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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let k be an algebraically closed field, \(R=k[[x,y]]\), \(m=(x,y)\) the maximal ideal of R and \(h(I)(z)=\sum h_ i(I)z^ i \) the Hilbert function of an ideal I of R of \(colength\quad n\) where \(h_ i(I)=\dim_ k(m^ i/((I\cap m^ i)+m^{i+1}))\). For a fixed polynomial h with nonnegative integer coefficients and \(h(1)=n\) the ideals I with Hilbert function \(h(I)=h\) are parametrized by a locally closed subscheme \(Z_ h\) of the punctual Hilbert scheme \(Hilb^ nR\) and give a stratification \(Hilb^ nR=\cup_{h(1)=n}Z_ h \) [\textit{A. A. Iarrobino}, Mem. Am. Math. Soc. 188 (1977; Zbl 0355.14001), Bull. Am. Math. Soc. 78, 819-823 (1972; Zbl 0268.14002) and \textit{J. Briançon}, Invent. Math. 41, 45-89 (1977; Zbl 0353.14004)].
The author constructs a cellular decomposition of the strata \(Z_ h\) and computes their Betti numbers by modifying the cellular decomposition of \(Hilb^ n{\mathbb{P}}_ 2\) given by \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)]. Hilbert stratum; Hilbert function Göttsche L., Manuscripta Math. 66 pp 253-- (1990) Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Betti numbers for the Hilbert function strata of the punctual Hilbert scheme in two variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a complex, integral, locally planar curve \(C\), let \(C^{[n]}\) denote the Hilbert scheme of \(n\) points on \(C\).
\textit{J. V. Rennemo} [J. Eur. Math. Soc. (JEMS) 20, No. 7, 1629--1654 (2018; Zbl 1409.14011)] built on work of \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] and
\textit{I. Grojnowski} [Math. Res. Lett. 3, No. 2, 275--291 (1996; Zbl 0879.17011)] to construct a Weyl algebra
acting on the Borel Moore homology \(V = \oplus_{n \geq 0} H_* (C^{[n]})\) and described \(V\) in terms of
representation theory of Weyl algebras [J. Eur. Math. Soc. (JEMS) 20, 1629--1654 (2018; Zbl 1409.14011)].
The author extends Rennemo's approach to describe \(V = \bigoplus_{n \geq 0} H_* (C^{[n]}, \mathbb Q)\) when
\(C\) is reduced and locally planar, but not necessarily integral.
If \(C\) has \(m\) irreducible components, he constructs a bigraded action of
\[
A=A_m = [x_1, \dots, x_m,
\partial_{y_1}, \dots \partial_{y_m}, \sum_{i=1}^m y_i, \sum_{i=1}^m \partial_{x_i}]
\]
on \(V=\bigoplus V_{n,d}\) graded by the number of points \(n\) and homological degree \(d\).
After showing that the total spaces of relevant Hilbert flag schemes are smooth, he constructs operators on \(V\)
to set up the bivariant Borel-Moore homology formalism of \textit{W. Fulton} and \textit{R. MacPherson} [Categorical framework for the study of singular spaces. Providence, RI: American Mathematical Society (AMS) (1981; Zbl 0467.55005)]
and proves the commutation relations.
The end of the paper is devoted to the simplest case when \(m=2\) and \(C\) is the union of two
intersecting lines in \(\mathbb P^2\).
Starting with the geometric description of \(C^{[n]}\) given by \textit{Z. Ran} [J. Algebra 292, No. 2, 429--446 (2005; Zbl 1087.14005)], the author computes the
\(A\)-action, showing that \(\displaystyle V \cong \frac{\mathbb Q [x_1,x_2,y_2,y_2]}{\mathbb Q [x_1,x_2,y_1+y_2](x_1-x_2)}\) as an \(A\)-module. Hilbert scheme of points; representation theory Parametrization (Chow and Hilbert schemes) Hecke correspondences for Hilbert schemes of reducible locally planar curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth separated scheme over a Noetherian base ring \(\mathbb{K}\). The decomposition we are interested in is an isomorphism
\[
\text{R}{\mathcal H}\text{om}_{{\mathcal O}_{X\times_\mathbb{K} X}}({\mathcal O}_X,{\mathcal O}_X)\cong\bigoplus_q\bigl(\bigwedge_{{\mathcal O}_X}^q{\mathcal T}_{X/\mathbb{K}}\bigr)[-q]
\]
in the derived category \(\text{D}(\text{Mod }{\mathcal O}_{X\times_\mathbb{K} X})\). Here \({\mathcal T}_{X/\mathbb{K}}\) is the relative tangent sheaf. Upon passing to cohomology sheaves such an isomorphism recovers the Hochschild-Kostant-Rosenberg Theorem. If \(\mathbb{K}\) has characteristic 0 there is a decomposition that relies on a particular homomorphism of complexes from poly-tangents to continuous Hochschild cochains. We discuss sheaves of continuous Hochschild cochains on schemes and show why this approach to decomposition fails in positive characteristics. Our main result is a proof of the decomposition valid for any Gorenstein Noetherian ring \(\mathbb{K}\) of finite Krull dimension, regardless of characteristic. The proof is based on properties of minimal injective resolutions. smooth separated schemes; relative tangent sheaves; cohomology sheaves; continuous Hochschild cochains; injective resolutions; Hochschild complexes; derived categories Amnon Yekutieli, Decomposition of the Hochschild complex of a scheme in arbitrary characteristic, Canad. J. Math. 54 (2002), no. 4, 866 -- 896. Amnon Yekutieli, The continuous Hochschild cochain complex of a scheme, Canad. J. Math. 54 (2002), no. 6, 1319 -- 1337. (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Resolutions; derived functors (category-theoretic aspects), Syzygies, resolutions, complexes in associative algebras Decomposition of the Hochschild complex of a scheme in arbitrary characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems No review copy delivered. Algebraic coding theory; cryptography (number-theoretic aspects), Varieties over finite and local fields, Applications to coding theory and cryptography of arithmetic geometry, Analysis of algorithms, Cryptography On the optimal pre-computation of window \(\tau\mathrm{NAF}\) for Koblitz curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The number of positive solutions to a system of two polynomials in two variables defined over the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. The main result of this paper is the construction of a system as above having 7 positive solutions. This is achieved using tools developed in tropical geometry. When the corresponding tropical hypersurfaces intersect transversally, one can easily estimate the positive solutions to the system using the classical combinatorial patchworking for complete intersections. We apply this generalization to construct a system as above having 6 positive solutions. We also show that this bound is sharp. Consequently, our main result is proved using non-transversal intersections of tropical curves. real algebraic geometry; tropical geometry; solving polynomial systems; fewnomial theory El~Hilany, B.: Tropical geometry and polynomial systems. Ph.D. thesis, Comunauté Université Grenoble Alpes (2016). https://www.math.uni-tuebingen.de/user/boel/Thesis.pdf , Solving polynomial systems; resultants, Topology of real algebraic varieties Constructing polynomial systems with many positive solutions using tropical geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author works over the complex field \({\mathbb{C}}\). - Theorem 1: Let f,g: \(C\to {\mathbb{P}}^ 1\) be two coverings of \({\mathbb{P}}^ 1\) by a smooth irreducible curve C of genus \(\geq 1\). Assume that both have simple branching and that the branching occurs over the same set \(\Gamma \subset {\mathbb{P}}^ 1\). If \(\Gamma\) is sufficiently general, then f and g are isomorphic as coverings of \({\mathbb{P}}^ 1.\)
This theorem was previously known if \(d<(g-1)/2\) [\textit{E. Arbarello} and \textit{M. Cornalba}, Math. Ann. 256, 341-362 (1981; Zbl 0454.14023)]. - Examples showing that the conditions: genus\((C)\geq 1\) and \(\Gamma\) sufficiently general are necessary, are given. - The authors show that this theorem implies that the Tyrell conjecture (saying that the 40 elliptic curves tangent to 6 general concurrent lines are pairwise nonisomorphic) is true.
In the last section some geometric results on the monodromy of the covering \({\mathcal H}_{d,g}\to {\mathcal P}_ b\) (where \({\mathcal H}_{d,g}\) is the Hurwitz scheme of degree \(d\) branched covers of \({\mathbb{P}}^ 1\) by curves of genus \(g\geq 1\) and \({\mathcal P}_ b\) is the moduli space of b- pointed rational curves with \(b=2d-2+2g)\) are presented. Geometric interpretations of a Cohen result for \(d=3\) [\textit{D. B. Cohen}, J. Algebra 32, 501-517 (1974; Zbl 0343.20002)] and a Maclachlan remark for \(d=4\) [\textit{C. Maclachlan}, Mich. Math. J. 25, 235-244 (1978; Zbl 0366.20032), last paragraph] are given. isomorphic coverings; Tyrell conjecture; monodromy of the covering; Hurwitz scheme 10. Eisenbud, D., Elkies, N., Harris, J., Speiser, R.: On the Hurwitz scheme and monodromy. Compositio Math. 77, 95--117 (1991) Coverings of curves, fundamental group, Topological properties in algebraic geometry On the Hurwitz scheme and its monodromy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:S\to B\) be a finite cyclic covering fibration of a fibered surface. We study the lower bound of the slope \(\lambda_f\) when the relative irregularity \(q_f\) is positive. Fibrations, degenerations in algebraic geometry, Families, moduli of curves (algebraic), Surfaces of general type Slope inequarities for irregular cyclic covering fibrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth algebraic variety, \(E\) a vector bundle on \(X\) and let \(Y\) be a smooth subvariety of \(X\).
In characteristic 0, if \(V \subset H^0 E\) generates \(E\), then a general member of \(V\) is transversal to \(Y\); this fact is often used to compute the dimension of 0-loci of sections and their singular loci. In characteristic \(p > 0\), transversality may fail. The main result of this paper shows that, on the other hand, if we make the additional hypothesis that \(V\) also generates the sheaf of principal parts of \(E\), then transversality holds in any characteristic. The author uses his result to prove that the classification of arithmetically Buchsbaum curves, found by \textit{M. C. Chang}, works in positive characteristic, too. vector bundle; dimension of 0-loci; singular loci; classification of arithmetically Buchsbaum curves Walter, C, Transversality theorems in general characteristic and arithmetically Buchsbaum schemes, Internat. J. Math., 5, 609-617, (1994) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Finite ground fields in algebraic geometry Transversality theorems in general characteristic and arithmetically Buchsbaum schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a system of (complex, multivariate) polynomial equations \(f_1=\cdots=f_s=0\), with \(f_1,\ldots,f_s\) generating the ideal \(I=\langle f_1,\ldots,f_s\rangle\subseteq R={\mathbb C}[x_1,\ldots,x_n]\), and with finitely many solutions \(V(I)\subseteq{\mathbb C}^n\), numerical eigenvalue methods for root finding rely on different variants of the eigenvalue theorem. Classically, for a given \(g\in{\mathbb C}[x_1,\ldots,x_n]\) one considers the multiplication map \(m_g:R/I\rightarrow R/I\) given by \(f+I\mapsto gf+I\) whose eigenvalues are \(g(z)\in {\mathbb C}\) for \(z\in V(I)\), and if one has a vector space basis \(b_1,\ldots, b_d\in R/I\) and the ideal \(I\) is radical, then the eigenvectors of \(m_g\) are the tuples \((b_1(z),\ldots,b_d(z))\in{\mathbb C}^d\) for each zero \(z\in V(I)\), and the coordinates of the solutions can be recovered from these eigenvectors.
In previous work, \textit{S. Telen} [J. Pure Appl. Algebra 224, No. 9, Article ID 106367, 26 p. (2020; Zbl 1442.14187)] studied numerically robust algorithm for sparse homogeneous polynomials systems taken from the Cox ring of a compact toric variety \(X\). The corresponding algorithm computes homogeneous coordinates of the solutions from the eigenvalues of a multiplication map in certain degrees of the Cox ring, requiring that the solutions of the system have multiplicity one and lie in the simplicial part of the toric variety \(X\).
The main contribution of the present paper extends the use of eigenvalue computations to solve polynomial systems on a toric variety \(X\) allowing that the equations may have isolated singularities which need not belong to the simplicial part of \(X\). In order to do that, they introduce a notion of regularity for these zero-dimensional systems, similar to the classical Castelnuovo--Mumford regularity for projective space and let the degrees to vary in the class group of \(X\). The first main result shows that for a homogeneous ideal \(I\subseteq \text{Cox}(X)\) in the Cox ring of a toric variety \(X\) defining a zero-dimensional subscheme \(V_X(I)\subseteq X\) of degree \(\delta^+\) and for any regularity pairs \(\alpha,\alpha_0\in\text{Cl}(X)\), if \(\phi\) is a regular rational function on \(V_X(I)\), the multiplication by \(\phi\) map on \((\text{Cox}(X)/I)_{\alpha+\alpha_0}\) has eigenvalues \(\phi(\zeta_i)\) of algebraic multiplicity \(\mu_i\), for \(\zeta\in V_X(I)\) and where \(\delta^+=\mu_1+\cdots+\mu_{\delta}\). Using this result and its proof which characterize the eigenvectors of the multiplication map, the authors obtain a numerical algorithm for computing homogeneous coordinates of the points in \(V_X(I)\). This algorithm depends on the regularity of the pair \((\alpha+\alpha_0)\), and the second main result gives a criterion for extending a regular degree \(\alpha\) to a regularity pair when \(X\) is a complete intersection. solving polynomial systems; sparse polynomial systems; toric varieties; Cox rings; eigenvalue theorem; symbolic-numeric algorithm Toric varieties, Newton polyhedra, Okounkov bodies, Numerical computation of roots of polynomial equations, Numerical computation of solutions to systems of equations Toric eigenvalue methods for solving sparse polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Der Einsatz der Public-Key Kryptografie zur elektronischen Signatur und Verschlüsselung ist aus unserer heutigen, vernetzten Welt nicht mehr wegzudenken. Aufgrund der niedrigen Schlüssellängen erhalten Kryptosysteme, die auf Jacobischen von Kurven basieren, erhöhte Aufmerksamkeit und werden im Fall elliptischer Kurven bereits in Mobiltelefonen, Reisepässen und in vielen anderen Bereichen eingesetzt.
Will man ein Kryptosystem auf Jacobischen von Kurven aufbauen, ist es wichtig, dass die Gruppenordnung der \(k\)-rationalen Punkte der Jacobischen über dem endlichen Körper \(k\) einen großen Primfaktor enthält. Das diskrete Logarithmusproblem kann sonst mit Hilfe des chinesischen Restsatzes zu schnell gelöst werden, was das Kryptosystem unbrauchbar machen würde. Wir benötigen also Methoden zur Bestimmung der k-rationalen Punkte der Jacobischen einer Kurve. Für Kurven von Geschlecht 1 bis 3 und hyperelliptische Kurven von Geschlecht 4 gibt es zahlreiche effiziente Methoden, deren Ansatz entscheidend von der Charakteristik \(p\) des Grundkörpers \(k\) beeinflußt wird.
Diese Methoden lassen sich leider nicht auf großes \(p\) und nichthyperelliptische Kurven von Geschlecht 4 übertragen. Wir stellen in dieser Arbeit einen Algorithmus zur Bestimmung der \(\mathbb F_p\)-rationalen Punkte der Jacobischen von allgemeinen Modulkurven vor, der auf der Berechnung des Hecke-Operators \(T_p\) basiert und mit seiner linearen Komplexität in Zeit- und Speicheraufwand gerade für die zahlreichen nichthyperelliptische Modulkurven von Geschlecht 4 effizient ist. Wegen des Index-Calculus Angriffs von C. Diem gelten Kurven von Geschlecht 4 erst ab 172-Bit großer Gruppenordnung als sicher. Wir zeigen in expliziten Beispielen, dass unser Algorithmus in dieser Grössenordnung praktikabel ist. Der Algorithmus ist voll parallelisierbar und in allen Eingabegrößen simultanisierbar.
C. Paar, J. Pelzl und T. Wollinger schlagen hyperelliptische Kurven von Geschlecht 4 über Körpern mit 32-Bit für die Implementation von Kryptosystemen für ARM Prozessoren vor. Unser Algorithmus kann Beispiele dafür leicht berechnen und daher fügen wir dieser Arbeit im Anhang eine ganze Reihe davon hinzu. Analytic computations, Hecke-Petersson operators, differential operators (several variables), Arithmetic aspects of modular and Shimura varieties, Cryptography, Applications to coding theory and cryptography of arithmetic geometry Point-counting algorithms for the Hecke operator and applications to modular curves of genus 4 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the paper is the computation of the Artin-Greenberg function of a planar branch. As a corollary it is shown that two planar branches have the same topological type if and only if they have the same multiplicity and the same Artin-Greenberg function. plane curve germs; topological type Hickel ( M. ) , Calcul de la fonction d'Artin-Greenberg d'une branche plane , Pacific Journal of math. 213 , p. 37 - 47 ( 2004 ), et Preprint Université Bordeaux I N^\circ 145 (Avril 2002 ). Zbl 1054.14036 Singularities of curves, local rings, Étale and flat extensions; Henselization; Artin approximation, Plane and space curves Computation of the Artin-Greenberg function of a planar branch | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A ``good \(\text{mod } \ell\) model'' \(X^ A\) for the etale topological type of a commutative ring \(A\) is found in certain cases. Thus,
(1) If \(A= \mathbb{Z}[1/ \ell]\) for an odd regular prime \(\ell\), \(X^ A= \mathbb{R} P^ \alpha\vee S^ 1\).
(2) If \(A\) is a coordinate ring of a suitable affine curve over \(\mathbb{F}_ q\), then \(X^ A\) is a fibration over \(S^ 1\) with fiber the \(p\)- completion of a finite wedge of circles.
(3) If \(A\) is a generalized local field of transcendence degree \(r\) over \(\mathbb{F}_ q\), \(X^ A\) is a fibration over \(S^ 1\) with fibre the \(p\)- completion of a product of \(r\) circles.
Secondly, the etale \(K\)-theory space of \(X^ A\) in case (1) has a cohomology which injects into the cohomology of the algebraic \(K\)-theory space of \(A\). Corollaries depending on the validity of the Lichtenbaum- Quillen conjecture are then derived. etale topology; topological model of ring; etale cohomology of ring; algebraic \(K\)-theory; \(\text{mod } \ell\) model; affine curve; local field; etale \(K\)-theory; Lichtenbaum-Quillen conjecture Dwyer, William G.; Friedlander, Eric M., Topological models for arithmetic, Topology, 33, 1, 1-24, (1994), MR1259512 (95h:19004) Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Algebraic \(K\)-theory of spaces, Relations of \(K\)-theory with cohomology theories, Étale and other Grothendieck topologies and (co)homologies Topological models for arithmetic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the first part of this work [\textit{N. D. Duong} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18, No. 3, 977--1032 (2018; Zbl 1453.14056)], we studied affine group schemes over a discrete valuation ring (DVR) by means of Néron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of \(\mathcal{D}\)-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of ``infinite type,'' Néron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of \(\mathcal{D}\)-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Group schemes, Linear algebraic groups over adèles and other rings and schemes On the structure of affine flat group schemes over discrete valuation rings. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) be an elliptic curve defined over \({\mathbb{F}}_{2^n}\). The inverse operation of point doubling, called point halving, can be done up to three times as fast as doubling. Some authors have therefore proposed to perform a scalar multiplication by an ``halve-and-add'' algorithm, which is faster than the classical double-and-add method.
If the coefficients of the equation defining the curve lie in a small subfield of \({\mathbb{F}}_{2^n}\), one can use the Frobenius endomorphism \(\tau \) of the field extension to replace doublings. Since the cost of \(\tau \) is negligible if normal bases are used, the scalar multiplication is written in ``base \(\tau \)'' and the resulting ``\(\tau \)-and-add'' algorithm gives very good performance.
For elliptic Koblitz curves, this work combines the two ideas for the first time to achieve a novel decomposition of the scalar. This gives a new scalar multiplication algorithm which is up to 14.29\% faster than the Frobenius method, without any additional precomputation. Koblitz curves; scalar multiplication; point halving; \(\tau \)-adic expansion; integer decomposition Avanzi R., Ciet M., Sica F.: Faster scalar multiplication on Koblitz curves combining point halving with the Frobenius endomorphism. In: Bao F., Deng R.H., Zhou J. (eds.) Public Key Cryptography--PKC 2004, 7th International Workshop on Theory and Practice in Public Key Cryptography, Singapore, March 1--4, 2004, Lecture Notes in Comput. Sci., vol. 2947, pp. 28--40. Springer (2004). Cryptography, Applications to coding theory and cryptography of arithmetic geometry Faster scalar multiplication on Koblitz curves combining point halving with the Frobenius endomorphism | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be an adjoint semisimple algebraic group over an algebraically closed field. \textit{C. De Concini} and \textit{C. Procesi} [in: Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] proved in characteristic zero the following fundamental result:
Theorem. \(G\) admits a canonical smooth completion \(\overline{G}\) such that:
(i) the action of \(G\times G\) by left and right multiplication on \(G\) extends to \(\overline{G}\),
(ii) the boundary \(\overline{G}\setminus G\) is the union of smooth irreducible divisors intersecting transversely along the unique closed \(G\times G\)-orbit, and
(iii) the partial intersections of these boundary divisors are exactly the orbit closures.
Subsequently this result was extended in positive characteristic by \textit{E. S. Strickland} [see Math. Ann. 277, 165-171 (1987; Zbl 0595.14037)]. In both papers the method was via representation theory. In the paper under review the author obtains algebro-geometric realizations of \(\overline{G}\) as follows:
Let \(P\) be a parabolic subgroup of \(G\) and set \(X:=G/P\). The group \(G\times G\) acts on the Hilbert scheme \(\text{Hilb}(X\times X)\) via the natural action of \(G\times G\). When the action of \(G\) on \(X\) is faithful, then the \(G\times G\)-orbit of the diagonal \(\Delta_X\) (regarded as a point of \(\text{Hilb}(X\times X)\)) is isomorphic to \(G\).
Then the author proves that \(\overline{G}\) is isomorphic to the closure of this orbit. completion of algebraic group; action of algebraic group; Hilbert scheme Michel Brion, Group completions via Hilbert schemes, J. Algebraic Geom. 12 (2003), 605-626. Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations Group completions via Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We investigate the representation theory of domestic group schemes \(\mathcal{G}\) over an algebraically closed field of characteristic \(p > 2\). We present results about filtrations of induced modules, actions on support varieties, Clifford theory for certain group schemes and applications of Clifford theory for strongly group graded algebras to the structure of Auslander-Reiten quivers. The combination of these results leads to the classification of modules belonging to the principal block of the group algebra \(k\mathcal{G}\). induced modules; group schemes; Auslander-Reiten quivers; group algebra \(k\mathcal{G}\) Representation type (finite, tame, wild, etc.) of associative algebras, Group schemes, Modular Lie (super)algebras, Ordinary representations and characters Classification of indecomposable modules for finite group schemes of domestic representation type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field of characteristic zero, \(X\) a non-singular variety over \(k\) and \(\mathcal{I}\) a non-zero ideal sheaf over \(X.\) The problem of principalization of \(\mathcal{I}\) is to get a proper birational morphism \(\phi:Y\to X\), by means of a finite sequence of monoidal transforms such that \(Y\) is non-singular and \(\mathcal{IO}_{Y}\) is locally principal. In the paper under review, the author gives an algorithm for principalization of a locally monomial ideal sheaf on a 3-fold. Then he shows how this algorithm can be used to construct a toroidalization of a locally toroidal morphism of 3-folds. principalization; toroidalization; resolution of morphisms \(3\)-folds, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Local structure of morphisms in algebraic geometry: étale, flat, etc. A principalization algorithm for locally monomial ideal sheaves on 3-folds with an application to toroidalization | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X_0\) be a smooth Kähler threefold with trivial canonical bundle. In the paper under review, the author uses Kuranishi theory to produce a construction of the local analytic Hilbert scheme of curves on \(X_0\), at a fixed smooth irreducible curve \(Y_0\), as a gradient scheme. More precisely, a versal deformation \(s:X\to X'\) of \(X_0\) over an analytic polydisc \(X'\) is considered, then an analytic neighborhood \(Y'\) of \(Y_0\) in the relative Hilbert scheme of proper curves in \(X/X'\). The universal curve \(p:Y\to Y'\) induces a map \(\pi:Y'\to X'\), which can be extended to a smooth morphism \(U'\to X'\), where \(U'\) is an analytic polydisc, containing \(Y'\) as a closed analytic subscheme. The dimension of \(U'\) is \(\dim X'+h^0(N_{Y_0/X_0})+1\). Finally \(\tilde X'\) denotes the manifold obtained by removing the zero-section from the total space of the line bundle on \(X'\) \(s_*\Omega^3_{X/X'}\), and \(\tilde U', \tilde Y'\) are obtained from \(U',Y'\) via base change by \(\tilde X'/X'\).
The author constructs a potential function \(\Phi\) on \(\tilde U'\) such that the relative Hilbert scheme \(\tilde Y'\) is given by the gradient ideal of \(\Phi\), that is, its germ is the zero-scheme of the section \(d_{\tilde U'/\tilde X'}\Phi\) of \(\Omega^1_{\tilde U'/\tilde X}\). Moreover, using a result of Donagi--Markman, a new formulation is given of the Abel--Jacobi map into the intermediate Jacobian.
The last two sections of the paper, written respectively by Richard Thomas and Claire Voisin, contain analogous results in two related settings, i.e. the deformation problem of the pair \((E_0,X_0)\), where \(E_0\) is a holomorphic bundle on \(X_0\), and that of the triple \((X_0, S_0,L_0)\), where \(S_0\) is a smooth very ample divisor on \(X_0\) and \(L_0\) is a line bundle on \(S_0\) satisfying some suitable conditions. Calabi-Yau threefold; local analytic deformation theory; gradient scheme; Abel-Jacobi map Clemens H.: Moduli schemes associated to K-trivial threefolds as gradient schemes. J. Algebraic Geom. 14(4), 705--739 (2005) Families, moduli, classification: algebraic theory, Parametrization (Chow and Hilbert schemes), Calabi-Yau manifolds (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(3\)-folds Moduli schemes associated to \(K\)-trivial threefolds as gradient schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using a reduction to the Beauville systems, a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism, is obtained. The structure of the paper is as follows: In Section 2 the relation between the Jacobians of two curves which are linked by a ramified cyclic covering of prime order is studied. In Section 3 a space of polynomial matrices of size \(p\) is introduced and its automorphisms of order \(p\) are studied, with particular attention to the fixed point set of such an automorphism. In Section 4, both the space and its fixed point set are related to the corresponding spectral curves, upon using the momentum map and the results of Section 2. Beauville's result also used to describe the fibers of the algebraic completely integrable systems under construction. The Hamiltonian structure of the space of polynomial matrices, its fixed point sets and their quotients (by the adjoint action) are considered in Section 5. In particular a multi-Hamiltonian structure of the newly constructed phase spaces is also obtained. The algebraic integrability this system is proved in Section 6. integrable systems; Jacobians; algebraic integrability; curves with automorphisms Poisson manifolds; Poisson groupoids and algebroids, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties Algebraic integrable systems related to spectral curves with automorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors prove the following theorem: The field of formal-rational functions \(\hat K\) along a connected closed subscheme of positive dimension in a Grassmannian Grass(n,r) is exactly the field of rational functions on Grass(n,r). The proof uses a reduction to the special case of \({\mathbb{P}}^ 1\) in \({\mathbb{P}}^ n\) (where the result is known) and a lemma which states \(\Phi((X))\cap L(X)=\Phi(X)\) for fields \(\Phi\subset L\) and a finite set of indeterminates X. formal-rational functions; Grassmannian Babakhanian, A., Hironaka, H.: Formal functions over Grassmannians. Ill. J. Math. 26, 201--211 (1982) Grassmannians, Schubert varieties, flag manifolds, Formal neighborhoods in algebraic geometry Formal functions over Grassmannians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) be an algebraically closed field, and let \(F\{\omega\}\) be the Puiseux field over it, that is, the field of formal fractional power series over \(F\). The order of an element of \(F\{\omega\}\) is the exponent of the lowest order non-zero term. Let \(P\) be a polynomial of degree \(D\) over \(F\{\omega\}\). For simplicity, assume that \(P\) is monic and that all its coefficients are ordinary power series with non-negative order. (The paper deals with the general case.) A root \(\rho\) of \(P\) is said to be regular if \(\rho_0\), the 0-order term of \(\rho\), is a simple root of \(P_0\), the polynomial over \(F\) whose coefficients are the 0-order terms of the coefficients of \(P\). H. T. Kung and J. F. Traub [J. Assoc. Comput. Mach. 25, 245-260 (1978; Zbl 0371.68019)] showed that \(O(m^2D)\) operations over \(F\) suffice to compute a regular root to order \(m\). They also explained how to reduce the general problem of computing roots of \(P\) to the regular case, but did not give a complexity analysis of the process.
In this paper, a modification of their Newton polygon method is announced that obtains all roots of an arbitrary \(P\) to order \(m/D\) using \(O(k^2m^2D^6)\) operations in \(F\) and fewer than \(D\) calls on a root oracle that provides a root of any given polynomial over \(F\). The factor \(k\) depends on the orders and fractional exponents of the coefficients of \(P\), and is 1 in the simple case described above. The process can be carried out in time \(O(kmD^3)\) using \(O(kmD^3)\) parallel processors.
The algorithm is described in detail; full proofs of the complexity analysis are given in a paper to appear elsewhere. Puiseux field; Newton polygon method Algorithmic information theory (Kolmogorov complexity, etc.), Computational aspects of algebraic curves Approximations and complexity for computing algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper investigates the topological recursion, related to the Witten-Kontsevich theorem and tau-function for the KdV hierarchy of the moduli space of stable curves, applied to spectral curve $xy^2 = \frac{1}{2}$, which the authors called Bessel curve. Then the authors obtained similar results to the studies on the Airy curve $y^2=x$, e.g., [\textit{B. Eynard} and \textit{N. Orantin}, J. Phys. A, Math. Theor. 42, No. 29, Article ID 293001, 117 p. (2009; Zbl 1177.82049)]. Witten-Kontsevich theorem; KdV hierarchy; Airy curve; Bessel curve Families, moduli of curves (algebraic), Relationships between algebraic curves and physics, Groups and algebras in quantum theory and relations with integrable systems, Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Topological recursion on the Bessel curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We calculate the number of the isomorphism class of the finite flat models over the ring of integers of an absolutely ramified \(p\)-adic field of constant group schemes of rank two over finite fields by counting the rational points of a moduli space of finite flat models. group scheme; \(p\)-adic field Imai N., Finite flat models of constant group schemes of rank two, Proc. Am. Math. Soc. 138 (2010), 3827-3833. Group schemes, Varieties over finite and local fields, Algebraic moduli problems, moduli of vector bundles Finite flat models of constant group schemes of rank two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Around 1980 J. McKay discovered a remarkable bijection between isomorphism classes of non trivial irreducible representations of a finite subgroup \(G\) of \(\text{SL} (2,{\mathbb C})\) and irreducible components of the exceptional divisor of the minimal resolution of the (2-dimensional) quotient singularity \(X_G = {\mathbb C}^2/G\). The original presentation was rather formal (it used Dynkin diagrams). Later considerable work was done to describe the correspondence more geometrically, as well as trying to generalize it to other groups and higher dimension. To achieve this goal, different techniques were used: reflexive sheaves, \(G\)-Hilbert schemes of points of the complex plane (where \(G\) is a suitable finite subgroup of \(\text{SL}(2,{\mathbb C})\)), \(K\)-theory, derived categories, etc. In the two dimensional case, to obtain a generalized ``McKay correspondence'' it seems necessary to consider small subgroups of \(\text{GL}(2,{\mathbb C})\) (i.e., those acting freely on \({\mathbb C}^2\) away from the origin) and certain representations of \(G\), called special.
In this paper the author, a leading expert in this area, gives a brief survey of the theory, primarily in the two-dimensional case. Certain items are discussed in some detail, e.g., special representations and their associated special reflexive sheaves, \(G\)-Hilbert schemes and their applications. The paper is mainly expository, but it contains a new, more combinatorial description of a generalized McKay correspondence in the case where \(G\) is a finite cyclic subgroup of \(\text{GL}(2,{\mathbb C})\), as well as some examples. The article includes some comments about the three-dimensional case. quotient singularity; minimal resolution; irreducible representation Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) An introduction to the special 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 The McKay correspondence establishes a bijection between non-trivial irreducible representations of a finite subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb{C})\) and irreducible components of the exceptional divisor of the minimal resolution of the quotient surface singularity \(X_\Gamma:= \mathbb{C}^2/\Gamma\). This was discovered by J. McKay around 1980, using a rather formal approach. There was much subsequent work trying to give a more geometric description of the procedure. Among others, there were relevant contributions of Gonzalez-Springerg, Verdier, Esnault, Artin, working with suitable auxiliary reflexive sheaves. Later Itô and Nakamura used certain Hilbert schemes of zero-dimensional subschemes of the plane and Kapranov and Vasserot introduced suitable derived categories to describe the correspondence. Also attempts were made to generalize the theory. For instance, Riemenschneider conjectured that a similar correspondence could be obtained by using certain finite subgroups of \(\text{GL}(2,\mathbb{C})\) (namely, the small ones, i.e. those freely acting on \(\mathbb{C}^2-(0, 0)\)) and certain representations thereof (the so-called special ones). The conjecture was proved by \textit{A. Ishii} [J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057)], using Hilbert schemes and derived categories techniques.
In the present article, the author reviews these and other developments and shows that for a cyclic quotient singularity Ishii's theorem can be proved relatively easily [by using results of \textit{R. Kidoh}, Hokkaido Math. J. 30, No. 1, 91--103 (2001; Zbl 1015.14004)]. The author also presents a theorem giving several equivalent characterizations of the notion of special representation. These results had been obtained by the author several years ago, but had not been presented in a mathematical journal before.
The author also proposes a problem, namely to describe the deformation theory of \(X_\Gamma\) in terms of representations of \(\Gamma\). He announces some partial results in this direction. quotient surface singularity; reflexive sheaves; special representation; McKay correspondence; Hilbert scheme O. Riemenschneider, Special representations and the two-dimensional McKay correspondence, Hokkaido Math. J. 32 (2003), 317--333. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) Special representations and the two-dimensional 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 The McKay correspondence is a bijection involving certain simple Lie Algebras and irreducible representations of finite subgroups of \(\text{SL}(2,\mathbb C)\), observed by J. McKay in 1979. This has an interpretation in terms of the exceptional divisors of the minimal resolution of the (two-dimensional) singularity \(\mathbb C^2/G\). Since the appearance of \textit{J. McKay's} brief original paper [in: Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183--186 (1980; Zbl 0451.05026)] many works were produced attempting to geometrically explain this process, and to generalize it. The present expository article primarily deals with a three-dimensional generalization of the original McKay correspondence [see \textit{Y. Ito} and \textit{M. Reid}, in: Higher dimensional complex varieties. Proc. Int. Conf., Trento 1994, 221--240 (1996; Zbl 0894.14024)]. Namely, if \(G\) is a finite subgroup of \(\text{SL}(3,\mathbb C)\), \(X=\mathbb C^3/G\) and \(p:Y \to X\) is a crepant resolution of \(X\) and \(S\) is the set of conjugacy classes of \(G\) then \(\chi (Y) = \text{card} (S)\), where \(\chi\) denotes Euler characteristic and crepant means: \(K_Y = p^*(K_X)\). For a while the validity of this equality (sometimes called ``Vafa's formula'') was a conjecture, arising from the work of researchers in Mathematical Physics. Other authors (Markushevich, Roan) also made contributions in this direction. Among other things, the present paper includes a discussion of the classification of the finite subgroups of \(\text{SL}(3,\mathbb C)\), a proof of existence of crepant resolutions for singularities of the form \(X=\mathbb C^3/G\), with \(G\) a finite subgroup of \(\text{SL}(3,\mathbb C)\) which is not of monomial type (based on the mentioned classification) and the notion of age of an element \(g\) of \(G\) as above, which plays an important role in the proposed proof of Vafa's formula. There are also some examples, as well as a summary of known results on McKay correspondences, both in the case of surfaces and three-folds. This paper is a good introduction to the subject. resolution of singularities; crepant resolution; group representation; Euler characteristic Ito, Y.: The McKay correspondence--a bridge from algebra to geometry. In: European Women in Mathematics (Malta, 2001), pp. 127-147. World Scientific Publishing, River Edge (2003) Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients) The McKay correspondence -- a bridge from algebra to geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0614.00007.]
In characteristic 0, a rational double point is a \(quotient\quad X\) of \(k^ 2\) (k is the field) by a finite \(subgroup\quad G\) of SL(2,k). The McKay correspondence establishes a one-to-one correspondence between the irreducible representations of G and the vertices of the extended Dynkin diagram associated to the minimal desingularization of X. Moreover, it says how to read on the Dynkin diagram the tensor product of the standard representation \(G\subset SL(2,k)\) with any other one. We refer to it as the multiplicative structure. One may rephrase it in erms of irreducible reflexive modules on X, knowing that they are in one-to-one correspondence with the irreducible representations of G, and that \(G\subset SL(2,k)\) corresponds to the Kähler one \(forms\quad \Omega^ 1_ X\) (as a reflexive module).
In characteristic p\(>0\), where the group G no longer exists in general, the one-to-one correspondence between irreducible reflexive modules and vertices of the extended Dynkin diagram is still true (Gonzalez- Sprinberg, Verdier and Artin, Verdier). In the paper under review the authors complete the picture in characteristic \(p.\) Replacing \(\Omega^ 1_ X\) by \(\Omega\), the unique non trivial extension of the maximal ideal by \({\mathcal O}\) (which was also considered by other people, among them M. Auslander), they show that the multiplicative behavior remains true, except in few cases which are studied precisely. rational double point; McKay correspondence; Dynkin diagram; characteristic p Gonzalez-Sprinberg, G.; Verdier, J. -L.: Structure multiplicative de modules réflexifs sur LES points doubles rationnels. Travaux en cours 22, 79-100 (1987) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Structure multiplicative des modules réflexifs sur les points doubles rationnels. (Multiplicative structure of the reflexive modules on the rational double points) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0565.00006.]
For a finite subgroup G of SL(2,\({\mathbb{C}})\) the McKay correspondence is an isomorphism between its Dynkin diagram and its diagram of non trivial irreducible finite dimensional representations. The author gives here an approach to McKay's result via invariant theory by looking at the orbits of G in the projective space of the representation. In this way, he recovers by an explicit computation Gonzalez-Sprinberg's and Verdier's description of the McKay correspondence (obtained also by explicit computation), which assigns to each representation the first Chern class of the corresponding vectorbundle on the minimal desingularization of \({\mathbb{C}}^ 2/G\). Later on, several people gave more theoretical proofs of those facts, which partly apply in characteristic p\(>0\). rational double points; McKay correspondence; invariant theory Knörrer, H.: Group representations and the resolution of rational double points. Contemp. math. 45, 175-222 (1985) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Group actions on varieties or schemes (quotients) Group representations and the resolution of rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite subgroup G of SL(2,k) and char k\(=0\), the McKay correspondence is a bijection between the vertices of the Dynkin diagram and the non trivial representations of G. The multiplication table by the natural representation \(G\to SL(2,k)\) is ''given by'' the Dynkin diagram. All this may be seen geometrically. The first Chern class of a non trivial indecomposable reflexive module on \(S=k^ 2/G\) (k in any characteristic) is dual to some specific exceptional curve whereas the multiplication table by \(\Omega^ 1_ S=Ext^ 1_{{\mathcal O}_ S}({\mathfrak m},{\mathcal O}_ S)\) (\({\mathfrak m}\) is the maximal ideal) is ''given by'' the Dynkin diagram. In this announcement, the authors study in characteristic p\(>0\) the multiplication by \(Ext^ 1_{{\mathcal O}_ S}({\mathfrak m},{\mathcal O}_ S)\). The Dynkin diagram still describes the multiplication in good characteristic (i.e. rank of the module is prime to the p). In the remaining cases, the authors give explicitly the correction term. rational double points; McKay correspondence; Dynkin diagram; reflexive module; characteristic p Singularities in algebraic geometry, Finite ground fields in algebraic geometry Sur la règle de McKay en caractéristique positive. (On McKay's rule in positive characteristic) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Starting from McKays observation on the description of (an essential part of) the representation theory of binary polyhedral groups \(\Gamma\) in terms of extended Coxeter-Dynkin-Witt diagrams \(\widetilde {\underline\Delta} (\Gamma)\) and working in the differential geometric framework of hyper-Kähler-quotients \textit{P. B. Kronheimer} [C. R. Acad. Sci., Paris, Sér. I 303, 53-55 (1986; Zbl 0591.53057) and J. Differ. Geom. 29, No. 3, 665-683 (1989; Zbl 0671.53045)] was able to give a new construction of the semiuniversal deformations of the Kleinian singularities \(X=\mathbb{C}^2/ \Gamma\) as well as of their simultaneous resolutions. As far as the deformations were concerned, he already gave a purely algebraic geometric formulation of his results in terms of representations of certain quivers naturally attached to the diagrams \(\widetilde {\underline\Delta} (\Gamma)\). By making use of the invariant-theoretic notion of ``linear modification'' (cf. section 6) and applying it to Kronheimer's quiver construction we show here how to obtain a purely algebraic geometric simultaneous resolution as well (section 7). On the way, we remind the reader of various facts about Kleinian singularities (section 1), McKay's observation (section 2), symplectic geometry (section 3), Kronheimer's work (section 4), and quivers (section 5). -- This article covers the main results of the doctoral dissertation by \textit{H. Cassens} (``Lineare Modifikation algebraischer Quotienten, Darstellungen des McKay-Köchers und Kleinsche Singularitäten'', Hamburg 1995; Zbl 0842.14036). linear modification; Kleinian singularities; simultaneous resolutions; quivers Cassens, H.; Slodowy, P., On Kleinian singularities and quivers, (Singularities. Singularities, Oberwolfach, 1996. Singularities. Singularities, Oberwolfach, 1996, Progr. Math., vol. 162, (1998), Birkhäuser: Birkhäuser Basel), 263-288 Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Global theory and resolution of singularities (algebro-geometric aspects) On Kleinian singularities and 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 The McKay correspondence links the theory of singularities in algebraic geometry with noncommutative algebra and group theory. This article gives a generalization of McKays original correspondence based on the concept of canonical orders.
One of the classical versions of the McKay correspondence is the following: Consider a finite subgroup \(G<SL_2\). Then \(\text{Spec}\,k[[s,t]]^G\) is a Kleinian singularity and the McKay correspondence states that there is a bijection between the set of exceptional curves in a minimal resolution for \(\text{Spec}\,k[[s,t]]^G\) and the set of indecomposable reflexive \(k[[s,t]]^G\)-modules not isomorphic to \(k[[s,t]]^G\). This ``numerical equivalence'' is seen as a consequence of a derived equivalence: The \textit{order} \(A:=k[[s,t]]\ast G\) is reflexive Morita equivalent to \(k[[s,t]]^G\), and is thus interpreted as a noncommutative resolution of \(\text{Spec}\,k[[s,t]]^G\). \textit{M. Kapranov} and \textit{E. Vasserot} [Math. Ann. 316, No. 3, 565--576 (2000; Zbl 0997.14001)] showed that there is a derived equivalence between \(A\) and the minimal resolution \(\text{Spec} k[[s,t]]^G\).
There is a version of Mori's minimal model program for orders on surfaces. As in the original Mori program, there are noncommutative analogues for canonical surfaces, called dubbed canonical orders, that includes the Kleinian singularities as well as their associated skew group rings \(k[[s,t]]\ast G\). Unlike other noncommutative generalizations of Kleinian singularities, there is a notion of minimal resolution for a canonical order \(B\). This minimal resolution is a certain order on a resolution of the center of \(B\). The author proved that any canonical order is reflexive Morita equivalent to another canonical order of the form \(A:=\varepsilon k[[s,t]]\ast G^\prime\) where \(G^\prime\) is a finite group and \(\varepsilon\) is a central primitive idempotent of \(k[[s,t]]\ast G^\prime\). This order is interpreted as a noncommutative resolution of the original canonical order. It is natural to ask if \(A\) is derived equivalent to the minimal resolution of the original order.
This article proves the following theorem in detail: Given a canonical order \(B\) of any ramification type other than \(DL_n\), there is a reflexive Morita equivalent order of the form \(A=\varepsilon k[[s,t]]\ast G^\prime\) such that \(A\) is derived equivalent to the minimal resolution of \(B\).
To prove this theorem it is necessary to study the minimal resolution of a canonical order. Up to now, only ramification data are known about the resolution, but the author proves that minimal resolutions can always be constructed as noncommutative cyclic covers. In the process, it is seen that in most cases, minimal resolutions of canonical orders can also be realized as skew group rings. This allows one to use an equivariant version of the classical ``commutative'' McKay correspondence to prove the above result.
Another question to ask is whether or not one can extract a numerical form of the McKay correspondence for canonical orders using the above result. It is known that if \(B\) denotes a canonical order, the set of indecomposable reflexive \(B\)-modules can be partitioned into certain orbits containing one or two modules. Then the number of these orbits is one more than the number of exceptional curves in a minimal resolution of \(B\). This paper explains this numerical McKay correspondence for types other than \(DL_n\) by using the derived equivalence of the main theorem. The result it the following:
Theorem. Let \(E_1,\dots E_r\) be the exceptional curves of the minimal resolution of a canonical order \(B\). The number of indecomposable reflexive \(B\)-modules up to isomorphism is \(n_0+n_1+\cdots+n_r\), where
{\parindent=7mm
\begin{itemize}\item[(i)]\(n_0=2\) if \(B\) is of type \(A_{1,2,\zeta},BL_n\) or \(B_n\) and is \(1\) otherwise.
\item[(ii)]\(n_i=1\) if \(E_i\) is of type \(0,I_{1,e},C_2\) or \(X_2\).
\item[(iii)]\(n_i=2\) if \(E_i\) is of type \(I_{2,e}\).
\end{itemize}}
The author reminds the reader of the definition of terminal and canonical orders in terms of ramification data. He shows how to construct terminal and canonical orders as skew group rings in the complete local case, and defines minimal resolutions of canonical orders and determines when they can be constructed via skew group rings. Then he describes an equivariant version of the derived McKay correspondence and a procedure for extracting the numerical McKay correspondence in the cases of interest. Also, minimal resolutions can all be described, up to Morita equivalence, as noncommutative cyclic covers, and the author gives a case-by-case analysis depending on the ramification type of the canonical order by constructing for each type the minimal resolution as a noncommutative cyclic cover and, where possible, as a skew group ring. In the cases where there is a description via a skew group ring, he applies the results concerning the equivariant derived McKay correspondence to prove the main theorem and to elucidate the numerical McKay correspondence.
Terminal and canonical orders are certain orders on surfaces which arise naturally in the minimal model program for orders over surfaces. They are defined essentially in terms of geometric ramification data. Given terminal or canonical ramification data, the author shows how to construct a skew group ring with that ramification data in important cases.
The construction of orders via noncommutative cyclic covers is reviewed, and terminal orders in the complete local case are constructed. The author proves that minimal resolutions of canonical orders can always be constructed via noncommutative cyclic covers. This is uniform, unlike the case-by-case analysis required for the skew group ring construction.
The article demands the basics of canonical orders, but besides this fact, it is more or less self contained. The proofs are detailed and stringent, and the results give a good understanding of the concept of McKay correspondence. canonical order; McKay correspondence; minimal resolution; cyclic cover Chan, D., Mckay correspondence for canonical orders, Trans. Amer. Math. Soc., 362, 4, 1765-1795, (2010) Singularities in algebraic geometry, Representations of orders, lattices, algebras over commutative rings, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Twisted and skew group rings, crossed products, Noncommutative algebraic geometry, McKay correspondence McKay correspondence for canonical orders | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For \(G\subset \text{SL}(2,{\mathbb C})\) a finite group, the quotient variety \(X={\mathbb C}^2/G\) is called a Klein quotient singularity. The resolution of singularities \(Y\rightarrow X\) has exceptional locus consisting of \(-2\)-curves \(E_i\) (i.e. isomorphic to \({\mathbb P}_{{\mathbb C}}^1\), with self-intersection \(E_i^2=-2\)), and whose intersections \(E_iE_j\) are given by one of the Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\) or \(E_8\). The classical McKay correspondence begins in the late 1970s with the observation that the same graph arises in connection with the representation theory of \(G\), i.e. there is a one-to-one correspondence between the components of the exceptional locus of \(Y\rightarrow X\) and the nontrivial irreducible representations of \(G\subset \text{SL}(2,{\mathbb C})\). The paper explains this coincidence in several ways, and discusses higher dimensional generalizations. group action; \(K\)-theory; derived category; quotient variety; resolution of singularity; motivic integration; McKay correspondence; Hilbert schemes of \(G\)-orbits; crepant resolution; discrepancy divisor; Klein quotient singularity Reid, Miles, La correspondance de McKay, Astérisque, 276, 53-72, (2002) Global theory and resolution of singularities (algebro-geometric aspects), Linear algebraic groups over arbitrary fields, Homogeneous spaces and generalizations McKay's correspondence | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The classical McKay correspondence for finite subgroups \(G\) of \(\mathrm{SL}(2, \mathbb{C})\) gives a bijection between isomorphism classes of nontrivial irreducible representations of \(G\) and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity \(\mathbb{A}_{\mathbb{C}}^{2}/G\). Over non algebraically closed fields \(K\) there may exist representations irreducible over \(K\) which split over \(\overline{K}\). The same is true for irreducible components of the exceptional divisor. In this paper, we show that these two phenomena are related and that there is a bijection between nontrivial irreducible representations and irreducible components of the exceptional divisor over non algebraically closed fields \(K\) of characteristic 0 as well. Arcs and motivic integration, Arithmetic problems in algebraic geometry; Diophantine geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] McKay correspondence over non algebraically closed fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a finite dimensional complex vector space and \(G\subset \text{SL}(V)\) a finite subgroup. Let \(X:=V/G\). Then two natural questions arise: 1) when does \(X\) admit a crepant resolution of singularities \(f\colon Y\to X\), and 2) if such a resolution exists, what can be said about the homology \(H_*(Y,\mathbb Q)\)? In dimension \(2\) J. McKay proved that there always exists a crepant resolution \(f\colon Y\to X\) such that the fiber \(f^{-1}(0)\) over the singularity of \(X\) is a rational curve whose components are numbered by the conjugacy classes of \(G\); moreover, the homology classes of these components freely generate \(H_2(X,\mathbb Q)\) and \(H_i(X,\mathbb Q)=0\) for \(i>0\) and \(i\neq 2\). This is the so-called McKay correspondence [cf. \textit{J. McKay}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183--186 (1980; Zbl 0451.05026)]. If \(\text{dim}(V)=3\) the first question was solved affirmatively by several people independently, while the second question was solved by \textit{Y. Ito} and \textit{M. Reid} who proved that the same result holds true as in dimension \(2\) [in: Higher dimensional complex varieties. Proceedings of the International Conference, Trento, Italy, June 15--24, 1994, 221--240 (1996; Zbl 0894.14024)]. In the paper under review the author imposes an additional assumption of the pair \((V,G)\), namely he assumes that \(V\) has a nondegenerate symplectic form and the inclusions \(G\subset \text{Sp}(V)\subset \text{SL}(V)\) preserve not only the volume form in \(V\) but also the symplectic form. Under these hypotheses he proves a higher-dimensional analogue of the McKay correspondence. Crepant resolutions; finite subgroups of \(\text{SL}(V)\) Kaledin, D.: McKay correspondence for symplectic quotient singularities. Invent. Math. 148(1), 151--175 (2002) Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Singularities in algebraic geometry McKay correspondence for symplectic quotient singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The McKay correspondence [cf. \textit{J. McKay}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026)] is a bijection between the irreducible representations of a given finite subgroup G of SL(2,\({\mathbb{C}})\), the vertices of the extended diagram associated to the minimal desingularization \(f: \tilde X\to X\) of the rational double point \(X={\mathbb{C}}^ 2/G\), and the indecomposable reflexive modules on X. In order to understand geometrically this correspondence \textit{G. Gonzalez-Springberg} and \textit{J.-L. Verdier} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 409-449 (1983; Zbl 0538.14033)] and \textit{M. Artin} and \textit{J.-L. Verdier} [Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)] show that a reflexive module on X is determined by its first Chern class on \(\tilde X.\) - More generally, a surface singularity is a quotient if and only if it has finitely many indecomposable reflexive modules. Therefore it is a natural question (posed by H. Knörrer and J. L. Verdier) to ask whether those modules are determined by their first Chern class. In this note we give a negative answer. To this aim we give a numerical criterion in terms of vertical divisors of f for rank one reflexive modules and - along the same line - study properties of higher rank reflexive modules. McKay correspondence; minimal desingularization; rational double point; indecomposable reflexive modules; first Chern class; surface singularity; quotient singularities Hélène Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362 (1985), 63 -- 71. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Representation theory for linear algebraic groups, Linear algebraic groups over global fields and their integers Reflexive modules on quotient surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this article is to generalize a construction by \textit{H. Cassens} and \textit{P. Slodowy} [Prog. Math. 162, 263--288 (1998; Zbl 0957.14004)] of the semiuniversal deformations of the simple singularities of types \(A_r\), \(D_r\), \(E_6\), \(E_7\) and \(E_8\) to the singularities of inhomogeneous types \(B_r\), \(C_r\), \(F_4\) and \(G_2\) defined in [Math. Z. 158, 157--170 (1978; Zbl 0352.58009)] by \textit{P. Slodowy}. Let \(\Gamma\) be a finite subgroup of \(\mathrm{SU}_2\). Then \(\mathbb C^2/\Gamma\) is a simple singularity of type \(\Delta(\Gamma)\). By studying the representation space of a quiver defined from \(\Gamma\) via the McKay correspondence, and a well chosen finite subgroup \(\Gamma'\) of \(\mathrm{SU}_2\) containing \(\Gamma\) as normal subgroup, we will use the symmetry group \(\Omega=\Gamma'/\Gamma\) of the Dynkin diagram \(\Delta(\Gamma)\) and explicitly compute the semiuniversal deformation of the singularity \((\mathbb C^2/\Gamma,\Omega)\) of inhomogeneous type. The fibers of this deformation are all equipped with an induced \(\Omega\)-action. By quotienting we obtain a deformation of a singularity \(\mathbb C^2/\Gamma'\) with some unexpected fibers. Deformations of singularities, Representation theory for linear algebraic groups, Root systems Inhomogeneous Kleinian singularities and 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 Based in part on the authors' preface: The Kleinian singularities \({\mathbb C}^2/G\) associated to finite subgroups \(G \subset SL_2({\mathbb C})\) are important in algebraic geometry, singularity theory and other branches of mathematics. New remarkable properties continue to be discovered, one such being the McKay correspondence and its interpretation by Gonzalez-Springberg and Verdier in terms of the minimal resolution \({\mathbb C}^2//G\). Their results give identifications
\[
K_0({\mathbb C}^2//G) \simeq Rep(G) \simeq \widehat{\mathbf h}_Z
\]
where \(K_0\) is the Grothendieck group, Rep is the representation ring and \(\widehat{\mathbf h}_Z\) is the root lattice of the affine Lie algebra (of type A-D-E) associated to \(G\). The paper under review first extends these results by describing the derived category of coherent sheaves on \({\mathbb C}^2//G\), rather than just \(K_0\). The approach is a refinement of techniques by Kronheimer and Nakajima. Then the authors define an Euler-characteristic version \({\mathbb H}\) of the Hall algebra of the category of coherent sheaves on \({\mathbb C}^2//G\), and exhibit a subalgebra in \({\mathbb H}\) isomorphic to \(U(\text{g}_G^+)\), where \(\text{g}_G^+\) is the nilpotent part of the finite dimensional Lie-algabra (of type A-D-E) corresponding to G. As a consequence, taking the intersection graph of the \({\mathbb P}^1_i\) as a Dynkin graph, they get a possibly infinite-dimensional Kac-Moody Lie algebra, and prove that the positive part of this algebra acts in the space of functions on isomorphism classes of coherent sheaves in S. This partly extends results of Nakajima to a wider geometric context. Kleinian singularities; McKay correspondence; Lie algebras; derived category of coherent sheaves; Hall algebra; Kac-Moody Lie algebra M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and hall algebras, Math. Ann. 316 (2000), no. 3, 565-576. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities of surfaces or higher-dimensional varieties, Derived categories, triangulated categories, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Kleinian singularities, derived categories and Hall algebras | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If G is a small finite subgroup of \(SL_ 2({\mathbb{C}})\), McKay computed that the associated diagram to the table of multiplication of representations of G is the extended Dynkin diagram of G. This was explained geometrically later on by Gonzalez-Sprinberg, Verdier, Artin, Knörrer and myself: if (X,0) is the corresponding rational double point, \(\pi: \tilde X\to X\) the minimal desingularization, the first Chern class of \(\pi^*(M)/torsion\), where M is the reflexive hull of the flat module on X-0 associated to an irreducible representation of G, is dual to one and only one exceptional divisor of \(\pi\). The multiplication is computable with Chern classes.
If G now is a small finite subgroup of \(GL_ 2({\mathbb{C}})\), the McKay correspondence does not hold: to one possible Chern class there might correspond several reflexive modules. The author constructs for each exceptional divisor \(E_ i\) a module \(M_ i\) whose corresponding first Chern class is dual to \(E_ i\). He shows how to modify the multiplication. Shortly, if \(M=M_ i\), it behaves as in the rational double point case, if \(M\neq M_ i\), it does not. The formulae are given. As a corollary, the Chern character does not determine M. Further the author gives a complete description in the cyclic case (also through an approach based on invariant theory) and computes some examples of first Chern classes in the general case. two-dimensional quotient singularities; Dynkin diagram; McKay correspondence; reflexive modules; Chern class Wunram, J.: Reflexive Moduln auf zweidimensionalen Quotientensingularitäten. Dissertation Fachber. Math. Univ. Hamburg (1986) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Characteristic classes and numbers in differential topology Reflexive Moduln auf zweidimensionalen Quotientensingularitäten. (Reflexive modules on two dimensional 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 McKay correspondence gives a bijection between the conjugacy classes of finite subgroups of \(SU(2)\) and the simply laced complex simple Lie algebras which are classified by the Dynkin diagrams of ADE type.
The paper under review gives a uniform description of this correspondence and generalize it to the case of all simple Lie algebras. Moreover, it is shown that the generalized correspondence is related to the \textit{triality} of the quaternions \(\mathbb{H}\). The main result of the paper is to establish the following natural bijection:
\[
\bigg\{\text{equivalence classes of the pairs } (\widetilde{\Gamma},O_v) \bigg\}\leftrightarrow \bigg\{\text{the pairs } (\mathfrak{g},\tau) \bigg\}.
\]
Here \(\widetilde{\Gamma}\) is finite subgroup of \(\mathrm{SU}(2)\) and \(O_v\) is an outer automorphism of \(\widetilde{\Gamma}\) induced by \(Ad(v)\) for a \(v\in \mathrm{SU}(2)\). \(\mathfrak{g}\) is a complex simple Lie algebra and \(\tau\) is an outer automorphism of \(\mathfrak{g}\). Note that any non-simply laced simple Lie algebra can be obtained form a suitable pair \((\mathfrak{g},\tau)\) where \(\mathfrak{g}\) is simply laced simple Lie algebra by taking the fixed part of \(\mathfrak{g}\) under the action of \(\tau\). McKay correspondence; quaternions; triality McKay correspondence, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Non-simply laced McKay correspondence and triality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities McKay correspondence describes the classical geometry of the resolution of \(Y\to [\mathbb{C}^3/G]\) where \(G\) is a finite subgroup of \(\mathrm{SL}(3)\) in terms of the representation theory of \(G\). Here \(Y\) is the preferred resolution \(G\text{-Hilb}^{|G|}(\mathbb{C}^3)\). In the case that \(G \subset \mathrm{SU}(2)\) or \(G\subset \mathrm{SO}(3)\), \textit{J. Bryan} and \textit{A. Gholampour} in [Invent. Math. 178, No. 3, 655--681 (2009; Zbl 1180.14010)] gave a description of the quantum geometry of the preferred resolution \(Y\). In particular they found a closed formula for the Gromov-Witten invariants of \(Y\) in terms of the ADE root system attached to \(G\). By means of the Crepant Resolution Conjecture, Bryan and Gholampour formulated a conjectural closed formula for the genus 0 Gromov-Witten invariants of the orbifold \([\mathbb{C}^3/G]\) in terms of the ADE root system. This conjecture is proven for type A by Coates et al., and for \(G=\mathbb{Z}_2\times \mathbb{Z}_2 \text{ and } A_4 \subset \mathrm{SO}(3)\) by Bryan and Gholampour. Higher genus analogs of the conjecture for the type A are also proven by \textit{D. Maulik} [``Gromov-Witten theory of A-resolutions'', \url{arxiv:0802.2681}] and \textit{J. Zhou} [``Crepant resolution conjecture in all genera for type A singularities'', \url{arXiv:0811.2023}].
The paper under review proves the conjecture above for the type \(D\). The method of the proof is to, by using WDVV equations and reduction to normal subgroups, reduce the computation of the invariants for infinitely many type D subgroups to some known type A invariants. The conjecture for all the type D subgroups is then obtained by proving a polynomiality property for the remaining integrals and using the reduction result above. Gromov-Witten invariants; Hurwitz-Hodge integral; McKay correspondence; crepant resolution conjecture; orbifolds Ke, H.-Z., Zhou, J.: Quantum McKay correspondence for disc invariants of orbifold vertex. arXiv:1410.4374 Multiplier ideals, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The quantum McMay correspondence for singularities of type D | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 finite subgroup \(\mathrm{SL}_3(\mathbb{C})\), work of Bridgeland-King-Reid constructs an equivalence between the \(G\)-equivariant derived category of \(\mathbb{C}^3\) and the derived category of the crepant resolution \(Y=G\mathrm{-Hilb}\mathbb{C}^3\) of \(\mathbb{C}^3/G\). When \(G\) is abelian, we show that this equivalence gives a natural correspondence between irreducible representations of \(G\) and exceptional subvarieties of \(Y\), thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid's recipe and extends earlier work from [\textit{S. Cautis} and \textit{T. Logvinenko}, J. Reine Angew. Math. 636, 193--236 (2009; Zbl 1245.14016); erratum ibid. 689, 243--244 (2014)] and [\textit{T. Logvinenko}, J. Algebra 324, No. 8, 2064--2087 (2010; Zbl 1223.14018)] which dealt only with the case when \(\mathbb{C}^3/G\) has one isolated singularity. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], McKay correspondence, Derived categories, triangulated categories Derived Reid's recipe for abelian subgroups of \(\mathrm{SL}_3(\mathbb{C})\) | 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 Given a group \(G\) acting on a variety \(M\), the McKay Correspondence studies the relation between resolutions of singularities \(\phi:Y\to M/G\) and certain moduli spaces of \(G\)-equivariant objects in \(M\). When \(M=\mathbb{C}^3\) and the group \(G\) is a finite subgroup of \(\mathrm{SL}_3(\mathbb{C})\) the celebrated result of \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] asserts that \(G\)-Hilb\((\mathbb{C}^3)\), the fine moduli space of \textit{\(G\)-clusters} (or scheme-theoretic orbits), is actually a crepant resolution.
The paper under revision studies the case when \(G\subset \mathrm{GL}_3(\mathbb{C})\) where instead of looking for crepant resolutions, which no longer exist, one can ask about the ``closest resolution to being crepant''. This approach leads to the notion of \textit{economic resolutions} (also known as the Danilov resolution in the abelian case), which are obtained as a sequence of weighted blowups. In this case one has to consider the generalization of \(G\)-clusters know us \textit{\(G\)-constellations}, which are \(G\)-equivariant coherent sheaves \(\mathcal{F}\) on \(\mathbb{C}^n\) with \(H^0(\mathcal{F})\) isomorphic to the regular representation \(\mathbb{C}[G]\) as \(\mathbb{C}[G]\)-modules. For a generic stability parameter \(\theta\), it is know by \textit{A. Craw} et al. [Proc. Lond. Math. Soc. (3) 95, No. 1, 179--198 (2007; Zbl 1140.14046)] that the (possibly not normal) moduli space \(\mathcal{M}_\theta\) of \(\theta\)-stable \(G\)-constellations contains a unique irreducible component \(Y_\theta\) which is birational to \(\mathbb{C}^n/G\).
Following the spirit of \textit{I. Nakamura}'s \(G\)-graphs to construct \(G\)-Hilb\((\mathbb{C}^3)\) [J.\ Algebr.\ Geom.\ 10, No.\ 4, 757--779 (2001; Zbl 1104.14003)], the author introduces the notion of \textit{G-bricks} to show that the economic resolution of a 3-fold terminal quotient singularity \(\mathbb{C}^3/G\), that is \(G=\frac{1}{r}(1,a,r-a)\) with \(r\) coprime to \(a\), is isomorphic to the birational component \(Y_\theta\) of the moduli space \(\mathcal{M}_\theta\) for a suitable stability parameter \(\theta\). This result was conjectured by \textit{O. Kędzierski} [Tohoku Math. J. (2) 66, No. 3, 355--375 (2014; Zbl 1309.14010)]. In fact, \(G\)-bricks determines an affine open cover of \(Y_\theta\), which consists of a union of copies of \(\mathbb{C}^3\). The author finishes with the explicit example \(G=\frac{1}{12}(1,7,5)\), relating it with Kedzierski's calculations. McKay correspondence; terminal quotient singularities; economic resolutions Jung, S.-J., Terminal quotient singularities in dimension three via variation of GIT, J. algebra, 468, 354-394, (2016) McKay correspondence, Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory Terminal quotient singularities in dimension three via variation of GIT | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\text{SL}(n,\mathbb{C})\). The generalized McKay correspondence aims to relate the geometry of crepant (i.e. with trivial canonical divisor) resolutions of singularities of the quotient \(\mathbb{C}^n/G\) to the representations of the group \(G\). This paper deals with the natural candidate given by the Hilbert scheme of \(G\)-regular orbits introduced by \textit{I. Nakamura} [J. Algebr. Geom. 10, No.~4, 757--779 (2001; Zbl 1104.14003)], parametrizing generalized \(G\)-orbits on \(\mathbb{C}^n\), denoted by \(G\text{-Hilb}(\mathbb{C}^n)=:Y\). By a theorem of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], for \(n=3\) this provides the required resolution of singularities. The McKay correspondence is realized as follows: there exists a natural integral basis of the Grothendieck group \(K(Y)\) given by natural bundles \(\mathcal{R}_k\) indexed by the irreducible representations of \(G\). This provides, through Chern character, a rational basis of the cohomology \(H^*(Y,\mathbb{Q})\) in one-to-one correspondence with the irreducible representations of \(G\). It is still an open problem to get a similar correspondence for the integral cohomology \(H^*(Y,\mathbb{Z})\). Reid conjectured that some ``cookery'' with the Chern classes of these bundles \(\mathcal{R}_k\) should provide an integral basis. This paper establishes explicitly this integral McKay correspondence for all abelian subgroups \(A\) in \(\text{SL}(3,\mathbb{C})\) (Theorem 1.1). The method follows the recipe introduced by Reid, uses previous work of \textit{Y. Ito, H. Nakajima} [Topology 39, No.~6, 1155--1191 (2000; Zbl 0995.14001)] and an explicit algorithm of computation of \(A\text{-Hilb}(\mathbb{C}^3)\) already described by \textit{A. Craw} and \textit{M. Reid} [in: Geometry of toric varieties. Lect. summer school. Grenoble. 2000, Sémin. Congr. 6, 129--154 (2002; Zbl 1080.14502)] and extending the initial work of Nakamura [loc.cit.], based upon a decoration of the toric fan of \(A\text{-Hilb}(\mathbb{C}^3)\) with the characters of the group \(A\). The integral basis of \(H^2(Y,\mathbb{Z})\) is then given by the first Chern classes of some \(\mathcal{R}_k\)'s indexed by specific non-trivial characters (Proposition 7.1). In order to base \(H^4(Y,\mathbb{Z})\), the author computes all relations between the line bundles (since the group is abelian) \(\mathcal{R}_k\) in \(\text{Pic}(Y)\), and introduces a family of virtual bundles \(\mathcal{V}_m\) indexed by the remaining non-trivial irreducible representations, whose second Chern classes will give the expected integral basis (Proposition 7.3). Hilbert scheme of orbits; toric geometry Craw, A., An explicit construction of the McKay correspondence for \(A\)-Hilb \({\mathbb{C}^3}\), J. Algebra, 285, 682-705, (2005) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(3\)-folds, Toric varieties, Newton polyhedra, Okounkov bodies, Classical real and complex (co)homology in algebraic geometry, Ordinary representations and characters An explicit construction of the McKay correspondence for \(A\)-Hilb \(\mathbb C^3\) | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma\) be a finite subgroup of \(SL(2, {\mathbb C})\), and \(X\) be the corresponding Kleinian singularity. Gonzalez-Springberg and Verdier showed that there is natural bijection between the irreducible components of the exceptional fibre in the minimal resolution of \(X\), and the non-trivial irreducible representations of \(\Gamma\). \textit{Y. Ito} and \textit{I. Nakamura} [see e.g. Proc. Japan Acad., Ser. A 72, No. 7, 135-138 (1996; Zbl 0881.14002)] found a beautiful new interpretation of this bijection, by using an interpretation of the minimal resolution of \(X\) as a subset of the Hilbert scheme of codimension \(|\Gamma|\) ideals in \({\mathbb C}[x,y]\).
In this paper the author gives a new proof of the result of Ito and Nakamura. This proof avoids a case by case analysis, contrary to the proof of Ito and Nakamura. Kleinian singularity; McKay correspondence; minimal resolution; Hilbert scheme Dlab, V.: Representations of valued graphs. In: Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 73. Presses de l'Université de Montréal, Montreal, Que (1980) Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) On the exceptional fibres of Kleinian singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{SL}(2,{\mathbb C})\) be a finite subgroup and \(Y\) the Hilbert scheme of \(G\)-orbits (\(G\)-Hilb) introduced by Nakamura. \textit{Y. Ito} and \textit{I. Nakamura} [in: New trends in algebraic geometry. Proc. Warwick 1996, Lond. Math. Soc. Lect. Note Ser. 264, 151--233 (1999; Zbl 0954.14001)] studied \(I/mI\) as representations of \(G\) for ideals \(I\) in the exceptional locus of \(Y\). They proved that
1) \(Y\) is the minimal resolution of \({\mathbb C}^2/G\) and
2) there is a bijection of irreducible exceptional curves \(\{E_1,\ldots,E_n\}\) with non-trivial irreducible representations \(\{\rho_1,\ldots,\rho_n\}\) such that \(I_y/(mI_y+a)\) is isomorphic either with \(\rho_i\oplus\rho_j\) if \(y\in E_i\cap E_j\), or with \(\rho_i\) if \(y\in E_i\) and \(y\not\in E_j\) for \(j\not =i\), where \(a=(m^G){\mathcal O}_{{\mathbb C}^2}\) (this bijection coincides with the original McKay's correspondence).
The aim of this paper is to show that the above result holds if \(G\) is a finite subgroup of GL\((2,{\mathbb C})\) choosing some \textit{special} representations defined by Wunram. This was conjectured by \textit{O. Riemenschneider} [On the two-dimensional McKay correspondence, Hamb. Beitr. Math. 94 (2000), \texttt{http://www.math.uni-hamburg.de/meta/preprints/ms/ms2000094.html]}. Hilbert scheme of \(G\)-orbits; exceptional curves; irreducible representations Ishii, A., On the mckay correspondence for a finite small subgroup of \(\operatorname{GL}(2, \mathbb{C})\), J. Reine Angew. Math., 549, 221-233, (2002), MR 1916656 Group actions on varieties or schemes (quotients), Geometric invariant theory, Global theory and resolution of singularities (algebro-geometric aspects), Representation theory for linear algebraic groups, Complex surface and hypersurface singularities, Global theory of complex singularities; cohomological properties On the McKay correspondence for a finite small subgroup of \(\text{GL}(2,\mathbb{C})\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{SL}(n,\mathbb{C})\) be a finite group, without quasi-reflections, acting via its natural representation on \(\mathbb{C}^n,\) where \(n\in\{2,3\}.\) Let \(Y=\)\(G-\)\(\text{Hilb}(\mathbb{C}^n)\) be the \(G-\)Hilbert scheme, that is a moduli space of scheme-theoretical orbits of \(G.\) The McKay correspondence says that there is a close link between the \(G-\)equivariant geometry of \(\mathbb{C}^n,\) the geometry of \(Y\) and the representation theory of \(G.\) In particular, the natural map \(\pi:Y\longrightarrow \mathbb{C}^n/G\) is a resolution and the Fourier-Mukai transform \(\Phi: D(Y)\longrightarrow D^G(\mathbb{C}^n)\) given by the universal family on \(\mathbb{C}^n \times Y\) is an equivalence of categories (cf. \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14 , No. 3, 535--554 (2001; Zbl 0966.14028)]).
Moreover, \textit{G. Gonzalez-Sprinberg} and \textit{J.-L. Verdier} [Ann. Sci. Éc. Norm. Supér. (4) 16, 409--449 (1983; Zbl 0538.14033)] proved that for \(n=2\) the exceptional curves on \(Y\) are in one-to-one correspondence with the non-trivial representations of \(G.\) The paper under review generalizes this result for \(n=3\) and \(G\) a finite, abelian group. Let \(\Psi\) be the inverse of \(\Phi,\) given by the dual of the universal family. The main theorem states that the complex \(\Psi(\mathcal{O}_0\otimes \rho),\) where \(\mathcal{O}_0\) is the skyscraper sheaf supported at \(0\in\mathbb{C}^3\) and \(\rho\) is an irreducible representation of \(G\), is quasi-isomorphic to a shift of a coherent sheaf on \(Y.\) Moreover, this sheaf is expressed explicitly in terms of tautological bundles on \(Y\) and structure sheaves of toric exceptional divisors (and theirs intersections). This result gives a homological explanation for Reid's recipe, which assigns characters of \(G\) to toric curves and toric surfaces in the exceptional set of \(\pi,\) see \textit{A. Craw} [J. Algebra 285, No. 2, 682--705 (2005; Zbl 1073.14008)]. McKay correspondence; Reid's recipe; Fourier-Mukai transform; integral transform; quotient singularities; derived categories T. Logvinenko, Reid's recipe and derived categories, J. Algebra 324 (2010), no. 8, 2064-2087. McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients) Reid's recipe 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 \(G\) be a finite subgroup of \(\mathrm{SL}_d(\mathbb{C})\) acting naturally on \(\mathbb{C}^d\). The classical McKay correspondence relates the representation theory of \(G\) to the geometry of a crepant resolution of \(\mathbb{C}^d/G\).
The paper under review studies such a correspondence for a finite cyclic subgroup \(G\subset \mathrm{SL}_d(k)\) when the characteristic of the perfect field \(k\) is equal to \(|G|=p\). This is the special case of the so called wild McKay correspondence. For a finite dimensional \(G\)-representation \(V\), let \(V=\bigoplus_{\lambda=1}^l V_{d_\lambda}\) be the decomposition into indecomposable representations where the index \(d_\lambda\) denotes the dimension of \(V_{d_\lambda}\). Let \(D_V=\sum_{\lambda=1}^l d_\lambda(d_\lambda-1)/2\). For \(D_V\geq p\) the stringy motivic invariant of the quotient variety \(X=V/G\) denoted by \(M_{\mathrm{st}}(X)\) is defined by means of a motivic integration over the arc space of \(X\). The main result of the paper under review proves the following the formula
\[
M_{\mathrm{st}}(X)=\mathbb{L}^d+\frac{\mathbb{L}^{l-1}(\mathbb{L}-1)(\sum_{s=1}^{p-1}\mathbb{L}^{s-\sum_{\lambda=1}^l\sum_{i=1}^{d_\lambda-1}[ij/p]})}{1-\mathbb{L}^{p-1-D_V}}.
\]
If \(X\) admits a crepant resolution of singularities \(Y\) then \(D_V=p\) and this gives a simple formula for \([Y]=M_{\mathrm{st}}(X)\) by the virtue of a result of \textit{V. V. Batyrev} [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Singapore: World Scientific. 1--32 (1998; Zbl 0963.14015)]. In particular this shows that \(Y\) has topological Euler characteristic \(p\) as conjectured by Reid and proven by \textit{V. V. Batyrev} [J. Eur. Math. Soc. (JEMS) 1, No. 1, 5--33 (1999; Zbl 0943.14004)] in characteristic 0. The paper under review also proves a Poincaré duality for the stringy motivic invariant of the projectivization of \(X\). The proofs use the the motivic integration over the space of twisted arcs in quotient stacks. McKay correspondence; stringy motivic integration; positive characteristic Yasuda, Takehiko, The \(p\)-cyclic McKay correspondence via motivic integration, Compos. Math., 150, 7, 1125-1168, (2014) McKay correspondence The \(p\)-cyclic McKay correspondence via motivic integration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 note is an appendix to the article of the same title by \textit{M. Artin} and \textit{J.-L. Verdier} [Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)]. There is established geometrically a one-to-one correspondence between irreducible non trivial reflexive modules and exceptional curves for a rational double point in characteristic zero. This is one part of the so-called McKay correspondence, the other part being the multiplicative behavior of those modules. We prove here by geometrical methods the corresponding multiplication formula. quotient singularities; reflexive modules; rational double point; McKay correspondence; multiplication formula H. Esnault and H. Knörrer : Reflexive modules over rational double points , Math. Ann. 272 (1985) 545-548. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Reflexive modules over rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be any affine complete rational surface singularity, \(\hat{X}\) be the minimal resolution of \(\mathrm{Spec}\,R\), and \(\mathrm{End}_R(\bigoplus M)\) be the reconstruction algebra, where the sum is taken over all indecomposable special Cohen--Macaulay \(R\)-modules. In the paper under review the following results are obtained. The quiver of the reconstruction algebra can be computed combinatorially from the dual graph of \(\hat{X}\) labelled with self-intersection numbers. As a consequence it is shown that the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Applying the first result to quotients of \(\mathbb{C}^2\) by finite subgroups of \(\mathrm{GL}(2,\mathbb{C})\) the author of the paper obtains that for any finite subgroup \(G\) of \(\mathrm{GL}(2,\mathbb{C})\) the endomorphism ring of the special Cohen--Macaulay \(\mathbb{C}[[x,y]]^G\)-modules can be used to build the dual graph of the minimal resolution of \(\mathbb{C}^2/G\). Thus the result of \textit{J. McKay} [in: Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183--186 (1980; Zbl 0451.05026)] for finite subgroups of \(\mathrm{SL}(2,\mathbb{C})\) is extended to all finite subgroups of \(\mathrm{GL}(2,\mathbb{C})\). special Cohen-Macaulay module; reconstruction algebra; singularity Wemyss, M., The \(\operatorname{GL}(2, \mathbb{C})\) McKay correspondence, Math. Ann., 350, 3, 631-659, (2011) Arcs and motivic integration, Rings arising from noncommutative algebraic geometry, Cohen-Macaulay modules, Global theory and resolution of singularities (algebro-geometric aspects) The \(\mathrm{GL}(2,{\mathbb{C}}\)) 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 Let \(G\) be a nontrivial finite subgroup of \(SL_n (\mathbb{C})\). Suppose that the quotient singularity \(\mathbb{C}^n/G\) has a crepant resolution \(\pi:X\to \mathbb{C}^n/G\) (i.e. \(K_X= {\mathcal C}_X)\). There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of \(X\) and the representations (or conjugacy classes) of \(G\) with a ``certain compatibility'' between the intersection product and the tensor product. The purpose of this paper is to give more precise formulation of the conjecture when \(X\) can be given as a certain variety associated with the Hilbert scheme of points in \(\mathbb{C}^n\). We give the proof of this new conjecture for an abelian subgroup \(G\) of \(SL_3(\mathbb{C})\). group action; homology group; quotient singularity; crepant resolution; McKay correspondence; Grothendieck group; intersection product; Hilbert scheme of points Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology, 39, 1155--1191 (2000) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, \(K\)-theory of schemes McKay correspondence and Hilbert schemes in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite subgroup \(\Gamma\subset \mathrm{SL}(2,\mathbb{C})\) and \(n\geqslant 1\), we construct the (reduced scheme underlying the) Hilbert scheme of \(n\) points on the Kleinian singularity \(\mathbb{C}^2/\Gamma\) as a Nakajima quiver variety for the framed McKay quiver of \(\Gamma\), taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed
McKay quiver by removing an arrow and then `cornering', and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of the stability parameter. Hilbert scheme of points; quiver variety; Kleinian singularity; preprojective algebra; cornered algebra Representations of quivers and partially ordered sets, Parametrization (Chow and Hilbert schemes), McKay correspondence, Singularities in algebraic geometry Punctual Hilbert schemes for Kleinian singularities as 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 The paper under review is an equivariant extension of the functorial construction of the moduli space of coherent sheaves on a projective scheme from [\textit{L. Álvarez-Cónsul} and \textit{A. King}, Invent. Math. 168, No. 3, 613--666 (2007; Zbl 1137.14026)] (see also [\textit{L. Álvarez-Cónsul} and \textit{A. King}, Lond. Math. Soc. Lect. Note Ser. 359, 212--228 (2009; Zbl 1187.14016)] for an expository survey).
Recall that the (non-equivariant) construction consists of a linearisation from the geometric moduli problem to a moduli problem for so called Kronecker modules, which are modules for a certain explicit finite-dimensional algebra. In the case of a finite group action on a projective scheme, the main new ingredient is a further translation of the problem from equivariant Kronecker modules to so called Kronecker-McKay modules. These are modules for another explicit finite-dimensional algebra, whose construction is inspired by the McKay quiver for a finite group.
In section 2 the definitions from the Álvarez-Cónsul-King paper are generalised to the equivariant setting, and in section 3 it is shown that the notions of semistability in the linearisation process are preserved. Section 4 discusses the notion of Kronecker-McKay modules, for which one can use the explicit GIT constructions of moduli of representations for finite-dimensional algebras. This allows the authors to give a functorial construction of the moduli space of equivariant sheaves on a projective scheme in section 5, by exhibiting a closed embedding in the moduli space for the Kronecker-McKay modules. In the final section systems of coordinates on these moduli spaces are described, as done by Álvarez-Cónsul-King in the non-equivariant case. moduli of equivariant sheaves; moduli of representations S. Amrutiya and U. Dubey, Moduli of equivariant sheaves and Kronecker-McKay modules, Internat. J. Math. \textbf{26} (2015), 38. Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets Moduli of equivariant sheaves and Kronecker-McKay 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 explicit McKay correspondence, as formulated by \textit{G. Gonzalez-Sprinberg} and \textit{J. L. Verdier} [Ann. Sci. Éc. Norm. Supér. (4) 16, 409--449 (1983; Zbl 0538.14033)], associates to each exceptional divisor in the minimal resolution of a rational double point, a matrix factorization of the equation of the rational double point. We study deformations of these matrix factorizations, and show that they exist over an appropriate ``partially resolved'' deformation space for rational double points of types A and D. As a consequence, all simple flops of lengths 1 and 2 can be described in terms of blowups defined from matrix factorizations. We also formulate conjectures which would extend these results to rational double points of type E and simple flops of length greater than 2. Curto, C.; Morrison, D. R., Threefold flops via matrix factorization, J. Algebraic Geom., 22, 4, 599-627, (2013) Minimal model program (Mori theory, extremal rays), McKay correspondence Threefold flops via matrix factorization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 abelian subgroup of \(GL_n(K)\), where \(K\) is an algebraically closed field whose characteristic does not divide \(r=| G| \). Its McKay quiver is the directed graph whose vertices are the isomorphism classes of the irreducible representations of \(G\), with an arrow from \(\rho\rho_i\) to \(\rho\) for all irreducible constituents \(\rho_1,\ldots,\rho_n\) of the defininig representation of \(G<GL_n(K)\). The \(r\)-dimensional torus \(T\) acts by base change on the space of representations of this quiver with dimension vector \((1,\ldots,1)\).
The authors consider a \(T\)-stable closed subscheme \(Z\) of this \(rn\)-dimensional affine space defined by quadratic equations; the moduli spaces \(Z//_{\theta}T\) (where \(\theta\) belongs to the rational vector space of weights) play a crucial role in the McKay correspondence. The authors prove that there is a unique irreducible component \(V\) of \(Z\) that is not contained in any of the coordinate hyperplanes, and describe the coordinate ring of \(V\) as the semigroup algebra of the semigroup generated by the columns of an integer matrix encoding the McKay quiver (as a labeled directed graph). They show that the Geometric Invariant Theory quotient \(Y_{\theta}=V//_{\theta}T\) is a not-necessarily-normal toric variety that admits a projective birational morphism \(Y_{\theta}\to K^n/G\) (where \(K^n/G\) is the affine quotient of the affine space \(K^n\) by \(G\)) obtained by variation of GIT quotient. For generic parameter \(\theta\), the variety \(Y_{\theta}\) is the unique irreducible component of \(Z//_{\theta}T\) containing the \(T\)-orbit closures of the points of \(Z\cap(K^{\times})^{nr}\). This yields as a special case a new construction of Nakamura's \(G\)-Hilbert scheme.
The authors describe also the toric fan of \(Y_{\theta}\), and describe explicitly the set of \(\theta\)-semistable McKay quiver representations corresponding to the points of \(Y_{\theta}\). crepant resolution; McKay correspondence; moduli space; quiver representation; Hilbert scheme Craw A., Maclagan D., Thomas R.R.: Moduli of McKay quiver representations I: the coherent component. Proc. Lond. Math. Soc. 95(1), 179--198 (2007) arXiv:0505115 Toric varieties, Newton polyhedra, Okounkov bodies, Effectivity, complexity and computational aspects of algebraic geometry, Representations of quivers and partially ordered sets Moduli of McKay quiver representations. I: The coherent component | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{GL}(n,\mathbb C)\) be a finite group and consider the corresponding quotient singularity \(X=\mathbb A^n/G\). Inspired by the classical McKay correspondence, many interesting connections between resolutions \(Y\to X\) of the quotient singularity and the representation theory of \(G\) have been discovered. In particular, for \(G\subset \text{SL}(3,\mathbb C)\) abelian, every projective crepant resolution is given by a moduli space of \(G\)-constellations or, equivalently, a moduli space of representations of the McKay quiver of \(G\) of dimension vector \((1,\dots,1)\); see \textit{A. Craw} and \textit{A. Ishii} [Duke Math.\ J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)]. Given a resolution \(Y\to \mathbb A^3/G\) with \(G\not\subset \text{SL}(3,\mathbb C)\) one may ask whether \(Y\) can be identified with a moduli space of representations of the McKay quiver too.
The author studies this question for the Danilov resolution (also known as the economic resolution) of the terminal quotient singularity of type \(\frac 1r(1,a,r-a)\) for coprime numbers \(a\), \(r\). This means that \(G\subset \text{GL}(3,\mathbb C)\) is the cyclic group of order \(r\) acting diagonally with eigenvalues \(\zeta\), \(\zeta^a\), and \(\zeta^{r-a}\) where \(\zeta\) is a primitive \(r\)-th root of unity. The Danilov resolution \(Y\to X=\mathbb A^3/G\) is a toric resolution given by a series of weighted blow-ups.
For \(G\subset \text{GL}(n,\mathbb C)\), a \(G\)-constellation is defined as a \(G\)-equivariant sheaf \(F\) on \(\mathbb A^n\) whose global sections \(\Gamma(F)\) are given by the regular representation \(R\) of \(G\). A stability parameter for \(G\)-constellations is given by a \(\mathbb Z\)-linear map \(\theta: \text R(G)\to \mathbb Q\) from the representation ring to the rational numbers such that \(\theta(R)=0\). A \(G\)-constellation \(F\) is called \(\theta\)-stable if for every non-trivial \(G\)-subsheaf \(E\subset F\) we have \(\theta(\Gamma(F))>0\). Due to more general results of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)], there is a fine moduli space \(\mathcal M_\theta\) of \(\theta\)-stable \(G\)-constellations. As structure sheaves of free \(G\)-orbits do not have any non-trivial \(G\)-subsheaves, they are \(\theta\)-stable for every parameter \(\theta\). The irreducible component \(Y_\theta\) of \(\mathcal M_\theta\) containing the structure sheaves of free orbits is called the coherent component. Let \(f: Y\to X\) be a resolution of the singularities. A family \(\mathcal F\) of \(G\)-clusters over \(Y\) is called a gnat-family (short for \(G\)-natural) if \(\mathcal F_y\) is supported on the \(G\)-orbit \(f(y)\) for every \(y\in Y\). If \(\mathcal F\) is in addition \(\theta\)-stable, this means that the classifying morphism \(Y\to\mathcal M_\theta\) factorises over the coherent component and commutes with \(f\) and the canonical morphism \(Y_\theta\to X=\mathbb A^n/G\) given by sending a \(G\)-constellation to the \(G\)-orbit supporting it. For \(G\) abelian, \textit{T. Logvinenko} [Doc. Math., J. DMV 13, 803--823 (2008; Zbl 1160.14025)] gave a characterization of all gnat-families on \(Y\) in terms of \(G\)-Weyl divisors.
In the paper under review, toric divisors on the Danilov resolution \(Y\) are used in order to construct a gnat-family \(\mathcal F\). Then the cone of stability parameters \(\theta\) for which \(\mathcal F\) is \(\theta\)-stable is computed. Furthermore, it is shown that the fibres of \(\mathcal F\) are pairwise non-isomorphic. It follows that the classifying morphism \(Y\to Y_\theta\) is bijective and consequently that \(Y\) is the normalisation of \(Y_\theta\). It is conjectured that \(Y_\theta\) is normal so that \(Y\cong Y_\theta\). McKay correspondence; resolutions of terminal quotient singularities; Danilov resolution; moduli of quiver representations Kȩdzierski, O.: Danilov's resolution and representations of the mckay quiver. Tohoku math. J. (2) 66, No. 3, 355-375 (2014) McKay correspondence, Representations of quivers and partially ordered sets, Geometric invariant theory Danilov's resolution and representations of the McKay quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein \(3\)-fold quotient singularities \(\mathbb{A}^3/G\) with the representation theory of the group \(G\). The first crepant resolution studied in depth was the \(G\)-Hilbert scheme \(G\text{-Hilb}\,\mathbb{A}^3\), which is also a moduli space of \(\theta \)-stable representations of the McKay quiver associated to \(G\). As the stability parameter \(\theta\) varies, we obtain many other crepant resolutions. In this paper we focus on the case where \(G\) is abelian, and compute explicit inequalities for the chamber of the stability space defining \(G\text{-Hilb}\,\mathbb{A}^3\) in terms of a marking of exceptional subvarieties of \(G\text{-Hilb}\,\mathbb{A}^3\) called Reid's recipe. We further show which of these inequalities define walls. This procedure depends only on the combinatorics of the exceptional fibre and has applications to the birational geometry of other crepant resolutions. wall-crossing; McKay correspondence; Reid's recipe; quivers McKay correspondence, Toric varieties, Newton polyhedra, Okounkov bodies, Representations of quivers and partially ordered sets Walls for \(G\)-Hilb via Reid's recipe | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 given finite small binary dihedral group \(G\subset\mathrm{GL}(2,\mathbb{C})\) we provide an explicit description of the minimal resolution \(Y\) of the singularity \(\mathbb{C}^{2}/G\). The minimal resolution \(Y\) is known to be either the moduli space of \(G\)-clusters \(G\)-Hilb\((\mathbb{C}^{2})\), or the equivalent \(\mathcal{M}_{\theta}(Q,R)\), the moduli space of \(\theta\)-stable quiver representations of the McKay quiver. We use both moduli approaches to give an explicit open cover of \(Y\), by assigning to every distinguished \(G\)-graph \(\Gamma\) an open set \(U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)\), and calculating the explicit equation of \(U_{\Gamma}\) using the McKay quiver with relations \((Q,R)\). mckay correspondence; \(G\)-Hilbert scheme; quiver representations Á. Nolla de Celis, Dihedral \({G}\)-Hilb via representations of the McKay quiver , Proc. Japan Acad. Ser. A 88 (2012), 78-83. McKay correspondence, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Global theory and resolution of singularities (algebro-geometric aspects) Dihedral \(G\)-Hilb via representations of the McKay quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite subgroup \(\Gamma \subset \text{SL}(2, \mathbb{C})\) and for \(n \ge 1\), we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of \(n\) points on the minimal resolution \(S\) of the Kleinian singularity \(\mathbb{C}^2 / \Gamma\). It is well known that \(X := \text{Hilb}^{[n]} (S)\) is a projective, crepant resolution of the symplectic singularity \(\mathbb{C}^{2n} / \Gamma_n\), where \(\Gamma_n = \Gamma \wr \mathfrak{S}_n\) is the wreath product. We prove that every projective, crepant resolution of \(\mathbb{C}^{2n} / \Gamma_n\) can be realised as the fine moduli space of \(\theta\)-stable \(\Pi \)-modules for a fixed dimension vector, where \(\Pi\) is the framed preprojective algebra of \(\Gamma\) and \(\theta\) is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of \(\theta \)-stability conditions to birational transformations of \(X\) over \(\mathbb{C}^{2n} / \Gamma_n\). As a corollary, we describe completely the ample and movable cones of \(X\) over \(\mathbb{C}^{2n} / \Gamma_n\), and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to \(\Gamma\) by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of \textit{H. Nakajima} [Duke Math. J. 76, No. 2, 365--416 (1994; Zbl 0826.17026)] in the case where the quiver variety is smooth. Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Minimal model program (Mori theory, extremal rays), Singularities of surfaces or higher-dimensional varieties, Geometric invariant theory, Algebraic moduli problems, moduli of vector bundles Birational geometry of symplectic quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group which acts on a smooth complex variety \(X\) via holomorphic maps. Generally speaking, a McKay correspondence establishes an equivalence between invariants of the orbifold \((X,G)\) and invariants of crepant resolutions of the quotient \(X/G\).
In the present writing, the author establishes such an equivalence between equivariant elliptic genera for open varieties with torus actions, building on work of \textit{L. Borisov} and \textit{A. Libgober} [Ann. Math. (2) 161, No. 3, 1521--1569 (2005; Zbl 1153.58301)] in the non-equivariant closed case. The localization techniques of \textit{M. F. Atiyah} and \textit{R. Bott} [Topology 23, 1--28 (1984; Zbl 0521.58025)] are used to extend the earlier results to the new setting.
As corollaries, the author proves an equivariant analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula and reproves the equivariant elliptic genus analogue of the classical McKay correspondence from his earlier paper [Pac.~J. Math. 235, No. 2, 345--377 (2008; Zbl 1203.58007)]. elliptic genus; equivariant cohomology; McKay; localization; Hilbert scheme Waelder R 2008 Equivariant elliptic genera and local McKay correspondences \textit{Asian J. Math.} 12 251--84 Birational geometry, Equivariant homology and cohomology in algebraic topology Equivariant elliptic genera and local 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 The McKay correspondence describes connections among finite subgroups \(G\) of \(\mathrm{SL}_2(\mathbb{C})\), classical Lie groups of simply-laced type, and resolutions of singularities of \(\mathbb{C}^2/G\). There have been several attempts to generalize this correspondence to \(\mathrm{SL}_n(\mathbb{C})\). In the paper under review, the notions of the generalized McKay quiver and the generalized Cartan matrix are considered. Some properties of the generalized Cartan matrices are studied. The complete list of generalized McKay quivers for \(\mathrm{SL}_3(\mathbb{C})\) is given. McKay correspondence; McKay quivers; generalized Dynkin diagrams McKay correspondence, Other groups of matrices, Discrete subgroups of Lie groups Generalized McKay quivers of rank three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Translated from the author's introduction: Binary polyhedral groups, \(\Gamma \subset \text{SL} (2, \mathbb{C})\), are associated with Kleinian singularities, \(\mathbb{C}^2/ \Gamma\), and McKay quivers. In this work we describe an invariant-theoretic interpretation of a differential-geometric construction of P. B. Kronheimer that explains the deformation of Kleinian singularities and their simultaneous resolution by means of families of representations of McKay quivers. In the construction of the resolution a new invariant-theoretic method, the linear modification of affine algebraic quotients, is developed. We discuss the basics of this method, give some applications to the representation of oriented CDW-graphs and finally connect it to the invariant theory of McKay quivers. Connections with the theory of abstract root systems and simple Lie algebras of corresponding CDW-type \(\Delta (\Gamma)\) also turn up in the representation theory of oriented CDW-graphs. Kleinian singularities; representations of McKay quivers; linear modification; invariant theory Homogeneous spaces and generalizations, Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Linear modification of algebraic quotients, representations of the McKay quiver and 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 The paper deals with the following problem: Given a finitely generated associative algebra over an algebraically closed field, the algebraic variety of \(d\)-dimensional modules is endowed with a natural \(\text{Gl}_d\)-action. Various problems associated with this action are studied in the paper. Some strong results are obtained for algebras of finite representation type.
The author considers pointed varieties of the form \((\overline {O(m)},n)\) where \(\overline {O(m)}\) is the closure of the orbit of a module \(M\) and \(n\) is a minimal degeneration of \(M\) (see the paper for definitions). He calls the pointed varieties of the form \((\overline {O(m)}, n)\) minimal singularities.
The main theorem asserts that all minimal singularities occurring in representations of Dynkin quivers are very smoothly equivalent to \((D(p,q),0)\) where \(D(p,q)\) is the set of \(p \times q\) matrices with rank \(\leq 1\). The theorem at the end of section 1 relates degenerations of two distinct finite dimensional modules. It is a fundamental result used in the rest of the paper.
With this result the author derives the famous result of \textit{H. Kraft} and \textit{C. Procesi} on minimal singularities of conjugacy classes of nilpotent matrices [which appeared in Invent. Math. 53, 227-247 (1979; Zbl 0434.14026)] and states that in this setting any minimal singularity is equivalent to the subregular singularity inside the set of nilpotent matrices of some smaller size or to the singularity at 0 inside the set of all nilpotent matrices of rank at most one.
Section 4 deals with tilting modules, in particular corollary 1 shows a very close relation between the \(\text{Gl}_d\)-stable subsets of \(Y(\underline {d})\) and \(\text{Gl}_e\)-stable subsets of \(Y(\underline {e})\). Here a tilting module \(Y(\underline {d})\) consists of the category of torsion \(A\)-modules of vector dimension \(\underline {d}\). \(Y\) is the subcategory of \(B\)-mod corresponding to \(\tau\) (the torsion free part), and \(Y(\underline {e})\) is the full subcategory of \(Y\) whose objects have vector dimension \(\underline {e}\).
If \([M] = \underline {d}\) then \(\underline {e} = [\text{Hom} (T,M)] - [\text{Ext}^1 (T,M)]\). Theorem 3 and its corollary are very beautiful applications to tilting theory. Section 5 studies possible reduction of the underlying Gabriel quiver. Under reduction of the Gabriel quiver with some technical hypotheses the author gets associated pointed varieties which are very smoothly equivalent. Section 6 studies the minimal degeneration in the cases where the partial orders \(\leq\) and \(\leq_{\text{Ext}}\) are equivalent. This equivalence of the partial orders \(\leq_{\text{Ext}}\) and \(\leq\) is valid for preprojective modules; the author uses this fact and gets that any minimal degeneration of representations of a Dynkin quiver is of codimension one and then gets the main result which is Theorem 6. The proof of this result is very technical and complex.
The entire paper uses techniques of algebraic geometry applied to the representation theory of algebras. This paper is certainly a very nice one, although it requires from the reader a good knowledge of representation theory and algebraic geometry. finitely generated algebras; algebraic variety; \(\text{Gl}_ d\)-actions; algebras of finite representation type; pointed varieties; minimal degenerations; minimal singularities; representations of Dynkin quivers; nilpotent matrices; tilting modules; tilting theories; Gabriel quivers; preprojective modules Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575--611 (1994) Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry Minimal singularities for representations of Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G< \mathrm{GL}_2(\mathbb{C})\) be finite small subgroup. By a result of the first author [J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057)], the \(G\)-Hilbert scheme \(Y\) gives the minimal resolution of the quotient singularity \(\mathbb{A}^2/G\). If \(G<\mathrm{SL}_2(\mathbb{C})\) it is known the resolution is crepant and it induces the equivalence of derived categories (McKay Correspondence):
\[
\Phi:D^b(\text{coh } Y)\to D^b(\text{coh }[\mathbb{A}^2/G]).
\]
If \(G\not <\mathrm{SL}_2(\mathbb{C})\), then \(Y\) is no longer a crepant resolution and \(\Phi\) is no longer an equivalence but is a full and faithful embedding with admissible essential image, which is generated by \(\{\mathcal{O}_{\mathbb{A}^2}\otimes\rho\}\) where \(\rho\) runs over the special irreducible representations of \(G\).
The main result of the paper under review proves that for a suitable fully faithful functor
\[
\Phi':D^b(\text{coh } Y)\to D^b(\text{coh }[\mathbb{A}^2/G])
\]
there is an exceptional collection \(E_1,\dots, E_n \in D^b(\text{coh }[\mathbb{A}^2/G]) \) and a semi-orthogonal decomposition
\[
D^b(\text{coh }[\mathbb{A}^2/G])=\langle E_1,\dots,E_n, \Phi'(D^b(\text{coh } Y))\rangle,
\]
where \(n\) is the number of non-special irreducible representations of \(G\). If \(G\) is cyclic then one can take \(\Phi'=\Phi\). This result can be viewed as a continuation of the result of Craw-Wemyss that describes \(D^b(\text{coh } Y)\) as the derived category of modules over the path algebra of the special McKay quiver. From this, the paper under review deduces the global version in which \(\mathcal{X}\) is the canonical stack associated to a surface \(X\) with at worst quotient singularities. The proof of the main result consists of several steps. It is first proven for cyclic subgroups. Next, for \(G_0=G\cap \mathrm{SL}_2(\mathbb{C})\) which is a normal subgroup of \(G\), and \(A=G/G_0\) one obtains an equivalence
\[
\Phi_0:D^b(\text{coh } [Y_0/A])\to D^b(\text{coh }[\mathbb{A}^2/G])
\]
where \(Y_0\) is the \(G_0\)-Hilbert scheme. The stack \([Y_0/A]\) is obtained by the iterated root constructions from a canonical stack whose coarse moduli space has a minimal resolution isomorphic to a (non-minimal resolution) of \(\mathbb{A}^2/G\). The theorem is proven by finding the orthogonal decompositions the derived categories of these various intermediate stacks and resolutions. McKay correspondence; exceptional collections Ishii, A., Ueda, K.: The special McKay correspondence and exceptional collections. Tohoku Math. J. (2) \textbf{67}(4), 585-609 (2015) McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The special McKay correspondence and exceptional collections | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(0\) denote a complete local algebra of dimension 2 with maximal ideal \({\mathfrak m}\) over an algebraically closed field k, such that \(0/{\mathfrak m}\simeq k\). Assume that the spectrum of \(0\) is a rational double point. The object of this note is to describe the isomorphism classes of reflexive \(0\)-modules which are indecomposable. In characteristic zero, such a ring is the ring of invariants of a finite subgroup \(G\subset SL_ 2(k)\), operating linearly on a power series ring k[[x,y]]. In this case, the McKay correspondence [cf. \textit{J. McKay}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026) and \textit{G. Gonzalez-Sprinberg} and \textit{J.-L. Verdier}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 409-449 (1983; Zbl 0538.14033)] establishes bijective correspondences between the three sets: (i) isomorphism classes of indecomposable reflexive modules; (ii) vertices of the extended Dynkin diagram associated to \(0\) and to G, (iii) isomorphism classes of irreducible representations of G. - Since the group G is not always available in characteristic p, it seems of some interest to derive the correspondence (i)\(\leftrightarrow (ii)\) directly, without assumption on the residue characteristic. rational double point; McKay correspondence; indecomposable reflexive modules; characteristic p Artin M., Verdier J.-L.: Reflexive modules over rational double points. Math. Ann. 270, 79--82 (1985) Singularities in algebraic geometry Reflexive modules over rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{V. Ginzburg} and \textit{D. Kaledin} [Adv. Math. 186, No. 1, 1--57 (2004; Zbl 1062.53074)] posed the problem of comparing the McKay correspondence, the dual McKay correspondence and the multiplicative McKay correspondence for a finite dimensional \(\mathbb C\)-vector space \(V\), with an action of a finite subgroup \(G\) of SL\((V)\). The vector space \(V\) is assumed to be equipped with a symplectic form, which is preserved by \(G\). Moreover a crepant resolution of singularities \(Y\to V/G\) is fixed. Ginzburg and Kaledin proposed to compute explicitly the Poincaré isomorphism and the Chern character isomorphism (see Problems 1.4 and 1.5 of the above quoted article).
In the paper under review, the author solves the problem in the special case when \(V=\mathbb C^n\otimes \mathbb C^2\), with the action by permutations of the symmetric group \(S_n\) and the canonical symplectic form. In this situation \(Y\) is the Hilbert scheme Hilb\(^n(\mathbb C^2)\). The author gives explicit formulae for the Poincaré and the Chern character isomorphisms and uses them to prove the main theorem of the paper (Theorem 1.2). It says that the McKay correspondence is compatible with the topological filtration of the Grothendieck group \(K(\text{Hilb}^n(\mathbb C^2))\) and with the decreasing filtration of the space of symmetric functions \(\Lambda^n\). It is then proved that the graded McKay correspondence so obtained coincides with both the multiplicative and the dual McKay correspondence. symmetric functions; equivariant cohomology; Macdonald polynomials S. Boissière, On the McKay correspondences for the Hilbert scheme of points on the affine plane, arXiv:math.AG/0410281. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Symmetric groups, Equivariant homology and cohomology in algebraic topology, Global theory and resolution of singularities (algebro-geometric aspects) On the McKay correspondences for the Hilbert scheme of points on the affine plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For any finite subgroup \(\Pi\subset\text{SU}(2,\mathbb{C}),\) the classical McKay correspondence links the \(\Pi-\)equivariant geometry of \(\mathbb{C}^2\) (that is the representation theory of \(\Pi\)) with the geometry of the minimal resolution of the singularity \(\mathbb{C}^2 / \Pi.\) The paper under review generalizes the classical set up to the case of any cocompact discrete subgroup \(\Gamma\subset\text{PSU}(1,1)\) of signature \((0;e_1,\ldots,e_n).\) The analog of minimal resolution turns out to be a smooth surface \(\widetilde{V}_\Gamma\) with trivial canonical class, defined as follows. Let \(V_\Gamma\) be a normal surface, equal to the affine spectrum of the graded algebra of automorphic forms with respect to \(\Gamma\)
\[
A(\Gamma)=\bigoplus_{k=0}^{\infty}H^0(\mathbb{D},T_{{\scriptstyle{ \mathbb{D}}}}^{*\otimes k})^\Gamma,
\]
where \(\mathbb{D}=\{z\in\mathbb{C}^2\; :\;|z|<1\}\) and \(T^{*}_{\scriptstyle{\mathbb{D}}}\) is the cotangent bundle. There exists a unique minimal smooth normal crossing compactification \(\overline{V}_\Gamma\) of \(V_\Gamma,\) preserving the \(\mathbb{C}^*-\)action defined by the grading of \(A(\Gamma).\) The surface \(\overline{V}_\Gamma\) is equal to the disjoint sum \((V_\Gamma-\{0\}) \cup \text{\textbf{E}}_0 \cup \text{\textbf{E}}_\infty,\) where \(0\) is the unique closed \(\mathbb{C}^*-\)orbit in \(V_\Gamma.\) The component \(\text{\textbf{E}}_0\) is an exceptional curve of the minimal resolution of the singular point \(0\) in \(V_\Gamma.\) The component \(\text{\textbf{E}}_\infty\) is is equal to the union of smooth rational curves with self--intersection \(-2,\) which dual graph is a star--shaped tree with \(n\) arms of length \(e_1,\ldots,e_n\). The surface \(\widetilde{V}_\Gamma\) is equal to \((V_\Gamma-\{0\})\cup \text{\textbf{E}}_\infty.\)
On the other side of the correspondence, the author considers representations of the group \(\Pi,\) which is defined as a preimage of \(\Gamma\) in the universal cover of \(\text{SU}(1,1).\) A unitary representation of \(\Pi\) is called admissible if its irreducible summands remain irreducible after restriction to any subgroup of finite index. Let \(\Gamma'\) be a surface subgroup of \(\Gamma\) of genus \(g\) with finite quotient \(G=\Gamma/\Gamma'\) and let \(C=\mathbb{D}/\Gamma'.\) The author proves that there exists a natural bijective correspondence between admissible \(r-\)dimensional representations of \(\Pi\) of level \(m\) (i.e. with the order of the image of the center of \(\Pi\) equal to \(m\)) such that \(m\) divides \(2g-2\) and \(m.G-\)linearized stable rank \(r\) vector bundles on \(C,\) where \(m.G\) is some central extension of \(\Pi.\) Moreover, every such vector bundle defines a vector bundle on the surface \(\widetilde{V}_\Gamma.\)
The author describes explicitly the Grothendieck group \(K_0(\widetilde{V}_\Gamma)\) and conjectures that it is isomorphic to the Grothendieck group based on the set of admissible representations of \(\Pi.\) McKay correspondence; Fuchsian groups I. Dolgachev, ''McKay's correspondence for cocompact discrete subgroups of SU(1,1),'' in: Groups and Symmetries (CRM Proc. Lect. Notes, Vol. 47, J. Harnard and P. Winternitz, eds.), Amer. Math. Soc., Providence, R. I. (2009), pp. 111--133. McKay correspondence, Vector bundles on curves and their moduli, Fuchsian groups and their generalizations (group-theoretic aspects), Representation theory of groups McKay's correspondence for cocompact discrete subgroups of SU\((1,1)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities There exists an extensive literature on the deformation theory of Kleinian singularities, i.e. quotient surface singularities of embedding dimension 3 alias rational double points. E.g. \textit{P. B. Kronheimer} [J. Differ. Geom. 29, No. 3, 665-683 (1989; Zbl 0671.53045)] gave an invariant-theoretic construction of the (monodromy trivializing finite covering of the) versal deformation and the simultaneous resolution of this family. A quiver-theoretic approach has been proposed by him and was carried out by others [see \textit{H. Cassens}, ``Lineare Modifikationen algebraischer Quotienten, Darstellungen des McKay-Köchers und Kleinsche Singularitäten'' (Thesis 1995; Zbl 0842.14036)].
As one can easily check in simple examples this construction has no straigthforward generalization to quotient singularities \(X=\mathbb{C}^2/\Gamma\), \(\Gamma\) a (small) finite subgroup of \(\text{GL}(2,\mathbb{C})\) (not in \(\text{SL}(2,\mathbb{C})\), i.e. the case of Kleinian singularities) and its associated McKay- or Auslander-Reiten-quiver. The purpose of the present note is to restate the quiver construction for the \(A_n\)-singularities in such a manner that it can be generalized to yield (up to a smooth factor) the monodromy covering of the Artin-component for all cyclic quotients. Kleinian singularities; quotient surface singularities; rational double points Riemenschneider, O., Cyclic quotient surface singularities: constructing the Artin component via the mckay-quiver, Surikaisekikenkyusho Kokyuroku, 1033, 163-171, (1998), Singularities and complex analytic geometry (Japanese) (Kyoto, 1997) Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients) Cyclic quotient surface singularities: constructing the Artin component via the McKay-quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 give an explicit geometric description to some of H. Nakajima's quiver varieties. More precisely, let \(X = {\mathbb{C}}^2\), \(\Gamma \subset \text{SL}({\mathbb{C}}^2)\) be a finite subgroup, and \(X_\Gamma\) be a minimal resolution of \(X/\Gamma\). The main result states that \(X^{\Gamma [n]}\) (the \(\Gamma\)-equivariant Hilbert scheme of \(X\)) and \(X_\Gamma^{[n]}\) (the Hilbert scheme of \(X_\Gamma\)) are quiver varieties for the affine Dynkin graph corresponding to \(\Gamma\) via the McKay correspondence with the same dimension vectors but different parameters.
In section two, basic concepts such as the definition of quivers, quiver varieties, representation of quivers and the construction of Crawley-Boevey were reviewed. In section three, the author reproduced in a short form a geometric version of the McKay correspondence based on investigation of \(X_\Gamma\), and proved a generalization of certain result of \textit{M. Kapranov} and \textit{E. Vasserot} [Math. Ann. 316, No. 3, 565--576 (2000; Zbl 0997.14001)]. The main result mentioned above was verified in section four. In particular, it follows that the varieties \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) are diffeomorphic. In section five, \(({\mathbb{C}}^* \times {\mathbb{C}}^*)\)-actions on \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) for cyclic \(\Gamma \cong {\mathbb{Z}}/d {\mathbb{Z}}\) were considered. The author proved the combinatorial identity \(UCY(n, d) = CY(n, d)\) where \(UCY\) and \(CY\) denote the number of uniformly colored diagrams and the number of collections of diagrams respectively. quiver varieties; Hilbert schemes; McKay correspondence; moduli space Kuznetsov, A.: Quiver varieties and Hilbert schemes. Moscow Math. J. \textbf{7}, 673-697 (2007). arXiv:math.AG/0111092 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Quiver varieties and Hilbert schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The principal aim of this article is to introduce the reader to the wild McKay
correspondence, that is, the McKay correspondence in arbitrary characteristics.
What is interesting about this subject is that it forms a new bridge between the
singularity theory and the number theory. The ``McKay correspondence'' is now
used as a generic term for a series of phenomena that an algebraic invariant of a
linear representation of a finite group coincides with a geometric invariant of the
corresponding quotient variety. Such coincidences give rise to interaction between
geometry and algebra, and one may expect that a difficult problem of one side
would be translated to an easy one on the other side or that we would get a
new insight into the relevant algebraic or geometric object. In characteristic zero,
there have been a lot of studies on the McKay correspondence about various kinds
of invariants. In this article, we discuss the McKay correspondence in terms of
motivic integration and stringy invariants, following \textit{V. V. Batyrev} [J. Eur. Math. Soc. (JEMS) 1, No. 1, 5--33 (1999; Zbl 0943.14004)] and \textit{J. Denef} and \textit{F. Loeser} [Compos. Math. 131, No. 3, 267--290 (2002; Zbl 1080.14001)]. Thus, to explain stringy invariants and motivic integration is our secondary
aim. Arcs and motivic integration, Stacks and moduli problems, McKay correspondence, Singularities in algebraic geometry, Varieties over finite and local fields, Ramification and extension theory The wild McKay correspondence via motivic integration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a version of three talks given by the author at the workshop ``Schémas de Hilbert, algèbre noncommutative et correspondence de McKay'' held at CIRM in Luminy (France), October 27--31, 2003. The author is thus given the opportunity to explain the subject in a more complete way than what is usual, and the result is a really nice treatment of the subject.
Non-commutative algebras have been used to associate to it canonical (partial) resolutions \(Y\twoheadrightarrow \mathbb{C}^d/G\) of quotient singularities (\(G\) a finite group) by using the skew group algebra \(\mathbb{C}[x_1,\dots,x_d]\#G\) which is an order with center \(\mathbb{C}[x_1,\dots,x_d]=\mathbb{C}[x_1,\dots,x_d]^G\) or deformations of it. In dimension \(d=2\) (the case of Kleinian singularities) this gives minimal resolutions, for \(d=3\) and \(G\) abelian, this gives crepant resolutions, but for general \(G\) one obtains at best partial resolutions with conifold singularities remaining. The author treats the general case for \(d>3\), ending up in a conjectured list of nice singularities for \(d>3\). Assuming that any quotient singularity \(X=\mathbb{C}^d/G\) has associated to it a nice order \(A\) with center \(R=\mathbb{C}[X]\), the coordinate ring of \(X\), such that there is a stability structure \(\theta\) with the scheme of all \(\theta\)-semistable representations of \(A\) being a smooth variety, then the associated moduli space will be a partial resolution
\[
\text{moduli}^\theta_{\alpha}A\twoheadrightarrow X=\mathbb{C}^d/G,
\]
having a sheaf of smooth orders \(A\) over it, allowing the singularities to be controlled. If \(A\) is a smooth order over \(R=\mathbb{C}[X]\) then its non-commutative variety \(\text{max} A\) of maximal twosided ideals is birational to \(X\) away from the ramification locus. If \(P\) is a point of the ramification locus \(\text{ram} A\) then there is a finite cluster of infinitesimal close non-commutative points lying over it. The local structure of the noncommutative variety \(\text{max} A\) near this cluster can be summarized by a (marked) quiver setting \((Q,\alpha)\) which in turn allows us to compute the étale local structure of \(A\) and \(R\) in \(P\). The central singularities which appear in this way have been classified up to smooth equivalence, giving explicit lists of singularities.
The author explains the terms given in the introduction through the rest of the text. Starting of with the necessary theory of noncommutative algebra. One argument for using this theory is the McKay correspondence: \(G\)-equivariant geometry of \(\mathbb{C}^d\) already knows about the crepant resolution \(Y\twoheadrightarrow\mathbb{C}^d/G\) for a reductive group \(G\). This principle may then be generalized.
The non-commutative candidates for the resolution are \(R\)-orders \(A\) in a central simple \(K\)-algebra \(\Sigma\), where \(K\) is the function field of \(X\). These terms are defined in an explicit way and easy examples are given. Even the skew group ring is defined. The properties smooth and nonsingular are studied for \(R\)-orders, and these concepts turns out to be non-equivalent. Orders are constructed by descent: Both (the noncommutative) Zariski and étale toplogy is defined resulting in the Zariski and étale twisted forms. Then Azumaya algebras are ready to be introduced as particular \(R\)-orders. The reflexive Azumaya algebras are the most suitable orders for this project, in particular to characterize the singularities of \(X\). Cayley-Hamilton algebras are noncommutative algebras which are the level \(n\) generalization of the category of commutative algebras, and which contain all \(R\)-orders. With is lined out, the author defines non-commutative geometry. To any Cayley Hamilton algebra \(A\) of order \(n\) the variety \(\text{max} A\) is a noncommutative manifold if \(A\) is a smooth order. The author explains very well why a noncommutative geometry is needed. He gives several nice examples, and the marked quivers and Morita settings makes classification of singularities (étale local types) possible. Finally, the last section is named ``non-commutative desingularizations''. Maybe this is the correct title of the article? Anyway, not all quotient singularities have smooth orders over their coordinate ring, and the stability structure is introduced to repair this, that is, the moduli of stable and semi stable representations have the ``usual'' properties and give at least partial resolutions. A lot more could be said about this article. It is easy to read, and one understands how to move from non-commutative algebra to non-commutative geometry and vice versa. The commutative results given by the noncommutative are of course of utmost importance. One of the most entertaining articles on the market. noncommutative geometry; noncommutative \(k\)-algebras; partial desingularizations Le Bruyn, L.: Non-commutative algebraic geometry and commutative desingularizations. In: Noncommutative algebra and geometry, Volume 243 of Lect. Notes Pure Appl. Math., Boca Raton, FL: Chapman \& Hall/CRC, 2006, pp. 203--252 Noncommutative algebraic geometry, Relevant commutative algebra, Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc. Non-commutative algebraic geometry and commutative desingularizations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(G\le \operatorname{SL}(n, \mathbb C)\) be a finite group. The McKay correspondence describes the phenomenon that many properties of crepant resolutions of the quotient variety \(\mathbb C^n/G\) are determined by the representation theory of \(G\). One explanation for this is that crepant resolutions are often moduli spaces of \(G\)-equivariant sheaves on \(\mathbb C^n\). Indeed, for \(n=2\), the crepant resolution is unique (it is the minimal resolution), and it agrees with the \(G\)-Hilbert scheme; see [\textit{Y. Ito} and \textit{I. Nakamura}, Lond. Math. Soc. Lect. Note Ser. 264, 151--233 (1999, Zbl 0954.14001)]. The \(G\)-Hilbert scheme is the moduli space of \(G\)-clusters, that is \(G\)-invariant subschemes \(Z\subset \mathbb C^n\) with \(H^0(\mathcal O_Z)\cong \mathbb C\langle G\rangle\), where \(\mathbb C\langle G\rangle\) denotes the regular representation of \(G\). In dimension 3, [\textit{A. Craw} and \textit{A. Ishii}, Duke Math. J. 124, No. 2, 259--307 (2004, Zbl 1082.14009)] proved that every crepant resolution of \(\mathbb C^3/G\), where \(G\le \operatorname{SL}(3, \mathbb C)\) is a finite abelian subgroup, is a moduli space of \(\theta\)-stable \(G\)-constellations for some stability parameter \(\theta\in \operatorname{Hom}_{\mathbb Z}(R(G), \mathbb Q)\) with \(\theta(\mathbb C\langle G\rangle)=0\). Here, a \(G\)-constellation is a \(G\)-equivariant coherent sheaf \(F\) on \(\mathbb C^3\) with \(H^0(F)\cong \mathbb C\langle G\rangle\), and it is called \(\theta\)-stable, if for every \(G\)-equivariant subsheaf \(E\subset F\), we have \(\theta(E) >0\).
For a finite subgroup \(G\le \operatorname{GL}(3,\mathbb C)\) which is not contained in \(\operatorname{SL}(3,\mathbb C)\), the appropriate replacement for a crepant resolution of the quotient \(\mathbb C^3/G\) is that of a relative minimal model. The paper under review makes the following conjecture: For every finite subgroup \(G\le \operatorname{GL}(3,\mathbb C)\) and every relative minimal model \(Y\to \mathbb C^3/G\), there exists a stability parameter \(\theta \in \operatorname{Hom}_{\mathbb Z}(R(G), \mathbb Q)\) such that \(Y\) is isomorphic to an irreducible component of the moduli space of \(\theta\)-stable \(G\)-constellations. This conjecture is proven for a quite large class of cyclic subgroups, not contained in \(\operatorname{SL}(3,\mathbb C)\).
The proofs heavily use toric geometry. An important ingredient is the notion of \(G\)-bricks. These are certain finite sets of Laurent monomials defining \(\mathbb C\)-bases of torus invariant \(G\)-constellations. McKay correspondence; G-constellations; minimal models McKay correspondence, Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory On the Craw-Ishii conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Soit G un sous-groupe fini de SL(2,\({\mathbb{C}})\), \(G\neq \{1\}\). Il agit naturellement sur \({\mathbb{C}}^ 2\). Le quotient \({\mathbb{C}}^ 2/G=S\) est une surface avec un point singulier isolé, qui est un point double rationnel. Le graphe dual associé au diviseur exceptionnel D de la désingularisation minimale \(\tilde S\) de S est un diagramme de Dynkin de type \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\) suivant le groupe G. Par ailleurs, en considérant seulement le groupe G. \textit{J. McKay} a associé un diagramme à l'ensemble des classes de représentations irréductibles de G qui se trouve être le diagramme de Dynkin (complété) obtenu avec le graphe dual précédent. Ceci permet de définir de façon purement combinatoire une bijection entre les représentations irréductibles de G (non-triviales) et les composantes irréductibles du diviseur exceptionnel D.
On présente ici une description géométrique de cette correspondance de McKay. Plus précisément, à toute représentation irréductible non-triviale \(\rho\) de G, on associe un fibré vectoriel \({\mathcal F}_{\rho}\) sur \(\tilde S\) dont la classe de Chern \(c_ 1({\mathcal F}_{\rho})\) est un élément de la base canonique de Pic(\~S) en bijection avec les composantes irréductibles de D. quotient of group action; Dynkin diagram of desingularization; representation of linear group; exceptional divisor Gérard Gonzalez-Sprinberg & Jean-Louis Verdier, ``Construction géométrique de la correspondance de McKay'', Ann. Sci. Éc. Norm. Supér.16 (1983), p. 409-449 McKay correspondence, Group actions on varieties or schemes (quotients), Singularities of surfaces or higher-dimensional varieties, Representation theory for linear algebraic groups, Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry, Representations of finite symmetric groups Geometrical construction of the McKay correspondence | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The representation moduli of the McKay quiver, or the modulei of \(G\)-constellations, is known to be a crepant resolution of the quotient singularity \(\mathbb{C}^3/G\) for a finite subgroup \(G\) of \(SL(3,\mathbb C)\) The proof of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] uses derived category technique. In this note, we give local (toric) coordinates of the moduli when \(G\) is abelian, generalising \textit{I. Nakamura}'s [J. Algebr. Geom. 10, No. 4, 757--779 (2001; Zbl 1104.14003)] coordinates on \(G\)-Hilb . We also resume the chamber structure for stability from [\textit{A. Craw, A. Ishii}, Duke Math. J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)] briefly. A. Ishii and K. Ueda, The special McKay correspondence and exceptional collection , preprint, [math.AG] 1104.2381v1 Global theory and resolution of singularities (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles Representation moduli of the McKay quiver for finite abelian subgroups of \(SL(3,\mathbb C)\) | 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 This paper deals with derived McKay correspondence. Let \(G\) be a finite subgroup of \(SL_n({\mathbb C})\) and \(Y\) a crepant resolution of \({\mathbb C}^n / G\), if one exists. Then it is conjectured that there exists an equivalence \(D(Y) \to D^G({\mathbb C}^n)\) between the bounded derived category of coherent sheaves on \(Y\) and the bounded derived category of \(G\)-equivariant coherent sheaves on \({\mathbb C}^n\). In most cases where it is known such a correspondence to hold, the equivalence can be explicitly written as a Fourier-Mukai transform which sends point sheaves of \(Y\) to pure sheaves in \(D^G({\mathbb C}^n)\). The author gives then a sufficient condition for an object \(E\) in \(D^G (Y \times {\mathbb C}^n)\) to define such a transform.
Let \(Y\) be any irreducible separated scheme of finite type over \({\mathbb C}\) and \(G\) a finite subgroup of \(SL_n ({\mathbb C})\). Given a \textit{gnat}-family \(F\) on \(Y\), we can define a Hilbert-Chow morphism \(\pi_F: Y \to {\mathbb C}^n/G\). Consider \(Y\) and \(F\) such that \(\pi_F\) is birational and proper. Then the Fourier-Mukai functor \(\Phi_F\) with kernel \(F\) is an equivalence if the non-orthogonality locus of \(F\) is of high enough codimension. Using this criterium, the author gives the first known example of derived McKay correspondence for a non-projective crepant resolution of \({\mathbb C}^3/G\). In proving such results, the author shows how to compute \(\theta\)-stable families of \(G\)-constellations and their direct transforms. Fourier-Mukai transforms; gnat-families; non projective crepant resolution T. Logvinenko, Derived McKay correspondence via pure-sheaf transforms, Math. Ann. 341 (2008), no. 1, 137-167. Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli, classification: algebraic theory, Derived categories, triangulated categories Derived McKay correspondence via pure-sheaf transforms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 presents (in expository form) results and recent questions on the relation between simple singularities and the corresponding finite groups. The questions treated here arose from conjectures of Grothendieck, solved by Brieskorn and published first time with complete proofs and additional contributions by the author in his book ``Simple singularities and simple algebraic groups'', Lect. Notes Math. 815 (1980; Zbl 0441.14002). Let \(X=\mathbb{C}^ 2/F\), \(F\subset SU(2,\mathbb{C})\) a finite subgroup, then \(X\) is a surface with an isolated singularity, and the dual graph of its minimal resolution is a Dynkin diagram of type \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\) respectively, corresponding to the cyclic group of order \(n\), the binary dihedral group of order \(4n\), the binary tetrahedral, octahedral, icosahedral group, respectively. The 2-dimensional homology of the minimal resolution can be obtained from the corresponding root system \(A_ n\), \(D_ n\), \(E_ n\), its intersection form is given by the Cartan matrix \(C\). Conversely, if \(C\) is given corresponding to some Dynkin diagram, the group \(F\) can be expressed in terms of generators and relations by the elements of \(C\).
On the other hand, let \(G\) be a simply connected simple Lie group of type \(A_ n\), \(D_ n\), \(E_ n\), \(T\subset G\) a maximal torus, \(W=N_ G(T)/T\) the corresponding Weyl group, \({\mathcal X}:G\to T/W\) the quotient by the adjoint action of \(G\) on itself. Then the theorem of Brieskorn asserts: If \(S\) is a section transversal to the subregular unipotent orbit, then \(\mathcal X\) restricted to \(S\) is a versal deformation of the singularity of the same type. [Note that a different construction was recently found by \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002); it gives some surprising deformations of the singularities in characteristic 2 and 3.]
There arises the question of a more direct connection between the Lie group we started with and the finite group \(F\). A remark of \textit{J. McKay} [cf. Proc. Am. Math. Soc. 81, 153-154 (1981; Zbl 0477.20006)] relates \(F\) with the Dynkin diagram by means of representation theory: If \(N\) is the 2-dimensional representation \(F\subset SU(2,\mathbb{C})\), \(R_ 0,\ldots,R_ r\) (representants of) the irreducible representations of \(F\) and \(N\otimes R_ i=\oplus_ ja_{ij}R_ j\), then \(2E_{r+1}- (a_{ij})\) is the Cartan matrix of the extended Dynkin diagram associated to the group \(F\) [later efforts to understand McKay's correspondence are related with the socalled Auslander-Reiten theory, cf. \textit{M. Auslander} and \textit{I. Reiten}, Trans. Am. Math. Soc. 302, 87-97 (1987; Zbl 0617.13018); results of Artin, Verdier, Esnault, Knörrer, Buchweitz, Greuel, Schreyer and others concern the maximal Cohen-Macaulay modules over simple singularities, cf. e.g. \textit{H. Knörrer} in Representations of algebras, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 116, 147-164 (1986; Zbl 0613.14004)]. Let \({\mathcal X}_ i\) denote the character of the representation \(R_ i\), \(d_ i={\mathcal X}_ i(1)\), then \(d=(d_ 0,\ldots,d_ r)\) generates the kernel of \(2E_{r+1}-(a_{ij})\), \(d_ i\) are the coefficients of the maximal root in the corresponding root system and give the fundamental cycle of the minimal resolution of \(\mathbb{C}^ 2/F\). Further, \(\sum d_ i\) is the Coxeter number of the root system. Until now, the Dynkin diagrams of type \(B\), \(C\), \(F\) and \(G\) are missing; this is explained in the following way: They can be obtained by factorizing some of the homogeneous diagrams by a group \(\Gamma\) of symmetries. A simple singularity of type \(B_ n\), \(C_ n\), \(F_ 4\), \(G_ 2\), respectively is defined to be a couple \((X,\Gamma)\), where \(X\) is one of \(A_{2n-1}\), \(D_{n+1}\), \(E_ 6\), \(D_ 4\), respectively, such that \(\Gamma\) acts as a group of automorphisms. The author's results generalize the Brieskorn theorem to singularities of type \(B\), \(C\), \(F\), \(G\): If \(G\) is a simple Lie group of one of the above types, \(T\subset G\) a maximal torus, \(W\) the corresponding Weyl group, \({\mathcal X}:G\to T/W\) the adjoint quotient, a versal deformation of the singularity of the same type is obtained as follows: Take a transversal section \(S\) to the orbit of a unipotent subregular element \(x\in S\) such that \(S\) is stabilized by a reductive subset of \(Z_ G(x)\). Then \(\mathcal X\) restricted to \(S\) induces an equivariant versal deformation. \(S\cap\text{Uni}(G)\) is a singularity of type \(A_{2n-1}\), \(D_{n+1}\), \(E_ 6\), \(D_ 4\), respectively, and the action of \(\Gamma\) is given by a subgroup of \(Z_ G(x)\). A refined result is obtained considering the mentioned groups together with their automorphisms. -- Again, the preceding diagrams \(B\), \(C\), \(F\), \(G\) are considered from the viewpoint of representation theory of finite groups, this time in the relative case.
The author formulates as a dominating question the explanation of the relation between the finite group \(F\) and the corresponding Lie group, to show why the simple singularity \(\mathbb{C}^ 2/F\) appears on the unipotent variety of \(G\). finite subgroup of \(Sl(2,\mathbb{C})\); simple singularities; Dynkin diagram; homology of minimal resolution; root system; Cartan matrix; simple Lie group; McKay correspondence; Coxeter number Singularities in algebraic geometry, Representations of finite symmetric groups, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, General properties and structure of complex Lie groups On finite groups associated to 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 authors associate to a finite subgroup \(\Gamma\) of \(SL(2,\mathbb C)\) and an element \(\tau\) in the center of the group algebra \(\mathbb C[\Gamma]\) a category, which is interpreted as a noncommutative analog of the complex projective plane. This category arises as a quotient of the category of graded modules over a graded version of the deformed preprojective algebra introduced by \textit{W. Crawley-Boevey} and \textit{M. P. Holland} [Duke Math. J. 92, No. 3, 605--635 (1998; Zbl 0974.16007)].
On the other hand, the authors associate to \(\Gamma\), \(\tau\) and each pair of finite dimensional \(\Gamma\)-modules a quiver variety, defined as a geometric invariant theory quotient of a variety of certain triples of homomorphisms. This quiver variety is isomorphic to a Nakajima quiver variety via the McKay correspondence.
The first main result is that the quiver variety is naturally isomorphic to a space of isomorphism classes of certain torsion free objects. This is then used to prove a generalized version of a conjecture by Crawley-Boevey and Holland. noncommutative algebraic geometry; deformed preprojective algebra; McKay correspondence; Nakajima quiver variety Baranovsky, V.; Ginzburg, V.; Kuznetsov, A., Quiver varieties and a noncommutative \(\mathbb{P}^2\), Compos. Math., 134, 3, 283-318, (2002) Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry Quiver varieties and a noncommutative \(\mathbb P_2\) | 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 authors aim at two main purposes with the article. Firstly, they want to explain how to generalize the classical theorem of Grothendieck, Brieskorn and Slodowy on slices to the subregular nilpotent orbit in a simple Lie algebra to the case of arbitrary orbits. The idea is to put the problem in the framework of Poisson deformations. Secondly, new examples of singular symplectic surfaces, seen as higher dimensional analogous of the Kleiman or DuVal ADE-singularities, arise as slices to very special nilpotent orbits.
Throughout, \(\mathfrak g\) denotes a simple complex Lie algebra, \(G\) its adjoint group, and \(\phi:\mathfrak g\rightarrow\mathfrak g//G\) the quotient map. The nullfibre \(=\phi^{-1}(0)\) of the nilpotent elements in \(\mathfrak g\) is called the nilpotent cone and is an irreducible variety which decomposes into finitely many orbits. The dense orbit is called the regular orbit, denoted by \(\mathcal O_{\text{reg}}\), its complement \(N\setminus\mathcal O_{\text{reg}}\) is again irreducible, with dense orbit the subregular orbit \(\mathcal O_{\text{sub}}\).
Let \(x\in N\) be a nilpotent element. By the Jacobson-Morozov theorem, there exists elements \(h,y\in\mathfrak g\) such that \(x,h,y\) form an \(\mathfrak{sl}_2\)-triplet, i.e., \([h,x]=2x\), \([h,y]=-2y\), and \([x,y]=2\). Then the affine space \(S=x+\text{Ker}(\text{ad}(y))\) is a special transversal slice to the orbit through \(x\), a Slodowy slice.
The authors consider the following theorem, conjectured by Grothendieck, proved by Brieskorn and formulated for Lie algebras by Slodowy:
Theorem. Let \(\mathfrak g\) be a simple Lie algebra of type ADE. Let \(N\) be its nipotent cone, and let \(S\) be a slice to the orbit of a subregular nilpotent element in \(\mathfrak g\). Then the germ \((S\cap N,x)\) is a Kleinian surface singularity with the same Coxeter-Dynkin diagram as \(\mathfrak g\), and the restriction \(\phi|_S:(S,x)\rightarrow (\mathfrak g//G,0)\) of the characteristic map \(\phi\) is isomorphic to the semiuniversal deformation of the surface singularity \((S\cap N, x)\).
Deeper down in the orbit stratification where \(S_0=S\cap N\) is no longer an isolated singularity and there is no versal deformation theory, it is observed that \(S_0\) carries a natural Poisson structure. This means that \(\phi_S:=\phi|_S:S\rightarrow\mathfrak g//G\) can be considered as a deformation of Poisson varieties, and that the space of infinitesimal Poisson deformations is again finite dimensional. \(\phi_S\) carries a natural relative Poisson structure for the Slodowy slice to any nilpotent element. The structure is essentially induced by the Lie bracket on \(\mathfrak g\). it provides Poisson structures on each fibre of \(\phi_S\), and \(\phi_S\) is considered a Poisson deformation of \(S_0\) over the base \(\mathfrak g//G\).
Let \(\pi:\tilde N\rightarrow N\) denote the Springer resolution of the nilpotent cone, its fibre \(F_x=\pi^{-1}(x)\), the Springer fibre, is the variety of all Borel subalgebras \(\mathfrak b\subset\mathfrak g\) such that \(x\in\mathfrak b\). The main results of the article are the following:
Theorem. Let \(x\) be a non-regular nilpotent element. Then \(\phi_S:S\rightarrow\mathfrak g//G\) is the formally universal Poisson deformation of \(S_0\) if and only if the restriction map \(\rho_x:H^2(\tilde N,\mathbb Q)\rightarrow H^2(F_x,\mathbb Q)\) is an isomorphism. \vskip0,1cm
Theorem. Let \(x\) be a non-regular nilpotent element. Then the restriction map \(\rho_x\) is an isomorphism except in the following cases:
(\(B_n\)) the subregular orbit;
(\(C_n\)) orbits of Jordan types \([n,n]\) and \([2n-2i,2i],\) for \(1\leq i\leq n/2\);
(\(G_2\)) the orbits of dimension eight and ten;
(\(F_4\)) the subregular orbit.
\noindent In particular, \(\rho_x\) is an isomorphism for all non-regular nilpotent elements in a simply laced Lie algebra.
If \(x\) is a subregular nilpotent element in a Lie algebra of type \(B_n\), \(C_n\), \(F_4\), and \(G_2\), then \((S_0,x)\) is a surface singularity of type \(A_{2n-1}\), \(D_{n+1}\), \(E_6\) and \(D_4\), respectively. One can therefore construct its universal Poisson deformation as a Slodowy slice in the corresponding simply laced Lie algebra.
Many examples of singular symplectic varieties indicate that symplectic singularities tend to require large embedding codimensions. In particular, singular symplectic hypersurfaces should be rare phenomena. Previously known were only the Klein-DuVal surface singularities in \(\mathbb C^3\). In this article, the authors found the following new examples: Firstly, a series of four-dimensional symplectic hypersurfaces that are intersections \(S_0=N\cap S\) of the nilpotent cone \(N\) with Slodowy slices \(S\) to certain nilpotent orbits in \(\mathfrak{sp}_{2n}\):
\[
f=a^2x+2aby+b^2z+(xz-y^2)^n\in\mathbb C[ab,x,y,z].
\]
Secondly, a six-dimensional symplectic hypersurface presented in the same way.
Natural maps between deformation functors are made for resolutions of singularities, and the main theorems are explicitly proved using Poisson deformations. The examples are constructed by related methods. An interesting article with important and useful results. nilpotent orbits; symplectic singularities; symplectic hypersurfaces; Poisson deformations; regular orbit; subregular orbit; slodowy slices; Jacobson-Morozov theorem; semiuniversal deformation Lehn, M.; Namikawa, Y.; Sorger, Ch., \textit{slodowy slices and universal Poisson deformations}, Compositio Math., 148, 121-144, (2012) Deformations of singularities, Lie algebras of linear algebraic groups, Poisson algebras Slodowy slices and universal Poisson deformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The McKay correspondence and Schur-Weyl duality have inspired a vast amount of research in mathematics and physics. The McKay correspondence establishes a bijection between the finite subgroups of the special unitary 2-by-2 matrices and the simply laced affine Dynkin diagrams from Lie theory. It has led to the discovery of many other remarkable A-D-E phenomena. Schur-Weyl duality reveals hidden connections between the representation theories of two algebras that centralize one another in their actions on the same space. We merge these two notions and explain how this gives new insights and results. Our approach uses the combinatorics of walks on graphs, the Jones basic construction, and partition algebras. McKay correspondence; Schur-Weyl duality Benkart, G., Connecting the mckay correspondence and Schur-Weyl duality, (Proceedings of International Congress of Mathematicians Seoul 2014, vol. 1, (2014)), 633-656 McKay correspondence, Combinatorial aspects of representation theory, Group rings of finite groups and their modules (group-theoretic aspects) Connecting the McKay correspondence and Schur-Weyl duality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0619.00007.]
For a cyclic two dimensional quotient singularity (X,0) the author computes the Chern-classes of the vector bundles on the resolution \(\pi: X\to X\) that are of the form \(\pi\) *(M)/torsion, where M is an indecomposable reflexive module over \({\mathcal O}_{X,0}\). In this way [and in his paper in Math. Ann. 279, No.4, 583-598 (1988; Zbl 0616.14001)] he generalizes a result of \textit{G. Gonzales-Sprinberg} and \textit{J. L. Verdier} [Ann. Sci. Éc. Norm. Supér., II. Sér. 16, 409-449 (1983; Zbl 0538.14033)] who computed these Chern-classes for rational double points and in this way gave a geometric interpretation of the correspondance of \textit{J. McKay} [Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026)]. McKay correspondance; cyclic two dimensional quotient singularity; Chern- classes of the vector bundles; resolution; reflexive module J. Wunram, Reflexive modules on cyclic quotient surface singularities, in \(Singularities, Representation of Algebras, and Vector Bundles\) (Springer, Berlin, 1987), pp. 221-231 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Other special types of modules and ideals in commutative rings, Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients) Reflexive modules on cyclic quotient surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite subgroup \(\mathsf{G}\) of the special unitary group \(\mathsf{SU}_2\), we study the centralizer algebra \(\mathsf{Z}_k(\mathsf{G}) = \mathsf{End}_{\mathsf{G}}(\mathsf{V}^{\otimes k})\) of \(\mathsf{G}\) acting on the \(k\)-fold tensor product of its defining representation \(\mathsf{V} = \mathbb {C}^2\). These subgroups are in bijection with the simply laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of \(\mathsf{Z}_k(\mathsf{G})\) as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram. Benkart, G., Halverson, T.: Exceptional McKay centralizer algebras, in preparation Combinatorial aspects of representation theory, McKay correspondence, Ordinary representations and characters McKay centralizer algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subseteq SL(2,{\mathbb{C}})\) be a finite group and let V be a two- dimensional representation of G. Then V gives an action of G on \(S={\mathbb{C}}[[ X,Y]]\) as a group of \({\mathbb{C}}\)- algebra automorphisms. It is proved that if V is faithful having no pseudoreflections then the category \(R_ S\) of reflexive modules over the invariant \({\mathbb{C}}\)-algebra \(S^ G\) has almost split sequences and they can be derived from a fundamental exact sequence. Moreover, the Auslander-Reiten quiver of \(R_ S\) is the McKay graph of (G,V) and it is isomorphic to the desingularization graph of the associated singularity. Some of the results above remain valid in more general situations. action; group of \({\mathbb{C}}\)-algebra automorphisms; reflexive modules; almost split sequences; Auslander-Reiten quiver; McKay graph; desingularization graph; singularity Auslander M.: Rational singularities and almost split sequences. Trans. Am. Math. Soc. 293(2), 511--531 (1986) Representation theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Singularities of surfaces or higher-dimensional varieties, Regular local rings, Representation theory of associative rings and algebras, Vector and tensor algebra, theory of invariants Rational singularities and almost split 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 In this note we attempt to generalize a few statements drawn from the 3-dimensional McKay correspondence to the case of a cyclic group not in \(SL(3,\mathbb{C})\). We construct a smooth, discrepant resolution of the cyclic, terminal quotient singularity of type \({1\over r}(1,1,r-1)\), which turns out to be isomorphic to Nakamura's \(G\)-Hilbert scheme. Moreover we explicitly describe the tautological bundles and use them to construct a dual basis to the integral cohomology of the resolution. McKay correspondence; terminal singularities Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) Cohomology of the \(G\)-Hilbert scheme for \(\frac 1r(1,1,r-1)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide a group-theoretic realization of two-parameter quantum toroidal algebras using finite subgroups of \(\text{SL}_2(\mathbb{C})\) via McKay correspondence. In particular our construction contains the vertex representation of the two-parameter quantum affine algebras of ADE types as special subalgebras. two-parameter quantum affine algebra; finite groups; wreath products; McKay correspondence DOI: 10.1090/S0002-9947-2011-05284-0 Quantum groups (quantized enveloping algebras) and related deformations, McKay correspondence, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Two-parameter quantum vertex representations via finite groups 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 ''With any simple curve singularity (plane, complex, affine-algebraic) of Dynkin-type \(\Delta\) we associate the category of all finitely generated torsionfree modules over its complete local ring. For each of these module categories we calculate the Auslander-Reiten quiver. We suggest the construction of the ''twisted quiver'' of a quiver with involution and valuation of arrows which gives rise to a (purely combinatorial) one-to-one correspondence between the Auslander-Reiten quiver and the Dynkin diagram \(\Delta\).''
Let us briefly indicate the idea of the proof: It is shown in {\S}2 that for each Dynkin-diagram \(\Delta\), \(\Lambda_{\Delta}=K[[X,Y]]/(f_{\Delta}(X,Y))\), \(f_{\Delta}\) the polynomial of the simple plane curve singularity of type \(\Delta\), is a Gorenstein-order and the category of finitely generated torsion free \(\Lambda_{\Delta}\)-modules coincides with the category of \(\Lambda\)- lattices. - Therefore for each \(\Delta\) it is possible to construct the Auslander-Reiten quiver by using Auslander-Reiten sequences ({\S}3). Auslander-Reiten quiver; twisted quiver; Dynkin diagram; simple plane curve singularity Dieterich E., Wiedemann A.: The Auslander Reiten quiver of a simple curve singularity. Trans. Am. Math. Soc. 294, 455--475 (1986) Singularities of curves, local rings, Representation theory of associative rings and algebras, Singularities in algebraic geometry The Auslander-Reiten quiver of a simple curve 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 Let G be a finite group, k an algebraically closed field in which \(| G|\) is nonzero, and \(V_ 1,...,V_ n\) the irreducible kG-modules. For a finite k-dimensional kG-module V, there is the McKay quiver, which has \(V_ 1,...,V_ n\) as vertices, and \(t_{ij}\) arrows from \(V_ j\) to \(V_ i\), where \(t_{ij}\) is the multiplicity of \(V_ j\) in \(V\otimes_ kV_ i.\)
The main result is that if \([V:k]=2\), then the underlying graph of the separated version of the McKay quiver is a disjoint union of extended Dynkin diagrams, but it is never of this form for \([V:k]>2\). The first part generalizes an observation of \textit{J. McKay} [Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026)]. This result is connected with work of \textit{D. Happel}, \textit{U. Preiser} and \textit{C. M. Ringel} [Manuscr. Math. 31, 317-329 (1980; Zbl 0436.20005)] and with some work of \textit{M. Auslander} on skew group rings and reflexive modules over rings of group- invariant power series [Trans. Am. Math. Soc. 293, 511-532 (1986)]. Several interesting examples of McKay quivers are given. finite group; irreducible kG-modules; McKay quiver; disjoint union of extended Dynkin diagrams Auslander, M., Reiten, I.: McKay quiver and extended Dynkin diagrams. Trans. Amer. Math. Soc., 293, 193--301 (1986) Group rings of finite groups and their modules (group-theoretic aspects), Representation theory of associative rings and algebras, Singularities in algebraic geometry, Ordinary representations and characters McKay quivers and extended Dynkin diagrams | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities There are many generalizations of the McKay correspondence for higher dimensional Gorenstein quotient singularities and there are some applications to compute the topological invariants today. But some of the invariants are completely different from the classical invariants, in particular for non-Gorenstein cases. In this paper, the author discusses the McKay correspondence for 2-dimensional quotient singularities via ``special'' representations which gives the classical topological invariants, and gives a new characterization of the special representations for cyclic quotient singularities in terms of combinatorics. \(G\)-clusters; \(G\)-Hilbert schemes; quotient singularities; special representations Ito, Yukari, Special {M}c{K}ay correspondence, Geometry of Toric Varieties, Sémin. Congr., 6, 213-225, (2002), Soc. Math. France, Paris Classical groups (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Special 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 Let \(G \subset \text{SL}_2(\mathbb C)\) be a finite subgroup, and \(V\) be its natural representation. Denote by \(Y\) the minimal resolution of \(X:=V/G\) and by \(\mathcal X:=[V/G]\) the corresponding orbifold. The classical McKay correspondence gives a bijection between two bases and hence an isomorphism of the vector spaces \(H^*(Y)\cong H^*(|I\mathcal X|)\), where \(|I\mathcal X|\) is the coarse moduli space of the inertia stack \(I\mathcal X\).
Motivated by the crepant resolution conjecture the paper under review generalizes the above correspondence in two directions by taking multiplicative structures into account and also by considering Chow motives instead of cohomology. The main result of this paper is as follows:
Let \(\mathcal X\) be a smooth proper two-dimensional Deligne-Mumford stack with isolated stacky points and projective coarse moduli space \(X\) having Gorenstein singularities. Let \(Y \to X\) be the minimal resolution. Then there is an isomorphism of algebra objects in the category of Chow motives with complex coefficients between the motive of \(Y\) and the orbifold motive of \(\mathcal X\): \[ \eta(Y )_{\mathbb C}\cong \eta(\mathcal X )_{\mathbb C}. \] In particular, one obtains isomorphisms of \(\mathbb C\)-algebras between the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological \(K\)-theory) of \(Y\) and the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of \(\mathcal X\). multiplicative McKay correspondence; Chow motives McKay correspondence, (Equivariant) Chow groups and rings; motives, Singularities of surfaces or higher-dimensional varieties, Orbifold cohomology Motivic multiplicative McKay correspondence for surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group of automorphisms of a non-singular three-dimensional complex variety \(M\), whose canonical bundle \(\omega_M\) is locally trivial as a \(G\)-sheaf. We prove that the Hilbert scheme \(Y=G \text{-Hilb }{M}\) parametrising \(G\)-clusters in \(M\) is a crepant resolution of \(X=M/G\) and that there is a derived equivalence (Fourier-Mukai transform) between coherent sheaves on \(Y\) and coherent \(G\)-sheaves on \(M\). This identifies the K-theory of \(Y\) with the equivariant K-theory of \(M\), and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible. quotient singularities; McKay correspondence; derived categories; group of automorphisms; three-dimensional complex variety; Hilbert scheme; crepant resolution; Fourier-Mukai transform; equivariant K-theory T.~Bridgeland, A.~King, and M.~Reid. Mukai implies McKay: the McKay correspondence as an equivalence of derived categories. \(ArXiv Mathematics e-prints\), August 1999. Automorphisms of surfaces and higher-dimensional varieties, Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Equivariant \(K\)-theory, Group actions on varieties or schemes (quotients), \(3\)-folds The McKay correspondence as an equivalence of derived categories | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is intended to describe the explicit constructions of crepant resolutions of higher-dimensional orbifolds with Gorenstein quotient singularities, that is, the algebraic (or analytic) varieties such that the analytic type of each singular point is described as \({\mathbb C}^n / G\), where \(G\) is a (non-trivial) finite subgroup of the special linear group \(\text{SL}_n ({\mathbb C})\). For the 2-dimensional case, finite subgroups \(G\) of \(\text{SL}_2 ({\mathbb C})\) were classically classified into ADE series. It is well known that \({\mathbb C}^2 / G\) are always hypersurface singularities, and the minimal resolutions of them give the desired crepant resolutions. For the 3-dimensional case, the required crepant resolutions were found for all finite subgroups \(G \subset \text{SL}_3 ({\mathbb C})\) by virtue of \textit{Y. Ito} [Proc. Japan Acad., Ser. A 70, 131--136 (1994; Zbl 0831.14006)], \textit{D. Markushevich} [Math. Ann. 308, 279--289 (1997; Zbl 0899.14016)], \textit{S. S. Roan} [Int. J. Math. 5, 523--536 (1994; Zbl 0856.14005)] and \textit{S. S. Roan} [Topology 35, 489--508 (1996; Zbl 0872.14034)], which are depending on the classical result on the classification of finite subgroups of \(\text{SL}_3 ({\mathbb C})\) due to Miller-Blichfeldt-Dickson [\textit{Y. A. Miller, H. F. Blichfeldt} and \textit{L. E. Dickson}, ``Theory and application of finite groups''. New York, Wiley (1915; JFM 45.0255.12)]. But, for such a higher-dimensional case, the non-uniqueness of crepant resolutions happens due to the existence of certain kinds of codimension 2 birational operations known as flops. In order to understand higher-dimensional crepant resolutions qualitatively, the development has resulted in the theory of \(G\)-Hilbert schemes \(\text{Hilb}^G ({\mathbb C}^n)\) associated to the quotient singularities \({\mathbb C}^n / G\) as in \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad.,Ser. A 72, 135--138 (1996; Zbl 0881.14002)]: the crepant resolutions of \({\mathbb C}^n / G\) would be related to \(G\)-Hilbert schemes \(\text{Hilb}^G ({\mathbb C}^n)\) of \(G\)-stable 0-dimensional subschemes of \({\mathbb C}^n\) of length equal to the order \(| G | \) of \(G\). As a result, the structure of \(\text{Hilb}^G ({\mathbb C}^n)\) now yields that \(\text{Hilb}^G ({\mathbb C}^3)\) is a toric crepant resolution of \({\mathbb C}^3 / G\) for every finite abelian subgroup \(G \subset \text{SL}_3 ({\mathbb C})\) by virtue of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)], \textit{Y. Ito} and \textit{H. Nakajima} [Topology 39, 1155--1191 (2000; Zbl 0995.14001)] and \textit{I. Nakamura} [J. Alg. Geom. 10, 757--779 (2001; Zbl 1104.14003)]. Whereas, for the cases \(n \geq 4\), there are very few results concerning the crepant resolutions of \({\mathbb C}^n / G\) and the structure of \(\text{Hilb}^G ({\mathbb C})\) for finite subgroups \(G \subset \text{SL}_n ({\mathbb C})\). The authors restricted themselves to the case where \(G\) is the subgroup \(A_r (n)\) of \(\text{SL}_n({\mathbb C})\) consisting of all diagonal matrices of order \(r+1\). In an earlier paper [Int.. J. Math. Math. Sci. 26, 649--669 (2001; Zbl 1065.14018)], the authors studied the case of \(n=4\) and \(G=A_r (4)\), and obtained crepant resolutions of \({\mathbb C}^4 / A_r (4)\) through the detailed investigations of the structure of \(\text{Hilb}^{A_r (4)} ({\mathbb C}^4)\). In the present paper, the authors study the case of \(n=4, 5\) and \(r=1\). More precisely, for \(n=4\) and \(r=1\), they describe the toric variety structure of \(\text{Hilb}^{A_1 (4)} ({\mathbb C}^4)\) which is NOT crepant. Then, blowing-down the divisor \({\mathbb P}^1 \times {\mathbb P}^1 \times {\mathbb P}^1\) on \(\text{Hilb}^{A_1 (4)} ({\mathbb C}^4)\) onto \({\mathbb P}^1 \times {\mathbb P}^1\) in different ways, they obtain three different toric crepant resolutions of \({\mathbb C}^4 / A_1(4)\). These three crepant resolutions are related to each other by 4-fold flops. For the case \(n=5\) and \(r=1\) also, as in the 4-dimensional case just above, they describe the toric variety structure of \(\text{Hilb}^{A_1(5)} ({\mathbb C}^5)\) which is NOT crepant, and obtain twelve mutually different crepant resolutions of \({\mathbb C}^5 / A_1(5)\), all of which are dominated by \(\text{Hilb}^{A_1(5)}({\mathbb C}^5)\). These twelve crepant resolutions are related to each other by 5-fold flops. \(G\)-Hilbert schemes; \(G\)-clusters; toric crepant resolution Chiang, L.; Roan, S. -S.: Crepant resolutions of \(Cn/A1(n)\) and flops of n-folds for n=4,5, Fields inst. Commun. 38, 27-41 (2003) Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Parametrization (Chow and Hilbert schemes) Crepant resolutions of \(\mathbb{C}^n/A_1(n)\) and flops of \(n\)-folds for \(n=4,5\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R=P/(f)\) be an analytic hypersurface ring. The author investigates first the relation between maximal Cohen-Macaulay modules (MCM) over R and over \(R_ 1=P_ 1/(f+y^ 2)\), where \(P_ 1=P<y>\) (and \(P=k<x_ 0,...,x_ n>\), k algebraically closed and char(k)\(\neq 2)\). He proves in \(corollary^ 2.8\) that there are only finitely many isomorphism classes of indecomposable MCM's over R if and only if this is true for \(R_ 1\). In theorem 3.1 it is shown - in a more general frame - that there is a canonical bijection between the sets of isomorphism classes of MCM's over R and over \(R_ 2=P_ 2/(f+y^ 2+z^ 2)\) respectively, where \(P_ 2=P<y,z>.\)
Since the two-dimensional simple singularities, i.e. the rational double points, have only finitely many isomorphism classes of MCM's over their local rings [by \textit{M. Artin} and \textit{J.-L. Verdier}, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)], one gets by iterated application of corollary 2.8 the main result of this paper: There are only finitely many isomorphism classes of indecomposable MCM's over the local ring of an isolated simple hypersurface singularity \((A_ k, D_ k, E_ 6, E_ 7, E_ 8\) in Arnold's classification). The author also gives a conceptional description of the Auslander-Reiten quivers of the simple plane curve singularities in \(char(k)=0\), showing that these quivers coincide with certain graphs associated to representations of finite reflection groups in \(Gl(2,k).\)
[See also the following review.] analytic hypersurface ring; maximal Cohen-Macaulay modules; isolated simple hypersurface singularity; Auslander-Reiten quivers Knörrer, H., Cohen-Macaulay modules on hypersurface singularities, I. Invent. Math., 88, 153-164, (1987) 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. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper proposes a generalization of the well-known triality between simply-laced Dynkin diagrams, simple Lie algebras, and Kleinian groups (with associated quotient singularities), and provides an account of the results achieved in this direction by the author and others. The exposition is exhaustive while concise, and subtle points and open questions are emphasized throughout the text.
The idea is to attach a surface singularity and a Lie algebra to a regular system of weights, which denotes a system of four integers \(W:=(a,b,c,h)\). A number of requirements is imposed upon this singularity and this algebra, in order to make this a generalization of a usual triality picture described above and also of elliptic Lie algebra theory. A singularity is defined as a zero set of a generic polynomial \(f_W\) of degree \(h\) in a weighted projective space \(P(a:b:c)\). An algebra is constructed through an intermediate step, which includes a construction of the homotopy category of graded matrix factorizations for \(f_W\), establishing a strongly exceptional collection there, and taking its associated quiver to construct a Lie algebra.
On the ``geometrical side'', investigating the singularities obtained from regular weight systems, and their vanishing cycle lattices, the author presents a notion of \(*\)-duality for regular weight systems. On one hand, it is shown to underlie the ``strange'' duality of Arnold; on the other, the author cites \textit{A. Takahashi} [Commun. Math. Phys. 205, No. 3, 571--586 (1999; Zbl 0974.14005)] where it is proven that the \(*\)-duality is equivalent to mirror symmetry for Landau-Ginzburg models. regular system of weights; simple Lie algebras; elliptic Lie algebras; homotopy category of matrix factorizations; vanishing cycles; \(*\)-duality Saito, K.: Towards a categorical construction of Lie algebras, Adv. stud. Pure math. 50, 101-175 (2008) Categories in geometry and topology, Lie algebras and Lie superalgebras, Singularities of surfaces or higher-dimensional varieties, Homology and cohomology theories in algebraic topology, Simple, semisimple, reductive (super)algebras, Derived categories, triangulated categories, Calabi-Yau manifolds (algebro-geometric aspects) Towards a categorical construction on Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities F. Klein studied simple singularities, classifying them as quotients of $\mathbb{C}^2$ by the action of a finite subgroup $\Gamma \subseteq \mathrm{SU}_2$. P. Du Val showed that the exceptional divisors of the minimal resolution of the isolated singularity of such a quotient form an arrangement of projective lines whose dual graph is a simply-laced Dynkin diagram $\Delta(\Gamma)$; thus the quotient $\mathbb{C}^2/\Gamma$ is called a simple singularity of type $\Delta(\Gamma)$. P. Slodowy then extended the definition of a simple singularity to the non simply-laced types by adding a second finite subgroup $\Gamma' \subseteq \mathrm{SU}_2$ such that $\Gamma'\supseteq \Gamma$ as a normal subgroup; $\Gamma'/\Gamma =\Omega$ acts on $\mathbb{C}^2/\Gamma$ and this action can be lifted to the minimal resolution of the singularity, inducing an action on the exceptional divisors, which corresponds to a group of automorphisms of the Dynkin diagram of $\mathbb{C}^2/\Gamma$. \par A deformation of a simple singularity $(X_0,\Omega)$ is an $\Omega$-equivariant deformation of the singularity $X_0$ with a trivial action of the automorphism group $\Omega$ on the base space. Setting $\pi : X \rightarrow Y$ as a deformation of $X_0$, a deformation $\psi : X' \rightarrow Y'$ of $X_0$ is induced from $\pi$ by a morphism $\varphi : Y' \rightarrow Y$ if there exist a morphism $\Phi :X' \rightarrow Y'$ such that $\pi\circ \Phi= \varphi\circ\psi$, and given $X_0 \stackrel{i}{\hookrightarrow} X$ and $X_0\stackrel{j}{\hookrightarrow} X'$, $\Phi\circ j=i$. \par A semiuniversal deformation $\pi_0 : X \rightarrow Y$ of a simple singularity $(X_0,\Omega)$ is a deformation of $(X,\Omega)$ such that any other deformation $\psi : X' \rightarrow Y'$ of $(X,\Omega)$ is induced from $\pi_0$ by an $\Omega$-equivariant morphism $\varphi : Y' \rightarrow Y$ with a uniquely determined differential $d_{y'} \varphi :T_{y'} Y' \rightarrow T_y Y$. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$ by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type $X=D_s$, $E_6$ or $E_7$. \par Letting $\alpha : X_\Gamma \rightarrow \mathfrak{h}/W$ to be the semiuniversal deformation of a simple singularity of type $\Delta(\Gamma)=A_{2r-1}$ ($r \geq 2$), $D_{r}$ ($r \geq 4$) or $E_6$ obtained by the construction of H. Cassens and P. Slodowy, and $\mathfrak{h}$ and $W$ being the Cartan subalgebra and the associated Weyl group of the simple Lie algebra $\mathfrak{g}$ of the same type, respectively, they showed that $\Omega$ of the Dynkin diagram of $\mathfrak{g}$ acts on $X_\Gamma$ and $\mathfrak{h}/W$ such that $\alpha$ is $\Omega$-equivariant. Slodowy then showed that taking the restriction $\alpha^\Omega$ of $\alpha$ over the $\Omega$-fixed points of $\mathfrak{h}/W$ leads to a semiuniversal deformation of a simple singularity, which is inhomogeneous. As $\alpha$ is $\Omega$-equivariant, there is an action of $\Omega$ on every fiber of $\alpha^\Omega$, and the quotient leads to a new morphism $\overline{\alpha^\Omega}$, which is a non-semiuniversal deformation of a simple singularity of homogeneous type $\Delta(\Gamma')$. \par Let $\Gamma$ be a finite subgroup of $\mathrm{SU}_2$, $R$ its regular representation, $N$ its natural representation as a subgroup of $\mathrm{SU}_2$, and $\Delta(\Gamma)$ the associated Dynkin diagram. If $\Omega$ acting on $M(\Gamma)=(\mathrm{End}(R) \otimes N)^\Gamma$ is symplectic, then $\widetilde{\alpha}:X_{\Gamma}\times_{\mathfrak{h}/W}\mathfrak{h}\rightarrow \mathfrak{h}$ and $\alpha:X_{\Gamma}\rightarrow \mathfrak{h}/W$ can be made into $\Omega$-equivariant maps (Theorem 1.4, page 388): letting $M(\Gamma)$ to be the representation space of a McKay quiver built on a Dynkin diagram of type $A_{2r-1}$, $D_{r}$ or $E_6$, there exists a symplectic action of $\Omega=\Gamma'/\Gamma$ on $M(\Gamma)$, inducing the natural action on the singularity $\mathbb{C}^2/\Gamma$; this action then turns $\alpha$ into an $\Omega$-equivariant morphism. \par After A. Caradot shows that the morphism $\alpha^{\Omega} : X_{\Gamma, \Omega} \rightarrow (\mathfrak{h}/W)^{\Omega}$ is $\Omega$-invariant, it follows that $\Omega$ acts on each fiber of $\alpha^{\Omega}$, and the fibers can be quotiented. Furthermore it is known that $(\alpha^{\Omega})^{-1}(\overline{0}) = X_{\Gamma,0}=\mathbb{C}^2/\Gamma$, and thus $(\alpha^{\Omega})^{-1}(\overline{0})/\Omega=X_{\Gamma,0}/\Omega = (\mathbb{C}^2/\Gamma)/(\Gamma'/\Gamma) \cong \mathbb{C}^2/\Gamma'$, which is a simple singularity since $\Gamma'\subseteq \mathrm{SU}_2$ is finite. Thus, $\overline{\alpha^{\Omega}} : X_{\Gamma, \Omega} /\Omega \rightarrow (\mathfrak{h}/W)^{\Omega}$ is a deformation of the simple singularity $\mathbb{C}^2/\Gamma'$ of type $\Delta(\Gamma')$, where the deformation $\overline{\alpha^{\Omega}}$ is obtained through $\Delta(\Gamma)-\Delta(\Gamma,\Gamma')-\Delta(\Gamma')$-procedure. \par Caradot also studies the regularity of the fibers of $\overline{\alpha^{ \Omega}}$ (Theorem 2.3, page 390): assuming $\alpha^\Omega$ is the semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$, every fiber of the quotient $\overline{\alpha^\Omega}$ is singular. \par Finally, after stating a conjecture (Conjecture 3.1, page 396) that there exists a subset $\Theta$ of simple roots of the root system of type $\Delta(\Gamma')$ such that the Dynkin diagram associated to the singular configuration of any fiber of $\overline{\alpha^\Omega}$ is a subdiagram of the Dynkin diagram of type $\Delta(\Gamma')$ containing the vertices associated to $\Theta$, the author proves the conjecture for the types $A_3-B_2-D_4$, $A_5-B_3-D_5$, $D_4-C_3-D_6$, $D_4-G_2-E_6$, $D_4-G_2-E_7$, and $E_6-F_4-E_7$. (Theorem 3.2, page 397). deformations of simple singularities; simple root systems; simple singularities of inhomogeneous types; singular configurations Deformations of singularities, Root systems, Representation theory for linear algebraic groups Root systems and quotients of deformations of simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V/\Gamma\) be a symplectic quotient singularity, i.e. a quotient of a finite dimensional vector space \(V\) by a linear action of a finite subgroup \(\Gamma \subset \mathrm {Sp}(V)\). The article under review concerns \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), i.e. projective, crepant, birational morphisms \(\rho : Y \rightarrow V/\Gamma\) such that \(Y\) has only \(\mathbb{Q}\)-factorial terminal singularities, and symplectic resolutions of \(V/\Gamma\), i.e. smooth \(\mathbb{Q}\)-factorial terminalizations. By results of Namikawa, a symplectic quotient singularity admits finitely many \(\mathbb{Q}\)-factorial terminalizations, and if one of them is smooth then all are smooth.
The main result is a formula for the number of non-isomorphic \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), expressed in terms of the Calogero-Moser deformation of \(V/\Gamma\) and the Namikawa Weyl group associated to \(V/\Gamma\). From this theorem the author derives a more explicit formula or obtains a number of symplectic resolutions for all groups \(\Gamma\) such that \(V/\Gamma\) is known to admit a symplectic resolution. These are: the infinite series of wreath products \(\mathcal{S}_n \wr G\), where \(G \subset \mathrm {SL}(2,\mathbb{C})\), acting on \(V = \mathbb{C}^{2n}\), a 4-dimensional representation \(G_4\) of the binary tetrahedral group and a 4-dimensional representation of \(Q_8 \times_{\mathbb{Z}_2} D_8\). In the case of \(\mathcal{S}_n \wr G\) the result is a formula involving the description of the Weyl group associated to \(G\) via the McKay correspondence, and for the remaining two cases the author obtains 2 and 81 symplectic resolutions respectively. symplectic resolutions; symplectic reflection algebras; Orlik-Solomon algebras Bellamy, G., Counting resolutions of symplectic quotient singularities, Compos. Math., 152, 1, 99-114, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Deformations of associative rings, Poisson algebras Counting resolutions of symplectic quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article gives a summary of the author's unpublished Ph.D thesis. It is known that Dynkin diagrams can be separated in two classes: the simply laced (or homogeneous) ones \(A_k\) (\(k\geq 1\)), \(D_k\) (\(k\geq 4\)), \(E_6\), \(E_7\) and \(E_8\), and the non-simply laced (or inhomogeneous) ones \(B_k\) (\(k\geq 2\)), \(C_k\) (\(k\geq 3\)), \(F_4\) and \(G_2\).
The aim of the article is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the homogeneous simple singularities to the inhomogeneous ones.
To a homogeneous simple singularity, one can associate the representation space of a particular quiver.
This space is endowed with an action of the symmetry group of the Dynkin diagram associated to the simple singularity which allows the construction and explicit computation of the semiuniversal deformations of the inhomogeneous simple singularities.
By quotienting such maps, deformations of other simple singularities are obtained.
In some cases, the discriminants of these last deformations are computed. simple singularities; quiver representations; root systems; foldings Deformations of singularities, Representations of quivers and partially ordered sets, Root systems Deformations of inhomogeneous simple singularities and quiver representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Associated to each finite subgroup \(\Gamma\) of \(\text{SL}_2(\mathbb{C})\) there is a family of noncommutative algebras \(O^\tau(\Gamma)\), which is a deformation of the coordinate ring of the Kleinian singularity \(\mathbb{C}^2/\Gamma\). We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank
one projective modules over \(O^\tau\) and a certain class of quiver varieties associated to \(\Gamma\). We show that this bijection is naturally equivariant under the action of a ``large'' Dixmier-type automorphism group \(G\). Our construction leads to a completely explicit description of ideals of the algebras \(O^\tau\). deformations of coordinate rings; Kleinian singularities; finitely generated projective modules; quiver varieties; rings of invariants Eshmatov, F.: DG-models of projective modules and nakajima quiver varieties. (2006) Rings arising from noncommutative algebraic geometry, Differential graded algebras and applications (associative algebraic aspects), Representations of quivers and partially ordered sets, Deformations of associative rings, Formal methods and deformations in algebraic geometry, Noncommutative algebraic geometry, Deformations of singularities, Free, projective, and flat modules and ideals in associative algebras, Derived categories, triangulated categories, Abstract and axiomatic homotopy theory in algebraic topology DG-models of projective modules and Nakajima 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 The theta lift, which is basically the integration of a theta kernel against a modular form (i.e., an automorphic form on \(\mathrm{SL}_{2}\)) and producing an automorphic form on an orthogonal group, has already been used in many branches of mathematics, proving useful for many applications. One of them is the construction of Green functions for special divisors on orthogonal Shimura varieties. The point of view on this process that lies in the context of the present paper is when \(\mathrm{SL}_{2}\) and the orthogonal group are considered as a dual reductive pair inside a larger symplectic group. Very roughly, the aim of this paper is to generalize this theory for another dual reductive pair, in which \(\mathrm{SL}_{2}=\mathrm{Sp}_{2}\) is replaced by \(\mathrm{Sp}_{4}\), and produce the corresponding Green objects.
The classical theta lift requires a quadratic space over \(\mathbb{Q}\), of signature \((n,2)\) (in this normalization). We remark that the paper works in the context of a quadratic space \(V\) over a totally real field \(F\), whose signature is indeed \((n,2)\) in one real embedding of \(F\) but is positive definite in all the other embeddings. This generalization, using Whittaker forms, was defined and investigated in the \(\mathrm{SL}_{2}\) setting in [\textit{J. H. Bruinier}, J. Reine Angew. Math. 672, 177--222 (2012; Zbl 1268.11058)], and the current paper uses the same ideas concerning this generalization.
The main idea here is to define the relevant theta kernel (which includes in its data a function of some level \(K\) on the points of \(V\) over the finite adéles of \(F\)), taking values on differential forms of Hodge type \((1,1)\) over the (\(n\)-dimensional complex) symmetric space \(\mathbb{D}\) associated with our vector space \(V\), and investigate the results of the (regularized) theta lift with respect to appropriate Whittaker forms. More specifically, the author shows that up to images of the differentiation maps \(\partial\) and \(\overline{\partial}\) on the corresponding Shimura variety \(X_{K}\) of level \(K\), the resulting theta lift yields the same class as the Green current of a special cycle of codimension 2 (from the theory of the Kudla program). The relation with the images of higher Chow groups under the Beilinson regulator is also studied.
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The paper is divided into 4 sections, each of which consists of several subsections (called also sections in the paper), some of which are partitioned even further. We shall now describe these sections and subsections in more detail, but because of the length and the technical nature of the paper, some the results will be not described here in full detail. Subsection 1A of the Introduction includes the statement of the main results, while Subsections 1B and 1C contain an outline of the structure of the paper and some useful notation respectively. In Section 2, Subsection 2A reviews the theory of the relevant orthogonal Shimura varieties (both algebraically and analytically), including the definition of the relevant tautological line bundle, while Subsection 2B defines the special cycles that such varieties carry by the Kudla program. Section 3 is the technical heart of the paper, which includes 10 subsections, to be described in further details below. Finally, Section 4 considers the 2-dimensional example in which the Shimura variety \(X_{K}\) is the product of two Shimura curves associated with the same rational, indefinite, non-split quaternion algebra. Subsection 4A includes the relevant definitions of quaternion algebras and Shimura curves (with the relation between a GSpin group of dimension 4 and the multiplicative group of the quaternion algebra), while Subsection 4B, on its various parts, presents the explicit form that all the notions from Section 3 take in this case: The special divisors, the \((1,1)\) Green currents, and the resulting connections with the Hecke algebra.
It remains to describe the notions and arguments appearing in the 10 subsections of Section 3. Subsection 3A reviews the construction, due to the reference [\textit{T. Oda} and \textit{M. Tsuzuki}, Publ. Res. Inst. Math. Sci. 39, No. 3, 451--533 (2003; Zbl 1044.11033)], of spherical functions on \(\mathbb{D}\) (depending also on a complex parameter \(s\) from an appropriate right half-plane) from hyper-geometric series, and their properties (eigenfunctions of the Laplacian, growth, etc.). In Section 3B, the argument from [Bruinier loc. cit.] through which the aforementioned functions produce, via summation on \(\Gamma\)-translates (where \(\Gamma\) is an appropriate discrete subgroup of the automorphisms of the symmetric space \(\mathbb{D}\)) and taking the constant term of the expansion in \(s\) near the central point \(\rho_{0}=\frac{n}{2}\), Green functions for special divisors on \(X_{\Gamma}=\Gamma\backslash\mathbb{D}\) is presented. The definition of the latter function depends on a totally positive vector \(v \in V\), and Subsection 3C investigates similar functions that are based on two vectors in \(V\) that span a totally positive definite plane. This subsection shows that the new functions have properties that are similar to the ones from [Oda and Tsuzuki loc. cit.] and [Bruinier loc. cit.] (e.g., local integrability of them and their derivatives), that their restriction to the special divisor associated with \(v\) (which is an orthogonal Shimura variety of dimension one less) yields Green functions for divisors on that smaller Shimura variety, and presents them as Laplace transforms of appropriate Whittaker forms. Subsection 3D uses appropriate differential \((1,1)\)-forms arising through differentiating the functions from Subsection 3C and summation over appropriate one-sided quotients of \(\Gamma\), and shows, using some technical work, that they are integrable on \(X_{\Gamma}\). The connection between the constant terms of these functions of \(s\) at \(s_{0}=\frac{n-1}{2}\) and \((1,1)\)-currents on \(X_{\Gamma}\) arising from push-forwards of Green functions on the relevant special divisors are proven in Subsection 3E: They are the same modulo images of \(\partial\) and \(\overline{\partial}\) of smooth 1-forms on \(X_{\Gamma}\). Adapting these definitions to be valid, in a coherent manner, for \(X_{K}\), whose set of complex points can be presented as the disjoint union of several \(X_{\Gamma}\)s, is the content of Subsection 3F, which also investigates the simple properties of the resulting functions and their (simple) behavior under projections from varieties of higher level. Subsection 3G considers the function associated with a Schwartz function \(\varphi\) on the finite adélic points of \(V\) and a totally positive definite symmetric \(2\times2\) matrix \(T\) over \(F\), and shows that such functions are more natural, and also allow for a good definition of inverse limits with respect to the level. In part 1 of Subsection 3H the \((1,1)\)-forms-valued theta kernel associated with \(\varphi\) from above as well as an appropriate Schwartz function on the real points of \(V\) is defined, using the Weil representation of \(\mathrm{Sp}_{4}\). Using some Whittaker functions, part 2 of that subsection investigates the properties of the theta lift of certain Whittaker forms, and shows that they coincide, on an open dense subset of \(X_{K}\), with the differential forms from Subsection 3G, thus establishing the proof of the main result. Then Subsection 3I reviews the definitions of the higher Chow group \(CH^{2}(X_{K},1)\) (using roughly divisors on \(X_{K}\) and rational functions on them, up to some conditions and divided by some relations), the Beilinson regulator map \(r_{\mathcal{D}}\) from that group to \((1,1)\)-forms on \(X_{K}\), and characterizes the image of \(r_{\mathcal{D}}\) inside those differentials forms as precisely those combinations of classes that are \(dd^{c}\)-harmonic. Finally, Subsection 3J carries out some explicit calculations, in terms of the Kudla-Millson kernel function, Whittaker forms, and other theta kernels, and deduces some relations between the \((1,1)\)-currents arising from the various theta lifts in the theory. theta correspondence; orthogonal Shimura varieties; special cycles; Green currents; higher Chow groups; Beilinson regulators Theta series; Weil representation; theta correspondences, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties Regularized theta lifts and \((1,1)\)-currents on GSpin Shimura varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper a twisted version of the categorical McKay correspondence is conjectured. Recall that the (untwisted) categorical McKay correspondence conjecture was stated in \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14 (3), 535--554 (2001; Zbl 0966.14028)]. Let \(G\) a finite group acting on a smooth irreducible variety \(U\) over \(\mathbb{C}\) and \(X\) a crepant resolution of the quotient \(U/G\). If the action of \(G\) preserves the canonical bundle of \(U\) and the fixed point sets always have codimension \(>1\), then it is conjectured that the bounded derived category \(D_G(U)\) of \(G\)-equivariant coherent sheaves on \(U\) is equivalent to the bounded derived category \(D(X)\) of coherent sheaves on \(X\).
On one side, following \textit{A. Adem} and \textit{Y. Ruan} [Commun. Math. Phys. 237, No. 3, 533--556 (2003; Zbl 1051.57022)], given a class \(\alpha\) in \(H^2 (G, \mathbb{C}^{\ast})\), one can define the twisted equivariant derived category \(D_{G, \alpha}(U)\). On the other side, following \textit{A. Caldararu} [J. Reine Angew. Math. 544, 161--179 (2002; Zbl 0995.14012)], given a class \(\beta\) in the Brauer group Br\((X)\), one can define the bounded derived category \(D(X, \beta)\) of \(\beta\)-twisted coherent sheaves on \(X\). By constructing the subgroup \(B_G(U)\) of \(H^2(G, \mathbb{C}^{\ast})\) as the set of the classes contained in Br\((X)\), the authors can conjecture the equivalence between the categories \(D_{G,\alpha}(U)\) and \(D(X,\alpha)\) for \(\alpha\) in \(B_G(U)\). The authors prove that this two categories have periodic cyclic homology of the same dimension.
An explicit example of the group \(B_G(U)\) is given and some open problems are stated. Brauer group; derived category; McKay correspondence Baranovsky, P; Petrov, T, Brauer groups and crepant resolutions, Adv. Math., 209, 547-560, (2007) Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Brauer groups of schemes Brauer groups and 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 The paper is in some sense a follow-up to the paper ``Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes'' by the first author et al. [``Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes'', Preprint, \url{arXiv:0905.2191}]. In that paper, it was proved that every reduced excellent Noetherian scheme \(X\) of dimension at most two admits a resolution of singularities via finitely many ``permissible'' blowups. Their resolution procedure (called the CJS algorithm) is ``canonical'' in a precise sense; in particular it commutes with localization and completion. The proof of termination is by contradiction, which is unusual in this business.
The present paper introduces a new local invariant \(\iota(X, x)\) for schemes satisfying the above assumptions. In Theorem A, it is shown that this invariant decreases at each step of the CJS algorithm. This provides a new proof of termination of the CJS algorithm which is closer in spirit to the classical approach used in resolution theory. surface singularities; resolution of singularities; invariants for singularities; Hironaka's characteristic polyhedra Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A strictly decreasing invariant for resolution of singularities in dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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' introduction is a rather complete exposition of the results, the body of the article containing explicit definitions and proofs of the results. So this review is just a compressed version of the introduction and verbatim expressions occur.
The main issue of the article is to generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. This involves the statement of the standard resolution and the proof of their generalization.
Throughout, \(k\) is an algebraically closed field of characteristic \(0\), all rings are defined over \(k\), for an algebraic group \(G\), \(\widehat G\) denotes the set of isomorphism classes of irreducible \(G\)-representations.
For a normal noetherian domain \(S\), a non-commutative (crepant) resolution (NC(C)R) is an \(S\)-algebra of finite global dimension of the form \(\Lambda=\text{End}_S(M)\) where \(M\) is a non-zero finitely generated reflexive \(S\)-module. It is called crepant if \(S\) is Gorenstein and \(\Lambda\) is a maximal Cohen-Macaulay \(S\)-module.
The author remarks that the HPD-dual of a smooth projective variety, as defined by Kuznetsov, often is a NCR of a singular variety.
The NCCRs are the best NCRs as they have good homological properties. There is a connection to ordinary (commutative) crepant resolutions in algebaic geometry, and it is often possible to pass from one to the other by to a NCCR assigning a crepant resolution as a GIT moduli space of representations, and conversely a crepant resolution might give a NCCR obtained as \(R=\text{End}_X(T)\) for a tilting bundle \(T\) on a variety \(X\).
Also, under mild conditions, a NC(C)R yields a categorical crepant resolution of singularities: If \(S\) is a finitely generated integrally closed \(k\)-algebra with \(\Lambda=\text{End}_S(M)\) a NCR, then \(\Lambda\) is smooth as a DG-algebra. Also, if \(M\) is a projective \(\Lambda\)-module, the functor \(D(S)\rightarrow D(\Lambda):N\mapsto M\overset {L}\otimes_S N\) is fully faithful and yields a categorical resolution of singularities of \(S\). Moreover, if \(\Lambda\) is a NCCR then the categorical resolution is crepant.
In the following, a twisted NC(C)R of index \(u\) behaves as a NC(C)R except that it is generically a central simple algebra of index \(u\) rather than a matrix ring.
When a finite group \(G\) acts on a smooth affine variety \(X\) and \(U\) is a finite-dimensional \(G\)-representation, then \(M(U)=(U\otimes k[X])^G\) is called the corresponding \(k[X]^G\)-module of covariants. The abelian category \(\text{mod}(G,k[X])\) of \(G\)-equivariant finitely generated \(k[X]\)-modules has a projective generator \(U\otimes k[X]\) with \(U=\oplus_{V\in\widehat G}V\), and it follows that \(\text{mod}(G,k[X])\) is equivalent to \(\text{mod}(\Lambda)\) where \(\Lambda=\text{End}_{G,k[X]}(U\otimes k[X])=(\text{End}(U)\otimes k[X])^G=M(\text{End}_k(U))\) so that \(\Lambda\) is a \(k[X]^G\) algebra of covariants. Because \(\text{mod}(G,k[X])\) has finite global dimension, \(\text{gl\; dim} \Lambda<\infty\) and so \(\Lambda\) is a non-commutative resolution of \(k[X]^G=k[X\slash G]\) in a weak sense. If no element of \(G\) fixes a divisor then \(k[X]/k[X]^G\) is étale in codimension one so that \(\Lambda=\text{End}_{k[X]^G}(M(U))\), and then \(\Lambda\) is NCR of \(k[X]^G\). For \(V\) to be a NCCR one also needs \(k[X]^G\) to be Gorenstein, and that is obtained when \(X=\text{Spec}SW\) for a representation \(W\) and \(G\subset\text{SL}(W)\). For general reductive groups, especially when \(G\) is not finite, the generalization is more involved. However, this is handled by the authors.
The main body of the article looks for non-commutative resolutions of quotient singularities given by algebras of covariants. The first main theorem states that when \(G\) is a reductive group acting on a smooth affine variety \(X\), there exists a finite-dimensional \(G\)-representation \(U\) containing the trivial representation such that \(\Lambda=M(\text{End}(U)\otimes k[X])^G\) have \(\text{gl\;dim}<\infty\).
One has that \(\Lambda=M(\text{End}(U))\) is a \(k[X]^G\)-algebra, finitely generated as \(k[X]^G\)-module and with \(k[X]^G\subset Z(\Lambda)\) so that \(\Lambda\) is a weak non-commutative resolution of \(k[X]^G=k[X\slash G]\). As \(U\) contains a trivial direct summand, \(M(U)\) is a projective left \(\Lambda\)-module, one has that \(\text{End}_\Lambda(M(U))=k[X]^G\), and so one gets that if \(G\) is a reductive group acting on a smooth affine variety \(X\), \(k[X]^G\) has a categorical resolution of singularities given by an algebra of covariants.
Correspondingly, when \(X\) is projective, the authors prove a GIT-version of the main result, that is, for given \(G,\; X,\;\mathcal L\), there exists a finite-dimensional \(G\)-representation \(U\) containing the trivial representation such that the coherent sheaf of algebras \(\Lambda=M^{ss}(\text{End}(U))\) on \(X^{ss}\slash G\) has finite global dimension when restricted to affine opens.
A finite group \(G\) is said to be acting generically on a smooth affine variety \(X\) if \(X\) contains a point with closed orbit and stabilizer, and if \(X^s\subset X\) is the locus of points with that property, then \(\text{codim}(X-X^\ast)\geq 2\). Then a \(G\)-representation \(W\) is generic if \(G\) acts generically on \(\text{Spec}SW\cong W^\ast\). In this case, if \(G\) acts generically on \(X\), there exists a finite-dimensional \(G\)-representation \(U\) containing the trivial representation such that \(\Lambda=\text{End}_{k[X]^G}(M(U))\) is a NCR for \(k[X]^G\). Applying the corresponding techniques to abelian reductive groups, the authors prove that for \(S\subset\mathbb Z^n\) a finitely generated commutative positive normal semigroup, for \(n\in\mathbb N,\;n\gg 0\), the \(k[S]\)-module \(M=k[\frac{1}{n}S]\) defines a NCR for for \(k[S]\). This confirms that in characteristic \(p\) Frobenius twists provide canonical NCRs.
After the development of this theory, with the deep results indicated above, it is applied to some specific quotient singularities.
The first is the determinantal varieties and the result that the variety \(Y_{n,h}\), \(n<h\), of \(h\times h\)-matrices of rank \(\leq n\), \(k[Y_{n,h}]\) has a NCCR.
Considering Pfaffian varieties, letting \(Y_{2n,h}^-\), \(2n<h\), be the variety of skew-symmetric \(h\times h\)-matrices of rank \(\leq 2n\), then for odd \(h\), \(k[Y_{2n,h}^-]\) has a NCCR.
For determinantal varieties for symmetric varieties, for \(t<h\), let \(Y^+_{t,h}\) be the variety of symmetric \(h\times h\)-matrices of tank \(\leq t\). If \(t\) and \(h\) have opposite parity, then \(k[Y^+_{t,h}]\) has a NCCR. If the parity is equal, then \(k[Y^+_{t,h}]\) has a twisted NCCR of index \(2^{\lfloor h/2\rfloor}\).
Finally, the results for non-commutative resolutions for \(\text{SL}_2\)-invariants are stated, and trace rings are proven to be modules of covariants in a series of cases.
The introduction, which is rather exhaustive, ends by stating a number of sufficient conditions for the existences of NC(C)Rs, possibly twisted, for quotient singularities. Thus the necessary machinery is included and this article is a complete introduction to noncommuative resolutions, with a lot of nice examples. non-commutative resolutions; non-commutative crepant resolutions; quotients of reductive groups; HPD dual; DG algebra; twisted NC(C)R; tilting algebra; tilting module Špenko, Š.; Van den Bergh, M., Non-commutative crepant resolutions for some toric singularities I, (2017), preprint Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Derived categories and associative algebras Non-commutative resolutions of quotient singularities for reductive 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 This paper is concerned with the relationship between the generalised Kronheimer construction of Kähler metrics on crepant resolutions of \(\mathbb{C}^{3}/\Gamma\) and asymptotically locally flat Ricci flat Kähler metrics on the same space in the context of the gauge/gravity correspondence.
The generalised Kronheimer construction builds the crepant resolution \(Y^{\Gamma}\) of the quotient singularity \(\Gamma \subset \mathrm{SU}(3)\) by constructing a blowdown morphism \[ Y^{\Gamma} \longrightarrow \frac{\mathbb{C}}{\Gamma} \] The construction goes by building an associated quiver gauge theory, called the McKay quiver, via the representations of \(\Gamma\) and the embedding \(\Gamma \hookrightarrow \mathrm{SU}(3)\). The space \(Y^{\Gamma}\) and the Kähler metric are constructed via a Kähler quotient construction, the details of which are prescribed by the quiver. This leads to a metric on \(Y^{\Gamma}\) which coincides with the metric on the space of vacua of the quiver gauge theory.
One would like to understand how this relates to the IIB supergravity solution given by D3 branes with \(Y^{\Gamma}\) equipped with Ricci flat Kähler metric as the transverse space. They provide an explicit construction of such a solution in section 2. In the paper, they provide evidence for the following conjecture.
Conjecture 1.1. The Kronheimer Kähler metric \(\mathrm{ds}^{2}_{\mathrm{Kro}}[Y^{\Gamma}]\) and the Ricci-flat one \(\mathrm{ds}^{2}_{\mathrm{Ricflat}}[Y^{\Gamma}]\) on the same manifold, that has the same isomteries and is asymptotically locally ALE are different, yet they coincide on the exceptional divisor \(\mathcal{ED}\).
They study the cases \(\Gamma = \mathbb{Z}_{3}\) and \(\Gamma = \mathbb{Z}_{4}\).
In the \(\mathbb{Z}_{3}\) case, they are able to construct explicitly the Kähler potential for the Ricci flat metric on \(Y^{\Gamma} = \mathrm{tot} K_{\mathbb{P}^{2}} = \mathcal{O}_{\mathbb{P}^{2}}(-3)\) with the correct asymptotic behaviour and isometries via proving the following theorem.
Theorem 3.1. Let \(\mathcal{M}_{n}\) be a compact \(n\)-dimensional Kähler manifold admitting a dense open coordinate pacth \(z_{i},\, i=1,\dots,n\) which we can identify with the total space of the line bundle \(\mathcal{O}_{\mathbb{P}^{n-1}}(-n)\), the bundle structure being exposed by the coordinate transformation: \[z_{i}=u_{i}w^{1/n} \quad , \quad (i=1,\dots,n-1) \quad ; \quad z_{n} = w^{1/n} \] where \(u_{i}\) is a set of inhomogeneous coordinates for \(\mathbb{P}^{n-1}\). the Kähler potential \(\mathcal{K}_{n}\) of a \(\mathrm{U}(n)\) isometric Kähler metric on \(\mathcal{M}_{n}\) must necessarily be a real function of the unique real variable \(\Sigma = \sum_{i=1}^{n}|z_{i}|^{2}\). If we require that metric should be Ricci-flat, the Kähler potential is uniquely defined and is the following one: \[ \mathcal{K}_{n} = k + \frac{ (\Sigma^{n} + l^{n})^{-\frac{n-1}{n}} \left( (n-1)(\Sigma^{n} + l^{n}) - l^{n}(\Sigma^{-n}l^{n}+1)^{\frac{n-1}{n}} {}_{2}F_{1}(\frac{n-1}{n},\frac{n-1}{n},\frac{2n-1}{n};-l^{n}\Sigma^{-n}) \right) }{n-1} \] where \(k\) is an irrelevant additive constant and \(l>0\) is a constant that can be reabsorbed by rescaling all the complex coordinates by a factor \(l\), namely \(z_{i} \rightarrow l\tilde{z}_{i}\)
They compare this directly with the Kähler potential and metric induced by the generalised Kronheimer construction and show that they are not equal on the total space \(\mathrm{tot}K_{\mathbb{P}^{2}}\). They also compare the metrics restricted to the exceptional divisor \(\mathbb{P}^{2}\). They do so by moving to a set of convenient toric coordinates \(u,v,w\) (given by equation (3.39)) where the exceptional divisor is defined by \(|w|^{2}=0\). They show that, up to an irrelevant additive constant and multiplicative constant, the Kähler potentials agree.
In the \(\mathbb{Z}_{4}\) case, the total resolution is given by the canonical bundle of the second Hirzebruch surface \(\mathrm{tot}K_{\mathbb{F}_{2}}\) and a partial resolution is given by \(\mathrm{tot}K_{\mathbb{W}P[112]}\). They try to construct Ricci flat Kähler metrics on these total spaces that are asymptotically given by the cone over \(S^{5}/\mathbb{Z}_{4}\) and has isometries \(\mathrm{SU}(2)\times \mathrm{U}(1)\times \mathrm{U}(1)\). They start by building Kähler metrics with the right isometries which depend only on two real functions \(\Upsilon(s),P(t)\). They show that for the choice of functions \[ \Upsilon(s) = \sqrt{\frac{-s}{\tfrac{2}{3}s^{2} - s + 3}} \qquad P(t) = \frac{1}{\sqrt{-\tfrac{2}{3}t^{2}-t}} \] the induced metric is a Ricci flat metric on \(\mathrm{tot}K_{\mathbb{W}P[112]}\) with the correct asymptotics and they derive a closed form expression for the symplectic potential of this metric. The metric on the exceptional divisor \(\mathbb{W}P[112]\) is found by setting \(s=-3\).
Using previous results on the Kronheimer construction, they construct the Kähler potential on \(\mathbb{F}_{2}\), the exceptional divisor on the total resolution, in terms of the real variable \(t\) and a parameter \(\alpha\) describing the ratio of the volumes of the two cycles on \(\mathbb{F}_{2}\). In the degenerate limit \(\alpha = 0\), they show that you indeed recover the Kähler potential of the singular space \(\mathbb{W}P[112]\) derived from the Ricci flat metric on \(\mathrm{tot}K_{\mathbb{W}P[112]}\).
Finally, they derive the Monge-Ampére equation for a Ricci flat metric on the total space with the correct isometries. This is a partial differential equation in the Kähler potential or the symplectic potential and they show that one can define a series solution that is uniquely determined order by order in the value of the potential on the exceptional divisor. They find the series solution for the degenerate \(\alpha=0\) case, and show that the symplectic potential found for the Ricci flat metric on \(\mathrm{tot}K_{\mathbb{W}P[112]}\) satisfies the equation, hence giving the unique solution. While they are not able to produce an exact solution for general \(\alpha\) and hence the smooth space \(\mathrm{tot}K_{\mathbb{F}_{2}}\), they are able to find a series solution.
This work argues that one can find the Ricci flat metric on the crepant resolution of a quotient singularity \(\mathbb{C}^{3}/\Gamma\) via the Kronheimer construction by using the Monge-Ampére equation as above. This then implies that both the field theory and the dual gravity solution are determined by the McKay quiver which hence encodes the duality.
There is a typo on page 20 where `complexification \(\mathcal{F}_{\mathbb{Z}_{3}}\)' should read `complexification \(\mathcal{G}_{\mathbb{Z}_{3}}\)'. quotient singularities; crepant resolutions; Ricci-flat metrics; D3-brane solutions; IIB supergravity; gauge/gravity duality Arithmetic ground fields for curves, Singularities of surfaces or higher-dimensional varieties, Kähler manifolds, Special Riemannian manifolds (Einstein, Sasakian, etc.), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Topological field theories in quantum mechanics, Supersymmetric field theories in quantum mechanics, Supergravity Resolution à la Kronheimer of \(\mathbb{C}^3/\Gamma\) singularities and the Monge-Ampère equation for Ricci-flat Kähler metrics in view of D3-brane solutions of supergravity | 0 |
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