; Schottky problem Theta functions on moduli spaces of vector 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 This article is concerned with the relation between several classical and well-known objects: triangle Fuchsian groups, \(\mathbb{C}^\times\)-equivariant singularities of plane curves, torus knot complements in the 3-sphere. The prototypical example is the modular group \(\text{PSL}_2(\mathbb{Z})\): the quotient of the nonzero tangent bundle on the upper-half plane by the action of \(\text{PSL}_2(\mathbb{Z})\) is biholomorphic to the complement of the plane curve \(z^3- 27w^2= 0\). This can be shown using the fact that the algebra of modular forms is doubly generated, by \(g_2\), \(g_3\), and the cusp form \(\Delta= g^3_2- 27g^2_3\) does not vanish on the half-plane. As a byproduct, one finds a diffeomorphism between \(\text{PSL}_2(\mathbb{R})/\text{PSL}_2(\mathbb{Z})\) and the complement of the trefoil knot -- the local knot of the singular curve. This construction is generalized to include all \((p,q,\infty)\)-triangle groups and, respectively, curves of the form \(z^q+ w^p= 0\) and \((p,q)\)-torus knots, for \(p\), \(q\) co-prime. The general case requires the use of automorphic forms on the simply connected group \(\widetilde{\text{SL}}_2(\mathbb{R})\).
The proof uses ideas of Milnor and Dolgachev, which they introduced in their studies of the spectra of the algebras of automorphic forms of cocompact triangle groups (and, more generally, uniform lattices). It turns out that the same approach, with some modifications, allows one to handle the cuspidal case. triangle Fuchsian group; singularities of plane curves; torus knot complements Knots and links in the 3-sphere, Singularities of curves, local rings, Fuchsian groups and their generalizations (group-theoretic aspects), Automorphic forms in several complex variables Triangle groups, automorphic forms, and torus knots | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\xi\) be the torus bundle over the unit circle \(S^1\) which arises from the boundary of a one parameter degeneration of an abelian variety (of complex dimension \(n=2k-1\)) in algebraic geometry.
The goal of this paper is to calculate some singular defects and eta functions of the bundle \(\xi\). Let \((Y,\alpha)\) be a pair of a compact oriented smooth manifold \(Y\) of real dimension \(4k-1\) and a trivialization \(\alpha\) of the tangent bundle of \(Y\). \textit{F. Hirzebruch} [Enseign. Math., II. Sér. 19, 183-281 (1973; Zbl 0285.14007)] defined the signature defect \(\sigma(Y,\alpha)\) for a general framed manifold \((Y,\alpha)\) as the difference between the evaluation of an \(L\)-polynomial of relative Pontrjagin classes \(X\) in the fundamental class \([X,Y]\) and the signature on \(H^{2k}(X,Y,\mathbb{R})\). Also, the signature defect for a cusp singularity is defined. The eta invariant \(\eta(D,s)\) for a framed manifold \((Y,\alpha)\) can be defined using certain first order self-adjoint operators \(D\) induced by the flat connection. \textit{M. F. Atiyah, H. Donnelly} and \textit{J. M. Singer} [Ann. Math., II. Ser. 118, 131-177 (1983; Zbl 0531.58048)], proved using index theorems that \(\eta(D,0)=\sigma(Y,\alpha)\) for a cusp of a Hilbert modular variety.
The torus considered in the paper can be characterized in terms of a monodromy transformation \(T\) on \(H_1(X_t,\mathbb{Z})\). The cases that (I) the monodromy is unipotent and (II) the monodromy is finite are studied separately for technical reasons. In the first case it is shown that the eta function \(\eta(D,s)\) coincides with the Riemann zeta function up to a factor when \(n=1\) and identically vanishes when \(n\geq 2\). The signature defect \(\sigma(Y,\alpha)\) is explicitly calculated and it is shown that for \(n\geq 2\), the invariant \(\sigma(Y,\alpha)\) is an integer depending only on the dimension \(n\) and rank\((T-I)\). The corresponding result for the second case is also established. eta function; signature defect; torus bundle Ogata S., Saito M.-H.: Signature defects and eta functions of degenerations of Abelian varieties. Jpn. J. Math. 23, 319--364 (1997) Index theory and related fixed-point theorems on manifolds, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Signature defects and eta functions of degenerations of 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 A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristics, including the wild case, was recently formulated by the author in the case where the given finite group acts linearly on an affine space. In cases of very special groups and representations, the conjecture has been verified and related stringy invariants have been explicitly computed. In this paper, we try to generalize the conjecture and computations to more complicated situations such as nonlinear actions on possibly singular spaces and nonpermutation representations of nonabelian groups. McKay correspondence, Arcs and motivic integration, Ramification and extension theory Wilder McKay 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 Resolution of singularities is one of the central problems in algebraic geometry. \textit{H. Hironaka} proved, in a non-constructive way, that the singularities of a scheme over fields of characteristic zero can be resolved [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)]. Since the late eighties many constructive algorithms have appeared giving an explicit way to resolve singularities, e.g. \textit{O. E. Villamayor U.} [Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 1--32 (1989; Zbl 0675.14003)], \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207-302 (1997; Zbl 0896.14006)], \textit{S. Encinas} and \textit{O. E. Villamayor U.} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)].
The algorithms of resolution can be divided into two main steps. The first of them consists in constructing a sequence of monoidal transformations on smooth centers so as to achieve a simplification. This step is done in an inductive way, so as to obtain a monomial ideal supported on the exceptional locus in lower dimension. When this step is finished it is said that the scheme is within the \textit{monomial case}. The second step consists in an easy combinatorial argument that leads to resolution.
The first step relies on the existence of hypersurfaces of maximal contact and the existence of some inductive functions that allow to stratify the singular locus on smooth strata. It is well known that the existence of these hypersurfaces fails in positive characteristic [e.g. \textit{R. Narasimhan}, Proc. Am. Math. Soc. 89, 402--406 (1983; Zbl 0554.14010)]. In a previous paper of one of the authors [\textit{O. E. Villamayor U.}, Adv. Math. 213, No. 2, 687--733 (2007; Zbl 1118.14016)], the notion of restriction to hypersurfaces of maximal contact is replaced by transversal projections (elimination of variables) and elimination algebras.
The authors prove in this paper, using the elimination approach, a form to extend to arbitrary characteristic the inductive functions (in lower dimension) of the first step. Moreover, they prove that the values of these functions are independent of the chosen transversal projection. This result allow the authors to extend the first step of the algorithms to arbitrary characteristic and define a stratification of the singular locus. As a consequence, they also prove the ``reduction to the monomial case'' in positive characteristic. resolution of singularities; positive characteristic; Rees algebras; differential operators A. Bravo and O. Villamayor, Singularities in positive characteristic, stratification and simplification of the singular locus. Adv. Math. 224 (2010), 1349-1418. Global theory and resolution of singularities (algebro-geometric aspects) Singularities in positive characteristic, stratification and simplification of the singular locus | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \( G=Sp_{2n}(\mathbb{C})\), and \( \mathfrak{N}\) be Kato's exotic nilpotent cone. Following techniques used by \textit{R. Bezrukavnikov} [Represent. Theory 7, 1--18 (2003; Zbl 1065.20055)] to establish a bijection between \( \boldsymbol {\Lambda }^+\), the dominant weights for an arbitrary simple algebraic group \( H\), and \( \mathbb{O}\), the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution \( \widetilde {\mathfrak{N}}\) have vanishing higher cohomology, and compute their global sections using techniques of \textit{B. Broer} [Invent. Math. 113, No. 1, 1--20 (1993; Zbl 0807.14043)]. This allows us to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category \( D^b(Coh^G(\mathfrak{N}))\), and deduce that the resulting \( t\)-structure on \( D^b(Coh^G(\mathfrak{N}))\) coincides with the perverse coherent \( t\)-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of the \( t\)-structure on \( D^b(Coh^G(\mathfrak{N}))\). Lie algebras of linear algebraic groups, Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Equivariant coherent sheaves on the exotic nilpotent cone | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 improves on a result of the reviewer [Math. Nachr. 188, 321--340 (1997; Zbl 0907.14019)], by showing that the moduli space \({\mathcal A}_p\) of abelian surfaces with a polarisation of type \((1,p)\) for \(p\) prime is of general type if \(p=37\), \(43\), \(53\), \(61\) or \(67\) or \(p\geq 73\). In outline the present paper follows the reviewer's argument quite closely, and likewise relies on the 1994 Erlangen Ph.D. thesis of H.-J. Brasch (unfortunately never published). Brasch describes the singularities of a certain toroidal compactification of \({\mathcal A}_p\) in great detail. Not all of them are canonical: the reviewer used cusp forms vanishing to very high order at infinity in order to construct pluricanonical forms that would nevertheless extend to a resolution of singularities.
The improvement here (the reviewer had the bound \(p\geq 173\)) is brought about by considering a different toroidal compactification which is shown to have much milder singularities. Although a few non-canonical singularities are still present, they are so few as to be negligible for all but very small \(p\). It is still necessary to have cusp forms of weight \(3n\) vanishing to order~\(n\) at infinity, and this is achieved by taking a cusp form \(F_2\) of weight~\(2\) for the paramodular group associated with type \((1,p)\) and taking a weight \(3n\) form which is \(F_2^nF_n\) for some modular form \(F_n\) of weight \(n\). Such a weight \(2\) cusp form may be found, using a result of Gritsenko, by lifting a Jacobi form of weight \(2\) and level \(p\), and the primes for which such a Jacobi form exists are precisely those in the statement of the theorem. John McKay points out that these are also the primes that do not divide the order of the Monster. C. Erdenberger, The Kodaira dimension of certain moduli spaces of abelian surfaces, Math. Nachr., 274/275 (2004), 32--39.Zbl 1065.14058 MR 2092323 Algebraic moduli of abelian varieties, classification, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Fine and coarse moduli spaces The Kodaira dimension of certain moduli spaces of abelian 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 quiver \(Q\) is a directed graph having a finite number of vertices \(V\), a set of arrows \(A\), and two functions \(h,t:A\to V\). Any image of \(h\) is called the \textit{head} of the corresponding arrow, likewise any image of \(t\) is called a \textit{tail}. \(Q\) is said to be of \textit{ladder type} if \(V\) is the disjoint union of a set of heads \(H\) with a set of tails \(T\). With respect to a fixed dimension vector for \(Q\) (assigning to each vertex a natural number), there is a product of general linear groups with respect to \(H\) (denoted by \(G_H\)) and also with respect to \(T\) (denoted by \(G_T\)). The author likewise defines products of Grassmannians \(X_T\) and \(X_H\) where \(G_H\) and \(G_T\) respectively act. The main theorem of this paper states that the set of GIT quotients of \(X_T\) by \(G_H\) and the set of GIT quotients of \(X_H\) by \(G_T\) are in one-to-one correspondence. The proof comes down to the observation that the moduli space of representations of \(Q\), since \(Q\) is of ladder type, can be obtained by first taking a quotient by \(G_H\) resulting in \(X_H\) and then by \(G_T\), or by first taking the quotient by \(G_T\) resulting in \(X_T\) and then by \(G_H\). Later a similar theorem is described with respect to parabolic subgroups of the general linear group and with Grassmannians replaced by flag varieties. The remainder of the paper is devoted to algebraic and representation theoretic interpretations of these geometric results. GIT quotients; quotient correspondences; representation theory; branching rules Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Representations of quivers and partially ordered sets, Representation theory for linear algebraic groups Quivers, invariants and quotient correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is the topological study of the complement to irreducible sextics in the complex projective plane.
The author makes a very effective use of the lattice structure of \(H_2(\tilde X)\) (the intersection lattice of the resolution of the double cover of a sextic curve \(C\) with simple singularities) in order to study properties of the fundamental group of the complement of \(C\) in \(\mathbb P^2\).
In a recent paper, \textit{C. Eyral} and \textit{M. Oka} [J. Math. Soc. Japan 57, No. 1, 37--54 (2005; Zbl 1070.14031)] suggested the conjecture tha any irreducible sextic curve, which is not of torus type satisfies the following: its Alexander polynomial is trivial, and the fundamental group of its complement is abelian.
The author proves the first part of Oka's conjecture for simple singularities and disproves the second part, both for simple singularities and non-simple singularities.
Also, as a consequence of the techniques presented the author determines the rigid isotopy classes of sextics with simple singularities whose fundamental group factors through the dihedral group \(\mathbb D_{10}\) and \(\mathbb D_{14}\). This characterization brings up new examples of Alexander-equivalent Zariski pairs. Plane curves; Alexander polinomials; Zariski pairs; torus type sextics A Degtyarev, Oka's conjecture on irreducible plane sextics, J. Lond. Math. Soc. \((2)\) 78 (2008) 329 Coverings of curves, fundamental group, \(K3\) surfaces and Enriques surfaces, Computational aspects of algebraic curves, Families, moduli of curves (algebraic) Oka's conjecture on irreducible plane sextics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00004.]
The author describes a minimal embedded resolution of the affine variety given by the vanishing of the only nontrivial invariant polynomial for the fundamental linear representation of dimension 27 (resp. 56) of a simple Lie group G of type \(E_ 6\) (resp. \(E_ 7\)). We describe the results only in the case of \(E_ 6\). In this case the invariant polynomial is a famous Jordan-Cartan-Dickson cubic with each of its 45 monomials corresponding to tritangent planes of a nonsingular cubic surface. The variety is resolved by one blowing up of \({\mathbb{A}}^{27}\) at its vertex followed by the blowing up of the exceptional divisor \({\mathbb{P}}^{26}\) at the singular locus of the projectivized cubic. The latter is a nonsingular 16-dimensional variety isomorphic to G/P, where P is a maximal parabolic subgroup of G. The total inverse transform of the cubic is a normal crossing divisor with irreducible components of multiplicities 1, 3 and 2 corresponding to the proper inverse transform, the first and the second exceptional divisor respecively. minimal embedded resolution; \(E_ 6\); \(E_ 7\); Jordan-Cartan-Dickson cubic; total inverse transform Global theory and resolution of singularities (algebro-geometric aspects), Other algebraic groups (geometric aspects), Representation theory for linear algebraic groups, Singularities in algebraic geometry, Semisimple Lie groups and their representations The singularities of some invariant hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a two-dimensional rational singularity, classical theorems of \textit{J. Lipman} [Publ. Math., Inst. Hautes Étud. Sci. 36, 195--279 (1969; Zbl 0181.48903)] showed that every integrally closed ideal \(I\) satisfies \(I^2=IQ\) for every minimal reduction \(Q\) of \(I\), and that if \(I\), \(J\) are integrally closed ideals, then so is their product \(IJ\). The goal of this paper is to explore the ideal theory for a general two-dimensional normal non-rational singularity.
Let \((A,m)\) be an excellent normal local ring of dimension \(2\) such that \(A\) contains an algebraically closed field \(k\cong A/m\) and let \(f\): \(X\to\mathrm{Spec }A\) be a resolution of singularities with exceptional divisor \(E\). Suppose \(Z\neq 0\) is an anti-nef cycle of \(X\) such that the base locus of the linear system \(H^0(X, O_X(-Z))\) does not contain any component of \(E\). Then the authors show that \(h^1(X, O_X(-Z))\leq p_g(A)=h^1(X, O_X)\). \(Z\) is called a \(p_g\)-cycle if we have equality (and in this case \(O_X(-Z)\) is generated by global sections), and an integrally closed \(m\)-primary ideal \(I\) is called a \(p_g\)-ideal if \(I\) is represented by a \(p_g\)-cycle on some resolution (i.e., \(I=H^0(X, O_X(-Z))\)). If \(A\) is a rational singularity then every anti-nef cycle is a \(p_g\)-cycle and hence every integrally closed ideal is a \(p_g\)-ideal. The authors show that, in general, the class of \(p_g\)-ideal inherits nice properties of integrally closed ideals of rational singularities: for example \(I^2=IQ\) for every minimal reduction \(Q\) of \(I\), and if \(I\), \(J\) are \(p_g\)-ideals then so is their product \(IJ\).
The main result of the paper is that, for \((A,m)\) a two-dimensional normal local ring, there exists a resolution on which \(p_g\)-cycles exist (and thus \(p_g\)-ideals exist). Moreover, they used this to show that if \(A\) is non-regular Gorenstein, then it has good ideals, i.e., \(I^2=IQ\) and \(Q:I=I\). The paper also investigates many other applications of \(p_g\)-ideals, for example the number of minimal generators of certain integrally closed ideals, and (non)existence results on Ulrich ideals for certain elliptic singularity. \(p_g\)-cycles; \(p_g\)-ideals; rational singularities T. Okuma, K-i. Watanabe, and K. Yoshida, Good ideals and \(p_{g}\)-ideals in two-dimensional normal singularities, Manuscripta Math. 150 (2016), 499--520. Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Good ideals and \(p_g\)-ideals in two-dimensional normal 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 \(A\) be an elliptic curve. Consider a finite (non-cyclic) subgroup \(G<\mathrm{SL}(3,\mathbb{Z})\). Then \(G\) acts on \(A^3\cong\mathbb{Z}^3\otimes_{\mathbb{Z}}A\) by left multiplication on \(\mathbb{Z}^3\). Denote the orbit space from this action by \(Y=A^3/G\). Using the methods developed by \textit{M. Andreatta} and \textit{J. A. Wiśniewski} [Rev. Mat. Complut. 23, No. 1, 191--215 (2010; Zbl 1191.14049)], the author resolves the singularities of \(Y\) to obtain a projective variety \(\widetilde{Y}\) called a \textit{Kummer three-fold} (it is a \textit{crepant} resolution). The author notes that Kummer three-folds are \textit{Calabi-Yau}. In this paper the author classifies the Poincaré polynomials of all Kummer three-folds.
The paper begins by reviewing the classification of conjugate equivalent finite (non-cyclic) subgroups of \(\mathrm{SL}(3,\mathbb{Z})\); there are 16 of them. Two conjugate subgroups result in isomorphic Kummer three-folds. In each of these 16 cases, the author computes the (virtual) Betti numbers of \(Y\) and \(\widetilde{Y}\). In the former case, this can be accomplished by determining the dimension of the spaces of \(G\)-invariant forms on \(A^3\). To determine the Poincaré polynomial of \(\widetilde{Y}\), the author determines the contribution of the singularities in \(Y\) to the Betti numbers in \(\widetilde{Y}\) (the smooth strata of \(Y\) and \(\widetilde{Y}\) agree). There are 2 singular strata to consider; determined by the isotropy of the action. By comparing the virtual Poincaré polynomials of the strata of \(Y\) and \(\widetilde{Y}\), and using the additivity property of virtual Poincaré polynomials, the author is able to determine the virtual Poincaré polynomials for \(\widetilde{Y}\). However, the virtual Poincaré polynomial is the same as the Poincaré polynomial in this case since \(\widetilde{Y}\) is smooth and projective. Some of the calculations are performed with the computer algebra system GAP.
For example, there are 3 conjugation classes whose representatives are isomorphic to the symmetric group on four letters. In one such case, the Poincaré polynomial of \(Y\) is \(t^6+t^4+4t^3+t^2+1\) and the Poincaré polynomial of \(\widetilde{Y}\) is \(t^6+11t^4+8t^3+11t^2+1\). The other 15 cases are also presented. Kummer construction; Calabi-Yau variety; integral matrix groups; quotient singularity; virtual Poincaré polynomial Donten, M, On Kummer 3-folds, Rev. Mat. Complut., 24, 465-492, (2011) \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Calabi-Yau manifolds (algebro-geometric aspects), Integral representations of finite groups, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) On Kummer 3-folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a quiver of Dynkin type, it is natural to discuss if an orbit closure in the representation space is normal, and what singularity type it has. For types A and D, these questions were answered by \textit{S. Abeasis} et al. [Math. Ann. 256, 401--418 (1981; Zbl 0477.14027)], \textit{G. Bobiński} and \textit{G. Zwara} [Colloq. Math. 94, No. 2, 285--309 (2002; Zbl 1013.14011)] and \textit{V. Lakshmibai} and \textit{P. Magyar} [Int. Math. Res. Not. 1998, No. 12, 627--640 (1998; Zbl 0936.14001)]. The article at hand gives partial answers for type \(E\).
Let \(Q\) be a Dynkin quiver with source-sink orientation (that is, each vertex is either a source or a sink). For an orbit closure, a minimal free resolution is calculated by methods which generalize Lascoux's calculation of the resolutions of determinantal varieties.
The main result is stated in Theorem 3.5: If a representation \(V\) has an orbit closure \(\overline{\mathcal{O}}_V\) which admits a \(1\)-step desingularization, then \(\overline{\mathcal{O}}_V\) is normal and has rational singularities.
The methods of proof work hand in hand as follows:
Given an orbit closure with \(1\)-step desingularization, the author makes use of Reineke's construction of a desingularization of \(\overline{\mathcal{O}}_V\) (explained in Section 2.2). Via the pushforward \(\mathbb{F}_{\bullet}\) of the Koszul complex, \textit{J. M. Weyman}'s results [Cohomology of vector bundles and syzygies. Cambridge: Cambridge University Press (2003; Zbl 1075.13007)] on the terms of \(\mathbb{F}_{\bullet}\) (Theorem 2.2) can be made use of: if the terms \(\mathbb{F}_{i}\) vanish for negative \(i\) and if \(F_0\) has a certain structure (see Theorem 2.1), then \(\overline{\mathcal{O}}_V\) is normal and has rational singularities. The calculation of the terms of \(\mathbb{F}_{\bullet}\) uses \textit{A. Schofield}'s notion of an incidence variety [Proc. Lond. Math. Soc. (3) 65, No. 1, 46--64 (1992; Zbl 0795.16008)] and Bott's theorem (Theorem 2.4).
The article concludes by giving some examples in detail: in the cases \(A_4\), \(D_5\) and \(E_6\) concrete resolutions are calculated. orbit closures; resolutions; normality Sutar, K.: Orbit closures of source-sink Dynkin quivers, Int. math. Res. not. (2014) Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets Orbit closures of source-sink Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f\in {\mathbb{C}}[X_ 0,...,X_ d]\) be a polynomial such that the hypersurface defined by f has an isolated singularity. Denote by A the complete local ring \({\mathbb{C}}[[X_ 0,...,X_ d]]/(f)\). It is known that A has a simple singularity iff A has only finitely many isomorphism classes of indecomposable Cohen-Macaulay modules (Knörrer, Buchweitz- Greuel-Schreyer). In this case the Auslander-Reiten quiver Q(A) of A is determined and turns out to be closely related to the Dynkin diagram of A.
The author begins the study of the category of Cohen-Macaulay A-modules in the case when A has an isolated but not a simple singularity, proving the following results:
(1) Let C be the connected component of Q(A) containing A. Then \(Q(A)- C={\dot \cup}_{i\in I}A_{\infty}/<x^{n(i)}>,\) where I is an index set and \(n(i)\in\{1,2\}\). Moreover, if d is even, \(n(i)=1\) for all i.
(2) There exists an arithmetic sequence of positive integers r, 2r, 3r,... such that for each mr \((m\geq 1)\) there exists an infinite sequence \((M_{m,n})_{n\geq 1}\) of indecomposable pairwise non-isomorphic Cohen- Macaulay A-modules all of which of rank mr. simple singularity; Cohen-Macaulay modules; Auslander-Reiten quiver; Dynkin diagram Dieterich, E.: Reduction of isolated singularities. Comment. Math. Helvetici62, 654--676 (1987) Singularities in algebraic geometry, Representation theory of associative rings and algebras, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Reduction of 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 By the result of Dolgachev and Pinkham, the graded affine coordinate ring \(R\) of a normal quasi-homogeneous surface singularity \((X,o)\) is expressed as
\[
R=\bigoplus_{m\geq 0} H^0(U, L^{-m})^{\Gamma},
\]
where \(L\) is a line bundle over a simply connected Riemann surface \(U\), which is one of \(\mathbb C \roman{P}^1\), \(\mathbb C\) or the hyperbolic plane \(H\), together with an action of a discrete subgroup \(\Gamma \subset \roman{Aut}(U)\). \(X\) is said to be hyperbolic if \(U=H\).
The main theorem of this article is the following.
Theorem. If \(M\) is the link of a quasi-homogeneous hyperbolic \(\mathbb Q\)-Gorenstein surface singularity of index \(r\), then \(M\) is diffeomorphic to a coset space \(\Gamma_1\backslash G/\Gamma_2\), where \(G\) is the universal cover of the Lie group \(\roman{PSL}(2,\mathbb R)\), while \(\Gamma_1\) and \(\Gamma_2\) are discrete subgroups of the same level in \(G\), \(\Gamma_1\) is co-compact and the image of \(\Gamma_2\) in \(\roman{PSL}(2,\mathbb R)\) is a cyclic group of order \(r\). The converse also holds.
Note that in the Gorenstein case, it was shown by \textit{I. V. Dolgachev} [Math. Ann. 265, 529--540 (1983; Zbl 0506.14027)].
For the proof of the main theorem, the author uses an argument similar to that in [loc. cit.] and gives a characterization of quasi-homogeneous \(\mathbb Q\)-Gorenstein surface singularities in terms of the triple \((U,\Gamma,L)\). quasi-homogeneous surface singularity; \(\mathbb Q\)-Gorenstein surface singularity; link Pratoussevitch, A.: On the Link Space of a \(\mathbb{Q}\) -Gorenstein Quasi-Homogeneous Surface Singularity, In: Proceedings of the VIII Sao Carlos Workshop in Real and Complex Singularities in Luminy, pp. 311--325. Birkhäuser (2006) Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli On the link space of a \(\mathbb Q\)-Gorenstein quasi-homogeneous 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 This work presents a graphical interpretation of a certain type of hyperkähler manifolds which the author calls wild Hitchin moduli spaces -- loosely, hyperkähler manifolds attached to a Riemann surface with additional data. Wild Hitchin moduli spaces are similar to, but more complicated than, a large class of hyperkähler manifolds which are attached to graphs with additional data: the Nakajima quiver varieties. To better understand these moduli spaces, the author connects them to colored quivers through a new construction: multiplicative quiver varieties. This construction allows for a better understanding of the isomorphisms between the wild character varieties (wild Hitchin spaces).
Suppose \(\Gamma\) is a colored quiver with vertex set \(I.\) Let \(d\in \mathbb{Z}_{\geq0}^{I}\) \(q\in\left( \mathbb{C}^{\times}\right) ^{I}\;\)be an assignment of both a positive integer and a nonzero complex number to each node. Then \(\Gamma,q,\) and \(d\) determine an algebraic variety, denoted \(\mathcal{M}\left( \Gamma,q,d\right) .\)and called the multiplicative quiver variety Furthermore, contained in \(\mathcal{M}\left( \Gamma,q,d\right) \) is a canonical open subset of stable points \(\mathcal{M}^{\text{st}}\left( \Gamma,q,d\right) \) which, if nonempty, is a smooth symplectic algebraic variety. Both \(\mathcal{M}\left( \Gamma,q,d\right) \) and \(\mathcal{M} ^{\text{st}}\left( \Gamma,q,d\right) \) depend on \(d\) and \(q\) but not on their actual assignments. Furthermore, these varieties depend on the coloring: for example, if the underlying graph of \(\Gamma\) is a triangle, then there are two non-equivalent colorings of \(\Gamma,\) and in the case \(d=\left( 1,1,1\right) \) the varieties are not isomorphic.
To study the quiver varieties which relate to wild character varieties, and to study their isomorphisms, one needs to consider supernova graphs. So, suppose \(\Gamma\) is a simply laced supernova graph, vertex set \(I,\) which is colored in such a way so that the core is monochromatic. Then \(\mathcal{M}^{\text{st} }\left( \Gamma,q,d\right) \) is isomorphic to a wild character variety. Let \(s_{i}\in\Aut\left( \mathbb{Z}^{I}\right) \) and \(r_{i}\in\Aut\left( \left( \mathbb{C}^{\times}\right)^{I}\right) \) be the corresponding simple reflections generating the Kac-Moody Weyl group. If \(q_{i}\neq1\) for some \(i\in I\) then \(\mathcal{M}^{\text{st}}\left( \Gamma,q,d\right) \cong\mathcal{M}^{\text{st}}\left( \Gamma,r_{i}\left( q\right),s_{i}\left( d\right) \right) \) as smooth symplectic algebraic varieties. quiver variety; wild character variety; Weyl group; wild Riemann surface Boalch (P. P.).-- Global Weyl groups and a new theory of multiplicative quiver varieties. arXiv:1307.1033. Geometric invariant theory, Symplectic manifolds (general theory), Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain, Momentum maps; symplectic reduction Global Weyl groups and a new theory of multiplicative quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Over an algebraically closed field \(K\) of characteristic 0, simple singularities are classically known as hypersurfaces \(R=K[[X_1,\dots ,X_{d+1}]]/(f)\), where \(f\) is (up to contact equivalence) one of the equations ADE [cf. \textit{V. I. Arnol'd, S. M. Gusejn-Zade} and \textit{A. N. Varchenko}, Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts. Monographs in Mathematics, Vol. 82. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001)]. Among many others, those are equivalent conditions for a (formal) hypersurface:
(i) \(R\) is simple.
(ii) \(R\) is of finite CM-representation type (i.e. the number of isomorphism classes of indecomposable maximal Cohen-Macaulay modules is finite).
(iii) \(R\) is of finite deformation type.
This classification was extended by \textit{G.-M. Greuel} and \textit{H. Kröning} [Math. Z. 203, No. 2, 339--354 (1990; Zbl 0715.14001)] to arbitrary characteristic, where the equations \(f\) look slightly different in some cases.
The article under review gives a new characterization of simple singularities. In this context, \(R\) is \textit{simple} iff it is a hypersurface of finite CM-representation type.
The authors discuss the following condition: Let \(R\) be a local noetherian ring and \((\mathbf{mod}_R)\) the category of finitely generated \(R\)-modules. Let \({\mathcal G}(R)\) denote the full subcategory of modules \( M\in (\mathbf{mod}_R)\) such that there exists an exact sequence
\[
\dots \to F_n\to \dots \to F_1 \to \dots \to F_0 \to M\to 0
\]
with \(F_i\) f.g. free and such that the dual
\[
\dots \to F_n^*\to \dots \to F_1^* \to F_0^*
\]
of the complex \((F_i)_{i\in\mathbb N}\) is exact (\(^*\) denotes the algebraic dual \(\text{Hom} (- ,R)\)). Such modules are said to be of \textit{Gorenstein dimension} 0 (in the sense of Auslander-Bridger) or \textit{totally reflexive} (in the sense of Avramov-Martsinkovsky).
The authors show: Theorem A. Let \(R\) be complete. If the set of isomorphism classes of indecomposable modules in \({\mathcal G} (R)\) different from the class of \(R\) is finite and not empty, then \(R\) is simple.
There is, in fact, a close relation to the notion of MCM-representations. Thus it is reasonable to speak of \textit{finite Gorenstein representation type} in the above case.
A theorem of \textit{J. Herzog} [Math. Ann. 233, 21--34 (1978; Zbl 0358.13009)] states: If \(R\) is Gorenstein and of finite MCM-representation type then \(R\) is an abstract hypersurface, i.e. \(\hat{R} \cong A/xA\) with \(A\) regular, \(x\in m_A\).
On the other hand, if \(R\) is Gorenstein, \({\mathcal G} (R)= \mathcal{MCM}(R)\). Thus theorem A is a corollary of the following which does not make any assumption on the local noetherian ring \(R\).
Theorem B. Let \(R\) be a local noetherian ring. If the set of isomorphism classes of indecomposable modules in \({\mathcal G}(R)\) is finite, then \(R\) is Gorenstein or every module in \({\mathcal G}(R)\) is free.
In this general context, the known theory of CM-approximation is not appropriate any more, but it gives an idea how to prove the above result for the Gorenstein case. The authors develop an approximation theory of modules in \((\mathbf{mod}_R)\) with respect to \({\mathcal G}(R)\). More generally, it turns out necessary to do this for any \textit{reflexive subcategory} \(\mathcal B\) of \((\mathbf{mod}_R)\) i.e. for a subcategory \(\mathcal B\) with the following properties: Denote
\[
{\mathcal B}^\perp := \{ L\in (\mathbf{mod}_R) \mid \text{Ext}_R^i(B,L)=0 \;\;\forall B\in {\mathcal B}, i>0 \} .
\]
Then
\(\bullet\) \(R\in {\mathcal B} \cap {\mathcal B}^\perp \)
\(\bullet\) \(\mathcal B\) is closed under direct sums and direct summands.
\(\bullet\) \(\mathcal B\) is closed under syzygies.
\(\bullet\) \(\mathcal B\) is closed under algebraic duality.
\({\mathcal G} (R)\) is known to be the largest reflexive subcategory of \((\mathbf{mod}_R)\), and for any reflexive subcategory \(\mathcal B\) there are inclusions
\[
{\mathcal F} (R) \subseteq {\mathcal B} \subseteq {\mathcal G} (R),
\]
where \({\mathcal F} (R)\) denotes the full subcategory of free modules in \((\mathbf{mod}_R)\).
Using a result of Takahashi, conditions on the existence of covers are obtained: Assuming the Krull-Remak-Schmidt property for \((\mathbf{mod}_R)\), a module \(M\) has a \(\mathcal B\)-precover iff it has a \(\mathcal B\)-approximation.
There is the following relation to MCM-approximation:
\({\mathcal G}(R) \subseteq \mathcal{MCM}(R)\) iff \(R\) is Cohen-Macaulay.
Central part in the proof of Theorem B is Theorem C, which reduces Theorem B to showing that the residue field \(k\) of \(R\) has a reflexive hull.
Corollary. For a local ring \(R\) such that \({\mathcal G}(R)\) contains not only free modules, the following conditions are equivalent:
\(\bullet\) \(R\) is Gorenstein.
\(\bullet\) The residue field \(R/m\) has a \({\mathcal G}(R)\)-approximation.
\(\bullet\) All modules in \( (\mathbf{mod}_R)\) have a minimal \({\mathcal G} (R)\)-approximation. approximations; Cohen-Macaulay representation type; covers; Gorenstein dimension; precovers; simple singularity; totally reflexive modules Christensen, L. W.; Piepmeyer, G.; Striuli, J.; Takahashi, R., Finite Gorenstein representation type implies simple singularity, Adv. Math., 218, 1012-1026, (2008) Singularities in algebraic geometry, Relative homological algebra, projective classes (category-theoretic aspects), Cohen-Macaulay modules Finite Gorenstein representation type implies simple 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 For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed.
The paper is organized as follows: in Sect. 2, we first recall basic definitions and facts on quivers, their representations and on quiver moduli (Sect. 2.1). We also recall some results of \textit{M. Reineke} [from Invent. Math. 152, No. 2, 349-368 (2003; Zbl 1043.17010)] which will be used in the following (Sect. 2.3), thereby generalizing them to quivers possessing oriented cycles using a general purity result (Sect. 2.2).
Section 3 is devoted to the construction of the smooth models by a framing process. The objects parametrized by the smooth models are described (Proposition 3.6), and basic geometric properties are discussed. The section ends with an illustration of the general construction by several examples, which will be studied in more detail subsequently.
In Sect. 4, we first recall the Luna stratification of quiver moduli. This is used to define the stratification of the smooth models, whose geometric properties rely on the analysis of the fibres of the analogue of the Hilbert-Chow morphism in Theorem 4.1.
After recalling some Hall algebra techniques, Sect. 5 derives two formulas for Betti numbers of smooth models mentioned above by direct computations relying on the formulas of Sect. 2.3, and illustrates the use of the formulas in two examples.
Section 6 applies the techniques of the previous sections to the case of Hilbert schemes of path algebras, the methods being of a more combinatorial flavour.
The cell decompositions of Hilbert schemes of path algebras are constructed explicitly in Sect. 7. Section 8 compares the relevant combinatorial concepts and gives a typical example. moduli spaces; stable representations of quivers; quiver moduli; smooth varieties; Betti numbers; smooth models; moduli of vector bundles Engel, J., Reineke, M.: Smooth models of quiver moduli. Math. Z. \textbf{262}(4), 817-848 (2009). http://arxiv.org/abs/0706.4306 Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory Smooth models of quiver moduli. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors apply Shokurov's inductive method to a study of terminal and canonical singularities. As an easy consequence of the log minimal model program we show that for any three-dimensional log terminal singularity \((X\ni P)\) there exist some special, so called, plt (=``purely log terminal'') blow-ups \(Y\to X\). The authors discuss properties of them and construct some examples.
Roughly speaking a plt blow-up \(Y\to X\) is a blow-up with an irreducible exceptional divisor \(S\) such that \(S\) is normal and the pair \((S,\text{Diff}_S(0))\) is log terminal, where \(\text{Diff}_S(0)\) is the different [cf. \textit{V. V. Shokurov}, Russ. Akad. Sci., Izv., Math. 40, No. 1, 95-202 (1993) and 43, No. 3, 527-558 (1994); translation respectively from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105-203 (1992; Zbl 0785.14023) and 57, No. 6, 141-175 (1993; Zbl 0828.14027); \S 3]. These blow-ups are very useful for an inductive approach to the study of singularities and, more general, extremal contractions.
The authors also obtain necessary conditions for a log surface to be an exceptional divisor of a plt blow-up of a terminal singularity. Unfortunately a plt blow-up of a terminal singularity is never unique. However there are canonical singularities for which it is unique. These singularities are called weakly exceptional and have the most interesting and complicated structure. The authors obtain the criterion for a singularity to be weakly exceptional in terms of the exceptional divisor of some plt blow-up (theorem 4) and construct few examples. weakly exceptional singularities; canonical singularities; log minimal model program; three-dimensional log terminal singularity; plt blow-ups Prokhorov Yu.G., Blow-ups of canonical singularities, In: Proceedings of the International Algebraic Conference held on the occasion of the 90th birthday of A.G.Kurosh, Moscow, May 25--30, 1998, Walter de Gruyter, Berlin, 2000, 301--317 Minimal model program (Mori theory, extremal rays), \(3\)-folds, Singularities in algebraic geometry Blow-ups of canonical 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 book considers the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution and studies some topics in representation theory from this more general context. The first goal of the authors is to apply these considerations to quiver varieties and hyperbolic varieties, and also to other quantized symplectic resolutions. The first part is devoted to the theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology. In a second part, the authors define and study the category O for a symplectic resolution and observe that this category is often Koszul. Then they define the notion of a symplectic duality between symplectic resolutions. This definition leads to a conjectural identification of two geometric realizations due to Nakajima and Ginsburg-Mirkovic-Vilonen, of weight spaces of simple representations of simply-laced simple algebraic groups. An appendix by Ivan Losev provides a key step for the proof that the category O is of highest weight. quantizations; quantized symplectic resolutions; conical symplectic resolutions; Lie algebras; Harish-Chandra bimodules; cohomology; Koszul duality Braden, T.; Licata, A.; Proudfoot, N.; Webster, B., Quantizations of conical symplectic resolutions II: category O and symplectic duality, Astérisque, 384, 75-179, (2016) Research exposition (monographs, survey articles) pertaining to differential geometry, Deformation quantization, star products, Global theory and resolution of singularities (algebro-geometric aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Poisson algebras, Representation theory of associative rings and algebras Quantizations of conical symplectic resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(X_{\text{aff}} := \Gamma\setminus\mathbb{H}\), where in characteristic zero (resp. positive characteristic) \(\mathbb{H}\) is the complex upper half plane (resp. the Drinfeld's upper half plane) and where \(\Gamma\) is an arithmetic subgroup of \(GL_2(\mathbb{Q})\) (resp. \(GL_2(\mathbb{F}_q,(T))\), \(\mathbb{F}_q\) being a finite field). Let \(X_\Gamma\) be the canonical completion of \(X_{\text{aff}}\), then \(X_\Gamma - X_{\text{aff}}\) contains finitely many closed points, the cusps. The subgroup of the jacobian \({\mathcal J}_{X_\Gamma}\) that they generate is the cuspidal divisor class group \(C_\Gamma\) with respect to \(X_\Gamma\) or to \(\Gamma\). The author recalls the main properties of this group (finiteness when \(\Gamma\) is a congruence subgroup, main examples \dots), some of which are new in the positive characteristic case. This parallel makes clear the analogy and the difference between the two cases. An interesting problem is suggested: to find in positive characteristic examples of (non congruence) subgroup \(\Gamma\) such that the cuspidal divisor class group \(C_\Gamma\) is non finite (this is known in the characteristic zero case). modular curves; cuspidal divisor class group Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Drinfel'd modules; higher-dimensional motives, etc. Cuspidal divisor class groups of modular 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 It is well known that for even characteristic \(m\) the Thetanullwerte \(\tau\mapsto \vartheta_ m(\tau):= \vartheta_ m (\tau,0)\) are non-zero holomorphic functions on the Siegel upper half plane \(\mathbb{H}_ g\) of genus \(g\). These \(N+1:= 2^{g-1} (2^ g+1)\) functions give rise to injective holomorphic maps
\[
\theta: \Gamma_ g (4,8)\setminus \mathbb{H}_ g\to \mathbb{P}_ N \mathbb{C}, \qquad \theta^ 2: \Gamma_ g (2,4)\setminus \mathbb{H}_ g\to \mathbb{P}_ N \mathbb{C}, \qquad \theta^ 4: \Gamma_ g(2)\setminus \mathbb{H}_ g\to \mathbb{P}_ N \mathbb{C},
\]
where \(\theta^ k\) is induced by the map \(\tau\mapsto (\dots, \vartheta^ k(\tau), \dots)\). Clearly, for a natural number \(q\), \(\Gamma_ g(q)\) is the congruence subgroup of level \(q\) of the Siegel modular group \(\Gamma g:= \text{Sp}_ g(\mathbb{Z})\) and \(\Gamma_ g(q, 2q)\) the subgroup of \(\Gamma_ g (q)\) of those elements \({{a\;b} \choose {c\;d}}\) with \(\text{diag}^ t ab\equiv \text{diag}^ t cd\equiv 0(2q)\).
Now, by results in a forthcoming paper of the same author the maps \(\theta^ k\) extend in an equivariant manner to injective holomorphic maps \(\overline{\theta}^ k\) to the respective Satake compactifications (theorem 1) which are nothing other than the projective varieties associated to the graded ring \(A(\cdot)\) of corresponding modular forms. Furthermore the injectivity allows to calculate the cardinality of the fibres over points coming from rational boundary components of the natural projections
\[
\text{proj } A(\Gamma_ g (4,8))\to \text{proj } A(\Gamma_ g (2,4))\to \text{proj } A(\Gamma_ g(2)) \qquad (\text{theorems 2 and 3)}
\]
and to characterize the points of the respective Satake compactifications in terms of vanishing of Thetanullwerte.
In genus 3 the maps \(\theta^ 2\) and \(\overline{\theta}^ 2\) are immersions (theorem 5) whereas for \(g\geq 4\) these maps never share this property (theorem 6) contrasting the case \(g\leq 2\) where \(\theta^ 2\) and \(\theta^ 4\) are biholomorphic onto their respective images. The map \(\theta\) itself is always biholomorphic onto its image. Theta functions; Siegel modular forms; graded ring of modular forms; Thetanullwerte; Siegel modular group; injective holomorphic maps; Satake compactifications; projective varieties Salvati Manni, R.: On the projective varieties associated with some subrings of the ring of thetanullwerte. Nagoya Math. J.133, 71--83 (1994) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Theta functions and abelian varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the projective varieties associated with some subrings of the ring of 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 One of the classical problems in invariant theory is the study of binary quantics. The main object was to give an explicit description and study the geometric properties of \(SL_2\) quotients of the projective space for a suitable choice of linearization. The aim of this paper is to begin the study of a natural generalization of this classical question.
Let \(k\) be an algebraically closed field. Let \(G\) be a semisimple algebraic group over \(k\), \(T\) a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\), \(N\) the normalizer of \(T\) in \(G\), \(W= N/T\), the Weyl group. Consider the quotient variety \(N\setminus \setminus G/B\). In fact the aim is to study more generally the variety \(N\setminus \setminus G/Q\), where \(Q\) is any parabolic subgroup of \(G\) containing \(B\).
We study torus quotients of these homogeneous spaces. We classify the Grassmannians for which semi-stable = stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring of \(T\) invariants of the homogeneous coordinate ring of the Grassmannian \(G_{2,n}\) (\(n\) odd) over the ring generated by \(R_1\), the first graded part of the ring of \(T\) invariants.
We prove the following results:
(a) the varieties \(T\setminus \setminus G/Q\) and \(N\setminus \setminus G/Q\) are Frobenius split and as an application the vanishing theorems for higher cohomologies of these varieties;
(b) as a part of result (a), we prove the vanishing of the higher cohomology groups for the binary quantics;
(c) for the line bundle \(L\) on \(G_{r,n}\) associated to the fundamental weight \(\varpi_r\), \((G_{r,n})_T^{ss} (L)= (G_{r,n})_T^s(L)\) if and only if \(r\) and \(n\) are coprime;
(d) existence of smooth projective varieties as quotients of certain \(G/Q\) modulo a maximal torus \(T\) (in the case of \(G= SL_n)\);
(e) for \(n\) odd, a partial result about \(R_1\) generation of the graded ring \(k\widehat {[G_{2,n}]}^T= \bigoplus_{d\geq 0}R_d\). torus quotients of homogeneous spaces; Grassmannian; vanishing theorems Kannan, S. S., Torus quotients of homogeneous spaces, \textit{Proc. Indian Acad. Sci. (Math. Sci.)}, 108, 1, 1-12, (1998) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Vanishing theorems in algebraic geometry Torus quotients of homogeneous 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 Du Bois (or simply DB) singularities were first introduced by Steenbrink, based upon the work of Deligne and Du Bois, see [\textit{P. Du Bois}, Bull. Soc. Math. Fr. 109, 41--81 (1981; Zbl 0465.14009); \textit{J. H. M. Steenbrink}, Compos. Math. 42, 315--320 (1981; Zbl 0428.32017)].
In this paper, the author generalizes DB singularities to the context of pairs. He works in particular in the context \((X, \Sigma)\), where \(\Sigma\) is a subscheme of a variety \(X\). Suppose that
\[
D^{.} \to \underline{\Omega}_X^{0} \to \underline{\Sigma}_X^{0} \to D^{.}[+1]
\]
is a triangle in \(D^b_{\text{coh}}(X)\) where the map \(\underline{\Omega}_X^{0} \to \underline{\Sigma}_X^{0}\) is the natural one. In this setting, \((X, \Sigma)\) is called Du Bois if \(D^{.}\) is quasi-isomorphic to the ideal sheaf defining \(\Sigma\).
This definition differs from the more common definitions of pairs appearing throughout the minimal model program in that it is a relative notion. In particular, if the pair \((X, \Sigma)\) is DB, then \(X\) has Du Bois singularities if and only if \(\Sigma\) does (see Proposition 5.1 in the paper). However, it is not clear whether \((X, \Sigma)\) being DB implies that \(X\) is DB.
In Section 6 of this paper, the author proves several vanishing theorems for DB pairs. As an application, he proves that if \((X, \Delta)\) is a log canonical pair and \(\pi : \widetilde{X} \to X\) is a log resolution, and \(\widetilde{\Delta} = (\pi_*^{-1} \lfloor \Delta \rfloor + E)_{\text{red}}\), where \(E\) denotes the union of the exceptional non-klt places, then
\[
R^i \pi_* \mathcal{O}_{\widetilde{X}}(-\widetilde{\Delta}) = 0
\]
for \(i > 0\). Du Bois singularities; vanishing theorems; rational singularities; log canonical singularities Kovács, S. J., \textit{du bois pairs and vanishing theorems}, Kyoto J. Math., 51, 47-69, (2011) Singularities of surfaces or higher-dimensional varieties, Vanishing theorems in algebraic geometry Du Bois pairs and vanishing theorems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory developed in this paper arose from two main sources. They are the theory of varieties of group representations developed recently by \textit{M. Culler} and the second author [ibid. 117, 109-146 (1983; Zbl 0529.57005)] and Thurston's construction of a compactification of Teichmüller space. As an application of their ideas, the authors give a new construction of this compactification. As they state, their ''methods are drawn from the mathematical mainstream, and therefore help to explain Thurston's results by putting them in a wider framework.''
The central topic of the paper is a construction of compactifications of real and complex algebraic varieties. While there is an obvious way to compactify curves, which was used by Culler-Shalen, the problem of compactification of higher dimensional varieties is anything but routine. The authors' approach to this problem is motivated by the construction of Thurston's compactification of Teichmüller space. They consider an affine algebraic set V and an indexed family \((f_ j)_{j\in J}={\mathcal F}\) with countable index set J of functions which belong to the coordinate ring of V and generate it as an algebra. A compactification of V is canonically defined by \({\mathcal F}\) as follows. Let \({\mathcal P}\) be the quotient of \([0,\infty)^ J\setminus \{0\}\), where \([0,\infty)^ J\) is the Cartesian power, by the diagonal action of positive reals: \(\alpha (t_ j)_{j\in J}=(\alpha t_ j)_{j\in J}.\) Define a map \(\theta\) : \(V\to {\mathcal P}\) by \(\theta (x)=[\log (| f_ j(x)| +2)]_{j\in J}.\) Then the closure of \(\theta\) (V) in \({\mathcal P}\) is compact. This closure is the compactification in question. This compactification is studied on three different levels of generality.
The first one is that of a general variety. This is the theme of Chapter I. The points added to V are interpreted as valuations of the coordinate ring of V over a countable field of definition of V. These valuations are neither discrete nor of rank 1 in general. An important result says that there is a dense subset of added points consisting of discrete, rank 1 valuations.
At the second level V specializes to be the variety of characters X(\(\Gamma)\) of representations of a discrete group \(\Gamma\) in \(SL_ 2({\mathbb{C}})\). In this case there is a natural choice of \({\mathcal F}\). The corresponding \(f_ j\) are the values of characters on conjugacy classes in \(\Gamma\). Now the added points can be interpreted as actions of \(\Gamma\) on some generalized trees. On the vertices of an ordinary tree there is an integer-valued distance function. On generalized trees a similar distance function takes values in an ordered abelian group. The most important case is that of a subgroup of \({\mathbb{R}}\). The theory of such trees is developed from scratch up to a generalization of the well known Bass-Serre theory of trees associated to \(SL_ 2(F)\), where F is a field. While in the Bass-Serre theory a tree is associated with a discrete valuation of F, here the valuation can be nondiscrete. These trees are used for compactification. All this is the theme of Chapter II.
Finally, in Chapter III this theory is applied to the case \(\Gamma =\pi_ 1(S)\), where S is a surface. The Teichmüller space of S turns out to be a component of X(\(\Gamma)\) and the compactification of X(\(\Gamma)\) constructed in Chapter II leads to the Thurston's compactification of Teichmüller space.
In the second part of this paper, existing now in preprint form, these ideas are applied to the study of 3-manifolds. In particular, another important result of Thurston is re-proved from an entirely new point of view. representations of a dicrete group in \(SL_ 2({\mathbb{C}})\); actions on generalized trees; hyperbolic structures on surfaces; varieties of group representations; compactification of Teichmüller space; compactifications of real and complex algebraic varieties; affine algebraic set; valuations of the coordinate ring J. Morgan, P. Shalen. Valuations, trees, and degenerations of hyperbolic structures. I, \textit{Ann. of Math. } 120 (1984), 401--476. General low-dimensional topology, Abelian varieties and schemes, Valuations and their generalizations for commutative rings, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Compactification of analytic spaces, Group rings of finite groups and their modules (group-theoretic aspects), Classification theory of Riemann surfaces, Topology of Euclidean 2-space, 2-manifolds, Topology of general 3-manifolds Valuations, trees, and degenerations of hyperbolic structures. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a finite quiver \(\Gamma\) without loops or \(2\)-cycles, and \(x_{1} ,\dots,x_{r}\in\mathbb{Q}(X_1,\dots,X_r)\) algebraically independent, one can form a cluster algebra \(\mathcal{A}(x,\Gamma)\) where \(x=(x_1,\dots,x_r)\). Let \(\Lambda\) be the preprojective algebra corresponding to the quiver. Previously, the authors assigned to each nilpotent \(\Lambda\)-module a function in \(\mathbb{C}[N]\), where \(N\) is a pro-unipotent pro-group related to \(\Gamma,\) realizing cluster algebras as coordinate rings of unipotent cells of Kac-Moody groups.
The first part of this work is an attempt to compare (1) the authors' previous works, where such cluster variables of coordinate rings were shown to be expressible in terms of Euler characteristics of varieties of flags of submodules of preprojective algebra representations; with (2) the direction, starting with Caldero and Chapoton, which used Euler characteristics of Grassmannians of submodules of quiver representations to describe Laurent polynomial expansions of cluster variables under certain circumstances. This comparison is accomplished using the Chamber Ansatz of Berenstein, Fomin, and Zelevinsky, from which cluster algebras originated. Here, the Chamber Ansatz formulas are described in terms of representations of preprojective algebras, along with a generalization to Kac-Moody. Of particular interest is the relationship between the twist automorphisms and the Auslander-Reiten translations of the corresponding categories of modules over preprojective algebras.
These results are then used to study natural bases of cluster algebras containing the cluster monomials. By specializing to the a class of coefficient-free cluster algebras by specializing the coefficients of the cluster algebra structures on unipotent cells to 1, a description of such bases is given in terms of module varieties of endomorphism algebras of cluster-tilting modules. In particular, if the cluster algebra is cyclic, then Dupont's basis conjecture is proved. cluster algebra; flag variety; Grassmannian; Euler characteristic Geiss, C.; Leclerc, B.; Schröer, J., Generic bases for clusters algebras and Chamber ansatz, J. Amer. Math. Soc., 25, 1, 21-76, (2012) Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets, Kac-Moody groups Generic bases for cluster algebras and the chamber ansatz | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\text{SL}(3, \mathbb{C})\) acting with an isolated singularity on \(\mathbb{C}^3\). A crepant resolution of \(\mathbb{C}^3/G\) comes together with a set of tautological line bundles associated to each irreducible representation of \(G\). The author gives a formula for the triple product of the first Chern class of the tautological bundles in terms of both the geometry of the crepant resolution and the representation theory of \(G\). This is used to derive the way these triple products change when a flop is performed. Calabi-Yau orbifolds; McKay correspondence G. Di Cerbo and R. Svaldi, \textit{Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section}, arXiv:1608.02997. Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Rational and birational maps Flops of crepant resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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{E. Nart} and \textit{C. Ritzenthaler} [J. Number Theory 116, No. 2, 443--473 (2006; Zbl 1102.14019)] showed that the moduli space of the ordinary non-singular quartic curves over fields of characteristic \(2\) is isomorphic to a certain open subset of an affine variety, whose coordinate ring is given as the invariant algebra \(S[W'{^\ast}]^G\) of a certain \(6\)-dimensional \(\mathbb{F}_2\)-vector space \(W'{^\ast}\) provided of a linear action of the finite group \(G:=\text{GL}_3(\mathbb{F}_2)\).
In the paper under review the authors obtain a complete description of this invariant algebra by combining a theoretical analysis with the application of specially tailored computational techniques. They show that the module \(W'{^\ast}\) is a trivial source \(G\)-module and the algebra \(S[W'{^\ast}]^G\) is Cohen-Macaulay. An optimal set of primary invariants has degrees \(\{2,3,3,4,6,7\}\), and a corresponding minimal set of secondary invariants has cardinality \(18\); moreover, \(S[W'{^\ast}]^G\) is generated as an algebra by at most \(11\) invariants. J. Müller, C. Ritzenthaler, On the ring of invariants of ordinary quartics in characteristic two, J. Algebra, to appear. Computational aspects of algebraic curves, Curves over finite and local fields, Arithmetic ground fields for curves On the ring of invariants of ordinary quartic curves in characteristic 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author's approach [\textit{G. Faltings}, J. Reine Angew. Math. 483, 183-196 (1997; Zbl 0871.14012)] to resolve singularities of moduli spaces of Abelian varieties relies on some special features of the level structures considered there. Here the author tries a new approach which is more widely applicable, but gives weaker results. The paper is, as the reviewer supposes, not to be regarded as definitive but is rather in the nature of a progress report; but it introduces important ideas.
The singularities studied are those of the moduli space of degree \(p^g\) isogenies \(\phi: A\to B\) of \(g\)-dimensional Abelian varieties factored into a sequence of isogenies of degree \(p\). This moduli problem is representable over \({\mathbb Z}_p\). The local singularities are described, after Deligne and Pappas, in terms of the Hodge filtration on the universal sequence of Dieudonné modules. More concretely one fixes a discrete valuation ring \(V\) with uniformiser \(\pi\) and a complete flag
\[
N_g=\pi N\subset N_{g-1}\subset\ldots\subset N_0=N\cong V^g
\]
and examines the projective scheme \(X\) over \(V\) parametrising compatible families \(F_i\) of direct summands of \(N_i\) of rank \(a\). By studying suitable arrangements of lattices the author constructs a space \(Y\) which (for \(a=2\) at least) has toroidal singularities, and is equipped with a birational morphism to \(X\). This \(Y\) is an example of what is here referred to as a minimal model (for the Deligne scheme). The arrangements fit into a universal family over a suitable embedding of \(\text{PGL}(d)^r/\text{PGL}(d)\), endowed with a logarithmic structure induced by maps to a scheme with toroidal singularities: Unfortunately these maps, which coincide with those of Lafforgue, appear not to be smooth (contrary to a hope expressed, rather tentatively, by the author in the introduction). Shimura variety; toroidal resolution; minimal model Faltings, G.: Toroidal resolutions for some matrix singularities. In Moduli of Abelian Varieties (Texel Island, 1999), Volume 195 of Progr. Math., pp. 157-184. Birkhäuser, Basel (2001) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Algebraic moduli of abelian varieties, classification, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Toroidal resolutions for some matrix 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 an equivariant resolution of singularities of the null-fiber of a moment map related to the dual pair \((\text{Sp}_{2n}({\mathbb R}), \text{O}_{m_1, m_2}({\mathbb R}))\) in \(\text{Sp}_{2n(m_1+m_2)}(\mathbb R)\). Conormal bundles of some closed orbits in the Lagrangian Grassmannians are realized as categorical quotients. These bundles turn out to be resolutions of singularities of the closures of some nilpotent orbits.
The case of the dual pair \((\text{O}^*_{2n}, \text{Sp}_{n,n})\), \(n\) even, is discussed as well. resolution; null-fiber; conormal bundle; Lagrangian Grassmannian; flag variety; dial pair; nilpotent orbit Grassmannians, Schubert varieties, flag manifolds, Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Representation theory for linear algebraic groups Resolution of null fiber and conormal bundles on the Lagrangian 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 This paper seeks a method that determines the Stäckel class of any given \(2\)-dimensional non-degenerate second-order superintegrable system, i.e., for manifolds of arbitrary (including non-constant) curvature. A non-degenerate second-order maximally conformally superintegrable system in dimension \(2\) naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's Stäckel class can be obtained from this associated quadric. The Stäckel class of a second-order maximally conformally superintegrable system is its equivalence class under Stäckel transformations, i.e., under coupling-constant metamorphosis. This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with some preliminaries. The aim of Section 3 is to construct a certain variety that is invariant under Stäckel transform. It is encoded in a quadric, for a given (non-degenerate) \(2D\) second-order superintegrable system. Here the author relates this variety to the set of all flat realisations of a given Stäckel class, i.e., those non-degenerate \(2D\) second-order superintegrable systems inside the given Stäckel class that are realised on a flat geometry. Section 4 deals with further results and Section 5 with discussion and generalisations. Stäckel equivalence; quadrics; superintegrable systems Relationships between algebraic curves and integrable systems, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics, Conformal metrics (hyperbolic, Poincaré, distance functions) Stäckel equivalence of non-degenerate superintegrable systems, and invariant 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 Let \(X\) be a three-dimensional normal Gorenstein singularity admitting a crepant resolution \(\tilde X\rightarrow X\). Then one is interested in describing the bounded derived category \(\mathcal D\text{Coh}\tilde X\) of coherent sheaves on \(\tilde X\). Bridgeland shows that this category depends only on the singularity and not on the choice of crepant resolution. In many cases there exists a tilting bundle \(\mathcal X\in\mathcal D\text{Coh}\tilde X\) such that \(\mathcal D\text{Coh}\tilde X\) is equivalent as a triangulated category to the derived category of finitely generated \(A\)-modules \(\mathcal D\text{Mod}A\), where \(A=\text{End}\mathcal X\). This leads to Van den Bergh's introduction of a noncommutative crepant resolution (NCCR) of \(X\) which is a homologically homogeneous algebra of the form \(A=\text{End}(T)\), where \(T\) is a reflexive \(R\)-module, with \(R=\mathbb C[X]\) the coordinate ring of the singularity. However, a NCCR is not unique, and there are an infinite number of different noncommutative crepant resolutions.
To restrictions are made. The first is that \(X\) is a toric three-dimensional singularity. This implies the existence of a commutative crepant resolution. The second restriction is that the tilting bundle is a direct sum of nonisomorphic line bundles. From string theory (Franco, Hanany, Kennaway, Herzog, Vegh, Wecht), it follows that under these conditions the algebra \(A\) can be described using a dimer model on a torus. This means that \(A\) is the path algebra of a quiver \(Q\) with relations where \(Q\) is embedded in a two-dimensional torus \(T\) such that every connected piece of \(T\setminus Q\) is bounded by a cyclic path of length at least \(3\). The relations are given by demanding that for every arrow \(a\) that \(p=q\) where \(ap\) and \(aq\) are the bounding cycles that contain \(a\).
This description follows from the fact that the algebra \(A\) is a toric order, a special type of order compatible with the toric structure, and Calabi-Yau-3 (CY-3), i.e. it admits a self-dual bimodule resolution of length \(3\). It is known that every toric CY-3 order comes from a dimer model.
Under specific consistency conditions, a dimer model gives a noncommutative crepant resolution of its center. Quite different consistency conditions have been proposed. These includes cancellation, nonintersecting zig and zag rays, consistent R-charges and algebraic consistency. The aim of this article is to show that for dimer models on a torus, all these consistency conditions are equivalent. Also, the condition of being an order and the condition of being an NCCR are also equivalent to these consistency conditions. For the CY condition, the situation is less clear. The article contains an example of an infinite dimer model that is not cancellation but satisfies a suitable generalization of the CY-3 property to the infinite case. No finite counterexamples are known.
If one broadens the definition of a dimer model to allow other compact surfaces, the consistency conditions are no longer equivalent. The author discuss the differences for those cases.
The article contains a detailed introduction to quivers with relations and in particular to dimer models on a torus. Then the different consistency conditions are treated, and the known equivalences is stated and illustrated with examples. The concept of orders is treated in particular, and a main result of the article is that a Jacobi algebra of a dimer model on a torus is an order if and only if it is algebraically consistent. Finally, the author proves that the Jacobi algebra of a dimer model on a torus is a noncommutative crepant resolution of its center if and only if it is cancellation. A very precise summary of the different equivalences is given at the end. In addition to its aim of proving equivalences, the article is interesting because of its introduction to the different concepts and applications of dimer models. It triggers the reader to a further study on the subjects introduced. dimer model; quiver with relation; cancellation; consistent R-charge; consistency conditions; zig-zag rays R. Bocklandt, \textit{Consistency conditions for dimer models}, arXiv:1104.1592 [INSPIRE]. Toric varieties, Newton polyhedra, Okounkov bodies, Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry Consistency conditions for 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 In his paper [J. Differ. Geom. 61, No.1, 147--171 (2002; Zbl 1056.14021)], the author has studied aspects of the conjecture that birationally equivalent smooth projective varieties have equivalent derived categories iff they have equivalent canonical divisors. The article under review is dealing with singular instead of only smooth varieties and moreover with pairs of varieties and \(\mathbb Q\)-divisors, where at most log-terminal singularities are allowed:
Derived equivalence conjecture. Let \((X,B)\) and \((Y,C)\) be such pairs (and suppose some technical conditions). Denote \(\mathcal X\) and \(\mathcal Y\) the associated stacks, and assume there are proper birational morphisms \(\mu :W\to X\) and \(\nu :W\to Y\) such that \(\mu ^* (K_X+B)=\nu ^* (K_y+C)\). Then there exists an equivalence \(D^b({\mathcal X}) \to D^b({\mathcal Y})\) of triangulated categories .
After formulating the conjecture, first of all a converse statement is given. Main result of the article is a proof of the conjecture for toroidal varieties. The theorem generalizes an earlier result of the author [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 197--215 (2002; Zbl 1092.14023)]. It implies the McKay correspondence for abelian quotient singularities as a special case. derived category of coherent sheaves; derived equivalence conjecture Kawamata Y., Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo, 2005, 12(2), 211--231 Minimal model program (Mori theory, extremal rays), Derived categories, triangulated categories Log crepant birational maps 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 Let \((X,P)\) be a normal singularity, and let \(D=\sum d_i D_i\) be a divisor on \(X\) with real coefficients \(0 \leq d_i \leq 1\). The pair \((X,D)\) is said to be log canonical near \(P\) if the following conditions are satisfied:
(i) \(K_X +D\) is \({\mathbb{R}}\)-Cartier,
(ii) For some resolution \(f:Y\to X\) of \((X,P)\) with normal crossings for \(D\), if
\[
K_Y\equiv f^*(K_X+D)+\sum a(E,X,D) E
\]
where \(E\) runs over prime divisors on \(Y\), \(a(E,X,D)\in{\mathbb{R}}\) and \(a(D'_i,X,D)=-d_i\) for the strict transform \(D'_i\) of \(D_i\), then \(a(E,X,D) \geq -1\) for all \(E\).
A log canonical pair \((X,D)\) is said to be exceptional if there is at most one exceptional divisor \(E\) for which \(a(E,X,D)=-1\). The singularity \((X, P)\) is said to be exceptional if \((X,D)\) is exceptional for any \(D\) such that \((X,D)\) is log canonical. This notion was introduced by Shokurov in order to study the complements (i.e. good divisors in the multiple anticanonical systems, see definition 1.4 in the paper).
In the paper, the authors construct the first examples of 3-dimensional canonical exceptional singularities. The examples are the quotients of \({\mathbb{C}}^3\) by the action of Klein's simple group \(J_{168}\) of order 168 and of its central extension \(J'_{504}\) by the 3-rd roots of unity, of order 504. The proof uses the classical results of these groups. log canonical singularities; 3-dimensional canonical exceptional singularities; quotient singularities; finite group DOI: 10.1007/BF02367247 Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves Klein's group defines an exceptional singularity of dimension 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 paper contains a lot of material and information about maximal Cohen--Macaulay modules \(M\) over the local ring \(A\) of a surface singularity (\(\dim (A)=\text{depth}(M)\)). It starts with basic results and definitions as for instance the depth lemma and the Auslander--Buchsbaum formula, contains Matlis Duality, Grothendieck's Local Duality and a lot of other stuff from commutative algebra related to the study of maximal Cohen--Macaulay modules. Then general properties of maximal Cohen--Macaulay modules over surface singularities are presented as for instance the fact that in case \(A\) is normal \(M\) is maximal Cohen--Macaulay if and only if it is reflexive. It follows a section about maximal Cohen--Macaulay modules over two--dimensional quotient singularities containing the result that a normal surface singularity is a quotient singularity if and only if it has finite Cohen--Macaulay representation type.
The algebraic and the geometric approaches to McKay correspondence for quotient surface singularities as well as its generalization for simply elliptic and cusp singularities are described. A new proof of a result of \textit{R.-O. Buchweitz, G.-M. Greuel} and \textit{F.-O. Schreyer} [Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)] (the surface singularities \(A _{\infty}\) and \(D_{\infty}\) have countable Cohen--Macaulay representation type) is given. At the end one can find some conjectures concerning the Cohen--Macaulay representation type and an example for a \textsc{Singular}--computation. Cohen-Macaulay modules; surface singularity; quotient singularity Burban, I., Drozd, Y. (2008). Maximal Cohen--Macaulay modules over surface singularities, trends in representation theory of algebras and related topics.EMS Ser. Congr. Rep., Eur. Math. Soc.Zürich, pp. 101--166. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Cohen-Macaulay modules Maximal Cohen-Macaulay modules over 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 This paper introduces a general method for relating characteristic classes to singularities of a bundle map. The method is based on the notion of geometric atomicity. This is a property of bundle maps \(\alpha:E\to F\) which universally guarantees the existence of certain limits arising in the theory of singular connections. Under this hypothesis, each characteristic form \(\Phi\) of \(E\) or \(F\) satisfies an equation of the form
\[
\Phi=L+dT,
\]
where \(L\) is an explicit localization of \(\Phi\) along the singularities of \(\alpha\) and \(T\) is a canonical form with locally integrable coefficients. The method is constructive and leads to explicit calculations. For normal maps (those transversal to the universal singularity sets) it retrieves classical formulas of R. MacPherson at the level of forms and currents [see Part I, the authors, Asian J. Math. 4, 71--95 (2000; Zbl 0981.58003)]. It also produces such formulas for direct sum and tensor product mappings. These are new even at the topological level The condition of geometric atomicity is quite broad and holds in essentially every case of interest, including all real analytic bundle maps. An important aspect of the theory is that it applies even in cases of ``excess dimension,'' that is, where the singularity sets of \(\alpha\) have dimensions greater than those of the generic map. The method yields explicit calculations in this general context. A number of examples are worked out in detail. Global theory of complex singularities; cohomological properties, Singularities of differentiable mappings in differential topology, Characteristic classes and numbers in differential topology, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other connections Singularities and Chern-Weil theory. II: Geometric atomicity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Algebraic stacks are objects which generalize schemes from the viewpoint of fibered categories. As it has turned out, over the past decades, stacks are especially well adapted to the study of classification problems in algebraic geometry via geometric invariant theory.
In the present survey article, the author summarizes some general structure results about the particular class of Deligne-Mumford (DM) stacks of finite type over a field of characteristic 0, together with their applications to certain moduli spaces. After a brief introduction to stacks in general, a concrete description of smooth Deligne-Mumford stacks in dimension 1 is given, with special emphasis placed on orbifold curves over \(\mathbb{C}\) and their role in the classical moduli theory of vector bundles on compact Riemann surfaces. This is followed by a more detailed discussion of smooth Deligne-Mumford stacks, orbifolds, and quotient stacks of arbitrary dimension over a base field \(k\). Then the study turns to possibly singular Deligne-Mumford stacks, culminating in a new characterization of those stacks that are isomorphic to the stack quotient of a quasi-projective scheme by a reductive algebraic group acting linearly. This leads to the suggestion that a Deligne-Mumford stack over a field of characteristic 0 should be called ``(quasi-)projective'' if it admits a (locally) closed embedding into a smooth proper Deligne-Mumford stack with projective coarse moduli space. In this context, the class of (quasi-)projective Deligne-Mumford stacks is completely characterized in different ways. More precisely, it is outlined that a separated Deligne-Mumford stack with quasi-projective coarse moduli space over a field of characteristic 0 is quasi-projective (in the above sense) if and only if it enjoys several equivalent, well-studied properties, namely: being a quotient stack, satisfying the so-called resolution hypothesis, admitting a finite flat covering by a scheme, or possessing a generating sheaf. At the end of the paper, these general structure results are illustrated by various examples of concrete moduli stacks known from the recent literature.
Throughout the entire article, a basic reference is the fundamental work ``Brauer Groups and Quotient Stacks'' by \textit{D. Edidin, B. Hassett, A. Kresch} and \textit{A. Vistoli} [Am. J. Math. 123, No. 4, 761--777 (2001; Zbl 1036.14001)], together with numerous other, mostly very recent original research papers by various authors. algebraic stacks; Deligne-Mumford stacks; quotient stacks; orbifolds; moduli spaces; Brauer groups of schemes; geometric invariant theory Kresch, A., \textit{on the geometry of Deligne-Mumford stacks}, Algebraic geometry (Seattle 2005), part 1, 259-271, (2009), American Mathematical Society, Providence, RI Generalizations (algebraic spaces, stacks), Algebraic moduli problems, moduli of vector bundles, Brauer groups of schemes, Geometric invariant theory, Vector bundles on curves and their moduli, Homotopy theory and fundamental groups in algebraic geometry On the geometry of Deligne-Mumford 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 We study an analytically irreducible algebroid germ \((X, 0)\) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization \((\overline X, 0)\) of \((X, 0)\), centered at the point \(0\in\overline X\). If \((X, 0)\) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is a \(\mathbb{C}\)-algebra of finite type, namely the coordinate ring of a non-necessarily normal affine toric variety of the form \(\mathbb{Z}^\Gamma= \text{Spec\,}\mathbb{C}[\Gamma]\), and we show that the semigroup \(\Gamma\) is an analytical invariant of \((X, 0)\). This provides another proof of the analytical invariance of the normalized characteristic monomials of \((X, 0)\). If \((X, 0)\) is the algebroid germ of non necessarily normal toric variety, we apply the same method to prove a local version of the isomorphism problem for algebroid germs of non necessarily normal toric varieties (solved by Gubeladze in the algebraic case). DOI: 10.2996/kmj/1093351323 Complex surface and hypersurface singularities, Singularities in algebraic geometry Analytical invariants of quasi-ordinary hypersurface singularities associated to divisorial valua\-tions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article extends previous results by the author [Math. Ann. 302, No. 4, 601--608 (1995; Zbl 0842.14024); \textit{U. N. Bhosle} et al., Bull. Sci. Math. 138, No. 1, 41--62 (2014; Zbl 1288.14020)] on Narasimhan-Seshadri correspondences between (unitary) representations of the fundamental group of a nodal curve and (vector) Higgs/Hitchin bundles, to singular curves with cusps and ordinary \(r\)-points.
After the introduction in section 1, section 2 defines Hitchin pairs \((E, \phi)\) on an integral projective curve, where \(E\) is a torsion free sheaf on \(Y\) and \(\phi:E\rightarrow E\otimes L\), \(L\) being a torsion free rank \(1\) sheaf on \(Y\) (a Higgs bundle when \(E\) is locally free and \(L\) is the dualizing sheaf and invertible). The data of the singularities together with the normalization of \(Y\) yield the notions of generalized parabolic Hitchin pairs and modules (Definitions 2.5 and 2.7) providing the adequate moduli spaces for the problem to study (Theorem 2.13). Section 3 shows, for cuspidal curves, that the universal categorical quotient of unitary representations plus data coming from the singularities by the conjugation relation is identified with a dense subset of the moduli space of semistable torsion free sheaves of rank \(n\) and degree \(0\) (Theorem 3.7) and the analog for representations in the general linear group with the moduli of semistable generalized parabolic Hitchin bundles (Theorem 3.9). Section 4 extends these results to more general singularities (Theorems 4.6, 4.7, 4.9 and 4.10). fundamental group; singular curves; Hitchin pairs; stability Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Representations of the fundamental group and Higgs bundles on singular integral 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 Vinberg's theory of \(\theta\)-representations takes as data a Dynkin diagram \(X_n\) corresponding to a simple Lie algebra \({\mathfrak g}\) (of a Lie group \(G\)), a node \(x\in X_n\) and a simple root \(\alpha_k\) corresponding to \(x\). This defines a \({\mathbb Z}\)-grading on the root system: a root \(\beta\) is of degree \(d\) if \(\beta=d\alpha_k+\beta'\), where \(\beta'\) is an integral linear combination of simple roots \(\alpha_j\), \(j\neq k\). The Lie algebra then decomposes also, as \({\mathfrak g}=\bigoplus_{i\in{\mathbb Z}}{\mathfrak g}_i\), and we get an action of \(G_0=(G,G)\times {\mathbb C}^*\), the connected closed subgroup of \(G\) corresponding to \({\mathfrak g}_0\), on \({\mathfrak g}_1\). Here \((G,G)\) is the connected semi-simple Lie grup whose Dynkin diagram is got by deleting \(x\) from \(X_n\). This is called a type~I representation if \(G\) is finite dimensional, and a type~II representation if \(G\) is a Kac-Moody algebra, so \(X_n\) is an affine Dynkin diagram.
The aim is to understand the orbits on \(U={\mathfrak g}_1\). In the type~II case there will be infinitely many of them. In the type~I case there is a good theoretical answer due to Vinberg, but it is not easy to compute with. For type~II, one can study the non-nilpotent (i.e.\ stable) \(G\)-orbits in \(U\) as follows. Pick a parabolic subgroup \(P<G\). By the Borel-Weil theorem one can write \(U=H^0({\mathcal U})\) for some homogeneous bundle \({\mathcal U}\) on \(G/P\): the construction is not unique in general. The action of \(P\) on the fibre \({\mathcal U}(y)\) at \(y\in G/P\) has finitely many orbits, and in fact one can retain this orbit structure while reducing to the action of a reductive group \(G'<P\) (the Levi subgroup of the stabiliser of \(y\)); in other words, to a type~I representation.
The orbit closure under \(G'\) on each fibre glue to give subvarieties \({\mathcal Y}\subset{\mathcal U}\). Given \(\nu\in H^0(G/P,{\mathcal U})\) we consider the subvarieties \(Y=\nu(G/P)\cap{\mathcal Y}\). Using local information in the form of free resolutions for the coordinate ring and some other modules for \(\nu(G/P)\cap {\mathcal Y}\cap{\mathcal U}(y)\), one can hope to calculate the canonical sheaf \(\omega_Y\) and the cohomology \(H^i({\mathcal O}_Y)\).
In this largely but not entirely expository article, several examples are worked out in detail. In all the cases considered, \(P\) may be chosen so that one of the degeneracy loci \(Y\) is a torsor for an abelian variety. In some cases this can also be seen via classical constructions of projective geometry. The details of the computations are carried out using Macaulay2, and some code is supplied. Vinberg \(\theta\)-group; Eagon-Northcott generic perfection; degeneracy locus Laurent Gruson, Steven V. Sam & Jerzy Weyman, Moduli of abelian varieties, Vinberg \(\theta\)-groups, and free resolutions, Commutative algebra, Springer, 2013, p. 419-469 Syzygies, resolutions, complexes and commutative rings, Abelian varieties and schemes, Semisimple Lie groups and their representations Moduli of abelian varieties, Vinberg \(\theta\)-groups, and free resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a series of papers, \textit{T. Milanov and Y. Ruan} [``Gromov-Witten theory of elliptic orbifold \(\mathbb{P}^1\) and quasi-modular forms'', Preprint, \url{arXiv:1106.2321}], \textit{M. Krawitz} and \textit{Y. Shen} [``Landau-Ginzburg/Calabi-Yau correspondence of all genera for elliptic orbifold \(\mathbb{P}^1\)'', Preprint, \url{arXiv:1106.6270}] have verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple elliptic singularities \(E^{(1,1)}_N \; (N=6,7,8)\). As a by-product it was also proved that the orbifold Gromov-Witten invariants of the orbifold projective lines \(\mathbb{P}^1_{3,3,3}\), \(\mathbb{P}^1_{4,4,2}\) and \(\mathbb{P}^1_{6,3,2}\) are quasi-modular forms on an appropriate modular group. While the modular group for \(\mathbb{P}^1_{3,3,3}\) is \(\Gamma(3)\), the modular groups in the other two cases were left unknown. The goal of this paper is to prove that the modular groups in the remaining two cases are, respectively, \(\Gamma(4)\) and \(\Gamma(6)\). simple elliptic singularities; orbifold Gromov-Witten invariants Milanov, T., Shen, Y.: The modular group for the total ancestor potential of Fermat simple elliptic singularities. Commun. Number Theory Phys. \textbf{8}, 329-368 (2014) Singularities of surfaces or higher-dimensional varieties, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Structure of modular groups and generalizations; arithmetic groups, Singularities in algebraic geometry, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) The modular group for the total ancestor potential of Fermat 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 This book appears at the time Robert Langland is given the Abel prize. It covers much more, but the Langlands program is included.
The book consists of three parts, the first preparatory, and is called Basics. It starts by explaining model examples, that is the simplest functors arising in algebraic geometry, number theory and topology. The functors take values in \(C^\ast\)-algebras, and are the noncommutative tori. The noncommutative torus \(\mathcal A_\theta\) is an algebra over \(\mathbb C\) on a pair of generators \(u,v\) satisfying \(vu=e^{2\pi i\theta}\) where \(\theta\) is a real constant. There are several more or less equivalent definitions of noncommutative tori. The original is the geometric, which involves a deformation of the commutative algebra \(C^\infty(T^2)\) of smooth complex valued functions on a two-dimensional torus \(T^2\). After this comes the analytic definition, and finally the algebraic definition. The different definitions give a good illustration of the connections, and the reason for the more abstract algebraic definition of a torus. Some basic facts about the torus \(\mathcal A_\theta\) are considered. The author gives the identity for when \(\mathcal A_\theta\) is Morita equivalent to \(\mathcal A_{\theta^\prime}\) in the geometric and analytic case, which says when the to tori are equivalent in the setting of noncommutative algebraic geometry. Also, complex and real multiplication, the relation to elliptic curves with Weiserstrass uniformation and Jacobi normal forms are considered. The functor \(F:\mathcal E_\tau\rightarrow\mathcal A_\tau\) is explained, and this makes the definition of the Sklyanin algebra \(S(\alpha,\beta,\gamma)\) natural. The functor \(F:\mathcal E_\tau\rightarrow\mathcal A_\theta\) is intertwined with the arithmetic of elliptic curves, and the ranks of elliptic curves are related to an invariant of algebras \(\mathcal A_{RM}\). The first chapter ends with a classification of surface automorphisms.
Via the basics of category theory, the book contains a thorough definition of \(C^\ast\)-algebras: A \(C^\ast\)-algebra is an algebra \(A\) over \(\mathbb C\) with a norm \(a\mapsto\parallel a\parallel\) and an involution satisfying \(\parallel ab\parallel\leq \parallel a\parallel\parallel b\parallel\) and \(\parallel a^\ast a\parallel=\parallel a\parallel^2\) and such that \(A\) is complete with respect to the norm. The properties of \(C^\ast\)-algebras are recalled, and the \(K\)-theory of \(C^\ast\)-algebras is given. After this it is possible to define the noncommutative tori and the almost finite (AF) algebras introduced by Bratteli, which are classified by their Bratteli diagrams. Particular cases of these are generic AF algebras, stationary AF-algebras, and the original case: Uniformly hyper-finite \(C^\ast\)-algebras (UHF-algebras). Also, the \(K\)-groups of the Cuntz-Krieger algebras are computed.
The second part of the book studies noncommutative invariants, first in the case of topology. This is done by constructing functors arising in the topology of surface automorphisms, fibre bundles, knots, links, etc. The functors have their images in the category of AF-algebras, Cuntz-Krieger algebras, cluster \(C^\ast\)-algebras and so forth, defining a set of homotopy invariants of the the corresponding topological space. Some invariants are new, and some are known: Torsion in fibre bundles, Jones and HOMFLY polyniomials, and more. For the classification of surface automorphisms of compact oriented surfaces of genus \(g\geq 1\), the text considers Pseudo-Anasov automorphisms of a surface, the Jacobians of measured foliations, Anosov maps of the torus and its numerical invariants, etc.
In the study of torsion in the torus bundles \(M_\alpha\), the study includes the Cuntz-Krieger functor and specific noncommutative invariants of torus bundles.
In the study of the obstruction theory for Anosov's bundles, a functor \(F\) from the category of mapping tori of the Anosov diffeomorphisms \(\phi:M\rightarrow M\) of a smooth manifold \(M\), the Anosov bundles, to a category of stable homomorphisms between corresponding AF-algebras is constructed. This is used to introduce an obstruction theory for continuous maps between Anosov's bundles built on the noncommutative invariants derived from the Handleman triple \((\Lambda,[I],K)\) attached to a stationary AF-algebra. Specific examples are given in dimension 2, 3 and 4.
Now follows the definition, properties and applications of cluster \(C^\ast\)-algebras and knot polynomials in the topological setting. The author gives a representation of the braid groups in the cluster \(C^\ast\)-algebra associated to a triangulation of the Riemann surface \(S\) with one or two cusps, and it is proved that the Laurent polynomials coming from the \(K\)-theory of such an algebra are topological invariants of the closure of braids. Jones and HOMFLY polynomials are special cases of the construction corresponding to the \(S\) being a sphere with two cusps and a torus with one cusp respectively. One should mention that the text covers the Birman-Hilden theorem and a lot of explicit examples.
The book turns over to noncommutative invariants in algebraic geometry: The setup is to look at \(\mathsf{CRng}\) as the category of coordinate rings of projective varieties, and to consider a functor \(F:\mathsf{CRng}\overset{\text{GL}_n} {\rightarrow}\mathsf{Grp}\hookrightarrow\mathsf{Grp-Rng}\). It is proved in the text that when \(\mathsf{CRng}\) are the coordinate rings of elliptic curves, the category \(\mathsf{Grp-Rng}\) are the noncommutative tori. Also, if \(\mathsf{CRng}\) are the rings of algebraic curves of genus \(g\geq 1\), then \(\mathsf{Grp-Rng}\) are the toric AF-algebras. If \(\mathsf{CRng}\) are coordinate rings of projective varieties of dimension \(n\geq 1\), then \(\mathsf{Grp-Rng}\) consists of the Serre \(C^\ast\)-algebras. Notice also that elliptic curves over the field of \(p\)-adic numbers are considered, and in this case \(\mathsf{Grp-Rng}\) consists of the UHF-algebras. Finally in this chapter on noncommutative invariants in algebraic geometry, it is proved that the mapping class group of genus \(g\geq 2\) are linear: They admit a faithful representation into the matrix group \(\text{GL}_{6g-6}(\mathbb Z)\). This chapter includes Elliptic curves, algebraic curves of genus \(g\geq 1\), Tate curves and UHF-algebras and the mapping class group. It should be mentioned that the link between topology and algebraic geometry is explored.
In number theory, noncommutative invariants appear via a restriction of a functor \(F:\mathsf{CRng}\rightarrow\mathsf{Grp-Rng}\) to the arithmetic schemes. An important example is when \(\mathcal E_{\text{CM}}\) is an elliptic curve with complex multiplication. Then \(F(\mathcal E_{\text{CM}})=\mathcal A_{\text{RM}}\) where \(\mathcal A_{\text{RM}}\) is a noncomutative torus with real multiplication. This is used to relate the rank of \(\mathcal E_{\text{CM}}\) to an invariant of \(\mathcal A_{\text{RM}}\). This invariant is called arithmetic complexity, and is used to prove that the complex number \(e^{2\pi i\theta+\log\log\varepsilon}\) is algebraic whenever \(\theta\) and \(\varepsilon\) are algebraic numbers in a real quadratic field. Also, the invariant is used to find generators of the abelian extension of a real quadratic number field. The text includes the definition of an \(L\)-function \(L(\mathcal A_{\text{RM}},s)\) of \(\mathcal A_{\text{RM}}\) and proves that this coincides with the Hasse-Weil function \(L(\mathcal E_{\text{CM}},s)\) of \(\mathcal E_{\text{CM}}\). This localization functor tells that the crossed products is an analogue of the prime ideals used in algebraic geometry. The function \(L(\mathcal A_{\text{RM}},s)\) is extended to the even-dimensional noncommutative tori \(\mathcal A_{\text{RM}}^{2n}\), and an analogue of the Langlands conjecture for such tori is sketched. The number of points of a projective variety \(V(\mathbb F_q)\) over a finite field \(\mathbb F_q\) is computed in terms of the invariants of the Serre \(C^\ast\)-algebra \(F(V_{\mathbb C})\) of the complex projective variety \(V_{\mathbb C}\). Also, this chapter consider isogenies of elliptic curves, symmetry of complex and real multiplication, ranks of elliptic curves, transcendental number theory, class field theory, noncommutative reciprocity, Langlands conjecture for the \(\mathcal A_{\text{RM}}^{2n}\), and finally, projective varieties over finite fields.
The final part of the book gives a survey of Noncommutative algebaic geometry (NCG). The start is the finite geometries and the axioms of projective geometry, including Desargues and Pappus axioms. Then continuous geometries in the weak geometry, von Neumann geometry. In particular, Connes' geometries is considered more deeply, including classification of type III factors, Connes' invariants, noncommutative differential geometry and Connes' index theorem. Chapter 10 contains a survey of Index theory: The Atiyah-Singer theorem and Fredholm operators, the index theorem, \(K\)-homology and Atiyah's realization, Kasparov's \(KK\)-theory. The applications of the index theory include the Novikov conjecture, Baum-Connes conhecture, positive scalar curvature and finally the coarse geometry. After this chapter follows a treatment of Jones polynomials, Braids and the trace invariant. The next to final chapter in the book is about quantum groups,and this includes examples of Hopf algebras already in the introduction to this chapter. Also Manin's quantum plane, Hopf algebras in general, operator algebras and quantum groups are treated. The final chapter of the book is a survey of noncommutative algebraic geometry. This chapter includes the most common theories of Artin, Van den Bergh (turning slightly into derived schemes). It includes the Serre isomorphism, twisted homogeneous coordinate rings, the Sklyanin algebras, and it even mentions the noncommutative algebraic geometry of O.A. Laudal.
Ok, so there is one more chapter, 14, indicating some trends in NCG, but this is most likely separated into its own chapter just because of tridecrafobia.
The book is a good survey of noncommutative geometry, and is an excellent starting point for doing good research in the field. Each section of the book ends with a list of references for going deeply into each subject, and so this book gives a framework for the field of noncommutative geometry. noncommutative tori; Anasov automorphisms; C*-algebras; K-theory; cluster C*-algebras; Sklyanin algebras; AF-algebras; UHF-algebras; Hecke eigenform; continuous geometries; Connes geometries; index theory; Kasparov KK-theory; Jones polynomials; quantum groups; Hopf algebra; noncommutative algebraic geometry; deformation quantization Research exposition (monographs, survey articles) pertaining to algebraic geometry, Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Categories in geometry and topology Noncommutative geometry. A functorial approach | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 following review is extracted from the introduction of the article.
``In this paper, we investigate a local-to-global question concerning quotient singularities. Recall that a variety \(X\) over a field \(k\) is said to have \textit{tame (abelian) quotient singularities} if it is étale locally the quotient of a smooth variety by a finite (abelian) group (whose order is relatively prime to the characteristic of \(k\). Every variety of the form \(U/G\), where \(U\) is smooth and \(G\) is a finite group of order relatively prime to the characteristic of \(k\), has at worst tame quotient singularities. The motivation for this paper is whether the converse is true, a question suggested to us by William Fulton:
Question 1.1. Let \(k\) be an algebraically closed field. If \(X\) is a variety over \(k\) with tame quotient singularities, does there exist a smooth variety \(U\) over \(k\) with an action of a finite group \(G\) such that \(X=U/G\)?
We show that the answer is ``yes'' when \(X\) is quasi-projective and globally a quotient of a smooth variety by a torus. In particular, every quasi-projective toric variety (with tame quotient singularities) is a global quotient by a finite abelian group. These results are immediate consequences of Theorem 1.2, which characterizes quotients of smooth varieties by finite abelian groups.
Theorem 1.2. Let \(X\) be a quasi-projective variety with tame abelian quotient singularities over an algebraically closed field \(k\). The following are equivalent. {\parindent=0.6cm\begin{itemize}\item[--] \(X\) is a quotient of a smooth quasi-projective variety by a finite abelian group. \item[--] \(X\) is the geometric quotient of a smooth quasi-projective variety by a torus acting with finite stabilizers. \item[--] \(X\) has Weil divisors \(D_1,\dots, D_r\) whose images generate Cl\(({\hat{\mathcal{O}}}_{X,x})\) for all closed points \(x\) of \(X\). \item[--] the canonical stack over \(X\) is a stack quotient of a quasi-projective variety by a torus.
\end{itemize}}
We emphasize that even when \(X\) is a toric variety, the answer to Question 1.1 is not obvious. We show in Proposition 4.1 that if \(X\) is the blow-up of weighted projective space \(\mathbb{P}(1,1,2)\) at a smooth torus-fixed point, it is not possible to present \(X\) as a toric quotient by a finite group. Nevertheless, we show in \S 3 that when \(X\) is a toric variety, the proof of our main theorem gives a procedure for constructing \(U\) as a (non-toric) slice of a higher-dimensional toric variety (see Theorem 3.1). We demonstrate this procedure for the example of \(\mathbb{P}(1,1,2)\) blown-up at a smooth torus-fixed point in \S 4, obtaining it as an explicit quotient of a smooth variety by \(\mathbb{Z}/2\). Sections 3 and 4 are not needed for the proof of Theorem 1.2, but they show that the proof is constructive and that the construction can be described completely without stacks, even though the proof itself relies on stack-theoretic techniques.
There are several variants of Question 1.1 that one can pose. For example, we have the following question.
Question 1.6. If \(X\) is a variety over an algebraically closed field with tame abelian quotient singularities, then is it of the form \(U/G\) with \(U\) a smooth variety and \(G\) a finite abelian group?
We show that the answer is ``no'' with Theorem 1.7. The key input is the equivalence of (1) and (4) in Theorem 1.2, which shows that if the canonical stack over \(X\) is not a stack quotient by a torus, then \(X\) cannot be expressed as a scheme quotient of a smooth scheme by a finite abelian group.
Theorem 1.7. Let \(k\) be an algebraically closed field with \(\mathrm{char}(k)\nmid 60\), and let \(V\) be an irreducible 3-dimensional representation of the alternating group \(A_5\). {\parindent=0.6cm\begin{itemize}\item[--] \(X=\mathbb{P}(V)/A_5\) has abelian quotient singularities, but is not a quotient of a smooth variety by a finite abelian group. \item[--] Moreover, no open neighborhood of a singular point of \(X\) is a quotient of a smooth variety by a finite abelian group.
\end{itemize}}
There are other variants of Question 1.1 in the literature whose answers are known to be positive. If one modifies Question 1.1 by dropping the assumption that \(G\) be finite, then the answer is ``yes'': if \(X\) is a variety with quotient singularities over a field of characteristic 0, then \(X=U/G\), where \(U\) is a smooth scheme and \(G\) is a linear algebraic group.
If one modifies Question 1.1 in a different direction, requiring a finite surjection \(U\to X\) with \(U\) smooth, but no group action, then the answer is also ``yes'': it follows that for an irreducible quasi-projective variety \(X\) with quotient singularities over a field \(k\), there is a finite surjection from a smooth variety to \(X\).
Question 1.1 therefore asks if there is a common refinement of these two variants.'' quotient stacks; torus quotients; toric varieties; quotient singularities Geraschenko A., Satriano M.: Torus Quotients as Global Quotients by Finite Groups. arXiv:1201.4807. Group actions on varieties or schemes (quotients), Stacks and moduli problems, Toric varieties, Newton polyhedra, Okounkov bodies Torus quotients as global quotients by finite groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For complex algebraic surfaces, this paper provides the final step in the proof of the conjecture (due to John Nash, about 30 years ago) that the singularities of an algebraic variety can be resolved by a finite succession of proper birational maps \(\mu\) : \(\bar S\to S\), \(\mu^*\Omega_{S/k}/\)torsion locally free of rank dim(S) and \(\mu\) universal with respect to that property (so called ``Nash transformations''). It is known by \textit{A. Nobile} [Pac. J. Math. 60, 297-305 (1975; Zbl 0324.32012)] that (in characteristic 0) \(\mu\) is an isomorphism iff S is nonsingular, and in particular, for plane curve singularities the conjecture is true. \textit{G. González-Sprinberg} [Ann. Inst. Fourier 32, No.2, 111-178 (1982; Zbl 0469.14019)] proved for complex surfaces, that normalized Nash transformations resolve rational double points and cyclic quotient singularities. By a theorem of H. Hironaka, any surface singularity is transformed by a finite succession of Nash transformations into ``sandwiched singularities'', i.e. singularities of a surface which birationally dominates a nonsingular surface.
The main theorem of the article under review states that sandwiched singularities are resolved by normalized Nash transformations, thus completing the proof of the above mentioned conjecture for \(\dim(S)=2\), \(k={\mathbb{C}}.\)
Chapter II of the paper gives a classification of sandwiched singularities, a problem which is shown to be equivalent to the classification of plane curve singularities, complete ideals in 2- dimensional regular local rings or valuations with center in a regular 2- dimensional local ring [partially, this is an overview of results of the same author, cf. Am. J. Math. 112, No.1, 107-156 (1990; Zbl 0716.13003)].
In chapter III, the main theorem is proved in two steps: First of all, minimal singularities are considered, i.e. here: rational surface singularities with reduced fundamental cycle. If \((S,\xi)\) is such a singularity, \(\Gamma\) its dual graph and \(\mu:S'\to S\) the normalized Nash transformation, \(\Gamma '_ i\) the dual graphs of the singularities of \(S'\), then \(\#\{\)vertices of \(\Gamma'_ i\}\leq \#\{\)vertices of \(\Gamma\},\) i.e. such procedure terminates after finitely many steps. The proof is completed by showing that the problem can be reduced to minimal singularities, i.e. any sandwiched singularity is transformed into a minimal one after finitely many normalized Nash transformations. resolution of singularities; Nash transformations; surface singularity; classification of sandwiched singularities; classification of plane curve singularities Mark Spivakovsky, ``Sandwiched singularities and desingularization of surfaces by normalized Nash transformations'', Ann. Math.131 (1990) no. 3, p. 411-491 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, Singularities of curves, local rings Sandwiched singularities and desingularization of surfaces by normalized Nash transformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Each infinitesimally faithful representation of a reductive complex connected algebraic group \(G\) induces a dominant morphism \(\Phi\) from the group to its Lie algebra \(\mathfrak g\) by orthogonal projection in the endomorphism ring of the representation space. The map \(\Phi\) identifies the field \(Q(G)\) of rational functions on \(G\) with an algebraic extension of the field \(Q(\mathfrak g)\) of rational functions on \(\mathfrak g\). For the spin representation of \(\text{Spin}(V)\) the map \(\Phi\) essentially coincides with the classical Cayley transform. In general, properties of \(\Phi\) are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find \(\text{Harm}(G)\) so that for the coordinate ring \(A(G)\) of \(G\) we have \(A(G)=A(G)^G\otimes\text{Harm}(G)\). As a consequence of a partial solution to this problem and a complete solution for \(\text{SL}(n)\) one has in general the equality \([Q(G):Q({\mathfrak g})]=[Q(G)^G:Q({\mathfrak g})^G]\) of the degrees of extension fields. Among other results, \(\Phi\) yields (for the complex case) a generalization, involving generic regular orbits, of the result of Richardson showing that the Cayley map, when \(G\) is semisimple, defines an isomorphism from the variety of unipotent elements in \(G\) to the variety of nilpotent elements in \(\mathfrak g\). In addition if \(G\) is semisimple the Cayley map establishes a diffeomorphism between the real submanifold of hyperbolic elements in \(G\) and the space of infinitesimal hyperbolic elements in \(\mathfrak g\). Some examples are computed in detail. infinitesimally faithful representations; reductive complex connected algebraic groups; Lie algebras; representation spaces; fields of rational functions; Cayley transforms; coordinate rings; regular orbits; varieties of unipotent elements Kostant, B.; Michor, P.; Christian, Duval, The generalized Cayley map from an algebraic group to its Lie algebra, \textit{Prog. Math.}, 213, 259-296, (2003), Birkhäuser, Boston, MA Representation theory for linear algebraic groups, Simple, semisimple, reductive (super)algebras, Lie algebras of linear algebraic groups, Classical groups (algebro-geometric aspects), Linear algebraic groups over the reals, the complexes, the quaternions, Representations of Lie and linear algebraic groups over local fields The generalized Cayley map from an algebraic group to its Lie algebra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The existence of crepant resolution for a quotient singularity by a finite subgroup of \(\mathrm{SL}(n,\mathbb{C})\) is known if \(n\) is equal to or smaller than \(3\), and also for complete intersection singularities in any dimension.
In this paper the author shows the existence of a projective crepant resolution for several families of quotient singularities by abelian noncyclic and non complete intersection type subgroups of \(\mathrm{SL}(4,\mathbb{C})\). Toric methods are essential for the proof.
The first sections are devoted to remind the notation and necessary previous results. In section \(2\) the main notions about toric quotient singularities are reminded, such as any toric variety admits an equivariant resolution of singularities. In section 3 a criterion for quotient singularities (by a finite abelian subgroup of \(\mathrm{SL}(n,\mathbb{C})\)) to be complete intersection is reminded.
Finally, in section \(4\), the main result is proven using toric techniques. crepant resolution; quotient singularities; dimension four Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Existence of crepant resolution for abelian quotient singularities by order \(p\) elements in dimension 4 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors establish the following theorem: Let X be a Noetherian scheme, and let \(\gamma\) be an element of \(H^ 2(X,{\mathcal O}^*_ X)\). Then there is a scheme Y and a proper birational morphism \(\alpha: Y\to X\) such that the cohomology class \(\alpha^*(\gamma)\) is represented by an Azumaya algebra on Y. - The proof is by a relatively brief inductive argument, which the authors describe as a simple version of a proof due to \textit{O. Gabber}.
When second cohomology is a birational invariant (for example, for smooth projective varieties), the theorem comes very close to identifying second cohomology and the Brauer group, and in fact does so when second cohomology is finite. As the authors remark, both these conditions obtain for a nonsingular projective model X of V/G where G is a finite group and V is a faithful complex representation of G (using the result of the first named author that in this case \(H^ 2(X,{\mathcal O}^*)\) is isomorphic to the finite group \(H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}}))\). cohomology class as Azumaya algebra; second cohomology; Brauer group Bogomolov, F. A.; Landia, A. N., \(2\)-cocycles and Azumaya algebras under birational transformations of algebraic schemes, Algebraic geometry (Berlin, 1988), Compositio Math., 0010-437X, 76, 1-2, 1-5, (1990) (Co)homology theory in algebraic geometry, Algebraic cycles, Brauer groups of schemes, Realizing cycles by submanifolds 2-cocycles and Azumaya algebras under birational transformations of algebraic schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of \(G\)-bundles \(\mathrm{Bun}_G\) for an arbitrary reductive group \(G\), and their relationship to the Langlands dual group \(\check{G}\) of \(G\).
The article consists of two parts. In the first and main part, we study the monodromy action on the nearby cycles sheaf along the principal degeneration of \(\mathrm{Bun}_G\) and relate it to the Langlands dual group \(\check G\). We describe the weight-monodromy filtration on the nearby cycles and generalize the results of [37] from the case \(G=\mathrm{SL}_2\) to the case of an arbitrary reductive group \(G\). Our description is given in terms of the combinatorics of the Langlands dual group \(\check G\) and generalizations of the Picard-Lefschetz oscillators found in [37]. Our proofs in the first part use certain local models for the principal degeneration of \(\mathrm{Bun}_G\) whose geometry is studied in the second part.
Our local models simultaneously provide two types of degenerations of the Zastava spaces; these degenerations are of very different nature, and together equip the Zastava spaces with the geometric analog of a Hopf algebra structure. The first degeneration corresponds to the usual Beilinson-Drinfeld fusion of divisors on the curve. The second degeneration is new and corresponds to what we call \textit{Vinberg fusion}: it is obtained not by degenerating divisors on the curve, but by degenerating the group \(G\) via the Vinberg semigroup. Furthermore, on the level of cohomology the degeneration corresponding to the Vinberg fusion gives rise to an algebra structure, while the degeneration corresponding to the Beilinson-Drinfeld fusion gives rise to a coalgebra structure; the compatibility between the two degenerations yields the Hopf algebra axiom. geometric representation theory; geometric Langlands program; moduli spaces of \(G\)-bundles; nearby cycles; Picard-Lefschetz theory; weight-monodromy theory; Vinberg semigroup; Langlands duality Geometric Langlands program (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems, Langlands-Weil conjectures, nonabelian class field theory, Vector bundles on curves and their moduli, Fibrations, degenerations in algebraic geometry Monodromy and Vinberg fusion for the principal degeneration of the space 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 We consider a homogeneous variety \(\mathbb{X}= G/P\), where \(G\) is a reductive group over an algebraically closed field \(k\) of characteristic zero and \(P\) is a parabolic subgroup of \(G\). We study the category of \(G\)-bundles of finite rank on \(\mathbb{X}\). This category is equivalent to the category \({\mathfrak p}\)-mod of finite dimensional integral \({\mathfrak p}\)-modules, where \({\mathfrak p}\) is the Lie algebra of \(P\). We describe this category via the category of all finite dimensional representations of some infinite quiver \(Q\) following \textit{A. I. Bondal}' and \textit{M. M. Kapranov} [in: Helices and Vector Bundles, Proc. Sem. Rudakov, Lond. Math. Soc. Lect. Note Ser. 148, 45-55 (1990; Zbl 0742.14011)]. The crucial problem is to find relations \({\mathcal R}\) in the path algebra of the quiver \(Q\) such that the category \((Q, {\mathcal R})\)-mod of finite dimensional representations of \(Q\) which satisfy the relations \({\mathcal R}\) is equivalent to \({\mathfrak p}\)-mod. The main result in this article states that these three equivalent categories are Koszul if and only if the unipotent radical \(P_u\) of \(P\) is abelian. Thus the relations \({\mathcal R}\) are quadratic. Koszul algebras; quiver; homogeneous variety Hille L.: Homogeneous vector bundles and Koszul algebras. Math. Nach. 191, 189--195 (1998) Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets, Linear algebraic groups over arbitrary fields Homogeneous vector bundles and Koszul 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 paper is devoted to Klein's resolvent problem for algebraic functions: Given an algebraic function \(\mathbf{z}\) on an irreducible variety \(X\), what is the smallest number \(k\) such that the function \(\mathbf{z}\) can be rationally induced from an algebraic function \(\mathbf{w}\) on some variety \(Y\) of dimension \(\leq k\)? The author completely solves the problem for algebraic functions unramified on \((\mathbb{C}\setminus\{ 0\} )^{n}\) using geometric and topological methods. In the first part of the paper, coverings over topological \(n\)-tori \((S^{1})^{n}\) are studied. The author introduces the notion of the topological essential dimension of a covering and then proves that the topological essential dimension of a covering over \(n\)-torus is equal to the rank of its monodromy group. In the second part of the paper these topological results are put in an algebraic context and used to prove that the algebraic essential dimension of an algebraic function unramified over the algebraic \(n\)-torus \((\mathbb{C}\setminus\{ 0\} )^{n}\) is equal to the rank of its monodromy group. This provides the solution of the starting problem. Furthermore, the author proves that the algebraic essential dimension of the universal algebraic function \(\mathbf{z}\) defined by the equation \(\mathbf{z}^{n}+x_{1}\mathbf{z}^{n-1}+\cdots+x_{n}=0\) is at least \(\lfloor n/ 2\;\rfloor\). This paper is a nice example of how results obtained by methods and techniques of one mathematical branch can be successfully applied to a problem of another mathematical branch, which is fruitful for both of them. Coverings in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Homotopy groups Coverings over tori and topological approach to Klein's resolvent problem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 quotient singularities of \(\mathbb{C}^n\) by a finite matrix group \(G\). In particular, they find that the Hirzebruch class of these singularities essentially agree with the Molien series of the group \(G\). They work with the equivariant Hirzebruch class (with respect to the torus \(\mathbb{C}^{\ast}\)-action) and use a Lefschetz-Riemann-Roch-type theorem from [\textit{S. E. Cappell} et al., Commun. Pure Appl. Math. 65, No. 12, 1722--1769 (2012; Zbl 1276.14009)]. As special case of the McKay correspondence proved for the elliptic class in [\textit{L. Borisov} and \textit{A. Libgober}, Ann. Math. (2) 161, No. 3, 1521--1569 (2005; Zbl 1153.58301)], they show that the equivariant Hirzebruch class of a crepant resolution is a combination of the Molien series of the centralizers of elements of \(G\). A certain positivity result is proved for local equivariant Hirzebruch and Chern-Schwartz-MacPherson classes of \(\mathbb{C}^n/G\) quotient varieties. Explicit examples including Du Val singularities and four dimensional symplectic singularities are discussed. quotient singularities; Hirzebruch class; Molien series; McKay correspondence; Du Val singularities; symplectic singularities McKay correspondence, Linear algebraic groups over the reals, the complexes, the quaternions, Global theory and resolution of singularities (algebro-geometric aspects), Algebraic cycles Equivariant Hirzebruch classes and Molien series of quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In [\textit{E. Looijenga}, Invent. Math. 61, 1-32 (1980; Zbl 0436.17005)], a family \(A_ H\) of Abelian varieties, associated with an affine root system and parametrized by the complex upper half plane \(H\) was introduced and investigated. The natural action of the modular group \(SL_ 2(\mathbb{Z})\) lifts to that family, as well as to the ample linear bundle \({\mathcal L}_ H\) on \(A_ H\).
The purpose of the present paper is to study the invariant theory of \(A_ H\) and \({\mathcal L}_ H\) with respect to the action of the modular group (or a subgroup \(\Gamma\) of finite index) as well as of the Weyl group \(W\) of the corresponding root system. As a first step, the quotient by \(\Gamma\) is considered and the extension of the family \(A_ H/\Gamma\to H/\Gamma\) is obtained in a natural and explicit fashion, using a toroidal embedding technique of the author [Acta Math. 157, 159- 241 (1986; Zbl 0635.14015)]. Then the invariants are studied. They form a bi-graded algebra over the ring of modular forms, graded by weight (referring to the behaviour with regard to \(\Gamma\)) and index (referring to the appropriate power of \(\mathcal L\)), and are called Jacobi forms since in the special case of the root system of type \(A_ 1\) they reduce to (weak) Jacobi forms in the sense of \textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms (Birkhäuser, 1985; Zbl 0554.10018)]. The algebra of invariants is determined for all types of root systems excluding \(E_ 8\). It turns out to be a polynomial algebra over the ring of modular forms, with generators that do not depend on the particular choice of the group \(\Gamma\).
The result has an application in singularity theory to deformation of fat points in the plane with defining ideal \((x^ 2-y^ 3, y^ k)\) or \((x^ 2-y^ 3, xy^{k-1})\), \(k\geq 3\) (this will appear elsewhere). invariant theory; modular group; Weyl group; root system; toroidal embedding; Jacobi forms; polynomial algebra; deformation of fat points Wirthmüller, K., Root systems and Jacobi forms, Comp. Math., 82, 293, (1992) Simple, semisimple, reductive (super)algebras, Relationship to Lie algebras and finite simple groups, Jacobi forms, Singularities in algebraic geometry, Formal methods and deformations in algebraic geometry Root systems and Jacobi forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A non-commutative crepant resolution (NCCR) of a Gorenstein algebra \(R\) is an endomorphism ring \(\text{End}_R(M)\) of a maximal Cohen-Macaulay \(R\)-module \(M\) such that \(\text{End}_R(M)\) is a maximal Cohen-Macaulay \(R\)-module of finite global dimension. The three main problems are the following {\parindent=6mm \begin{itemize}\item[(1)] Construct a NCCR of \(R\) and characterize the tilting object \(M\). \item[(2)] Construct a derived equivalence between the NCCR and a commutative crepant resolution. \item[(3)] Construct a (commutative) crepant resolution as a moduli space of modules over the NCCR.
\end{itemize}} Let \(V=\mathbb C^N\) be a complex vector space of dimension \(N\geq 2\). Consider the subset \(B(1)\) of \(\text{End}_{\mathbb C}(V)\) given by \(B(1)=\{X\in\text{End}_{\mathbb C}(V)|X^2=0,\;\dim\text{Ker}X=N-1\}\) which is a nilpotent orbit aof type \(A\). Then \(\overline{B(1)}\) is normal, and has only symplectic singularities. Thus its coordinate ring \(R\) is normal and Gorenstein. Also, since \(\text{codim}_{\overline{B(1)}}(\partial B(1))\geq 2\), \(R\simeq H^0(B(1),\mathcal O_{B(1)})\) as \(\mathbb C\)-algebras.
The main objective in this article is to study NCCRs of a minimal nilpotent orbit closure \(\overline{B(1)}\) of type \(A\).
Let \(H\) be a subgroup of \(\text{SL}_N\) such that \(\text{SL}_N/H\simeq B(1)\). Then \(H\) is isomorphic to a particular subgroup of \(\text{SL}_N\) on which it can be defined a character \(H\ni A\mapsto c^{-a}\). Let \(\mathcal M_a\) be the homogeneous line bundle on \(B(1)\) corresponding to this character, and set \(M_a=H^0(B(1),\mathcal M_a).\) It is proved in this article that a direct sum of \(R\)-modules \((M_a)_a\) gives a NCCR of \(R\). More precisely, each \(M_a\) is a Cohen-Macaulay \(R\)-module for \(N+1\leq a\leq N-1\), and for \(0\leq k\leq N-1\), the \(R\)-module \(\bigoplus_{a=-N+k+1}^kM_a\) gives a NCCR \(\text{End}_R(\bigoplus_{a=-N+k+1}^kM_a)\) of \(R\). This result is proved by \textit{tilting bundles} on the known (commutative) crepant resolutions \(Y\) and \(Y^+\) of \(\overline{B(1)}\) that are the total spaces of the cotangent bundles on \(\mathbb P(V)\) and \(\mathbb P(V^\ast)\) respectively. Consider the projections \(\pi: Y\rightarrow\mathbb P(V)\) and \(\pi^\prime: Y\rightarrow\mathbb P(V^\ast)\). It is proved that the bundles \(\mathcal T_k=\bigoplus_{a=_n+k+1}^k\pi^\ast\mathcal O_{\mathbb P(V)}(a)\) and \(\mathcal T_k^+=\bigoplus_{a=_n+k+1}^k{\pi^\prime}^\ast\mathcal O_{\mathbb P(V^\ast)}(a)\) are tilting bundles on \(Y,\;T^+\). Also there is a canonical isomorphism of \(R\)-algebras \(\Lambda_k=\text{End}_Y(\mathcal T_k)\simeq\text{End}_{Y^+}(\mathcal T^+_{n-k-1})\simeq \text{End}_R(\bigoplus_{a=-N+k+1}^kM_a).\) By the theory of tilting bundles, there is an equivalence of categories \(D^b(Y)\simeq D^b(\mathsf{mod}(\Lambda_k))\). The author provides another NCCR \(\Lambda^\prime\) of \(R\) that is not isomorphic to \(\Lambda_k\) but which is derived equivalent to \(\Lambda_k\). This is interesting in itself (by proving that reconstruction of algebras are not possible).
The article describes a NCCR \(\Lambda_k\) as the path algebra of the \textit{double Beilinso quiver} with explicitly given relations. Similar results for non-commutative resolutions of determinantal varieties are obtained by Buchweitz, Leuschke, and Van den Bergh, and also by Weyman and Zhao.
Let \(S=\text{Sym}^\bullet(V\otimes_{\mathbb C} V^\ast)\), let \(v_1,\dots,v_N\) be the standard basis of \(V=\mathbb C^N\), and let \(f_1,\dots,f_N\) be the dual basis of \(V^\ast\). Regarding \(x_{ij}=v_j\otimes f_i\in S\) as the variables of the affine coordinate ring of the affine variety \(\text{End}_{\mathbb C}(V)\simeq V^\ast\otimes_{\mathbb C}V\), it is clear that because \(\overline{B(1)}\) is a closed subvariety of \(\text{End}_{\mathbb C}(V)\), \(R\) is a quotient of \(S\).
The author proves that as \(S\)-algebra, the non-commutative algebra \(\Lambda_k\) is isomorphic to the path algebra \(S\widetilde{\Gamma}\) of the double Beilinson quiver \(\widetilde{\Gamma}\) with \(N\) vertices, and with relations \(v_iv_j=v_jv_i,\;f_if_j=f_jf_i,\;v_jf_i=f_iv_j=x_{ij}\) for all \(1\leq i,j\leq N\), \(\sum_{i=1}^Nf_iv_i=0=\sum_{i=1}^Nv_if_i.\)
By using this result, the crepant resolutions \(Y\) and \(Y^+\) are recovered from the quiver \(\widetilde{\Gamma}\) as moduli spaces of representations. The article contains a characterization of the simple representations of the quiver. It is proved that a simple representation corresponds to a point of a crepant resolution that lies over a non-singular point of \(\overline{B(1)}\).
The relations between a crepant resolution \(Y (Y^+)\) and a NCCR \(\Lambda_k\) is seen as a generalization of the \textit{McKay correspondence}. Classically, this states that for a finite group \(G\subset\text{SL}_2\) there are relations between the geometry of a quotient variety \(\mathbb C^2/G\) and the representations of the group \(G\). The generalization is understood as general relationships between the crepant resolution \(\widetilde{\mathbb C^2/G}\) of \(\mathbb C^2/G\) and a quotient stack \([\mathbb C^2/G].\) \(\widetilde{\mathbb C^2/G}\) is called a geometric resolution of \(\mathbb C^2/G\), and because a coherent sheaf on a quotient stack \([\mathbb C^2/G]\) is canonically identified with a module over the skew group algebra \(\mathbb C[x,y]\sharp G\), a smooth stack \([\mathbb C^2/G]\) is said to be an algebraic resolution of \(\mathbb C^2/G\). This leads to interpret the McKay correspondence as a correspondence between geometric and algebraic resolutions: A geometric resolution of \(\overline{B(1)}\) is \(Y\) or \(Y^+\), and an algebraic resolution is the NCCR \(\Lambda_k\).
The diagram of two crepant resolutions \(Y\mathop{\longrightarrow}\limits^{\phi}\overline{B(1)}\mathop{\longleftarrow}\limits^{\phi^+} Y^+\) is a local model of a class of flops called \textit{Mukai flops}. By blowing up, Kawamata and Namikawa defines functors \(\text{KN}_k:D^b(Y)\rightarrow D^b(Y^+)\) and \(\text{KN}_k^\prime:D^b(Y^+)\rightarrow D^b(Y)\) which by composing with \(\psi_k\) and \(\psi_{N-k-1}\) gives an equivalence \(D^b(Y)\rightarrow D^b(Y^+)\). It is a main result that this composition coincides with the Kawamata-Namikawa functors.
Finally, the author constructs \textit{multi-mutation} functors as an analogue to the Iyama-Wemyss's mutation functor. It is proved that a composition of two multi-mutation functors correspond to a P-twist \(P_k\in\text{Auteq}(D^b(\text{mod}(\Lambda_{N+k})))\).
In the case of three-dimentional flops, Donovan and Wemyss proved that a compostion of two IW functors corresponds to a spherical twist. The results in this article say that in the case of Mukai flops, a composition of many IW mutations corresponds to a P-twist.
The article is sufficiently detailed and gives a very good introduction to non-commutative algebraic geometry. Furthermore, the results are very nice and generalizes known results in the derived setting. derived category; Mukai flop; non-commutative crepant resolution; (NCCR); quiver representation; Gorenstein algebra; tilting object; tilting bundle; total spaces; double Beilinson quiver; path algebra; McKay correspondence; group representaitons; geometric resolution; algebraic resolution; multi-mutation functors Noncommutative algebraic geometry, Singularities in algebraic geometry, McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0632.00003.]
Consider a simple surface singularity X (or Du Val singularity). There is a very nice correspondence between the dual graph of the resolution of singularities of X and indecomposable Cohen-Macaulay modules over A [See \textit{M. Artin} and \textit{J.-L. Verdier}, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)]. Following this direction the authors study MCM (= maximal Cohen-Macaulay) modules over a Gorenstein ring A and sometimes over a hypersurface in order to give more precise results. The question of existence of MCM with given data \((n,m)\) (\(n\)= minimal number of generators and \(m=\text{rank}(M))\) is considered. The main technique is to study relations of MCM and Bourbaki sequences \((0\to F\to M\to I\), where F is A-free and I a codimension 2 CM-ideal or \(I=A)\). Then the problem of MCM is translated into an ideal problem.
Another important fact is the Rao correspondence: If R is a normal Gorenstein domain then there exists a bijection between stable isomorphism classes of orientable MCM-modules and even linkage classes of codimension 2 CM-ideals. maximal Cohen-Macaulay module; reflexive module; simple surface singularity; Du Val singularity; Gorenstein ring; Bourbaki sequences; Rao correspondence; linkage classes Herzog, J.; Kühl, M., Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki-sequences, Adv. Stud. Pure Math., 11, 65-92, (1987) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki- sequences | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider a space of binary forms \(R_ d\) of degree \(d\) in the variables \(t_ 0\), \(t_ 1\) over the field \(\mathbb{C}\) with the natural action of the group SL(2). A projective three-dimensional SL(2)-invariant submanifold \(X\) in \(\mathbb{P} (R_ d)\) will be said to be primitive if it is normal and for a general point \([x] \in X\) the form \(x \in R_ d\) does not have multiple factors. On the manifold \(X\), the open SL(2)-orbit \(\text{SL} (2) \cdot [x]\) exists. The complement of \(\text{SL} (2) \cdot [x]\) will be called a boundary and denoted by \(F(X)\). The stabilizer of a general point \(x\in X\) is called the stabilizer of \(X\), denoted by \(\Gamma (X)\) or \(\Gamma\). The smooth primitive manifolds are completely classified earlier. -- The primitive submanifolds \(X \subset \mathbb{P}(R_ d)\) can be classified, first of all, by the stabilizers \(\Gamma\) of their general points. The finite subgroup \(\Gamma \subset \text{SL} (2)\) is conjugate to one of the following groups:
1) \(\Gamma = \mathbb{I}\), the icosahedron group, \(| \Gamma | = 120\); 2) \(\Gamma = \mathbb{O}\), the octahedron group, \(| \Gamma | = 48\); 3) \(\Gamma = \mathbb{T}\), the tetreahedron group, \(| \Gamma | = 24\); 4) \(\Gamma = \mathbb{D}_ n\), a dihedral group, \(| \Gamma | = 4n\); 5) \(\Gamma = \mathbb{M}_ n\), a cyclic group, \(| \Gamma | = n\).
Theorem. Let \(X\) be a primitive SL(2)-invariant submanifold in \(\mathbb{P} (R_ d)\). Then \(X\) is an \(\mathbb{O}\)-factorial Fano manifold with log- terminal singularities and \(\text{Pic} (X) = \mathbb{Z} \cdot \mathbb{O} (1)\). The manifold \(X\) has the form \(\overline {\text{SL} (2) \cdot [x]}\) for a semi-invariant \(x \in R_ d\) of the group \(\Gamma\). action of linear algebraic group; three-dimensional invariant submanifold; space of binary forms Geometric invariant theory, Group actions on varieties or schemes (quotients), Fano varieties Invariant submanifolds in spaces of binary forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors consider the isomonodromy problems for flat \(G\)-bundles over punctured elliptic curves \(\Sigma_\tau= \mathbb{C}/(\mathbb{Z}\oplus\tau\mathbb{Z})\) with logarithmic singularities of connections at the marked points. The Lie group \(G\) is simple and complex. The bundles are classified by their characteristic classes. These classes are elements of the second cohomology group \(H_2(\Sigma_\tau,Z(G))\), where \(Z(G)\) is the center of \(G\). For any complex simple Lie group \(G\) and arbitrary class they define the moduli space of flat bundles, and in this way construct the monodromy preserving equations in the Hamiltonian form and their Lax representations. In particular, they include the Painlevé VI equation, its multicomponent generalizations and elliptic Schlesinger equations. The general construction is described for punctured curves of arbitrary genus. The authors extend the Drinfeld-Simpson (double coset) description of the moduli space of Higgs bundles to the case of flat connections. This local description allows them to establish the symplectic Hecke correspondence for a wide class of the monodromy preserving equations classified by characteristic classes of underlying bundles. In particular, the Painlevé VI equation can be described in terms of \(\mathrm{SL}(2,\mathbb{C})\)-bundles. Since \(Z(\mathrm{SL}(2,\mathbb{C}))=Z_2\), the Painlevé VI has two representations related by the Hecke transformation:
1) as the well-known elliptic form of the Painlevé VI for trivial bundles.
2) as the non-autonomous Zhukovsky-Volterra gyrostat for non-trivial bundles. monodromy-preserving deformations; Painlevé equations; flat connections; Schlesinger systems; Higgs bundles Levin, A.; Olshanetsky, M.; Zotov, A., Classification of isomonodromy problems on elliptic curves, Russ. Math. Surv., 69, 35, (2014) Vector bundles on curves and their moduli, Isomonodromic deformations for ordinary differential equations in the complex domain, Relationships between algebraic curves and integrable systems, Applications of Lie algebras and superalgebras to integrable systems Classification of isomonodromy problems 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 The purpose of this paper is to study minimal resolutions of quotient singularities with trivial canonical sheaf, more precisely the minimal resolution of the 3-dimensional quotient singularity by the isocahedral group and some higher dimensional abelian quotients.
Let \(\pi:V \to X\) the canonical projection from the vector space \(V=\mathbb{C}^n\) to the quotient space \(X=V/G\), where \(G\) is a finite subgroup of \(SL_n(\mathbb{C})\). The author wants to compare the Euler number \(\chi(X)\) of some minimal resolution of \(X\) with the ``orbifold Euler characteristic'' \(\chi(V,G)\). In a first part the author considers a double cover \(p:X\to Y\) of a smooth variety defined by some \(s \in \Gamma (Y, {\mathcal K}^2_Y)\), where \({\mathcal K}_Y\) is the canonical line bundle on \(Y\). Let \(B\) the ramification locus of \(p\) and assume \(\text{Sing} (B)=\bigcup^k_{j=1} C_j\), where each \(C_j\) is a non-singular codimension 2 subvariety of \(Y\). Then \(X\) is a normal variety with trivial canonical sheaf and \(\text{Sing} (X)=p^{-1} (\text{Sing} (B))\). -- By induction we can construct a modification \(\widetilde \sigma : \widetilde Y\to Y\), as the composition of blow-up of a curve \(C_j\), such that the double cover \(\widetilde X\) of \(\widetilde Y\) ramified at the strict transform of \(B\) on \(\widetilde Y\) gives an explicit resolution \(\widetilde \tau: \widetilde X \to X\) with trivial canonical bundle.
Then the author studies the space \(X=\mathbb{C}^3/I\) defined as the quotient 3-space by the icosahedral group \(I\). If we denote by \(J\) the subgroup of \(GL_3 (\mathbb{C})\) generated by \(I\) and \(-\text{id}\), then we get a double cover \(p : X \to Y= \mathbb{C}^3/J\). The quotient 3-space \(Y\) is smooth and the singular set of the branch locus \(B\) is the union of three smooth curves. Then we can apply the previous procedure and construct a resolution \(\widetilde X\) of \(X\) and we can calculate its Euler number \(\chi (\widetilde X)\) and we get the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^3,I) = 5\).
In the same way we can consider the quotient \(n\)-space \(X=\mathbb{C}^n/SD\), where \(SD=D \cap SL_n (\mathbb{C}) \simeq (\mathbb{Z}/2)^{n-1}\) and \(D\) is the subgroup of \(GL_n (\mathbb{C})\) consisting of all the diagonal order 2 elements, as a double cover of the smooth variety \(Y=\mathbb{C}^n/D\). We get a minimal resolution \(\widetilde X\) of \(X\) with trivial canonical bundle and we have the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^n,SD)=2^{n-1}\). minimal resolutions; 3-dimensional quotient singularity S.-S. Roan: On cx - 0 resolution of quotient singularity. Intern. Jour, of Math.,5, 523-536 (1994). Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, \(3\)-folds On \(c_ 1=0\) resolution of quotient singularity | 0 |
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