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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities the author shows in a very synthetic manner some results concerning algorithmic resolution in characteristic zero (in the sense of \textit{S. Encinas} and \textit{O. Villamayor} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)], and \textit{A. Bravo, S. Encinas} and \textit{O. Villamayor} [Rev. Mat. Iberoam. 21, No. 2, 349--458 (2005; Zbl 1086.14012)]) and their relationship with the weak equivalence introduced by Hironaka in the latest 70's, see for example [\textit{H. Hironaka}, ``Idealistic exponents of singularity'', Algebraic geometry, The Johns Hopkins centen. Lect., Symp. Baltimore/Maryland 1976, 52--125 (1977; Zbl 0496.14011)]. It simplifies some arguments and gives a very simple way to prove the ``naturality'' of the main step of the proof: the inductive step. singularities; marked ideals; blowing-up; permissible centers; resolution; equivalence O. Fujino, \textit{Semipositivity theorems for moduli problems}, preprint, arXiv:1210.5784v2 [math.AG] (2012). Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry, Local theory in algebraic geometry, Birational geometry On the use of naturality in algorithmic resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Generalizing the main results of the first author [Math. Ann. 373, No. 3--4, 1103--1133 (2019; Zbl 1416.15030)] for the case \(G = \mathrm{SL}_n\), this work studies the rays of the eigencone -- also called the tensor cone -- for an arbitrary connected complex semisimple groups \(G\). Upon fixing a maximal torus and Borel subgroup of \(G\), the finite-dimensional irreducible \(G\)-modules \(V(\lambda)\) are parametrized by dominant weights \(\lambda\). The tensor semigroup consists of all solutions \((\lambda_1,\lambda_2,\hdots,\lambda_s)\) to the problem: \(V(\lambda_1)\otimes \cdots\otimes V(\lambda_s)\) contains nonzero \(G\)-invariants; the closure of this semigroup is a convex polyhedral cone \(\Gamma(s,K)\) called the \textit{eigencone}. (The notation \(K\) stands for a maximal compact subgroup of \(G\) and hearkens to an original definition of the cone involving conjugacy classes in \(\mathfrak{k}\); if \(G = \mathrm{SL}_n\) then this eigencone parametrizes solutions to the Horn problem: possible sets of eigenvalues \(\lambda(A_1), \hdots, \lambda(A_s)\) of Hermitian matrices satisfying \(A_1+\hdots+A_s = 0\).)
The inequalities determining \(\Gamma(s,K)\) have been well-studied and determined without redundancy by several contributions, in particular, due to Horn, Klyachko, Knutson-Tao, Belkale, Berenstein-Sjamaar, Kapovich-Leeb-Millson, Belkale-Kumar and Ressayre. Namely, to certain data in Schubert calculus of any \(G/P\), one associates a regular facet \(\mathcal{F}\) of \(\Gamma(s,K)\). In the present paper, the authors reduce the problem of producing extremal rays to finding all the extremal rays on such a face \(\mathcal{F}\). They first construct a list of ``type I'' rays \(\delta_1,\hdots,\delta_q\) from the Schubert calculus data of \(\mathcal{F}\), making use of a rigidity result in the cohomology of certain line bundles proved by Belkale-Kumar-Ressayre. These rays project to an orthogonal basis of a subspace of \(\mathbb Q\Gamma(s,K)\), hence are linearly independent, and moreover there is a linear decomposition
\[
\prod_b \mathbb Z_{\ge 0} \delta_b \times \mathcal{F}_2 \simeq \mathcal{F}.
\]
They identify \(\mathcal{F}_2\) as the image of a smaller eigencone \(\Gamma(s,K(L^{ss}))\) -- here \(L\) is the standard Levi subgroup of \(P\) -- under an explicit linear map \(\mathrm{Ind}\). Therefore the ``type II'' extremal rays of \(\mathcal{F}_2\) are images of extremal rays of \(\Gamma(s,K(L^{ss}))\), and proceeding inductively, one can in principle find all the rays of \(\mathcal{F}\), hence of \(\Gamma(s,K)\).
The paper concludes with a formula for the dimension of the kernel of \(\mathrm{Ind}\) and with several examples. Minimal model program (Mori theory, extremal rays), Grassmannians, Schubert varieties, flag manifolds, Geometric invariant theory, Linear algebraic groups over arbitrary fields, Hermitian, skew-Hermitian, and related matrices, Positive matrices and their generalizations; cones of matrices, Inequalities involving eigenvalues and eigenvectors, Lie algebras of linear algebraic groups, Representation theory for linear algebraic groups, Homogeneous spaces Extremal rays in the Hermitian eigenvalue problem for arbitrary types | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 divided in two sections. In the first one the following theorem is proved:
Let \((R,{\mathfrak m}, K)\) be an excellent henselian Cohen-Macaulay local ring, with isolated singularity \(\text{Spec}(R)\) containing a field such that \(K\) is perfect or \([K:K^p] < \infty\) if \(\text{ch} (K)=p<0\). Let \(x=\{x_1, \dots, x_s\}\) be a system of parameters in \(R\) and \(R_t=R/(x^t_1, \dots, x_s^t)\), \(t\in\mathbb{N}\). Let \(L_{R_2}\) be the category of all finitely generated \(R_2\)-modules such that \(((x_1, \dots, x_{j-1}) P:x_j)_P=(x_1, \dots, x_j)P\), for all \(j=1, \dots, s\). Let \(\text{MCM} (R)\) be the category of maximal CM-modules of \(R\). Then the base change functor \(F_1: \text{MCM} (R)\to \text{Mod} (R_1)\), induces a bijection from the isomorphism classes of \(\text{MCM} (R)\) to the isomorphism classes of \(R_1\)-modules of the type \(x_1\dots x_sP\) for \(P\in L_{R_2}\).
In the second section it is shown that the isomorphism classes of maximal Cohen-Macaulay \(R\)-modules and the isomorphism classes of modules of \(L_{R_2}\) can be seen as orbits of an affine group acting on a Grassmannian variety when the residue field of \(R\) is algebraically closed. isomorphism classes of finitely generated modules; isomorphism classes of maximal Cohen-Macaulay modules; isolated singularity; Grassmannian variety Popescu, D.: Maximal Cohen--Macaulay modules over isolated singularities. J. algebra 178, 710-732 (1995) Cohen-Macaulay modules, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Grassmannians, Schubert varieties, flag manifolds, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Maximal Cohen-Macaulay modules over isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author systematically develops an approach based on singularity theory to the investigation of various special functions admitting an integral representation and occurring in physics, integral geometry, partial differential equations, etc. His book is devoted to the study of four important classes of functions: the volume functions, the Newton-Coulomb potentials, the Green functions of hyperbolic equations, and the multidimensional hypergeometric functions of Gelfand-Aomoto types.
The book consists of the preface, introduction, eight chapters and bibliography including 189 references. In the preface main results described in the book are listed. Next is the introduction written in a didactic style and containing an explanation of key ideas and basic results with a series of nonformal comments and remarks.
In the first two chapters the classical Picard-Lefschetz theory and its relations with the singularity theory of functions are discussed at great length. In particular, the author covers the following topics: critical points and values, Milnor fibre, monodromy and variation operators, Gauss-Manin connexion, Picard-Lefschetz formula, Dynkin diagrams, \({\mathcal R}\)-classification of real smooth and complex isolated hypersurface singularities, stratifications, intersection homology theory, etc. These chapters are particularly apt for those who want to enter topological aspects of singularity theory.
The third and seventh chapters deal with properties of the volume function whose notion (in the planar case) goes back to Isaac Newton. In the third chapter the non-algebraicity is proved of the volume function defined by any convex compact hypersurface in the even-dimensional real space. In the odd-dimensional case there are many topological obstructions for such a function to be algebraic. This leads to the conjecture that the only example of algebraic volume function is defined by the ellipsoids in \({\mathbb R}^{2n-1}\) (the so-called Archimedes' example). In chapter VII the author studies the Newton-Coulomb potential of hyperbolic algebraic surfaces in \({\mathbb R}^n.\) Three centuries ago Newton found that the potential of a sphere is constant inside the sphere. One hundred twenty two years later, J. Ivory proved the analogous statement for ellipsoids. Recently, natural generalizations of these classical results to the case of arbitrary hyperbolic hypersurfaces in \({\mathbb R}^n\) have been obtained by \textit{V. I. Arnold} and \textit{V. A. Vassiliev} [Notices Am. Math. Soc. 36, No. 9, 1148--1154 (1989; Zbl 0693.01005)]. Against such a historical background the author examines the case of a smooth hyperbolic hypersurface in \({\mathbb R}^n\) and analyses the behavior of Newtonian potentials outside the hyperbolicity domain. As a result, he describes all cases when such potentials of general hyperbolic hypersurfaces are algebraic. Furthermore conditions under which the potential is algebraic outside the hypersurface are determined and investigated; an approach to the study of the odd-dimensional case is also discussed. The main idea of the author's observations is to use an integral representation of the potential function, easy considerations from the monodromy theory of complete intersections, and properties of homology groups with coefficients in a local system.
Chapters IV and V are devoted to the study of the lacuna problem for hyperbolic differential operators with constant coefficients. First the author recalls the classification of the singular points of wave fronts for hyperbolic operators with constant coefficients, the description of local lacunae close to nonsingular points of fronts (following A. M. Davydova and V. A. Borovikov) and to the singularities \(A_2\) and \(A_3\) [following \textit{L. Gårding}, Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, Suppl., 53--68 (1977; Zbl 0369.35062)]. Of course, these results are reformulated in terms of singularity theory. Then the local Petrovskii cycles of strictly hyperbolic operators are also expressed in such a manner and the equivalence of local regularity of fundamental solutions of hyperbolic PDEs and the topological Petrovskii-Atiyah-Bott-Gårding condition is proved. Finally, a combinatorial algorithm which enumerates local lacunae close to all simple (and many nonsimple) singular points of wave fronts is described in detail.
Chapter VI is devoted to the investigation of homology of local systems, twisted monodromy theory and regularization problem of improper integration cycles. It contains, among other things, a description of twisted vanishing homology of complete intersection singularities and extensions of results from the second chapter that are mainly based on stratified Picard-Lefschetz theory with twisted coefficients.
The final chapter is concerned with the theory of multidimensional hypergeometric functions in the sense of Gelfand and Aomoto; the ramification, singularities, resonance and integral representations of such functions are also investigated. In particular, the author proves a well-known theorem on the analytic continuation of the multidimensional hypergeometric functions and discusses related problems; under certain condition he also obtains a detail description of the case of real plane arrangements [\textit{V. A. Vassiliev}, \textit{I. M. Gelfand} and \textit{A. V. Zelevinskij}, Funkts. Anal. Prilozh. 21, No. 1, 23--38 (1987; Zbl 0614.33008)], and so on.
It should be remarked that the book under review can be considered as an expanded and revised version of [\textit{V. A. Vassiliev}, Ramified integrals, singularities and lacunas. Mathematics and its Application. 315 Kluwer (1995; Zbl 0935.32026)]; a significant part of new materials is based on recent results and ideas of the author and his collaborators. The book contains many clear examples, comments, remarks, open questions and problems illustrated by lot of tables, pictures, and diagrams as well as important applications with instructive references and historical background. With no doubt this book is comprehensible, interesting and useful for graduate students; the variety of topics covered makes it also highly valuable for researchers, lecturers, and practicians working in either of the above mentioned fields of mathematics and its applications. Picard-Lefschetz theory; monodromy theory; isolated singularities; Dynkin diagrams; Gauss-Manin connexion; intersection homology theory; volume functions; Newton-Coulomb potentials; Green functions; hyperbolic equations; lacuna problem; non-integrability of ovals; twisted vanishing homology; ramification of potentials; homology of complements of plane arrangements; Grassmannians; multidimensional hypergeometric functions and integrals Vassiliev, V.A.: Applied Picard-Lefschetz Theory. American Mathematical Society, Providence (2002a) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry, Integral representations, integral operators, integral equations methods in higher dimensions, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Continuation and prolongation of solutions to PDEs, Other hypergeometric functions and integrals in several variables, Shocks and singularities for hyperbolic equations Applied Picard-Lefschetz theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simply connected simple algebraic group over \(\mathbb{C}\), \(B\) and \(B_-\) its two opposite Borel subgroups, and \(W\) the associated Weyl group. It is shown that the coordinate ring \(\mathbb{C}[G^{u,v}] (u, v\in W)\) of the double Bruhat cell \(G^{u,v}=BuB\cap B_-vB_-\) is isomorphic to the cluster algebra \(\mathcal{A}(\mathbf{i})_{\mathbb{C}}\) and the initial cluster variables of \(\mathbb{C}[G^{u,v}]\) are the generalized minors \(\Delta(k;\mathbf{i})\) ([\textit{A. Berenstein} et al., Duke Math. J. 126, No. 1, 1--52 (2005; Zbl 1135.16013); \textit{K. R. Goodearl} and \textit{M. T. Yakimov}, ``The Berenstein-Zelevinsky quantum cluster algebra conjecture'', Preprint, \url{arXiv:1602.00498}]). In the case that a classical group \(G\) is of type \(\mathrm{B}_r\), \(\mathrm{C}_r\) or \(\mathrm{D}_r\), we shall describe the non-trivial last \(r\) initial cluster variables \(\{\Delta(k;\mathbf{i})\}_{(m-2)r<k\leq (m- 1)r}\) (\(m\) is given below) of the cluster algebra \(\mathbb{C} [L^{u,e}]\) [\textit{C. Geiß} et al., Adv. Math. 228, No. 1, 329--433 (2011; Zbl 1232.17035); \textit{K. R. Goodearl} and \textit{M. T. Yakimov}, ``The Berenstein-Zelevinsky quantum cluster algebra conjecture'', Preprint, \url{arXiv:1602.00498}] in terms of monomial realization of Demazure crystals, where \(L^{u,e}\) is the reduced double Bruhat cell of type \((u,e)\). The relation between \(\Delta(k;\mathbf{i})\) on \(G^{u,e}\) and on \(L^{u,e}\) is described in Proposition 6.3 below. We also present the corresponding results for type \(\mathrm{A}_r\) though the results for all initial cluster variables have been obtained in [\textit{Y. Kanakubo} and \textit{T. Nakashima}, SIGMA, Symmetry Integrability Geom. Methods Appl. 11, Paper 033, 32 p. (2015; Zbl 1329.13039)]. Cluster algebras, Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over arbitrary fields, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Group actions on varieties or schemes (quotients), \(p\)-adic cohomology, crystalline cohomology Cluster variables on double Bruhat cells \(G^{u,e}\) of classical groups and monomial realizations of Demazure crystals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 motivated by the construction of 3-dimensional Calabi-Yau (CY) algebras from dimer models, which was in turn motivated by theoretical physics. Previous work by Broomhead, Mozgovoy and Reineke, and by Davison had shown that certain dimer models give rise to 3-dimensional CY algebras via a quiver with potential construction. These algebras, called toric orders, are meant to be thought of as ``noncommutative toric resolutions'' of certain toric varieties. The article under review generalizes the idea of a dimer model to that of a weighted quiver polyhedron, and generalizes the idea of a toric order to that of a cancellation algebra.
An algebra is called a cancellation algebra if it is the category algebra of a category in which every morphism is both epic and monic; this gives rise to a cancellation law in the associated category algebra.
A quiver polyhedron is a strongly connected quiver (i.e., every vertex lies on an oriented cycle) along with two disjoint sets of cycles in the quiver, \(Q_2^+\) and \(Q_2^-\). These sets of cycles are required to satisfy:
(PO) every arrow of \(Q\) appears in exactly one cycle of \(Q_2^+\) and appears in exactly one cycle of \(Q_2^-\);
(PM) at each fixed vertex, the incidence graph of the cycles and arrows that meet there is connected.
The author shows that this is essentially the same as drawing a strongly connected quiver on a compact, orientable surface, in such a way that erasing the quiver leaves simply connected pieces bounded by cycles. A positive weighting on a quiver polyhedron is an assignment of a positive integer \(E_c\) to each cycle \(c\) in \(Q_2^+\cup Q_2^-\) such that \(E_c|c|>2\), where \(|c|\) denotes the number of arrows in \(c\). This roughly corresponds to putting some orbifold singularities in the surface described above. Now there is a natural superpotential associated to a weighted quiver polyhedron, obtained by taking the difference of the positive and negative cycles but with some additional technicalities from the weighting. A positive grading on a weighted quiver polyhedron is an assignment of a positive real number \(R_a\) to each arrow such that \(R_cE_c\) is the same for every cycle in \(Q_2^+\) and \(Q_2^-\). The existence of such a grading means that the superpotential is homogeneous; the results of the paper do not depend on the exact choice of grading.
A group acting on a weighted quiver polyhedron without fixing any vertices corresponds to a cover morphism between orbifolds and Galois cover on the level of Jacobi algebras. The author proves in Theorem 5.8 and Lemma 5.9 that the property of being a cancellation algebra, and that of admitting a positive grading are compatible with the notion of Galois covers. This is important because it allows one to reduce to the case of trivial weighting, with two exceptions (Theorem 5.10).
The main theorem then states that if a positively graded cancellation algebra is 3-CY, it necessarily comes from a graded weighted quiver polyhedron (Theorem 6.1). This is used to show in Section 7 that if we restrict our cancellation algebras to the case of toric orders, the underlying manifold of the associated quiver polyhedron must be a torus. In other words, a toric order which is 3-CY must come from a dimer model on a torus (Theorem 7.7). noncommutative resolutions; quivers; dimer models; Calabi-Yau algebras; toric orders; weighted quiver polyhedra; orientable surfaces; cancellation algebras; Galois covers Bocklandt, R.: Calabi-Yau algebras and quiver polyhedra Representations of orders, lattices, algebras over commutative rings, Representations of quivers and partially ordered sets, Global theory and resolution of singularities (algebro-geometric aspects), Noncommutative algebraic geometry, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), String and superstring theories; other extended objects (e.g., branes) in quantum field theory Calabi-Yau algebras and weighted quiver polyhedra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 motivation of the paper comes from the following result by \textit{M. Brion} [J. Reine Angew. Math. 421, 125--140 (1991; Zbl 0729.14015)]. Let a complex reductive group \(G\) act on a smooth projective complex variety \(X\), and let \(T\) be a maximal torus of \(G\) with Weyl group \(W\). Denote by \(X^{\mathrm{ss}}_G\) the \(G\)-semistable locus, and similarly for \(X^{\mathrm{ss}}_T\). Then there is a canonical isomorphism between equivariant cohomologies
\[
\phi: H^\bullet_G(X^{\mathrm{ss}}_G; \mathbb{Q}) \to H^\bullet_T(X^{\mathrm{ss}}_T; \mathbb{Q})^a
\]
where the superscript \(a\) means the \(W\)-antisymmetric part. Assume that \(X^{\mathrm{ss}}_G\) equals the stable locus \(X^{\mathrm{s}}_G\), then \(H^\bullet_G(X^{\mathrm{ss}}_G, \mathbb{Q})\) is isomorphic to the cohomology of the GIT quotient \(X /\!/ G\). Therefore, it makes sense to compare the pairings \(\int_{X/\!/ G} \sigma_1 \cup \sigma_2\) and \(\int_{X/\!/T} \phi(\sigma_1) \cup \phi(\sigma_2)\), for \(\sigma_1, \sigma_2 \in H^\bullet(X/\!/G; \mathbb{Q})\).
The analogue in symplectic geometry is settled by [\textit{S. Martin}, ``Symplectic quotients by a nonabelian group and by its maximal torus'', Preprint, \url{arXiv:math/0001002}], for a Hamiltonian action of a connected compact Lie group \(G \supset T\) on a symplectic compact manifold \(X\), whose moment map is proper with \(0\) being a regular value. Symplectic reductions replace GIT quotients here. Martin proved that the two pairings differ by the factor \(|W|\). This result may also be deduced from general nonabelian localization theorems such as \textit{L. C. Jeffrey} and \textit{F. C. Kirwan} [Topology 34, No. 2, 291--327 (1995; Zbl 0833.55009)].
The paper deals with the general algebraic set-up as follows. Let \(k\) be a field and \(G\) be a connected reductive \(k\)-group acting on a projective variety \(X\) equipped with a \(G\)-linearized ample line bundle, such that \(X^{\mathrm{s}}_T = X^{\mathrm{ss}}_T \neq \emptyset\). Chow \(0\)-cycles \(\sigma \in A_0(X/\!/G)_{\mathbb{Q}}\) of nonzero degree replace the equivariant cohomology classes here. Let \(T \subset G\) be a maximal torus with Weyl group \(W\). The author considers the proportionality factor
\[
r^{X,\sigma}_{G,T} = \int_{X/\!/T} c_{\mathrm{top}}(\mathfrak{g}/\mathfrak{t}) \cap \tilde{\sigma} \bigg/ \int_{X/\!/G} \sigma
\]
where \(\tilde{\sigma} \in A_0(G/\!/T)_{\mathbb{Q}}\) stands for a certain lift of \(\sigma\); we refer to the paper for other details. Theorem 0.3 asserts that \(r^{X, \sigma}_{G,T} = r_G\) is actually an invariant of the group \(G\). Furthermore, when the root system of \(G\) decomposes into a product of simple ones of type \(\mathsf{A}\), then \(r_G = |W|\).
It is actually possible to show that \(r_G = |W|\) unconditionally, by reducing to Martin's result over \(k = \mathbb{C}\). This is briefly explained in the section 6, which makes use of relative GIT to deform the base field. geometric invariant theory; symplectic geometry Geometric invariant theory, (Equivariant) Chow groups and rings; motives, Classical groups (algebro-geometric aspects) A ratio of integration between quotients in geometric invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0547.00007.]
Let \(M_ g\) (resp. \(A_ g)\) be the moduli space of complete smooth curves of genus g (resp. principally polarized abelian varieties of dimension g). Here we work only over the field of complex numbers. There is a canonical holomorphic map \(i: M_ g\to A_ g\) by associating a curve C with its Jacobian variety J(C), which is known to be injective (Torelli theorem). There arises naturally a problem to characterize the image of i, which is now called ''the Schottky problem'' (in a wide sense). Here the author gives a compact but very well organized review on recent development on this subject. According to \textit{D. Mumford}'s earlier work ''Curves and their Jacobians'' (1975; Zbl 0316.14010) the author classifies 4 approaches and gives recent results in each direction: [1] algebraic equations (to find the relations of theta constants characterizing the closure of Im i in \(A_ g)\); [2] d trisecants (to use special properties of the Kummer surface of the Jacobian); [3] geometry of the moduli space (to use a special embedding \(A_ g(2,4)\) in \({\mathbb{P}}^ N\) by theta constants); [4] rings of differential operators (to characterize the image as the set of points where a certain theta function with the corresponding modulus satisfies the so-called K-P equation (abbr. to Kadomtsev-Petviashvili-Novikov's conjecture). The author himself gives a main contribution to the case [3] with van Geemen. The most definite result is Shiota's solution for the Novikov's conjecture by using the theory of K-P hierarchy due to M. Mulase. Now almost all papers in the references have been published: e.g. \textit{B. van Geemen} [Invent. Math. 78, 329-349 (1984; Zbl 0568.14015)]; \textit{T. Shiota} [Invent. Math. 83, 333-382 (1986)] and \textit{G. E. Welters} [Ann. Math., II. Ser. 120, 497- 504 (1984; Zbl 0574.14027)]. moduli space of complete smooth curves; principally polarized abelian varieties; Torelli theorem; Schottky problem; Kummer surface of the Jacobian; theta constants; K-P equation Jacobians, Prym varieties, Families, moduli of curves (analytic), Theta functions and abelian varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Partial differential equations of mathematical physics and other areas of application, Families, moduli of curves (algebraic) The Schottky problem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathcal N}_0\) denote the subvariety of the moduli space \({\mathcal A}_g\) of principally polarized abelian varieties of dimension \(g\) whose theta divisor \(\Theta\) admits a singularity. It is known that for \(g\geq 5\), \({\mathcal N}\) consists of 2 components, the subvariety \(\theta_{\text{null}}\) where \(\Theta\) has a (vanishing even) thetanull and another component \({\mathcal N}_0'\). This paper concerns the analysis of the intersection \({\mathcal N}_0'\cap\theta_{\text{null}}\). In [Int. Math. Res. Not. 2007, No. 15, Article ID rnm045, 15 p. (2007; Zbl 1132.14026)] \textit{S. Grushevsky} and \textit{R. Salvati Manni} show that a thetanull is a limit of singularities which are not thetanulls, if and only if it is not an ordinary double point.
The main result of the paper to be reviewed is a different proof of this result. Whereas the proof of Crushevsky and Salvati Manni [loc. cit.] applies only to singularities of theta divisors, the new proof is a consequence of a more general result. In fact, it is shown that in any sufficiently general deformation of an even analytic function having an isolated singularity at \(x=0\) which is not an ordinary double point, there is a pair of distinct conjugate singularies \(\{x, -x\}\) converging to the singularity at \(x=0\). This immediately implies the above-mentioned statement. Finally, some applications and open problems are given. singularities of the theta divison; even thetanull R. Smith and R. Varley, A splitting criterion for an isolated singularity at \( x = 0\) in a family of even hypersurfaces, Manuscripta Math. 137 (2012), 233-245. Theta functions and curves; Schottky problem, Deformations of singularities A splitting criterion for an isolated singularity at \(x = 0\) in a family of even hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities To a quiver, \textit{H. Nakajima} [in: Representation theory of algebras and related topics. Proceedings of the workshop, Mexico City, Mexico, August 16--20, 1994. Providence, RI: American Mathematical Society. 139--157 (1996; Zbl 0870.16008)] introduced certain associated framed moduli spaces. The first author of this article proved that these framed moduli spaces, called quiver flag varieties, has a tilting bundle. This was done by generalizing Beilinson and Kapranov's construction.
In this article, the construction is prolonged to the toric case by showing that every toric quiver flag variety can be reconstructed as a fine moduli space of cyclic modules over the endomorphism algebra of the tilting bundle.
The authors recall the constrictions and basic geometric principles of quiver flag varieties, i.e., framed quiver moduli, as developed by Nakajima, Reineke and the first author [Duke Math. J. 156, No. 3, 469--500 (2011; Zbl 1213.14026)]. Let \(\Bbbk\) be an algebraically closed field of characteristic \(0\), \(Q\) a connected, acyclic quiver with a unique source and \(l+1\) vertices in \(Q_1\). Fix a dimension vector \(\underline r=(r_i)\in\mathbb N^{l+1}\) with \(r_0=1\). The group \(G=\prod_{i=0}^l\operatorname{GL}(r_i)\) acts by conjugation on \(\operatorname{Rep}(Q,\underline r)=\bigoplus_{a\in Q_1}\operatorname{Hom}(\Bbbk^{r_{t(a)}},\Bbbk^{r_{h(a)}})\), and the \textit{quiver flag variety} associated to the pair \((Q,r)\) is defined as the GIT quotient \(Y:=\text{Rep}(Q,\underline r)\slash_\chi G\) for the linearization \(\chi=(-\sum_{i=1}^l r_i,1,\dots,1)\in G^\vee\). The quotient is non-empty iff \(r_i\leq s_i:=\underset{h(a)=i}{a\in Q_1}\sum r_{t(a)}\) holds for \(i>0\), for which \(Y\) is a smooth Mori Dream space of dimension \(\sum_{i=1}^l r_i(s_i-r_i).\) It can be obtained from a tower of Grassmann bundles \(Y:=Y_l\rightarrow Y_{l-1}\rightarrow\cdots\rightarrow Y_1\rightarrow Y_0=\text{Spec}\Bbbk\), where \(Y_i\) is isomorphic to the Grassmannian of rank \(r_i\) quotients of a fixed locally free sheaf of rank \(s_i\) on \(Y_{i-1}.\)
Quiver varieties come with a collection of vector bundles \( \mathcal W_1,\dots,\mathcal W_l\) defining algebraic invariants. For \(i>0\) the Grassmann bundle \(Y_i\) over \(Y_{i-1}\) gives the tautological quotient bundle of rank \(r_i\) on \(Y\) obtained as the pullback under the morphism \(\pi_i:Y\rightarrow Y_i\) in the tower. The invertible sheaves \(\det(\mathcal W_i),\dots,\det(\mathcal W_l)\) give an integral basis for the Picard group of \(Y\). Then the results of Beilinson and Kapranov generalizes as follows: When \(\text{Young}(k,l)\) denotes the set of Young diagrams with no more that \(k\) columns and \(l\) rows, for each \(\lambda\in\text{Young}(k,l)\) and for any vector bundle \(\mathcal W\) of rank \(r\), there is obtained a vector bundle \(\mathbb S^\lambda\mathcal W\) whose fibre over each point is the irreducible \(\text{GL}(r)\)-module of highest weight \(\lambda\). The first author [loc. cit.] proved that the vector bundle on \(Y\) given as
\[
E=\underset{\lambda_i\in\text{Young}(s_i-r_i,r_i)}{1\leq i\leq l}\sum\mathbb S^{\lambda_1}\mathcal W_1\otimes\cdots\otimes\mathbb S^{\lambda_l}\mathcal W_l
\]
is a tilting bundle on \(Y\). This says that the bounded derived category of coherent sheaves on \(Y\) is equivalent to the bounded derived category of finite-dimensional modules over \(A=\text{End}_{\mathcal O_Y}(E)\).
With this thoroughly explained, the authors are able to describe their main result. \textit{A. Bergman} and \textit{N. J. Proudfoot} [Pac. J. Math. 237, No. 2, 201--221 (2008; Zbl 1151.18011)] compared a smooth projective variety with a tilting bundle to a fine moduli space of modules over the endomorphism algebra. The moduli space for the modules over the tilting algebra \(E\) above is given by listing the indecomposable summands as \(E_0,E_1,\dots,E_n\) with \(E_0\cong\mathcal O_Y\), and considering the dimension vector \(\mathbf{v}=(v_j)\in\mathbb N^{n+1}\) satisfying \(v_j=\text{rk}(E_j)\) for all \(0\leq j\leq n\). Then for a special choice of \(0\)-generated stability condition \(\theta\), the fine moduli space \(\mathcal M(A,\mathbf{v},\theta)\) of isomorphism classes of \(\theta\)-stable modules with dimension vector \(\mathbf{v}\) was constructed by \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)] using GIT. Because each bundle \(E_j\) is globally generated, the first author proves that there is a universal morphism \(f_E:Y\rightarrow\mathcal M(A,\mathbf{v},\theta)\) which in the present case is a closed immersion. The work of Bergman and Proudfoot [loc. cit.] proves that \(f_E\) identifies \(Y\) with a connected component of \(\mathcal M(A,\mathbf{v},\theta)\) because \(Y\) is smooth, \(E\) is a tilting bundle, and then finally because the stability condition \(\theta\) is great.
This article considers the particular case where \(r_i=1\) for all \(1\leq i\leq l\). Then \(G\) is an algebraic torus and \(Y\) a toric variety: A \textit{toric quiver flag variety}. The toric fan \(\Sigma\) is described directly, and \(Y\) is a tower of projective space bundles. The main result of the article is the following very precise result stated verbatim:
Theorem. Let \(Y\) be a toric quiver flag variety. The morphism \(f_E:Y\rightarrow\mathcal M(A,\mathbf{v},\theta)\) is an isomorphism.
This says that toric quiver flag varieties can be applied to reconstruct varieties from its tilting bundle. The special case when \(Y\) is projective space, recovers Beilinson's result that \(\mathbb P^n\) can be reconstructed from the tilting bundle \(\bigoplus_{0\leq i\leq n}\mathcal O_{\mathbb P^n}(i),\) and the result indicates that toric quiver flag varieties are multigraded analogues of projective space.
The article gives an explicit connection between tilting theory and invariant theory. The GIT quotients, i.e., the moduli spaces are explicitly given, and so are the tilting bundles and their bounded derived equivalences. An elegant article proving Beilinson's well-known results in an explicit and understandable way. toric variety; toric quiver flag variety; Beilinson's theory; endomorphism algebra; tilting bundle; framed quiver moduli; algebraic torus; Young diagram; fine moduli Noncommutative algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Representations of quivers and partially ordered sets, Derived categories, triangulated categories Reconstructing toric quiver flag varieties from a tilting bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C^0\subset \mathbb{C}^2\) be a reduced pure dimension one curve with one singular point \(p\). A conjecture of Oblomkov and Shende relates the HOMFLY polynomial of the link of the singularity at \(p\) with topological invariants of the punctual Hilbert scheme parametrizing zero-dimensional subschemes of \(C^0\) with topological support at \(p\). This conjecture has a natural physical interpretation in terms of the so-called large \(N\) duality for conifold transitions. An interesting question concerns the construction of counting invariants which would correspond to general knots via large \(N\) duality. The paper under review proposes such a construction.
To be more precise, let \(X\) be a projective Calabi--Yau threefold with a single conifold singularity \(q\), whose formal neighbourhood is isomorphic to the formal neighbourhood of the origin in the singular hypersurface \(xz-yw=0\) in \(\mathbb{C}^4\), assume that there exists a Weil divisor \(\Delta\cong \mathbb{P}^2\) containing \(q\) which is locally determined by the equation \(z=0\), and let \(\Gamma \subset X_0\) be a reduced irreducible plane curve contained in \(\Delta\) and passing through \(q\). Suppose furthermore that \(\Gamma\) is singular in \(q\) and smooth otherwise. Blowing up \(X_0\) along \(\Delta\) gives a crepant resolution \(X\to X_0\). Denote by \(C\) the strict transform of \(\Gamma\) and by \(C_0\) the exceptional \((-1,-1)\)-curve. The authors consider so-called \(C\)-framed stable pairs, which by definition are pairs \((F,s)\) consisting of a sheaf \(F\) on \(X\) topologically supported on \(C\cup C_0\) and a section \(s: \mathcal{O}_X \to F\) with zero-dimensional kernel subject to a further technical condition. There is a scheme \(P(X,C,r,n)\) parametrizing \(C\)-framed stable pairs such that \(\chi(F)=n\) and \(\text{ch}_2(F)=[C]+r[C_0]\) (\(r\geq 0\)). The main result of the paper expresses, roughly speaking, the generating function of the topological Euler numbers of the spaces \(P(X,C,r,n)\) as a product of the generating function of the similarly defined spaces \(P(X,C_0,r,n)\) and some further functions. The idea is to use the wall-crossing formalism developed by Joyce--Song and Kontsevich--Soibelman for a certain one-parameter family, depending on \(b\in \mathbb{R}\), of weak stability conditions on the heart \(\mathcal{A}\) of a certain perverse t-structure on \(D^b(X)\), the bounded derived category of coherent sheaves on \(X\). More precisely, the authors show that the results obtained for usual stable pairs by \textit{S. Toda} in [Saito, Masa-Hiko (ed.) et al., New developments in algebraic geometry, integrable systems and mirror symmetry. Papers based on the conference ``New developments in algebraic geometry, integrable systems and mirror symmetry'', Kyoto, Japan, January 7--11, 2008, and the workshop ``Quantum cohomology and mirror symmetry'', Kobe, Japan, January 4--5, 2008. Tokyo: Mathematical Society of Japan. Advanced Studies in Pure Mathematics 59, 389--434 (2010; Zbl 1216.14009)] and [Duke Math.\ J.\ 149, No.\ 1, 157--208 (2009; Zbl 1172.14007)] carry over to their framed setting.
Section 2 contains some facts concerning slope limit stability and the \(C\)-framed subcategory \(\mathcal{A}^C\) which is the analogue of the above \(\mathcal{A}\) in this setting. In Section 3 the main result is proved. The first step is a wall-crossing formula relating invariants for \(b\ll 0\) to small \(b>0\) invariants. This is in fact established in Appendix A.1. The second step is to find a connection between moduli spaces of stable \(C\)-framed objects for small \(b>0\) with the Hilbert scheme invariants. This is roughly done as follows. The authors construct a certain moduli stack \(Q(X,C,r,n)\) of decorated sheaves on \(X\) which turns out to be a \(\mathbb{C}^*\)-gerbe over a relative Quot-scheme where the latter is geometrically bijective with a certain nested Hilbert scheme. On the other hand, \(Q(X,C,r,n)\) is equipped with a geometric bijection to the moduli space of \(C\)-framed pairs. A stratification computation then concludes the proof.
Section 4 is concerned with motivic invariants. More precisely, the authors compare a certain motivic Hilbert scheme series with the motivic invariants one gets when a (partly) conjectural construction employing motivic vanishing cycles for formal functions due to Kontsevich and the third author is applied. plane curve singularities; stable pairs; framed sheaves; conifold transitions; Donaldson-Thomas invariants; wall-crossing Jiang, Y.: The moduli space of stable coherent sheaves via non-archimedean geometry. arXiv:1703.00497 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Calabi-Yau manifolds (algebro-geometric aspects), Representations of quivers and partially ordered sets, Singularities of curves, local rings HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of the paper under review establishes a Gromov-Witten/Pairs (GW/P) correspondence for Fano or Calabi-Yau 3-folds that are complete intersections in products of projective spaces.
Let \(X \subset \mathbb P^{n_1}\times \cdots \mathbb P^{n_m}\) be a Fano or Calabi-Yau complete intersection and \(\gamma_i\in H^{2*}(X,\mathbb Q)\) be even classes, and \(\beta\in H_2(X,\mathbb Z)\) with \(d_\beta=\int_\beta c_1(X)\), then GW/P correspondence can be expressed as follows:
Firstly, the generating series of class \(\beta\) stable pair invariants with descendents is a rational function, i.e.,
\[
Z_P \left(X;q \mid \tau_{\alpha_1-1}(\gamma_1)\cdots \tau_{\alpha_\ell-1}(\gamma_\ell)\right)_\beta\in\mathbb Q(q)
\]
and secondly there is a correspondence
\[
\begin{aligned} (-q)^{-d_\beta/2}Z_P \bigg(X;q \mid \tau_{\alpha_1-1}(\gamma_1)&\cdots \tau_{\alpha_\ell-1}(\gamma_\ell)\bigg)_\beta \\&=(-iu)^{d_\beta}Z'_{GW} \bigg(X;q \mid \overline{\tau_{\alpha_1-1}(\gamma_1)\cdots \tau_{\alpha_\ell-1}(\gamma_\ell)}\bigg)_\beta \end{aligned}
\]
under the change of variable \(-q=e^{iu}\). Here, the right hand side is the generating series of class \(\beta\) Gromov-Witten invariants with descendants (of possibly disconnected domain curves). The over line in the right hand side is a correspondence rule defined by means of a universal correspondence matrix indexed by partitions of positive size and constructed from the capped descendent vertex in an earlier work of the authors of this paper. In the case where all descendents are primary or stationary, the over line correspondence is the identity and one recovers the standard GW/P correspondence conjectured earlier. The proof of the main theorem is by means of the degeneration scheme established by Maulik-Pandharipande. To run this scheme the authors of this paper prove GW/P correspondences for relative and descendent insertions in several simpler geometries. Moreover, the degeneration scheme requires the study of relative theories of \(\mathbb P^1\)-bundles over surfaces \(S\) where \(S\) is either(i) a toric surface, (ii) a \(K3\) surface,(iii) or a \(\mathbb P^1\)-bundle over a higher genus curve \(C\).The authors prove the descendent correspondences for the relative surface geometries (i)-(iii) among which the most difficult ones to establish are the surface geometries (iii).
Pandharipande-Thomas' recent proof of the full Katz-Klemm-Vafa conjecture for the Gromov-Witten theory of \(K3\) surfaces uses the GW/P correspondences for nontoric hypersurface Calabi-Yau 3-folds established in this paper. stable pair invariants; Gromov-Witten invariants; complete intersection threefolds; descendent invariants PP R.~Pandharipande and A.~Pixton. \newblock Gromov-Witten/Pairs correspondence for the quintic 3-fold. \newblock \em J. Amer. Math. Soc., \newblock Vol. 30, pp. 389--449, 2017. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles on curves and their moduli Gromov-Witten/pairs correspondence for the quintic 3-fold | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 motivation for the article is to extend the construction of the moduli spaces of polarized abelian varieties \(A_g\) and their compactifications to a non-commutative setting. In particular, the toroidal compactification due to
\textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Mumford} and \textit{B. Saint-Donat} [``Toroidal embeddings. I'', Lect. Notes Math. 339 (1973; Zbl 0271.14017)] has an interpretation as the moduli space of projective stable pairs \((X,D)\), and the authors generalize this notion. Consider a given non-commutative connected reductive algebraic group \(G\) defined over an algebraically closed field \(k\) of characteristic \(0\). Let \(B\) and \(B^-\) be opposite Borel subgroups of \(G\) with unipotent radicals \(U\), \(U^-\). Then \(T=B\cap B^-\) is a maximal torus of \(G\). The corresponding Weyl group is \(N_G(T)/T\) where \(N_G(T)\) is the normalizer of \(T\). There exists a unique automorphism \(\theta\) of \(G\) such that \(\theta(t)=t^{-1}\) for all \(t\in T\). Then \(\theta\) is involutive; it exchanges \(B\) with \(B^-\), stabilizes \(N\) and \(T\) and fixes pointwise \(W\). \(G\times G\) is a connected reductive group; \(B^-\times B\) and \(B\times B^-\) are opposite Borel groups with common torus \(T\times T\). \(\Theta:(g_1,g_2)\mapsto ((\theta(g_2),\theta(g_1))\) is an involutive automorphism of \(G\times G\), and the diagonal subgroups \(\text{diag}~G\), \(\text{diag}~T\), \(\text{diag}~N\) are \(\Theta\)-invariant, and the induced map on \(\text{diag}~W\) is the identity. The group \(G\) comes equipped with two commuting \(G\)-actions by left and right multiplication. The authors define a class of reductive varieties consisting of irreducible varieties with a \(G\times G\)-action. This class contains the group \(G\) and all of its equivariant embeddings, and is closed under (flat) degenerations with reduced and irreducible fibers. Also a ``toroidal'' compactification is constructed. The adjoint of a \(G\times G\)-variety \(X\) is the \(G\times G\)-variety \(\Theta(X)\) with action twisted by \(\Theta\). A \(G\times G\)-variety \(X\) is self adjoint if it admits an involutive automorphism \(\Theta_X\) such that \(\Theta_X(\gamma x)=\Theta(\gamma)\Theta_X(x)\) for all \(\gamma\in G\times G\) and \(x\in X\). This condition plays a fundamental role throughout the article, in the classification of the stable reductive varieties. This article concentrates on affine stable reductive varieties. They are all classified in terms of combinatorial data; these turn out to be complexes of cones in the weight space of \(G\), invariant under the action of the Weyl group \(W\). All their one parameter degenerations are described: They arise as in the toric case from certain reductive varieties for the larger group \(G_m\times G.\) From the classification it follows that stable reductive varieties are in bijective correspondence with stable toric varieties equipped with a compatible \(W\)-action. For the reductive varieties there exists an associative multiplication law, making each of them into an algebraic semigroup. The authors name these reductive semigroups. The authors finally study families of stable reductive varieties (in the same manner as
\textit{E. B. Vinberg} [in: Lie groups and Lie algebras: E. B. Dynkin's Seminar, Transl., Ser. 2, Am. Math. Soc. 169, 145--182 (1995; Zbl 0840.20041)] in his construction of the Vinberg family) and, more generally, varieties with reductive group action. Given a reductive variety \(X\), the authors constructs a family of such varieties with general fiber \(X\), and they prove that any (reduced and irreducible) degeneration of \(X\) arises in that way. They construct a fine moduli space for families of \(G\)-subvarieties of a fixed \(G\)-module. This applies to all multiplicity-finite \(G\)-varieties, and is an application and a generalization of the multigraded Hilbert scheme of \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No.4, 725--769 (2004; Zbl 1072.14007)]. This result plays a fundamental role in the follow up article of this, where the moduli space of projective pairs \((X,D)\) is constructed. The article is important, and proves a deep application of Lie groups and geometric invariant theory. Also, the generalization of the classification of the abelian varieties is successful. non-abelian group action; moduli of abelian varieties; adjoint group schemes Alexeev, V.; Brion, M., \textit{stable reductive varieties I: affine varieties}, Invent. Math., 157, 227-274, (2004) Noncommutative algebraic geometry, Geometric invariant theory, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies Stable reductive varieties. I: Affine varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(B\) be an indefinite quaternion algebra over \(\mathbb Q\), and \(G\) be the `standard' split unitary group of 2 variables over \(B\). When \(B\) is split, \(G\) is the usual symplectic group of 4 variables. For any integer \(N >0\), the principal congruence group \(\Gamma(N)\) acts on the Siegel upper half space, and let \(Y(N)\) be the corresponding nonsingular moldular 3-fold (after adding cusps and resolving the singularities).
In this paper, the author proves that \(Y(N)\) is of general type when \(N d(B)\) is sufficiently large and \(d(B) >1\), where \(d(B)\) is the discriminant of \(B\). An explicit bound is also given. The case \(d(B) =1\) was proved by \textit{T. Yamazaki} using similar method [Am. J. Math. 98, 39-53 (1976; Zbl 0345.10014)]. The main idea is to use \textit{F. Arakawa}'s explicit formula [J. Math. Soc. Japan 33, 125-145 (1981; Zbl 0458.10023)] on the space of cusp forms for \(\Gamma (N)\) and the defect technique of \textit{F. Knöller} [Manuscr. Math. 37, 135-161 (1982; Zbl 0486.14009)] and \textit{S. Tsuyumine}'s [Invent. Math. 80, 269-281 (1985; Zbl 0576.14036)] on studying cusps. modular 3-folds of general type; quaternion modular forms; defects; quaternion unitary groups; quaternion algebra; Siegel modular groups Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Surfaces of general type, \(3\)-folds, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic aspects of modular and Shimura varieties Modular 3-folds obtained from quaternion unitary groups of degree 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 of the present paper are interested in the weighted projective space \(X=\mathbb{P}(1,1,1,3)\) and an anticanonical \(K3\) surface \(Y\) in it. More precisely, they relate mirror symmetry of the two in an intricate way.
In mirror symmetry, there is an \(A\)-side and a \(B\)-side. To the \(A\)-side are associated a symplectic geometry (such as \(X\) or \(Y\)) and its symplectic Gromov-Witten invariants. To the \(B\)-side are associated a complex-geometric structure (such as a family of mirror varieties) and its \(B\)-model invariants.
The mirror map is a transcendental function that relates the invariants of the \(A\)- and \(B\)-side. It is typically computed from explicit hypergeometric functions. Denote by \(w_d\) the Taylor coefficients of the mirror map of \(Y\). The first author previously conjectured that the \(w_d\) are computed from specific \(B\)-model invariants of \(X\). The authors prove this conjecture in the present article.
The main result of the present paper thus is an expression of \(w_d\) in terms of a \(B\)-model invariant of \(X\). This invariant is expressed as an intersection product on the \textit{I. Ciocan-Fontanine} and \textit{B. Kim} [Algebr. Geom. 1, No. 4, 400--448 (2014; Zbl 1322.14083)] moduli space of the 2-pointed quasimap invariants. In fact, the authors introduce a birational modification of this moduli space, which yields the same invariants, yet is easier to do intersection theory on. The authors describe their moduli space as a compact toric orbifold and compute its Chow ring.
The main theorem of the present paper then is the result of an impressive feat of computation, which is possible thanks to the toric description of the moduli space, and the explicit description of the mirror map. \(j\)-invariant; mirror symmetry; moduli space of quasimaps Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Moduli space of quasimaps from \(\mathbb{P}^1\) with two marked points to \(\mathbb{P}(1,1,1,3)\) and \(j\)-invariant | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Since Brieskorn gave the connection between simple Lie algebras and ADE singularities in 1970, it has become an important problem to establish connections between singularities and solvable (nilpotent) Lie algebras. Recently, we have constructed some new natural maps between the set of complex analytic isolated singularities and the set of finite dimensional solvable (nilpotent) Lie algebras. Furthermore, we use these new Lie algebras to obtain Torelli-type theorems of simple elliptic singularities. The main purpose of this paper is to summarize the results that we have obtained on new Lie algebras arising from isolated hypersurface singularities. isolated hypersurface singularity; simple elliptic singularities; isolated complete intersection singularities; fewnomial singularities; Lie algebra of derivations; solvable Lie algebras; semi-simple Lie algebras; Torelli theorems; Milnor algebras; Tjurina algebras; Yau algebras; hessian Lie algebras Singularities in algebraic geometry, Local complex singularities On derivation Lie algebras of singularities and Torelli-type theorems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In a celebrated preprint from 1968, \textit{J. F. Nash jun.} [cf. Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)] introduced the space of arcs on an isolated singularity of a surface \(X\), and he asked if there is a bijection between the irreducible components of the space of arcs on \(X\) and the exceptional components of a minimal resolution of singularities of \(X\). This problem was extended recently by \textit{S. Ishii and J. Kollár} [Duke Math. J. 120, 601--620 (2003; Zbl 1052.14011)]
to any dimension, they also gave a counterexample in dimension 4. But in dimension two there is no counterexample and until recently there were very few positive answers to this problem by \textit{A. J. Reguera} [Manuscr. Math. 88, 321--333 (1995; Zbl 0867.14012)] for minimal singularities, \textit{A. Reguera} and \textit{M. Lejeune-Jalabert} [Rev. Mat. Iberoam. 19, No.2, 581--612 (2003; Zbl 1058.14006)] for sandwich singularities, \textit{C. Plénat} [Ann. Inst. Fourier 55, No.3, 805--823 (2005; Zbl 1080.14021)] for the singularity \(D_n\) and the reviewer for a very large class of singularities depending only in the dual graph of the exceptional set in the minimal resolution of singularities.
In the paper under review, the author introduces a map from the set of fat arcs to the set of valuations. A fat arc is an arc which does not factor through any proper closed subvarieties. This map is a generalization of the Nash map and the map defined by \textit{L. Ein, R. Lazarsfeld} and \textit{M. Mustata} [Compos. Math. 140, No.5, 1229--1244 (2004; Zbl 1060.14004)]. This paper gives an affirmative answer for a non normal toric variety. Another interesting example is the arc determined by a conjugacy class of a finite group \(G\) which gives the quotient variety \(X=\mathbb C^n/G\). The restriction of the map defined by the author onto a subset of these arcs coincides with the McKay correspondence. singularity; toric variety S. Ishii, ''Arcs, valuations and the Nash map,'' J. Reine Angew. Math., vol. 588, pp. 71-92, 2005. Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves Arcs, valuations and the Nash map | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper provides a classification of the simple integrable modules of double affine Hecke algebras via perverse sheaves. Let \(\underline G\) be a simple connected simply connected linear algebraic group. Let \(\underline{\text{Lie}}\,\underline G\) denote the Lie algebra of \(\underline G\), let \(\underline{\text{Lie}}\,\underline H\subset\underline{\text{Lie}}\,\underline G\) be a Cartan subalgebra and let \(\underline{\text{Lie}}\,\underline B\subset\underline{\text{Lie}}\,\underline G\) be a Borel subalgebra containing \(\underline{\text{Lie}}\,\underline H\). Let \(\underline\Phi\) be the root system of \(\underline G\) and let \(\Phi^\vee\) be the root system dual to \(\underline\Phi\). Let \(\{\alpha_i:i\in\underline I\}\), \(\{\alpha_i^\vee:i\in\underline I\}\) be the set of simple roots and of simple coroots, respectively. Let \(I:=\underline I\sqcup\{0\}\). Let \(\underline W\), \(W\) be the Weyl group and the affine Weyl group of \(\underline G\). We identify \(\underline I\) (resp. \(I\)) with the set of simple reflections in \(\underline W\) (resp. \(W\)). Let \(s_i\in W\) be the simple reflection corresponding to \(i\in I\). For all \(i,j\in I\), let \(m_{ij}\) denote the order of the element \(s_is_j\) in \(W\). Let \(\underline X\) be the weight lattice of \(\underline\Phi\) and let \(Y^\vee\) be the root lattice of \(\underline\Phi v\). Let \(\{\omega_i:i\in\underline I\}\) be the set of fundamental weights. Consider the lattices \(Y=\bigoplus_{i\in I}\mathbb{Z}\alpha_i\subset X=\mathbb{Z}\delta\oplus\bigoplus_{i\in I}\mathbb{Z}\omega_i\), \(Y^\vee=\bigoplus_{i\in I}\mathbb{Z}\alpha_i^\vee\), where \(\delta\) is a new variable. There is unique pairing \(X\times Y^\vee\to\mathbb{Z}\) such that \((\omega_i:\alpha_j^\vee)=\delta_{ij}\) and \((\delta:\alpha_j^\vee)=0\).
The double affine Hecke algebra \(\mathbf H\) is the unital associative \(\mathbb{C}[q,q^{-1},t,t^{-1}]\)-algebra generated by \(\{t_i,x_\lambda:i\in I\), \(\lambda\in X\}\) modulo the following defining relations:
\[
x_\delta=t,\quad x_\lambda x_\mu=x_{\lambda+\mu}(t_i-q)(t_i+1)=0,
\]
\[
t_it_jt_i\cdots=t_jt_it_j\cdots\text{ if }i\neq j\;(m_{ij}\text{ factors in both products),}
\]
\[
t_ix_\lambda-x_\lambda t_i=0\text{ if }(\lambda:\alpha_i^\vee)=0,\quad t_ix_\lambda-x_{s_i(\lambda)}t_i=(q-1)x_\lambda\text{ if }(\lambda:\alpha_i^\vee)=1,
\]
for all \(i,j\in I\), \(\lambda,\mu\in X\).
One important step of the proof is the construction of a ring homomorphism from \(\mathbf H\) to a ring defined via the equivariant \(K\)-theory of an affine analogue \(\mathcal Z\) of the Steinberg variety. \(\mathcal Z\) is an ind-scheme of ind-infinite type. It comes with a filtration by subsets \({\mathcal Z}_{\leq y}\) with \(y\) in the affine Weyl group \(W\). The subsets \({\mathcal Z}_{\leq y}\) are reduced separated schemes of infinite type, and the inclusions \({\mathcal Z}_{\leq y'}\subset{\mathcal Z}_{\leq y}\) with \(y'\leq y\) are closed immersions. The set \(\mathcal Z\) is endowed with an action of a torus \(A\) which preserves each term of the filtration. For a well-chosen element \(a\in A\), the fixed point set \({\mathcal Z}^a\subset{\mathcal Z}\) is a scheme locally of finite type. Hence there is a convolution ring \(\mathbf K^A({\mathcal Z}^a)\): it is the inductive limit of the system of \({\mathbf R}_A\)-modules \(\mathbf K^A(({\mathcal Z}_{\leq y})^a)\) with \(y\in W\). (Here \({\mathbf R}_A\) means \({\mathbf K}_A(\text{point})\).) The author defines a ring homomorphism \(\Psi_a\colon{\mathbf H}\to\mathbf K^A({\mathcal Z}^a)_a\), where the subscript \(a\) means specialization at the maximal ideal \(J_a\subset{\mathbf R}_A\) associated to \(a\). The map \(\Psi_a\) becomes surjective after a suitable completion of \(\mathbf H\). It is certainly not injective. Using \(\Psi_a\), a standard sheaf-theoretic construction, due to Ginzburg in the case of affine Hecke algebras, provides a collection of simple \(\mathbf H\)-modules. These are precisely the simple integrable modules. -- The paper also give some estimates for the Jordan-Hölder multiplicities of induced modules. simple integrable modules; double affine Hecke algebras; perverse sheaves; linear algebraic groups; Lie algebras; Cartan subalgebras; Borel subalgebras; root systems; affine Weyl groups; simple reflections; pairings; equivariant \(K\)-theory; Jordan-Hölder multiplicities of induced modules Vasserot, Eric, Induced and simple modules of double affine Hecke algebras, Duke Math. J., 126, 2, 251-323, (2005) Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Grothendieck groups, \(K\)-theory, etc., Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Induced and simple modules of double affine Hecke 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 In classical invariant and moduli theory, it is in general impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in another, the orbit space cannot be given a scheme structure in a natural way. In this paper these difficulties are overcome by introducing a noncommutative algebraic geometry, where affine schemes are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory.
The geometry in the theory is represented by a swarm, i.e. a diagram of objects in e.g. the category of \(A\)-modules, satisfying reasonable conditions. The noncommutative deformation theory permits the construction of a presheaf of associative \(k\)-algebras, locally parametrizising the diagram.
Chapter 1 of the paper contains the homological preparations needed for this theory, and is also of interest in its self. The theory of \(\text{Ext}\) and Hochschild cohomology modules are considered, and also their interaction with the resolving functors of \(\varinjlim\).
Chapter 2 contains the deformation theory, which in some sense is the affine part in noncommutative geometry. This comes to its full right when the formal rings are replaced by their algebraizations. Chapter 3 contains the definition of the presheaves corresponding to the regular functions in the commutative situation. It contains the definition of noncommutative schemes and the most important results of the theory. Also some nice examples are presented.
Chapter 4 proves that the noncommutative scheme-theory is a good extension of classical commutative scheme-theory. In chapter 5, the theory is modeled on the category of \(A-G\)-modules, where \(G\) is a Lie-group. This is used to solve classical problems in geometric invariant theory by use of noncommutative theory in chapter 6. Several examples are given in chapters 7 and 8.
The paper is compactly written, and contains a theory and ideas that will prove essential in the years to come. deformation theory; formal methods; representation theory Laudal O., Algebraic Geometry 687 pp 31-- (1978) Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry Noncommutative algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with minimal models of quotient singularities by finite subgroups of \(\mathrm{SL}_n(\mathbb C)\). Let us recall that crepant resolutions have important roles in algebraic geometry. In this work, the author studies the existence of crepant resolutions of quotient singularities. He proves that a quotient singularity by a finite subgroup \(G\) has a crepant resolution if \(G\) is generated by junior elements and generalizes a result of Verbitsky. Then, he explains how to compute the corresponding Cox ring. Finally, he investigates the problem of smoothness of minimal models of some quotient singularities. quotient singularities; finite subgroups of \(\mathrm{SL}_n(\mathbb C)\); Cox ring; projective symplectic resolutions; crepant resolutions Yamagishi, R.: On smoothness of minimal models of quotient singularities by finite subgroups of \(SLn(C)\). (2016) Global theory of complex singularities; cohomological properties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On smoothness of minimal models of quotient singularities by finite subgroups of \(SL_n(\mathbb{C})\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper studies a resolution of singularities for Drinfeld's relative compactification for moduli stacks of vector bundles over a connected smooth projective curve. When the curve has genus zero, a resolution of singularities was constructed in [\textit{B. Feigin} et al., ``Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces'', \url{arXiv:alg-geom/9711009}] for the space of quasimaps. Here the author similarly takes the approach of Kontsevich's stable maps from nodal deformations of the curve into twisted flag varieties to obtain the desired resolution of singularities. As an application, the author shows that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic at points sufficiently antidominant relative to their defect. resolution of singularities; Drinfeld's compactification; stable maps Geometric Langlands program: representation-theoretic aspects, Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic) A resolution of singularities for Drinfeld's compactification by stable maps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 moduli space \(A_ g\) of principally polarized complex abelian varieties contains, as particular subvarieties, the locus of Jacobians of nonsingular algebraic curves of genus \(g\), the locus of Prym varieties of curves of genus \(2g+1\) and fixed-point-free involutions, and the locus of Prym varieties of curves of genus \(2g\) and involutions with exactly two fixed points. The Schottky problem and the Prym-Schottky problem consist in characterizing the closures of these loci in \(A_ g\) by algebraic equations in terms of the theta constants for their Riemann matrices in the Siegel upper-half space \(H_ g\). These classical problems are yet to be solved, at least with regard to their algebraic setting, but in the past ten years various approaches have led to partial answers, in particular by characterizing Jacobians and Prym varieties via geometric conditions or analytic equations in their theta constants. The present paper gives a survey on one of the most successful approaches to the Schottky problem (and to the Prym-Schottky problem, respectively), which has been essentially carried out by the author himself. This is the approach of using the algebro-geometric theory of non-linear evolution equations (soliton equations), studied by Burchnall-Chaundy-Baker, Krichever, Novikov, Dubrovin, and others, together with its link to data on algebraic curves and their associated abelian varieties (the so-called Krichever correspondence).
In section 1 the Krichever correspondence and the hierarchy of Kadomtsev- Petviashvili equations (KP hierarchy) are discussed. The formalism of the Hirota tau-function and its bilinear relations is explained in section 2, whereas its geometrical meaning via M. Sato's theory of infinite- dimensional Grassmannians is sketched in the following section 3.
In section 4 the author discusses various aspects of the famous Novikov conjecture, including its precise relationship to the Schottky problem and comments on the different approaches by Gunning, Welters, Arbarello- De Concini, and Mulase. This section ends with an outline of the author's spectacular proof of the Novikov conjecture in 1984 [cf. the author, Invent. Math. 83, 333-382 (1986; Zbl 0621.35097)].
In the final section 5, as a further application of the methods developed for proving the Novikov conjecture, a sketch of the basic ideas for a proof of an analogue of Novikov's conjecture for Prym varieties is given. This approach is related to the author's own recent work [cf. the author, ``BKP equation and Prym varieties'', Sugaku no Ayumi 29, 174-200 (1987) (Japanese)] and results by \textit{A. P. Veselov}, \textit{I. M. Krichever} and \textit{S. P. Novikov} [in: Geometry today, Int. Conf., Rome 1984, Prog. Math. 60, 283-301 (1985; Zbl 0575.35084)] and \textit{I. A. Tajmanov} [cf. Sov. Math., Dokl. 35, 420-424 (1987); translation from Dokl. Acad. Nauk SSSR 293, 1065-1068 (1987; Zbl 0685.14024)].
There is an appendix added to the English translation, in which the explicit expression for the so-called Akhieser-Baker functions on compact Riemann surfaces is derived. Altogether, this is a highly enlightening survey on an extremely difficult topic. Written by one of the outstanding experts in the field, it contains a lot of information about the philosophy, the strategy, the basic methods, and the still open problems concerning the Novikov approach to problems of Schottky type for special abelian varieties. In this regard, the article serves as an excellent introduction to the author's original work (cited above) and the related literature. moduli space of principally polarized complex abelian varieties; locus of Jacobians; Prym varieties; Schottky problem; Prym-Schottky problem; Krichever correspondence; Kadomtsev-Petviashvili equations; KP hierarchy; Hirota tau-function; Novikov conjecture; soliton Theta functions and curves; Schottky problem, Theta functions and abelian varieties, KdV equations (Korteweg-de Vries equations), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Jacobians, Prym varieties The KP equation and the Schottky problem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This monograph is essentially about the representation theory of a sheaf of reductive Lie algebras on a generalization \(J(X;L,d)\) of the classical Jacobian to smooth projective surfaces \(X\), whose closed points are pairs \((\mathcal{E}, [e])\), \(\mathcal{E}\) torsion-free sheaves of rank 2 on \(X\), \(c_1(\mathcal{E})=L\), \(L\) a fixed divisor on \(X\), \(c_2(\mathcal{E})=d \geq 0\), \(e\) a global section of \(\mathcal{E}\) with homothety class \([e]\), \(Z_e = (e=0)\), and the sheaf of Lie algebras in question is obtained from reductive Lie subalgebra \(\tilde{\mathcal{G}}(\mathcal{E}, [e])\) of \(\mathfrak{gl}(H^0(\mathcal{O}_{Z_e}))\). Among the many applications developed in this work (analog of a Lie algebraic aspect of the classical Jacobian, analog of a variation of Hodge structure à la Griffiths, analog of an infinitesimal Torelli theorem, toric geometry, action of affine Lie algebras on the direct sum of cohomology rings of Hilbert schemes, \dots) chief among those is probably the connection with Langlands Duality, and it is very much in this spirit that this work was initiated. It is also having Langlands duality in mind that we recommend reading this work lest the reader be quickly sidetracked by peripheral results that are certainly bold and tantalizing connections to various subfields of Algebraic Geometry and Representation Theory, but which unfortunately are not fully exploited and somewhat obscure the author's original goal, that of providing new insights into the Langlands program.
Two connected results of the author that are worthy of attention are for one thing that \(J(X;L,d)\) yields a finite collection \(\mathcal{V}\) of quasi-projective subvarieties of \(X^{[d]}\), every element \(\Gamma\) of which determines a finite collection of nilpotent orbits in \(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C})\), \(d[\Gamma] \leq d\) intrinsically associated to \(\Gamma\), and second that the same \(\Gamma\)'s determine a finite collection \( ^L R(\Gamma)\) of irreducible representations of the Langlands dual group \( ^L \mathfrak{sl}_{d[\Gamma]}(\mathbb{C}) = \mathrm{PGL}_{d[\Gamma]}(\mathbb{C})\). On a certain subset \(\breve{J}\) of \(J(X;L,d)\), \(H^0(\mathcal{O}_{Z_e})\) has a certain direct sum decomposition, and with the ring structure on \(H^0(\mathcal{O}_{Z_e})\), we get a reductive Lie subalgebra \(\tilde{\mathcal{G}}(\mathcal{E}, [e])\) of \(\mathfrak{gl}(H^0(\mathcal{O}_{Z_e}))\), the semisimple part of which is denoted by \(\mathcal{G}(\mathcal{E}, [e])\). We also have a morphism of schemes \(J(X;L,d) \rightarrow{\pi} X^{[d]}\) that sends a point \((\mathcal{E}, [e])\) to \([Z_e]\). One can attach a nilpotent element \(D^+(\nu)\) of \(\mathcal{G}(\mathcal{E}, [e])\) to every tangent vector \(\nu\) of \(\breve{J}\) along the fibers of \(\pi\), at a point \((\mathcal{E}, [e])\). If one denotes by \(T_{\pi}(\mathcal{E}, [e])\) the space of all such vectors of \(\breve{J}\) at \((\mathcal{E}, [e])\), one obtains a linear map
\[
D^+_{(\mathcal{E}, [e])}: T_{\pi}(\mathcal{E}, [e]) \rightarrow \mathcal{N}(\mathcal{G}(\mathcal{E}, [e])).
\]
The nilpotent cone of \(\mathcal{G}(\mathcal{E}, [e])\) being partitioned into a finite set one gets the first result.
The loop version of this map has values in the infinite Grassmannian \(\mathrm{Gr}(\mathcal{G}(\mathcal{E}, [e]))\) of \(\mathcal{G}(\mathcal{E}, [e])\) and one obtains a loop version of the first result where now \(\Gamma\)'s determine a finite collection of orbits in \(\mathrm{Gr}(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C}))\), and taking the intersection cohomology complexes of those orbits one passes to the category of perverse sheaves on \(\mathrm{Gr}(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C}))\), from which the second result follows after making use of the geometric version of the Satake isomorphism of [\textit{V. Ginzburg}, ``Perverse Sheaves on a Loop Group and Langlands Duality'', \url{arXiv:alg-geom/9511007}] and \textit{I. Mirkovic} and \textit{K. Vilonen} [Ann. Math. (2) 166, No. 1, 95--143 (2007; Zbl 1138.22013)].
One other result that is worthy of attention is the following. \(J(X;L,d)\) determines a finite collection \(\mathcal{P}(X;L,d)\) of perverse sheaves on \(X^{[d]}\), intersection cohomology complexes associated to local systems \(\mathcal{L}_{\lambda}\) on \(\Gamma\), \(\Gamma \in \mathcal{V}\), \(\mathcal{V}\) as in the first result above, \(\mathcal{L}_{\lambda}\) corresponding to a representation \(\pi_1(\Gamma, [Z]) \rightarrow \mathrm{Aut}(H^{\bullet}(B_{\lambda}, \mathbb{C}))\), \(B_{\lambda}\) a Springer fiber over the nilpotent orbit \(O_{\lambda}\) of \(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C})\), \(\lambda\) a partition of \(d[\Gamma]\). \(\mathcal{P}(X;L,d)\) gives rise to a distinguished collection \(C(X;L,d)\) of irreducible perverse sheaves on \(X^{[d]}\) and one denotes by \(\mathcal{A}(X;L,d)\) the abelian category of finite direct sums of \(C[n]\), \(n \in \mathbb{Z}\), \(C \in C(X;L,d)\). \(J(X;L,d)\) also comes equipped with a Cartier divisor \(\Theta(X;L,d)\) parametrizing pairs \((\mathcal{E}, [e])\), \(\mathcal{E}\) not locally free, and letting \(J^0(X;L,d) = J(X;L,d)\setminus \Theta(X;L,d)\), \(\mathcal{T}^*_{J^0(X;L,d) / X^{[d]}}\) the sheaf of relative differentials of \(J^0\) over \(X^{[d]}\), then the author proves there is a natural map \(H^0(\mathcal{T}^*_{J^0(X;L,d) / X^{[d]}}) \rightarrow \mathcal{A}(X;L,d)\), which one can see as a deformation theoretic result instrumental in reformulating the classical Langlands correspondence into the geometric Langlands correspondence. Jacobian; Hilbert scheme of points; period map; Torelli problem; Springer resolution; Langlands duality; perverse sheaves; Griffiths period domain; affine Lie algebras; variation of Hodge structure Parametrization (Chow and Hilbert schemes), Variation of Hodge structures (algebro-geometric aspects), Torelli problem, Families, moduli, classification: algebraic theory, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Nonabelian Jacobian of projective surfaces. Geometry and representation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a finite dimensional algebra over an algebraically closed field \(k\) and let \(Q\) be its associated quiver (the set \(Q_0\) of vertices of \(Q\) consists of isomorphism classes of indecomposable projective modules). Let mod-\(A\), resp. mod-\(kQ\), denote the abelian category of finite dimensional representations of \(A\), resp. \(Q\). Fix a dimension vector \(\alpha\in Z^{Q_0}\). Then isomorphism classes of representations of \(Q\) with dimension vector \(\alpha\) are in bijective correspondence with the orbits of the group \(\text{GL}(\alpha)\) in a certain vector space \({\mathcal R}(Q,\alpha)\). Let \(\Delta\) denote the kernel of the representation of \(\text{GL}(\alpha)\) in \({\mathcal R}(Q,\alpha)\). Let \(V_A(\alpha)\) denote the representations in \({\mathcal R}(Q,\alpha)\) associated to \(A\)-modules with dimension vector \(\alpha\); it is a closed \(\text{GL}(\alpha)\)-invariant subvariety of \({\mathcal R}(Q,\alpha)\). Any \(\theta\in Z^{Q_0}\) can be interpreted either as a character \(\chi_\theta\) of \(\text{GL}(\alpha)\) or as a homomorphism \(\theta:K_0(\text{mod-}kQ)\to Z\). The main purpose of this paper is to show that the notions of stability and semistability arising from Mumford's geometric invariant theory [see \textit{D. Mumford, J. Fogarty} and \textit{F. Kirwan}, Geometric invariant theory (Springer-Verlag, 3rd edition 1993; Zbl 0797.14004)] applied to the action of \(\text{GL}(\alpha)\) on \(V_A(\alpha)\) can be translated into algebraic, \(K\)-theoretical, properties of \(\text{mod-}A\). More concretely, a point \(x \in V_A (\alpha)\) corresponding to an \(A\)-module \(M\) is \(\chi_\theta\)-stable (or semistable) for the action of \(\text{GL} (\alpha)\) if and only if \(M\) is a \(\theta\)-stable (or semistable) module. Here \(M\) is \(\theta\)-semistable (resp., stable) if \(\theta (M) = 0\) and \(\theta (M') \geq 0\) for any \(M' \subseteq M\) (resp., semistable and \(\theta (M') = 0\) for \(M' \subseteq M\) implies \(M' = 0\) or \(M' = M\)). The algebraic quotient \({\mathcal M}_A (\alpha, \theta) := V_A (\alpha) // (\text{GL} (\alpha), \chi_\theta)\) is a coarse moduli space for families of \(\theta\)-semistable modules with dimension vector \(\alpha\) up to \(S\)-equivalence [see also \textit{C. S. Seshadri}, Ann. Math., II. Ser. 85, 303-336 (1967; Zbl 0173.230)] and is in fact a projective variety. finite dimensional algebras; indecomposable projective modules; finite dimensional representations; dimension vector; stability; semistability; actions; coarse moduli spaces; projective variety A. D. King, Moduli of representations of finite-dimensional algebras. \textit{Quart. J. Math. Oxford Ser}. (2) \textbf{45} (1994), 515-530. MR1315461 Zbl 0837.16005 Module categories in associative algebras, Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Representations of orders, lattices, algebras over commutative rings Moduli of representations of finite dimensional algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The moduli algebra \(A(V)\) of a hypersurface singularity \((V,0)\) is a finite-dimensional \(\mathbb{C}\)-algebra. In 1982, Mather and Yau proved that two germs of complex-analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. It is a natural question to ask for a necessary and sufficient condition for a complex-analytic isolated hypersurface singularity to be quasi-homogeneous in terms of its moduli algebra.
In this paper, we prove that \((V,0)\) admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite-dimensional nonnegatively graded algebra.
In 1983, Yau introduced the finite-dimensional Lie algebra \(L(V)\) of an isolated hypersurface singularity \((V,0)\). \(L(V)\) is defined to be the algebra of derivations of the moduli algebra \(A(V)\) and is finite-dimensional.
We prove that \((V,0)\) is a quasi-homogeneous singularity if (1) \(L(V)\) is isomorphic to a nonnegatively graded Lie algebra without center, (2) there exists \(E\) in \(L(V)\) of degree zero such that \([E,D_i]= iD_i\) for any \(D_i\) in \(L(V)\) of degree \(i\), and (3) for any element \(a\in m-m^2\), where \(m\) is the maximal ideal of \(A(V)\), \(aE\) is not in the degree zero part of \(L(V)\). quasi-homogeneous singularities; moduli algebra; hypersurface singularity Xu, Y.-J., Yau, S.S.-T.: Micro-local characterization of quasi-homogeneous singularities. Am. J. Math. 118, 389--399 (1996) Complex surface and hypersurface singularities, Graded Lie (super)algebras, Singularities of surfaces or higher-dimensional varieties Micro-local characterization of quasi-homogeneous 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 \(\mathbb{A}^ N\) be affine space with coordinates \((t_ 1,\dots,t_ N)\), and \(z_ 1,\dots,z_ n\) fixed real numbers. Let \(U\) be the complement of all hyperplanes \(t_ p = z_ q\), \(t_ p = t_ q\) in \(\mathbb{A}^ N\) and \(j: U \to \mathbb{A}^ N\) the inclusion. Let \(\mathcal L\) be a local system over \(U\) whose monodromy around hyperplanes \(t_ p = z_ q\) (resp. \(t_ p = t_ q\)) is \(\exp(-2\pi i(\alpha_ p,\Lambda_ q)/k)\) (resp. \(\exp(2\pi i(\alpha_ p,\alpha_ q)/k))\), where \(\Lambda_ 1,\dots,\Lambda_ n\), \(\alpha_ 1,\dots,\alpha_ r\) are vectors in the dual space of a finite-dimensional vector space \(\mathfrak h\) with nondegenerate symmetric bilinear form ( , ), \(k\) is a nonzero complex number. Complexes \(j_ !{\mathcal L}[N]\), \(Rj_ *[N]\) are perverse sheaves over \(\mathbb{A}^ N\) smooth along the stratification defined by the above hyperplanes. A construction which associates to any such sheaf a quiver is considered. The method of calculating the cohomology of a perverse sheaf in terms of its quiver is presented. Standard quivers based on a complex of a free Lie algebra with coefficients in tensor powers of its universal enveloping algebras are constructed. The main result of the paper says that these quivers are ``quasi-classical'' limits of quivers associated to \(j_ !{\mathcal L}[N]\), \(Rj_ *{\mathcal L}[N]\). Most of the results are only formulated without proofs. Hochschild homology; quasi-classical quiver; sheaves; cohomology Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Representation theory of associative rings and algebras Vanishing cycles and quantum groups. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the existence of symplectic resolutions of the symplectic reduction of a coregular representation of \(\mathrm{Sl}_{2}\). If \(G\) is any semisimple complex linear algebraic group and \(V\) is a finite dimensional complex vector space on which \(G\) acts, then \(V\oplus V^{*}\) has a natural symplectic form and \(G\) acts symplectically on it. The moment map \(\mu:(V\oplus V^{*})^{\oplus 2}\longrightarrow\mathfrak{g}^{*}\), where \(\mathfrak{g}=\mathrm{Lie}(G)\), is a \(G-\)equivariant map, hence \(\mu^{-1}(0)\subseteq (V\oplus V^{*})^{\oplus 2}\) is a \(G-\)invariant subset. The quotient \((V\oplus V^{*})///G:=\mu^{-1}(0)//G\) is the symplectic reduction of \(V\oplus V^{*}\). If \(\mathcal{I}_{\mu}\) is the ideal corresponding to \(\mu^{-1}(0)\), then \((V\oplus V^{*})///G\) is isomorphic as schemes to \(\mathrm{Spec}(\mathbb{C}[V\oplus V^{*}]/\mathcal{I}_{\mu})^{G}\). It is not necessarily a symplectic variety, but it is if \(\mu^{-1}(0)\) is normal, the stabilizer of every regular point of \(\mu^{-1}(0)\) is trivial, and \((\mu^{-1}(0)_{\mathrm{sing}}//G)\subseteq ((V\oplus V^{*})///G)_{\mathrm{sing}}\) (see Section 2).
Suppose that \(G=\mathrm{Sl}_{2}\) and that the representation \(V\) is coregular, i. e. the quotient \(V//G\) is smooth. Coregular representations for simple, connected and simply connected groups are classified by \textit{G. W. Schwarz} [Invent. Math. 50, 1--12 (1978; Zbl 0391.20033)], and for \(\mathrm{Sl}_{2}\) there are only the following cases: (1) \(\mathbb{C}^{2}\), \(\mathbb{C}^{2}\oplus\mathbb{C}^{2}\) and \(\mathbb{C}^{2}\oplus\mathbb{C}^{2}\oplus\mathbb{C}^{2}\); (2) \(S^{3}\mathbb{C}^{2}\) and \(S^{4}\mathbb{C}^{2}\); (3) \(\mathfrak{sl}_{2}\) and \(\mathfrak{sl}^{\oplus 2}_{2}\); (4) \(\mathfrak{sl}_{2}\oplus\mathbb{C}^{2}\).
The case (1) is treated in section 3: the author shows that \(\mathbb{C}^{2}\oplus\mathbb{C}^{2}///\mathrm{Sl}_{2}\) is non-reduced, hence it is not a symplectic variety, while \((\mathbb{C}^{2}\oplus\mathbb{C}^{2})^{\oplus 2}///\mathrm{Sl}_{2}\) is the union of two normal components, each admitting a symplectic resolution. Finally, \((\mathbb{C}^{2}\oplus\mathbb{C}^{2})^{\oplus 3}///\mathrm{Sl}_{2}\) is a symplectic variety which is isomorphic to the subscheme of \(\mathfrak{so}_{6}\) of matrices \(A\) such that \(A^{2}=0\) and \(\mathrm{Pf}(QA)=0\), where \(\mathrm{Pf}(QA)\) is the Pfaffian of the 15 skew-symmetric \(4\times 4-\)minors of \(A\). This corresponds to the closure of the nilpotent orbit \(\mathcal{O}_{[2^{2},1^{2}]}\), which is singular at the origin, and which admits two symplectic resolution: one by the cotangent bundle, and one by its dual. The method is to study the action of \(\mathrm{Sp}_{2n}\) on \((\mathbb{C}^{2n})^{\oplus 2m}\): the author gives a criterion for reducedness and normality of \(\mu^{-1}(0)\), and describes the quotient \(\mu^{-1}(0)//\mathrm{Sl}_{2}\) as the scheme \(Z\) of matrices \(A\in\mathfrak{so}_{2m}\) such that \(A^{2}=0\) and \(\mathrm{rk}(A)\leq \min\{2n,m\}\), with the reduced structure. Using the natural action of \(SO_{2m}\) on \(Z\), then \(Z\) is given by union of closures of nilpotent orbits. By \textit{B. Fu} [Invent. Math. 151, No. 1, 167--186 (2003; Zbl 1072.14058)], the author guarantees the existence of symplectic resolutions.
The cases (2) and (3) are classical: \((S^{3}\mathbb{C}^{2}\oplus S^{2}\mathbb{C}^{2})///\mathrm{Sl}_{2}\simeq\mathbb{C}^{2}/(\mathbb{Z}/4)\) is a symplectic variety having an \(A_{3}-\)singularity at \(\{0\}\), which can be resolved symplectically by two successive blow-ups, while \((S^{4}\mathbb{C}^{2}\oplus S^{4}\mathbb{C}^{2})///\mathrm{Sl}_{2}\) is a symplectic variety isomorphic to a subset of \((\mathbb{C}^{2})^{\oplus 3}/S_{3}\), which can be resolved symplectically by the Hilbert scheme \(\mathrm{Hilb}^{3}(\mathbb{C}^{2})\). The case \(\mathfrak{sl}_{2}^{\oplus 2}///Sl_{2}\) gives a symplectic variety with an \(A_{1}-\)singularity at the origin, which can be resolved by a single blow-up. The case \(\mathfrak{sl}_{2}^{\oplus 4}///\mathrm{Sl}_{2}\) is a symplectic variety isomorphic to the subscheme of \(\mathfrak{sp}_{4}\) of matrices \(A\in\mathfrak{sp}_{4}\) such that \(A^{2}=0\). By \textit{M. Lehn} and \textit{C. Sorger} [J. Algebr. Geom. 15, No. 4, 753--770 (2006; Zbl 1156.14030)], the cotangent bundle gives a symplectic resolution.
The case (5) is the last treated. First, the author computes the invariant ring \(\mathbb{C}[\mathfrak{sl}_{2}\oplus\mathbb{C}^{2}]^{\mathrm{Sl}_{2}}\), the generating invariants of the action of \(\mathrm{Sl}_{2}\) on \((\mathfrak{sl}_{2}\oplus\mathbb{C}^{2})^{\oplus 2}\) (Propositions 5.1 and 5.2) and the ideal \(\mathcal{I}_{\mu}\). Using \texttt{Singular}, in Proposition 5.3 it is shown that \(\mathbb{C}[\mu^{-1}(0)]^{\mathrm{Sl}_{2}}\simeq \mathbb{C}[z_{1},\dots,z_{8}]/(h_{1},\dots,h_{9})\), and a precise expression of the polynomials \(h_{1},\dots,h_{9}\) is given. Hence \(\mu^{-1}(0)//\mathrm{Sl}_{2}\) is a subvariety \(Z\) of \(\mathbb{C}^{8}\) whose ideal is \(V(h_{1},\dots,h_{9})\). Then the author shows that \(Z\) is a \(4-\)dimensional symplectic variety with a stratification \(\{0\}\subseteq Z_{\mathrm{sing}}\subseteq Z\), where \(Z_{\mathrm{sing}}\) is the singular locus of \(Z\) and \(\{0\}\) is the singular point of \(Z_{\mathrm{sing}}\). The blow-up \(\widetilde{Z}\) of \(Z\) along \(Z_{\mathrm{sing}}\) is a symplectic resolution of \(Z\). The interesting phenomenon pointed out by the author is that \(\pi^{-1}(0)\) is the union \(E_{1}\cup E_{2}\) of two projective planes intersecting at one point. One can then perform a Mukai flop in \(E_{1}\) or in \(E_{2}\): the author shows using \texttt{Singular} that the two obtained Mukai flops \(W_{1}\) and \(W_{2}\) are two symplectic resolutions of \(Z\) which are not isomorphic (and each one is not isomorphic to \(\widetilde{Z}\)), but which are equivalent in the sense of \textit{B. Fu} and \textit{Y. Namikawa} [Ann. Inst. Fourier 54, No. 1, 1--19 (2004; Zbl 1063.14018)]. T. Becker, \textit{On the existence of symplectic resolutions of symplectic reductions}, Math. Z. \textbf{265} (2010), no. 2, 343-363. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Momentum maps; symplectic reduction, Simple, semisimple, reductive (super)algebras, Global theory and resolution of singularities (algebro-geometric aspects) On the existence of symplectic resolutions of symplectic reductions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field and let \(Q\) be a (finite) quiver with set \(V\) of vertices. The group \(\operatorname{Aut}(Q)\) consists of quiver automorphisms, either covariant or contravariant. Let \(\operatorname{Rep}_{Q,d}\) be the affine space of all quiver representations with dimension vector \(d \in \mathbb N^{V}\) and let \(\mathbf G_{Q,d}\) be the associated (reductive) group of transformations of this space. Given a a stability parameter \(\theta \in \mathbb Z^V\), the GIT quotient construction gives moduli spaces \(\mathcal M_{Q,d}^{\theta-s}\), or \(\mathcal M_{Q,d}^{\theta-ss}\), of stable, or semistable, representations of \(Q\) over \(k\) of dimension \(d\), see [\textit{A. D. King}, Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)].
The authors study the action of a subgroup \(\Sigma\) of \(\operatorname{Aut}(Q)\) on \(\mathcal M_{Q,d}^{\theta-ss}\) assuming a natural compatibility with \(d\) and \(\theta\). The main result is the decomposition of the \(\Sigma\)-fixed locus in \(\mathcal M_{Q,d}^{\theta-s}\). The crucial tool is group cohomology. By analogous techniques, the authors construct branes in hyperkähler quiver varieties. algebraic moduli problems; geometric invariant theory Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Representations of quivers and partially ordered sets, Fine and coarse moduli spaces, Symplectic structures of moduli spaces Group actions on quiver varieties and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a quiver algebra \(A\) with relations given by a superpotential, the author defines a set of invariants of \(A\) counting framed cyclic \(A\)-modules. In the special case when \(A\) is the non-commutative crepant resolution of the \(3\)-fold ordinary double point, it is proved by using localization that the invariants count infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank-\(1\) Donaldson-Thomas partition functions of the commutative crepant resolution of the singularity and its flop.
In section one, basic definitions such as quiver algebras and superpotentials are reviewed. It is shown that the moduli space of framed cyclic modules over these quiver algebras admits a perfect obstruction theory. This allows the author to define new invariants, using K. Behrend's weighted Euler characteristics. In section two, the invariants and generating functions for the non-commutative conifold are studied. Based on torus actions and localizations, the computations are reduced to combinatorics and dimer configurations. In section three, the author discusses possible interpretations, speculations and generalizations. They include the space of \textit{T. Bridgeland}'s stability conditions of a variant of the derived category of coherent sheaves [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)] and the wall-crossing phenomena inside this space. quiver algebra; Donaldson-Thomas invariants; conifolds B. Szendroi, Non-commutative Donaldson-Thomas invariants and the conifold. \textit{Geom. Topol. }12(2008), no. 2, 1171--1202.MR 2403807 Zbl 1143.14034 154S. Govindarajan, A. J. Guttmann, and V. Subramanyan Calabi-Yau manifolds (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry Non-commutative Donaldson-Thomas invariants and the conifold | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 motivated by a construction of \textit{I. Gordon} and \textit{T. Stafford} [Adv. Math. 198, No.1, 222--274 (2005; Zbl 1084.14005)] concerning the representation theory of a symplectic reflection algebra \(H_c\) and the spherical subalgebra \(U_c\) of \(H_c.\) If \(G\) is a finite subgroup of \(\text{SL}_2\mathbb{C}\) and \(\Gamma=G\wr S_m\), the wreath product by the symmetric group, then \(\Gamma\) acts on \(V=(\mathbb{C})^m\) preserving the natural symplectic structure.
Let \(Y_{\Gamma,m}\) be the set of \(\Gamma\)-invariant ideals \(I\) in the Hilbert scheme of \(m| G| \) points in \(\mathbb{C}^2\) such that the quotient \(\mathbb{C}[x,y]/I\) is isomorphic to a direct sum of \(m\) copies of the regular representation of \(G\). Then a crepant resolution \(Y_{\Gamma,m}\to V/\Gamma\) exists. The algebra \(U_c\) has a filtration such that the associated graded algebra is isomorphic to \(\mathcal O(V/\Gamma),\) and Gordon and Stafford suggested that there is a suitable category completing the square
\[
\begin{tikzcd} U_c\text{-mod}\ar[r,"\cong"]\ar[d,"\mathrm{gr}" '] & ???\ar[d,"\mathrm{gr}"]\\ \mathcal O(V/\Gamma)\text{-mod}\ar[r] & \mathcal O_{Y_{\Gamma,m}}\text{-mod}\rlap{\,.}\end{tikzcd}
\]
They proved that this is possible in the case where \(G=1\), implying \(\Gamma=S_m\). When \(m=1\), implying \(G=\Gamma\), \(V/\Gamma\) is a Kleinian singularity and the algebras \(H_c\) is introduced. If \(\Gamma\) is cyclic, \(Y_{\Gamma,m}\) is the \(\Gamma\)-Hilbert scheme \(\text{Hilb}_{\Gamma}\mathbb{C}^2\) that parameterizes \(\Gamma\)-invariant ideals \(I\) of \(\mathbb{C}[u,v]\) such that \(\mathbb{C}[u,v]/I\) is isomorphic to the regular representation of \(\Gamma\). The purpose of the paper is to solve the problem of Gordon and Stafford when \(\Gamma\) is cyclic of order \(n\).
To state the main result: There is an action of \(\Gamma\) on the first Weyl algebra \(\mathbb{C}[\delta,y]\) and on the localization \(\mathbb{C}[\delta,y^{\pm}]\). For \(\mathbf{k}\in\mathbb{C}^{n-1}\), the author constructs \(U_{\mathbf{k}}\) and \(H_{\mathbf{k}}\) as subalgebras of the crossed product \(\mathbb{C}[\delta,y^{\pm}]\ast\Gamma.\) Then, for suitable \(\mathbf{k},\mathbf{k}^\prime\in\mathbb{C}^{n-1}\), \(U_{\mathbf{k}^\prime}\)-\(U_{\mathbf{k}}\)-bimodules \(B(\mathbf{k}^\prime,\mathbf{k})\) are constructed and a sufficient condition for these to induce a Morita equivalence is given. The bimodules \(B(\mathbf{k}^\prime,\mathbf{k})\) are assembled to form a Morita \(\mathbb{Z}^{n-1}\)-algebra \(R\) which is a \(\mathbb{Z}^{n-1}\times\mathbb{Z}^{n-1}\)-graded algebra without identity. The algebras \(U_{\mathbf{k}}\) and the bimodules \(B(\mathbf{k}^\prime,\mathbf{k})\) are contained in \(\mathbb{C}[\delta,y^{\pm}]\ast\Gamma\) and have a differential operator (order) filtration inherited to \(R\). The associated graded algebra of \(U_{\mathbf{k}}\) is isomorphic to \(\mathcal O(V/\Gamma).\) \(\text{Coh}(\text{Hilb}_{\Gamma}\mathbb{C}^2)\) denotes the category of coherent sheaves on \(\text{Hilb}_{\Gamma}\mathbb{C}^2.\) For a graded algebra \(\mathcal{R},\) \(\mathcal{R}\)-qgr denotes the quotient category of finitely generated graded \(\mathcal{R}\)-modules modulo torsion. The main result is then
If \(\mathbf{k}\) is dominant, then: (1) there is an equivalence of categories \(R-\text{qgr}\cong U_{\mathbf{k}-\text{mod}};\) (2) there is equivalence of categories \(\text{gr} R-\text{qgr}\cong U_{\mathbf{k}-\text{mod}}\cong\text{Coh}(\text{Hilb}_{\Gamma}\mathbb{C}^2).\)
To prove this theorem, \(\mathbb{C}^2/\Gamma\) has to be controlled. When \(\Gamma\) is cyclic of order \(n\), the author computes the Hilbert-Chow morphism (the crepant morphism discussed above) using toric varieties. The insight in global sections of the scheme in question involves computation of graded Poincare series and Picard schemes. To construct \(R\), a discussion of various categories over multi-homogeneous coordinate rings is needed. This is done, and is a beautiful application of quotient categories. Also a definition of ample systems is of importance. Then \(\mathbb{Z}^n\)-algebras and Morita theory for spherical subalgebras are discussed, and adding up, this is sufficient to prove the main theorem above in a very nice and readable matter. The deformation theory however, is present, but somewhat hidden in the general results about the Weyl algebra. Finally, the concluding remarks prove that the theory considered here is consistent with earlier results. Weyl algebra; torsion module; multigraded algebra I. Musson, Hilbert schemes and noncommutative deformations of type A Kleinian singularities, J. Algebra, 293 (2005), 102--129. math.RT/0504543. Deformations of singularities Hilbert schemes and noncommutative deformations of type A 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 Cohomological Field Theory (CohFT) was introduced by \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)] to axiomatize the structure of Gromov-Witten invariants and certain aspects of mirror symmetry. The latter can be viewed as an isomorphism of CohFTs coming from the so-called A and B models in string theory. The work of Saito and Givental led to the construction of CohFTs corresponding to the B-side of the Landau-Ginzburg (LG) model related to a quasihomogeneous isolated singularity \(w\). A CohFT corresponding to the A-side was recently constructed by Fan, Jarvis and Ruan using moduli spaces of solutions to certain PDE corresponding to linear perturbatrions of potential \(w\). They proved it to be isomorphic to the B-side CohFT for the simple ADE singularities.
In this paper, the authors present a purely algebraic construction of (conjecturally) a generalization of the Fan-Jarvis-Ruan (FJR) CohFT. The main tools are matrix factorizations, generalizations of differential complexes obtained by replacing \(d^2=0\) by \(d^2=w\), and the state space is built from the Hochschild homology of the dg-categories of equivariant matrix factorizations associated with the singularity \(w\) and its finite group of symmetries \(G\). The coefficients are in the representation ring of \(G\), and the specialization to \(\mathbb{C}\) conjecturally gives the FJR theory. This conjecture is proved for simple singularities only, the general case is obstructed by difficulties with verifying the dimension axiom. The ``\(w\)-curves'' of FJR, orbicurves with marked orbipoints and a collection of line bundles coming from the monomials of \(w\), are reinterpreted as attributes of \(G\) rather than \(w\). The notion of \(w\)-structure is reformulated using principal bundles, and leads to the moduli stack of spin curves which replace the \(w\)-curves. This removes a requirement of FJR that the group of the diagonal symmetries of \(w\) be finite.
The authors also obtain a weak categorification of their CohFT by constructing a collection of functors inducing the CohFT maps (up to rescaling) after passing to Hochschild homology. The functors are given by kernels, termed fundamental matrix factorizations, living on the product of moduli spaces of spin curves with affine spaces. The factorizations play a role similar to the virtual fundamental class of the Gromov-Witten theory. The factorization axiom of FJR holds at the categorical level after passing to the appropriate finite covers of the moduli. It is expected that a version of quantum \(K\)-theory can also be developed in this context. Fan-Jarvis-Ruan; Landau-Ginzburg model; Gromov-Witten invariants; mirror symmetry; matrix factorizations; Hochschild homology Polishchuk, Alexander and Vaintrob, Arkady, Matrix factorizations and cohomological field theories, Journal für die Reine und Angewandte Mathematik. [Crelle's Journal], 714, 1-122, (2016) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories and commutative rings, Relationships between surfaces, higher-dimensional varieties, and physics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories, triangulated categories, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Matrix factorizations and cohomological field theories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of \(\mathrm{SL}(2)\). Moreover, for a normal subgroup scheme \(\mathcal{N}\) of a finite group scheme \(\mathcal{G}\), we show that there is a connection between the ramification indices of the restriction morphism \(\mathbb P(\mathcal{V}_{\mathcal{N}}) \rightarrow \mathbb P(\mathcal{V}_{\mathcal{G}})\) between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quivers. Representations of quivers and partially ordered sets, Group schemes AR-components of domestic finite group schemes: McKay-quivers and ramification | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 additive Grothendieck-Springer resolution (see diagram (1.0.2)) serves as a resolution of singularities in the study of du Val singularities for algebraic surfaces. Given the stack of principal \(G\)-bundles over an elliptic curve with a simply connected structure group \(G\), this article describes the singularities and log resolutions coming from the elliptic Grothendieck-Springer resolution (see diagram (1.0.3)), which is a stacky version of the additive one.
When \(G\neq SL_2\), Theorem 1.0.2 (proven in section 2) shows the existence of an equivariant slice through subregular unstable bundles with good properties. Theorems 1.0.3 (proven in section 3) and 1.0.6 (proven in section 4) gives explicit descriptions of the pullbacks to these slices, depending on the Dynkin type of the subregular unstable \(G\)-bundle, and the singular surfaces appearing on the fibrations.
This paper contains the results of chapters 5 and 6 of the author's PhD thesis. singularities; principal bundles; elliptic curves Vector bundles on curves and their moduli, Singularities of surfaces or higher-dimensional varieties, Classical groups (algebro-geometric aspects), Other algebraic groups (geometric aspects) On subregular slices of the elliptic Grothendieck-Springer resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 smooth variety \(Z\), a classical question asks how many rational curves there are in \(Z\) that satisfy some conditions such as passing through a given number of fixed points. The modern deformation-invariant answer to that question is as a genus 0 Gromov-Witten invariant. This theory is built from the moduli space of genus 0 stable maps, a compactification of the moduli space of rational curves. There are several techniques to compute Gromov-Witten invariants, notably via degenerations of \(Z\) or via mirror symmetry of \(Z\).
For the first approach, consider a semistable degeneration where \(Z\) is deformed into the union of two pairs \((X_1,Y_1)\) and \((X_2,Y_2)\) glued together by identifying \(Y_1\) and \(Y_2\). Here \(X_1\) and \(X_2\) are smooth varieties, and \(D_1\) and \(D_2\) are smooth divisors. For such a degeneration, the absolute Gromov-Witten theory of \(Z\) is recovered from the relative Gromov-Witten theories of the pairs \((X_1,Y_1)\) and \((X_2,Y_2)\), where in addition the rational curves in each component satisfy matching tangency conditions (the degeneration formula sums the contributions of all possible configurations).
The second approach is via mirror symmetry of \(Z\). There are several incarnations of this. In the one due to \textit{A. Givental} [Prog. Math. 160, 141--175 (1998; Zbl 0936.14031)], the genus 0 Gromov-Witten invariants are packaged into a generating function, the so-called \(J\)-function. The mirror geometry then determines an explicit hypergeometric function called \(I\)-function. A Givental-style mirror theorem then states that the \(J\)-function equals the \(I\)-function after a transcendental coordinate change called the mirror map, which also is encoded by explicit hypergeometric functions.
The theory of quasimaps, as developed by Kim and Ciocan-Fontanine is based on a different, more efficient compactification of the moduli space of rational curves. The resulting invariants also satisfy a Givental-style mirror theorem, without passing through the mirror map, i.e. the \(I\)- and \(J\)-functions are equal on the nose, rendering it much for efficient for computation. The comparison between the genus 0 Gromov--Witten and quasimap theories then passes through wall-crossing. The drawback is that \(Z\) needs to be given as a GIT quotient.
The present paper combines these two approaches by building a relative quasimap theory for pairs \((X,Y)\), where \(X\) is a smooth toric variety and \(Y\) is a smooth very ample divisor. The latter condition is necessary in order to obtain the virtual fundamental class of relative quasimaps to \((X,Y)\) as the pull-back of the fundamental class of the moduli space of relative quasimaps to \((\mathbb{P}^N,H)\) for some \(N\) and hyperplane class \(H\). This approach is inspired by \textit{A. Gathmann}'s work [Duke Math. J. 115, No. 2, 171--203 (2002; Zbl 1042.14032)] on the analogous relative Gromov-Witten invariants. While the assumptions on \((X,Y)\) are quite restrictive, this wins the authors a beautiful recursive formula to compute the relative quasimap invariants. This is similar to a formula obtained by Gathmann for relative Gromov-Witten invariants, yet it is much more efficient in the quasimap formulation since unlike for stable maps there are no rational tails for quasimaps.
The authors prove that the generating function of maximal tangency relative quasimap invariants equals the relative \(I\)-function that encodes relative Gromov-Witten invariants of maximal tangency by work of Fan-Tseng-You [\textit{H. Fan} et al., Sel. Math., New Ser. 25, No. 4, Paper No. 54, 25 p. (2019; Zbl 1443.14057)]. In other words, in terms of Givental mirror symmetry, relative quasimap theory satisfies the same relations to relative Gromov-Witten invariants as do the respective absolute theories, at least in genus 0 and for maximal tangency invariants.
Another important result of the paper concerns quasimap quantum Lefschetz, which expresses the quasimap invariants of \(Y\) in terms of those of \(X\), as a consequence of the recursion formula.
All in all, this is the beginning of a beautiful chapter in enumerative geometry. This article is leading the way for a more general theory of quasimaps for \(X\) given as a GIT quotient with divisor \(Y\) not-necessarily very ample and possibly simple normal crossings. quasimaps; relative quasimaps; relative mirror symmetry; quasimap quantum Lefschetz Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects) Relative quasimaps and mirror formulae | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 extend our previous work [the authors, Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011)] on coherent-constructible correspondence for toric varieties to toric Deligne-Mumford (DM) stacks. Following \textit{L. A. Borisov} et al. [J. Am. Math. Soc. 18, No. 1, 193--215 (2005; Zbl 1178.14057)], a toric DM stack \(\chi_{\boldsymbol{\Sigma}}\) is described by a ``stacky fan'' \({\boldsymbol{\Sigma}}=(N,\Sigma,\beta)\), where \(N\) is a finitely generated abelian group and \(\Sigma\) is a simplicial fan in \(N_{\mathbb{R}}N\otimes_{\mathbb{Z}}\mathbb{R}\). From \({\boldsymbol{\Sigma}}\), we define a conical Lagrangian \(\Lambda_{\boldsymbol{\Sigma}}\) inside the cotangent \(T^\ast M_\mathbb{R}\) of the dual vector space \(M_\mathbb{R}\) of \(N_{\mathbb{R}}\), such that torus-equivariant, coherent sheaves on \(\chi_{\boldsymbol{\Sigma}}\) are equivalent to constructible sheaves on \(M_\mathbb{R}\) with singular support in \(\Lambda_{\boldsymbol{\Sigma}}\). The microlocalization theorem of \textit{D. Nadler} and the last author [Sel. Math., New Ser. 15, No. 4, 563--619 (2009; Zbl 1197.53116); J. Am. Math. Soc. 22, No. 1, 233--286 (2009; Zbl 1227.32019)] provides an algebro-geometrical description of the Fukaya category of a cotangent bundle \(T^\ast M_\mathbb{R}\) in terms of constructible sheaves on the base \(M_\mathbb{R}\). This allows us to interpret the main theorem stated earlier as an equivariant version of homological mirror symmetry for toric DM stacks. B. Fang, C.-C. M. Liu, D. Treumann, and E. Zaslow. The coherent-constructible correspondence for toric Deligne-Mumford stacks. International Mathematics Research Notices, 2014(4):914-954, 2012. Stacks and moduli problems, Generalizations (algebraic spaces, stacks), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The coherent-constructible correspondence for toric Deligne-Mumford stacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\subset \mathbb P_C^3\) be a canonically embedded complex curve of genus four, and given a two-torsion point \(\epsilon\in \mathrm{Jac}(C)[2]\), let \(\mathrm{Prym}(\hat C_\epsilon /C)\) be the Prym variety associated with the unramified double cover \(\hat C_\epsilon \to C\). It is a classical work that if \(C\) is generic, (1) there is a bijection between the (nontrivial) two-torsion points of \(\mathrm{Jac}(C)\) and the Cayley cubics \(\Gamma\subset\mathbb P^3\) containing \(C\); (2) using the strict projective dual of \(\Gamma_\epsilon\) corresponding to \(\epsilon\in \mathrm{Jac}(C)[2]\), we can construct a curve \(X_\epsilon\) of genus \(3\) such that \(\mathrm{Jac}(X_\epsilon) = \mathrm{Prym}(\hat C_\epsilon /C)\). The authors of the paper under review generalizes this classical work to its natural limit for curves that are not necessarily generic and that are defined over a field of characteristic not equal to two.
Let \(C/k\) be a proper smooth curve of genus four where \(k\) is a field of characteristic not equal to two. For this general case of curves, the correspondence is made to \textit{ cubic symmetroids} \(\Gamma\subset\mathbb P^3\). One way to visualize cubic symmetroids is as follows: \(\mathbb P^3\) parametrizes conics on \(\mathbb P^2\) in such a way that a cubic symmetroid \(\Gamma\) becomes the locus of singular conics on \(\mathbb P^2\). Evaluating conics parametrized by \(\mathbb P^3\), we obtain the \textit{ symmetrization} of \(\Gamma\), a map \(\mathfrak q : \mathbb P^2 \to \widehat {\mathbb P}^3\). The authors prove that \textit{ for every line bundle \(\epsilon\) of order two on \(C\), there is a natural construction of a cubic symmetroid \(\Gamma_\epsilon \subset \mathbb P^3\) containing \(C\) and a symmetrization \(\mathfrak q_\epsilon\) of \(\Gamma_\epsilon\). Also given a pair \((\Gamma, \mathfrak q)\), there is a reverse construction of the class of \(\epsilon\).} In Section 5, they demonstrate an explicit construction of the curve \(X_\epsilon\) of genus three, whose Jacobian is \(\mathrm{Prym}(\hat C_\epsilon /C)\). The authors also show that this correspondence can be built from a curve \(X\) of genus three, i.e., given a curve \(X\) of genus three, we can construct a curve \(C\) of genus four, a cubic symmetroid \(\Gamma\) and a symmetrization \(\mathfrak q\) of \(\Gamma\) such that they fit into the construction with a line bundle \(\epsilon\) of order two on \(C\); in particular, \(\mathrm{Prym}(\hat C_\epsilon /C) = \mathrm{Jac}(X)\).
The work presented in the paper under review was inspired by \textit{W. P. Milne}'s work on tritangents and bitangents [Proc. Lond. Math. Soc. (2) 21, 373--380 (1922; JFM 48.0736.02)], and in the last section the authors revisit his work in modern language. Prym varieties; curves of genus four Special algebraic curves and curves of low genus, Jacobians, Prym varieties, Plane and space curves Prym varieties of genus four curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities By considering the equivariant algebraic \(K\)-theory of an exotic version of the Steinberg variety, \textit{S. Kato} [Duke Math. J. 148, No. 2, 305--371 (2009; Zbl 1183.20002); Am. J. Math. 133, No. 2, 519--553 (2011; Zbl 1242.20056)] introduced the exotic nilpotent cone to extend the Kazhdan-Lusztig-Ginzburg geometrization of affine Hecke algebras from the 1-parameter to the multi-parameter setting. Kato established a Deligne-Langlands type classification of irreducible modules of affine Hecke algebras of type \(\mathsf{C}\) with only mild restrictions on the parameters. Letting \(\mathcal{N}\subseteq \mathfrak{gl}_{2m}(\mathbb{C})\) be the ordinary nilpotent cone of type \(\mathsf{A}\) and letting \(\mathcal{S} \subseteq \mathfrak{gl}_{2m}(\mathbb{C})\) denote the \(\text{Sp}_{2m}(\mathbb{C})\)-invariant complement of \(\mathfrak{sp}_{2m}(\mathbb{C})\subseteq \mathfrak{gl}_{2m}(\mathbb{C})\), the exotic nilpotent cone is the singular affine variety \(\mathfrak{N}=\mathbb{C}^{2m}\times (\mathcal{S}\cap \mathcal{N})\).
The authors study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from a combinatorial point of view, which includes an analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. The authors also compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type \(\mathsf{C}\) Weyl group, inducing Kato's exotic Springer representation on cohomology. Their results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra. Saunders and Wilbert thus pave a way in generalizing the geometric analogues of Khovanov's arc algebra to the exotic setting. exotic Springer fibers; Springer representations; one-row bipartitions; Temperley-Lieb algebra Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory for linear algebraic groups Exotic Springer fibers for orbits corresponding to one-row bipartitions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 of this interesting article are concerned with the study of coarse moduli spaces parameterizing families of stable coherent systems over nodal reducible curves, that is, pairs \((E,V)\) where \(E\) is a locally free sheaf and \(V\) is a subspace of global sections of \(E\). For fixed degree \(d=\text{deg}(E)>0\) and in the case when \(0<k=\text{dim}(V)<r=\text{rk}(E)\), the moduli spaces parameterize BGN extensions, and the authors generalize the notion of a BGN extension of type \((r,d,k)\) to nodal reducible curves in order to study the moduli spaces of stable coherent systems.
The analysis is depending on the choice of polarization \(\underline{w}\), and the authors use a notion of \textit{good polarization}, which in particular when the curve \(C\) is of compact type, good polarizations are exactly those for which the trivial bundle \(\mathcal{O}_C\) is \(\underline{w}\)-stable. For \((C, \underline{w})\) a polarized nodal curve with \(\underline{w}\) good and for \(0<k<r\), the irreducible components of the moduli space parameterizing coherent systems \((E,V)\) with \(E\) locally free are characterized. In particular, the main result of the article proves that any irreducible component of the moduli space is birational to a Grassmannian fibration over irreducible components of the moduli space of \(\underline{w}\)-semistable depth one sheaves whose \(\underline{w}\)-rank and \(\underline{w}\)-degree are \((r-k)\) and \(d\), respectively; these components are the only ones which contain coherent systems \((E,V)\) with \(E\) locally free. coherent system; nodal curve; polarization; stability; moduli space; BGN extension Vector bundles on curves and their moduli, Sheaves in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Coherent systems and BGN extensions on nodal reducible curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(T\) be a tilting bundle on a scheme \(X\). Then \(T\) gives an equivalence between the derived category of quasicoherent sheaves on \(X\) and the derived category of right modules of the algebra \(A=\text{End}_X(T).\) The author claims that this gives the possibility to consider homological properties in either the algebraic or the geometric setting, where the geometric setting is the one containing a scheme \(X\).
Deformation theory is controlled by homological data, and when a scheme is derived equivalent to an algebra, one may think that the deformations of the scheme should be derived equivalent to certain deformations of the algebra. The author anticipates that if a tilting bundle exists, it can be lifted to deformations of the scheme, and that the endomorphism algebra of the lifted tilting bundle should be deformations of the algebra. The result is proved both for deformations over complete local noetherian schemes and for \(\mathbb C^\ast\)-equivariant deformations over affine schemes with a unique \(\mathbb C^\ast\)-invariant closed point.
The results above is seen as intertwining two perspectives. On one hand, they produce noncommutative algebras derived equivalent to geometric deformations, and construct noncommutative counterparts to geometric phenomena. An example is the simultaneous resolution of the Artin component for rational surface singularities where such a singularity has a versal deformation and the Artin component of this has a simultaneous resolution after base-change which can be realized as the versal deformation of the minimal resolution. Tilting bundles on minimal resolutions of rational surface singularities were constructed by \textit{M. Wemyss} [Math. Ann. 350, No. 3, 631--659 (2011; Zbl 1233.14012)] while \textit{O. Riemenschneider} [Math. Ann. 209, 211--248 (1974; Zbl 0275.32010)] proposed the existence of a noncommutative algebra derived equivalent to the simultaneous resolution. The study of these constructions is one of the main applications in this article, and realizing the algebras as deformations makes it possible to actually calculate them in many examples.
The other perspective is to construct deformations of noncommutative algebras via geometric techniques, with the advantage to find exactly the deformations of noncommutative algebras that correspond to geometric deformations. Let \(G\) be a finite subgroup of \(\text{SL}_n(\mathbb C)\). The skew group algebra \(\mathbb C[x_1,\dots, x_n]\rtimes G\) provides a noncommutative crepant resolution of the quotient singularity \(\mathbb C^n/G\), and as Koszul algebra, the PBW deformation is classified by Braverman and Gaitsgory. In the case of a small, finite subgroup \(G\subset\text{GL}_2(\mathbb C)\), the minimal resolution exists, but the skew group algebra is not derived equivalent to the minimal resolution, and does not have PBW deformations unless \(G\subset\text{SL}_2(\mathbb C)\). In this case a better noncommutative resolution is the reconstruction algebra defined as the endomorphism algebra of a specific tilting bundle on the minimal resolution. Such an algebra is however not Koszul, and there is a lack of available technology to calculate their \textit{graded} deformations which are unknown in general. The main difficulty is to calculate deformations without the Koszul deformation theorems.
The results of this article apply outside the Koszul setting. Certain \(\mathbb C^\ast\)-equivariant deformations of the schemes are lifted to graded deformations of the algebras via tilting bundles. Exactly the graded deformations of the algebra which are derived equivalent to commutative deformations of the schemes are given. As applications the author calculates certain deformations of the reconstruction algebras.
The first, really nice, main result of the article states that if \(p:X_0\rightarrow\text{Spec}R_0\) is a projective morphism of noetherian schemes and \(\rho:X\rightarrow\text{Spec}D\) is a flat deformation of \(X_0\) over a complete local noetherian \(\mathbb C\)-algebra \(D\), then any tiling bundle \(T_0\) on \(X_0\) lifts uniquely to a tilting bundle \(T\) on \(X\). \(A=\text{End}_X(T)\) is a flat deformation of \(A_0=\text{End}_{X_0}(T_0).\)
This result makes it possible to construct deformations of algebras using deformations of quasi-coherent sheaves (i.e. of schemes), and it leads up to the second main result, stating exactly the same in the equivariant (graded) setting. The author manage to state this very advanced result in nice way, and to prove it. The result has several geometric applications, and the article gives a nice survey of minimal resolutions of rational surface singularities and surface quotient singularities, both leading to graded reconstruction algebras. Finally, the crepant resolutions of symplectic quotient singularities and symplectic reflections algebras are considered.
This article builds bridges between representation theory, algebraic geometry and deformation theory of algebras, and contributes to the development of all three fields. It contains very interesting results, complicated examples, and is sufficiently nice written. derived categories; tilting bundles; deformations of algebras; rational singularities; deformations of tilting bundles; lifting of tilting bundles; equivariant tilting bundles; equivariant deformations; geometric deformations; simultaneous resolution of the Artin comonent; minimal resolutions of singularities; graded deformations Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Deformations of associative rings, Formal methods and deformations in algebraic geometry, Deformations of singularities Deformations of algebras defined by tilting bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author begins by explaining in detail a method, due to \textit{V. G. Kac} and \textit{Eh. B. Vinberg} [Adv. Math. 30, 137-155 (1978; Zbl 0429.20043)], for the classification of orbits of a (complex) reductive group \(G\), acting in the space of a linear representation. Then he applies this method in special cases, recovering some known results in a unified way. If \(\theta\) is a semisimple automorphism (possibly inner) of \(G\), having finite order \(m\), denote by \(G_ 0\) the identity component of the subgroup fixed pointwise by \(\theta\). The differential \(d \theta\) induces a \(\mathbb{Z}_ m\)-grading of the Lie algebra of \(G\), stable under \(G_ 0\), and the author calls ``\(\theta\)-group'' the image of \(G_ 0\) in its action on the 1-component of the grading. The case \(m = 1\) corresponds to the adjoint action of \(G\), while \(m = 2\) yields the isotropy group of a symmetric space associated to \(G\). The orbit classification essentially reduces to the case of nilpotent orbits. Here the author shows how to recover results of H. Kraft and C. Procesi, and of H. G. Quebbemann, which involve the classification of pairs of linear transformations subject to suitable notions of equivalence. reductive groups; linear representations; semisimple automorphisms; Lie algebras; actions; orbit classification; nilpotent orbits; linear transformations Capparelli, S.: The Jordan canonical form in some \({\theta}\)-groups. Rend. mat. (7) 11, 777-808 (1991) Linear algebraic groups over the reals, the complexes, the quaternions, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Semisimple Lie groups and their representations The Jordan canonical form in some \(\theta\)-groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H\) be a complex Hilbert space. If \(H\) is finite dimensional, then it is a classical result that the finite dimensional irreducible representations of the general linear group \(GL(H)\) can be realized geometrically as the natural action of the group \(GL(H)\) on the space of global holomorphic sections of a holomorphic line bundle over a space of flags in \(H\). By choosing a basis of \(H\), one can identify this space of holomorphic sections with a space of holomorphic functions on \(GL(H)\) that are certain polynomial expressions in minors of the matrices corresponding to the elements of \(GL(H)\). Infinite dimensional analogues of some of these representations occur in quantum field theory, infinite dimensional Grassmann manifolds play an important role in the framework of integrable systems. The first person to realize this was Sato. In the paper under review the authors give an infinite dimensional analogue of all these representations. In a separable Hilbert space \(H\) they consider a collection of flags that generalizes the Grassmannian from \textit{A. Pressley's} and \textit{G. Segal's} book [Loop groups, Oxford: Clarendon Press (1986; Zbl 0618.22011)]. This flag variety carries a natural Hilbert space structure and there exist line bundles over it that are similar to the finite dimensional ones. This includes the determinant bundle and its dual form. In the ``dominant'' case the space of global holomorphic sections of such a line bundle turns out to be non-trivial. However, the action of the analogue of the general linear group can, in general, not be lifted to the line bundle under consideration and one has to pass to a central extension of this group. The paper under review consists of three sections. In the first section the authors give the definition of the flag variety and treat some properties of the flag variety. The second section is devoted to the construction of the holomorphic line bundles, to a description of the corresponding central extensions and to the analysis of the space of global holomorphic sections. As an application they show in the final section what role geometry plays in the context of some integrable systems. Hilbert space; finite dimensional irreducible representations; general linear group; Grassmann manifolds; Grassmannian; flag variety Helminck, A.G., Helminck, G.F.: The structure of Hilbert flag varieties, Publ. RIMS, Kyoto Univ. 30, 401--441 (1994) Infinite-dimensional Lie groups and their Lie algebras: general properties, Grassmannians, Schubert varieties, flag manifolds, Other completely integrable equations [See also 58F07], Analysis on other specific Lie groups, Infinite-dimensional Lie (super)algebras The structure of Hilbert flag 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 Suppose that \(S=\bigoplus_{d\geqslant 0}S_d\) is a graded associative algebra generated by \(S_1\) with an automorphism \(\sigma\) of \(S_1\) as a linear space which can be extended to an automorphism \(\sigma\) of \(S\). Then there is a twisted algebra \(S^\sigma\) with multiplication defined as \(a*b=a\cdot\sigma(b)\) for homogeneous elements \(a,b\in S\).
The author considers the case when \(S\) is the complex polynomial algebra on \(Y_1,\dots,Y_n\) and \(\sigma\) is defined as \(\sigma(Y_1)=Y_1\) and \(\sigma(Y_{i+1})=Y_{i+1}+Y_i\) for \(i>0\). There is given a classification of the primitive ideals in \(S^\sigma\) which do not contain \(Y_1\). In particular, every primitive ideal in \(S^\sigma\) is generated by a regular sequence of homogeneous elements \(g_1,\dots,g_t\) where each \(g_i\) is irreducible and \(\sigma\)-invariant modulo the ideal generated by the preceding elements. So there exists a one to one correspondence between primitive ideals of \(S^\sigma\) and symplectic leaves of the Poisson structure induced by \(\sigma\). This result can be generalized to the case when an automorphism of \(S\) has a single eigenvalue. In this case the leaves are algebraic and realizable by orbits of an algebraic group. deformations of algebras; primitive ideals; twisted homogeneous coordinate rings; symplectic leaves; Poisson manifolds; complex polynomial rings; Poisson structures; primitive spectra Deformations of associative rings, Ordinary and skew polynomial rings and semigroup rings, Simple and semisimple modules, primitive rings and ideals in associative algebras, Graded rings and modules (associative rings and algebras), Ideals in associative algebras, Derivations, actions of Lie algebras, Noncommutative algebraic geometry Primitive and Poisson spectra of single-eigenvalue twists of polynomial algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The singularity category \(D_{\text{sing}}(A)\) of an associative algebra is defined to be the quotient of \(D(A)\) by the subcategory of perfect complexes [\textit{R.-O. Buchweitz}, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. Hannover: University of Hannover, unpublished manuscript, \url{http://hdl.handle.net/1807/16682} (1987)]; it recaptures the finiteness of \(A\) in the sense that \(A\) has finite global dimension iff \(D_{\text{sing}}(A)\cong 0\). The category of Gorenstein projectives embeds into \(D_{\text{sing}}(A)\) [loc. cit.] and the quotient category is defined to be the Gorenstein defect category of \(A\) [\textit{P. A. Bergh} et al., Q. J. Math. 66, No. 2, 459--471 (2015; Zbl 1327.13041)]. Several inputs [\textit{X.-W. Chen}, Algebr. Represent. Theory 12, No. 2--5, 181--191 (2009; Zbl 1179.18004); Bull. Lond. Math. Soc. 44, No. 2, 271--284 (2012; Zbl 1244.18007); ``The singularity category of a quadratic monomial algebra'', Preprint, \url{arXiv:1502.02094}] [\textit{C. M. Ringel}, J. Algebra 385, 241--261 (2013; Zbl 1341.16010); \textit{M. Kalck}, Bull. Lond. Math. Soc. 47, No. 1, 65--74 (2015; Zbl 1323.16012); \textit{X. Chen} et al., Algebr. Represent. Theory 18, No. 2, 531--554 (2015; Zbl 1341.16011); ``The Gorenstein-projective modules over a monomial algebra'' Preprint, \url{arXiv:1501.02978}] have bee given towards an explicit determination of these categories in special cases or for specific classes of algebras. The present paper mixes the fruitful approach of [\textit{X.-W. Chen}, Algebr. Represent. Theory 12, No. 2--5, 181--191 (2009; Zbl 1179.18004); \textit{B.-L. Xiong} and \textit{P. Zhang}, J. Algebra Appl. 11, No. 4, Article ID 1250066, 14 p. (2012; Zbl 1261.16018)] that gives necessary and sufficient conditions for a triangular matrix algebra \(\Lambda\) to be Gorenstein, and the techniques of recollements of triangulated categories: the bridge between these two techniques is the classical result that proj-finite matrix algebras possess a recollement [\textit{Q. Chen} and \textit{Y. Lin}, Sci. China, Ser. A 46, No. 4, 530--537 (2003; Zbl 1215.18014)] relative to \(D_{\text{sing}}(A)\) and \(D_{\text{sing}}(B)\). Mimicking a result of Zhang the author proved in a previous paper [\textit{P. Liu} and \textit{M. Lu}, Commun. Algebra 43, No. 6, 2443--2456 (2015; Zbl 1330.18016)] that there exists a recollement of the singularity categories of \(A,B\) assuming both are artinian rings and \({}_AM_B\) is two-sided proj-finite. The present paper applies a similar strategy to the study of the Gorenstein defect category \(D_{\text{def}}(\Lambda)\) obtaining a clear categorical explanation of the fact that the Gorenstein property for \(A\) (resp. \(B\)) induces a triangle equivalence between the defect categry of \(\Lambda\) and the defect category of \(B\) (resp. \(A\)); this is an immediate consequence of the exactness properties of left recollements (Thm 3.6). The left recollement of Thm. 3.6 can in fact be promoted to a full recollement provided additional proj-finiteness assumptions are satisfied by \(M\) or by all hom-modules \(\hom(M,P)\) for \(P\) an indecomposable projective \(A\)-module. ``Splitting'' theorems where the singularity/defect categories of a triangular matrix algebra \(\Lambda\) are triangle equivalent to disjoint unions of singularity/defect categories of \(A,B\) are investigated in the case \(A,B\) arise as the path \(K\)-algebra (\(K\) a field) of suitable quivers called ``simple gluing quivers''. Gorenstein defect category; triangular matrix algebra; recollement; Gorenstein algebra Derived categories, triangulated categories, Localization of categories, calculus of fractions, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Gorenstein defect categories of triangular matrix algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is a sketch of our research work on the subject given by the above title. In the work \textit{G. Shimura} [Ann. Math. (2) 85, 58--159 (1967; Zbl 0204.07201)], it was proved that there exists a modular function (that is called canonical model) that enables to obtain a certain class field (the Shimura class field) of some kind of CM field. In this article we show that for the case of the CM field embedded into the quaternion albebra coming from a co-compact arithmetic triangle group we can determine the canonical model as a hypergeometric modular function in an explicit way. Moreover we give several examples of Hilbert class fields of such kind of CM fields coming from the triangle group \(\Delta(3,3,5)\). For our work, we use Shimura's reciplocity law and the existense of the canonical model together with the result by \textit{K. Takeuchi} [J. Math. Soc. Japan 29, 91--106 (1977; Zbl 0344.20035) and J. Fac. Sci., Univ. Tokyo, Sect. I A 24, 201--212 (1977; Zbl 0365.20055)]. To construct explicit examples we use the modular function for genus 4 pentagonal curves discovered by \textit{K. Koike} [J. Math. Soc. Japan 55, No. 1, 165--196 (2003; Zbl 1038.14010)]. The first author has written the same subject in the book [Shg] (chapter 8). There he made a detailed explanation of the modular function for \(\Delta(3,3,5)\). In contrast, here we tried to explain the back ground of our research work. By both of two explanations he expects that the readers will have a nicer perspective of the story. For the full argument with exact proofs refer the paper \textit{A. Nagano} and \textit{H. Shiga} [J. Number Theory 165, 408--430 (2016; Zbl 1401.11107)]. Hilbert class field; complex multiplication; moduli of abelian varieties; hypergeometric functions; theta functions Deformations of analytic structures, Complex multiplication and abelian varieties, Theta functions and abelian varieties, Classical hypergeometric functions, \({}_2F_1\) One visualization of Shimura's complex multiplication theorem via hypergeometric modular functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{C. A. Weibel} in [Lect. Notes Math. 966, 390--407 (1982; Zbl 0499.18012); Contemp. Math. 83, 461--488 (1989; Zbl 0669.18007)] and \textit{R. W. Thomason} and \textit{T. Trobaugh} in [Prog. Math. 88, 247--435 (1990; Zbl 0731.14001)] proved (under some assumptions) that algebraic \(K\)-theory with coefficients is \(\mathbb A^1\)-homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass-Quillen fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above \(\mathbb A^1\)-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic \(K\)-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic \(K\)-theory with coefficients of the du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic \(K\)-theory (without coefficients). \(\mathbb A^1\)-homotopy; algebraic \(K\)-theory; Witt vectors; sheaf of dg algebras; dg orbit category; cluster category; du val singularities; noncommutative algebraic geometry Tabuada, Gonçalo, \(\mathbb{A}^1\)-homotopy invariance of algebraic \(K\)-theory with coefficients and du Val singularities, Ann. K-Theory, 2, 1, 1-25, (2017) Noncommutative algebraic geometry, Singularities of curves, local rings, \(K\)-theory of schemes, Klein surfaces, Witt vectors and related rings \(\mathbb A^1\)-homotopy invariance of algebraic \(K\)-theory with coefficients and du Val 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 is concerned with isolated singularities of plane projections of generic knots in \({\mathbb{R}}^3\) which were listed in [\textit{J. M. S. David}, J. Lond. Math. Soc., II Ser. 27, 552-562 (1983; Zbl 0488.58003)]. The author establishes a remarkable relation between two basic invariants of such singularities which may be described as follows. Fix a generic smooth knot \(C\) in \({\mathbb{R}}^3.\) For a singularity type \(X,\) denote by \(\text{cod}(X)\) the codimension in \({\mathbb{R}}^3\) of centers of projection for which a singularity of type \(X\) appears in the plane projection of \(C,\) and by \(n(X)\) the maximum number of stable nodes into which the singularity \(X\) splits when the center of projection is moved. Finally, let \(r\) be the number of branches of the curve involved in a singularity of type \(X.\) Then \(\text{cod}(X) = n(X) - r + 1,\) with one exception: if \(X\) is the so-called quadruple point then \(\text{cod}(X) = n(X) - r.\) The author reveals that these relations are merely traces in \(\mathbb{R}\) of a corresponding complex phenomenon: the relation between the vanishing homology of stable perturbations of an unstable singularity and the number of parameters of its versal deformation. This observation makes the proof especially illuminating and also explains why the quadruple point provides an exception for a ``generic'' relation. plane projection; quadruple point; vanishing homology; stable perturbations; unstable singularity; radial projection; node; embedded curve; ramphoidal cusp; tacnode; conductor ideal; isolated quasi-homogeneous singularities David Mond, Looking at bent wires --- \?\?-codimension and the vanishing topology of parametrized curve singularities, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 2, 213 -- 222. Knots and links (in high dimensions) [For the low-dimensional case, see 57M25], Milnor fibration; relations with knot theory, Knots and links in the 3-sphere, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities of differentiable mappings in differential topology, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Curves in Euclidean and related spaces, Singularities of curves, local rings Looking at bent wires -- \(\mathcal{A}_ e\)-codimension and the vanishing topology of parametrized curve 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 give a systematic study of a very interesting and important phenomenon in the theory of arithmetic quotient spaces: given two bounded, Hermitian, symmetric domains \(D_1 = SP(4; \mathbb{R})/U (2)\), \(D_2 = U(3,1)/U(3) \times U(1)\) associated to different algebraic groups \(SP (4 ; \mathbb{R})\), \(U(3,1)\), their quotient spaces \(\Gamma_1 \backslash D_1\), \(\Gamma_2 \backslash D_2\) can lead to the same projective variety in their compactification \(\overline {\Gamma_1 \backslash D_1} = \overline {\Gamma_2 \backslash D_2}\). More precisely, the authors prove that there are isomorphisms between the following varieties:
\[
\text{(i)} \;\overline {X(1)} \cong \overline {Y(1)}, \quad \text{(ii)} \;\overline {X(2)} \cong \overline {Y (\sqrt {-3})}, \quad \text{(iii)} \;\overline {X(3)} \cong \overline {Y(2)}, \quad \text{(iv)} \overline {X(6)} \cong \overline {Y(2 \sqrt {-3})}.
\]
Here \(X(-)\), \(Y(-)\) refer to respectively the symplectic and unitary quotient spaces and the decoration 1, 2, 3, 6, \(\sqrt {-3}\), \(2 \sqrt {- 3}\) refer to different levels of congruence subgroups. In the paper these varieties with different arithmetic interpretation are called Janus-like and hence the title of the paper. arithmetic quotient spaces B. Hunt and S. H. Weintraub, Janus-like algebraic varieties, Jour. Diff. Geom. 39, (1994), 509-557. Homogeneous spaces and generalizations, Families, moduli, classification: algebraic theory Janus-like algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let V be a finite-dimensional vector space over the field \({\mathbb{C}}\) of complex numbers, and G a finite subgroup of GL(V). Let \(S(V)=S\) be the symmetric algebra over V and \(R=S^ G\) the subalgebra of G-invariants of S. It is well known [see \textit{E. Noether}, Math. Ann. 77, 89-92 (1916)] that R is finitely generated whence \(R={\mathbb{C}}[X_ 1,...,X_ m]/I\). Define two parameters of R, codimension and defect, by codim R\(=m-\dim R\) and \(d(R)=s-co\dim R\), where m and s are minimal numbers of generators for R and I, respectively, and dim R is the Krull dimension of R. These parameters are tested for properties I and II defined as follows. Let \(\{G_ i\}\) be a set of finite groups, \(\{\rho_{ij}: G_ i\to GL(V_ j)\}^ a \)set of linear representations and \(\{R_{ij}=S(V_ j)^{\rho_{ij}(G_ i)}\}\) the set of corresponding algebras of invariants. The author says that the set \(\{G_ i\}\) satisfies condition I [resp., II] with respect to the representations \(\{\rho_{ij}\}\) if for every natural number k there is a natural number \(k'\) such that codim \(R_{ij}>k\) [resp., \(d(R_{ij})>k]\) whenever \(\dim_{{\mathbb{C}}}V_ j>k'\). In other words, codim \(R_{ij}\to \infty\) [resp., \(d(R_{ij})\to \infty]\) as \(\dim_{{\mathbb{C}}}V_ k\to \infty.\)
In this paper the author considers the classes \(\{S_ n\}\) and \(\{A_ n\}\) of all symmetric and alternating groups with respect to two sets of representations, defined as follows. Let X [resp. \(X']\) be the set of all representations of the groups \(S_ n\) [resp., \(A_ n]\). Let (n-1,1) and \((1^ n)\) denote the irreducible representations of \(S_ n\) associated with these partitions. Put \(X_ i=\{k(n-1,1)\oplus l(1^ n)|\) n,k,l natural numbers and \(k>i\}\). The main result is the following theorem: The set \(\{S_ n\}\) [resp., \(\{A_ n\}]\) of all symmetric [resp., alternating] groups satisfies conditions I and II with respect to the set \(X-X_ 0\) [resp. \(X'-X'_ 0]\) and condition I, at least with respect to the representations in the set \((X-X_ 0)\cup X_ 1\) (resp., \((X'-X'_ 0)\cup X'_ 1].\)
The proof of the theorem is based on the so-called slice method [see \textit{V. G. Kac}, in Invariant theory, Lect. Notes Math. 996, 74-108 (1983; Zbl 0534.14004)], according to which the parameters of \(R=S^ G\) are no less than those for \(S^ H\) for every subgroup H, call it closed, that satisfies the condition: \(H=\{g\in G|\) \(g^ v=v\) for every \(v\in V^ H\}\). Setting \(G=S_ n\) and \(H=<(12)>\) the author proves that (i) H is closed in every irreducible representation \(\rho_{\lambda}: S_ n\to GL(V_{\lambda})\) associated with the partition \(\lambda\neq (n)\), \((1^ n)\), \((2,1^{n-2})\) and \(n>7\) and (ii) the involution \(\rho_{\lambda}((12))\) has at least n-3/2 eigenvalues -1. The theorem then follows from the observation that if \(\sigma\) is an involution in GL(V) with r eigenvalues -1, then codim \(S^{(\sigma)}=\left( \begin{matrix} r\\ 2\end{matrix} \right)\) and \(d(S^{(\sigma)})\geq \left( \begin{matrix} r\\ 4\end{matrix} \right)\). symmetric algebra; codimension; defect; minimal numbers of generators; Krull dimension; linear representations; algebras of invariants; symmetric and alternating groups; irreducible representations Representations of finite symmetric groups, Group actions on varieties or schemes (quotients), Vector and tensor algebra, theory of invariants Invariants of symmetric and alternating 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 An isolated hypersurface singularity is a polynomial \(f\in k[x_1,\dots,x_n]\) for which the Milnor number \(\mu(f)=\dim k[x_1,\dots,x_n]/(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n})\) is finite.
An unfolding of a hypersurface singularity is a family of hypersurface singularities parametrized by an affine space. This is the problem of deformations of the \(k[y]\)-algebra \(k[x_1,\dots,x_n]\) given by sending \(y\) to \(y=f(x_1,\dots,x_n).\)
This article studies noncommutative unfoldings of hypersurface singularities which means deformations of the \(k[y]\)-algebra \(A=k[x_1,\dots,x_n]\) in the category of associative algebras. This can be seen as the study of the Hochschild cochain complex of \(A\) considered as \(k[y]\)-algebra with the Kontsevich Formality theorem as the possible ideal answer.
The proof of the formality theorem shows that for a smooth algebra \(A\), there is a weak equivalence between the Hochschild complex to an algebra of polyvector fields. In the case of this article, there is a quasi-isomorphism between \(A\) and a smooth dg algebra over \(k[y]\), and the proof of the formality theorem can be, due to the authors, easily generalized to this setup. Thus the Hochschild cochain complex can be replaced with a certain algebra of polyvector fields which in this case is a dg algebra.
The first part of the article proves the following theorem: Let \(\mathbb Q\subset R\) be a commutative ring, and let \(A\) be a commutative smooth dg \(R\)-algebra that is non-positively graded, semifree over \(A^0\) which is smooth as \(R\)-algebra. Then the Hochschild cochain complex of \(A\) over \(R\) is equivalent to the dg algebra of polyvector fields as (homotopy) Gerstenhaber algebras.
For a smooth dg algebra \(A\), the algebra of polyvector fields is designed as \(S_A(T[-1])\) with \(T=\text{Der}_R(A,A)\). Because \(S_A(T[-1])\) is a Gerstenhaber algebra, its Harrison chain complex \(B_{\text{Com}^\bot}(S_A(T[-1])\) has structure of a dg Lie bialgebra. A homotopy Gerstenhaber Lie algebra structure on the Hochschild complex \(C(A)\) can be defined via a dg Lie bialgebra structure on \(F^\ast_{\text{Lie}}(C(A)[1])\). An equivalence between Gerstenhaber algebras of polyvector fields \(S_A(T[-1])\) and the Hochschild complex \(C(A)\) is presented on the level of these Lie algebra models. The authors present a dg Lie algebra \(\xi(A)\) and two weak equivalences \(\xi(A)\rightarrow B_{\text{Com}^\bot}(S_A(T[-1])\) and \(\xi(A)\rightarrow F^\ast_{\text{Lie}}(C(A))[1])\) of dg Lie bialgebras. The second weak equivalence is deduced from a dg version of the Hochschild-Kostant-Rosenberg theorem. The setup of dg smooth algebras makes this deduction quite nontrivial and is the main deviation from the proof of Dolgushev-Tamarkin-Tsygan. In this theory, the Poisson bracket appears as a representative of the first order deformation extendable to a second order deformation. Thus the following definition is given:
A quasiclassical datum of quantization of a \(B\)-algebra \(A\) is its deformation over \(k[h]/(h^2)\) extendable to \(k[h]/(h^3)\).
Thus a quasiclassical datum for a quantization of the ring of smooth functions on a manifold is precisely a Poisson bracket on the manifold.
Let \(f\in k[x_1,\dots,x_n]\) be an isolated singular hypersurface and let \(W\) be a vector subspace of \(k[x_a,\dots,x_n]\) complementary to the ideal \((\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n})\). The authors prove in this article that the quasiclassical data for an MC unfolding of an isolated singularity \(f\in k[x_1,\dots,x_n]\) are given by pairs \((p,S)\) where \(p\in W\) and \(S\) is a Poisson vector field satisfying the condition \([f,S]=0\).
With this, the main result of the article is the following:
Let \(f\in k[x,y,z]\) define an isolated surface singularity. Then any quasiclassical datum of NC unfolding of \(f\) can be quantized to a noncommutative unfolding over \(k[[h]].\)
The proof of the above results involve the the study of a Lie algebroid over a commutative \(k\)-algebra \(A\), such that \(S_A(T[-1])\) has a natural structure of Gerstenhaber algebra. The main thing to solve, is to bypass the fact that the Hochschild complex \(C(A)\) is not a (genuine) Gerstenhaber algebra, this structure is only up to homotopy. The solution comes with Koszul duality and standard homotopy theory for colored operads. The involved subjects are explained through the preliminaries, that is colored operads, Koszul duality, \(\mathcal O_\infty\)-algebras, \(\tilde B\)-algebras. The last gives an algebra structure on the Hochschild cochain complex, and the article contains a section on Lie bialgebras versus \(G\)-algebras (where the operad \(G\) is Koszul). Finally, the different Lie algebra models for the Hochschild cochain complex are classified, that is, their types of equivalence is discussed. For an example, it is proved that the \(\text{Lie}_\infty\)-structure on \(C([1])\) defined by the \(\tilde B\)-structure, coincides with the (strict) Gerstenhaber bracket. Other results proves and explains weak equivalences between the models.
Finally the different models and results, in particular the comparison of the different structures on the Hochschild complexes are used to give an application on noncommutative unfolding (of a singularity). This is done by the notion of ``quasiclassical datum'' (defined as deformations of the \(B\)-algebra \(A\) over \(k[h]/(h^2)\) extendable to \(k[h]/(h^3)\)).
The article gives nice examples and applications of the noncommutative theory. The stack-theoretic aspect is somehow hidden behind generalities, making the text reasonably easy to read. Also, the results are nice, proven with good techniques. Kontsevich formality theorem; DG algebras; operads; hypersurface singularity; noncommutative unfolding; Gerstenhaber bracket; Gerstenhaber algebra; algebroid; Groupoid; Lie algebroid; Harrison chain complex Hinich (V.), Lemberg (D.).-- Noncommutative unfolding of hypersurface singularity, J. Noncommut. Geom. 8, no. 4, p. 1147-1169 (2014). Deformations and infinitesimal methods in commutative ring theory, Noncommutative algebraic geometry, Complex surface and hypersurface singularities Noncommutative unfolding of hypersurface 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 Authors' abstract: The representation scheme \(\mathbf{rep_n}\, A\) of the 3-dimensional Sklyanin algebra \(A\) associated to a plane elliptic curve and \(n\)-torsion point contains singularities over the augmentation ideal \(\mathfrak{m}\). We investigate the semi-stable representations of the non-commutative blow-up algebra \(B=A\oplus \mathfrak{m}t\oplus \mathfrak{m}^2t^2\oplus \ldots\) to obtain a partial resolution of the central singularity \(\mathbf{proj}\, Z(B)\twoheadrightarrow \mathbf{spec}\, Z(A)\) such that the remaining singularities in the exceptional fiber determine an elliptic curve and are all of type \(\mathbb{C}\times \mathbb{C}^2/\mathbb{Z}_n\). Sklyanin algebras; noncommutative geometry; representation schemes; superpotentials 10.1007/s10468-014-9515-6 Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry The geometry of representations of 3-dimensional Sklyanin 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 \textit{A. B. Coble} [Algebraic geometry and theta functions (AMS Coll. Publ. 10) (1929; JFM 55.0808.02)] established by explicit computation a unique \(A_2\)-invariant quartic hypersurface in \(\mathbb{P}(V_2)= \mathbb{C}\mathbb{P}^7\) that is singular along \(\varphi_2(A)\). Here \(A\) is a complex abelian variety of dimension 3 with a principal polarization given by an ample line bundle \({\mathcal L}\) with \(\dim H^0(A,{\mathcal L})= 1\), \((A,{\mathcal L})\) indecomposable and \(V_2= H^0(A,{\mathcal L}^2)\). \(\varphi_2: A\to \mathbb{P}(V_2)\) is defined by global sections in \({\mathcal L}^2\), and \(A_2\) is the kernel of the multiplication by 2 in \(A\). There is an analogous result for \(A\) of dimension 2, with \({\mathcal L}^3\), multiplication by 3, and an \(A_3\)-invariant cubic hypersurface in \(\mathbb{P}(V_3)= \mathbb{C}\mathbb{P}^8\). The author's main observation is that a general result about representations of the Heisenberg group implies these facts. The following proposition is given.
Let \(n= 3\) or \(4\); \(\nu=3\) if \(n= 3\) and \(\nu= 2\) if \(n= 4\); let \((T_1,\dots, T_N)\) be a coordinate system on \(\mathbb{P}(V_\nu)\); then, for \(X= A_\nu\)-invariant subvariety in \(\mathbb{P}(V_\nu)\), the space of hypersurfaces of degree \(n-1\) containing \(X\) admits a basis \((\partial F_i/\partial T_j)\), where \(F_1,\dots, F_m\) are forms of degree \(n\) on \(\mathbb{P}(V_\nu)\) such that the hypersurfaces \(F_i= 0\) are \(A_\nu\)-invariant and singular along \(X\). Beauville, The Coble hypersurfaces, C. R. Math. Acad. Sci. Paris 337 pp 189-- (2003) Hypersurfaces and algebraic geometry The Coble hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this monograph, the authors study the following question: given a compact, connected complex \(1\)-orbifold \(X\) over \(\mathbb{C}\), what is an explicit description of the canonical ring of \(X\)? This question also arises in another language: what is the structure of the ring of automorphic forms over \(\mathbb{C}\)?
They consider presentations for canonical rings in a general context as follows. A stacky curve \(\mathcal{X}\) over a field \(k\) is a smooth proper geometrically connected Deligne-Mumford stack of dimension \(1\) over \(k\) with a dense open subscheme. A stacky curve is tame if its stabilizers have order not divisible by char \(k\). A log stacky curve \((\mathcal{X}, \Delta)\) is a stacky curve \(\mathcal{X}\) equipped with a divisor \(\Delta\) which is a sum of distinct points each with trivial stabilizer. The notion of the signature \((g; e_1, \ldots, e_r; \delta)\) of a tame log stacky curve \((\mathcal{X}, \Delta)\) extends in a natural way: \(g\) is the genus of the coarse space \(X\) of \(\mathcal{X}\), there are \(r\) stacky points with (necessarily cyclic) stabilizers of order \(e_i \in \mathbb{Z}_{\geq 2}\), and \(\delta=\deg \Delta \in \mathbb{Z}_{\geq 0}\). The Euler characteristic is defined by
\[
\chi(\mathcal{X}, \Delta)=2-2g-\delta-\sum_{i=1}^{r} \left(1-\frac{1}{e_i}\right)
\]
and \((\mathcal{X}, \Delta)\) is called hyperbolic if \(\chi(\mathcal{X}, \Delta)<0\).
The authors give an explicit presentation by generators and relations for the canonical ring of a log stacky curve in terms of its signature as follows: Let \((\mathcal{X}, \Delta)\) be a hyperbolic, tame log stacky curve over a perfect field \(k\) with signature \(\sigma=(g; e_1, \ldots, e_r; \delta)\), and let \(e=\max(1, e_1, \ldots, e_r)\). Then the canonical ring
\[
R(\mathcal{X}, \Delta)=\bigoplus_{d=0}^{\infty} H^0(\mathcal{X}, \Omega(\Delta)^{\otimes d})
\]
is generated as a \(k\)-algebra by elements of degree at most \(3e\) with relations of degree at most \(6e\). In particular, if \(g+\delta \geq 2\), then \(R(\mathcal{X}, \Delta)\) is generated in degree at most \(\max(3, e)\) with relations in degree at most \(2 \max(3, e)\).
On the other hand, for log stacky curves that are not hyperbolic, the canonical ring is isomorphic to \(k\) (when \(\chi>0\)) or a polynomial ring in one variable (when \(\chi=0\)). canonical rings; canonical embeddings; stacks; algebraic curves; modular forms; automorphic forms; generic initial ideals; Gröbner bases Computational aspects of algebraic curves, Holomorphic modular forms of integral weight The canonical ring of a stacky curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a Calabi-Yau 3-orbifold \(X\) with transverse \(A_n\) singularities, this paper investigates the relationship between its Gromov-Witten theory (which studies intersection numbers on the moduli stack of orbifold stable maps to \(X\)) and its Donaldson-Thomas theory (which studies intersection numbers on the Hilbert scheme of substacks in \(X\)). The two generating functions of intersection numbers on both sides are supposed to coincide after an explicit change of variables.
In the toric case, both the GW and DT generating functions can be decomposed into contributions defined locally at each torus fixed point, the so-called orbifold topological vertex, which reduces global phenomena to the local setting. This paper conjectures an evaluation of local contributions on GW side (three-partition cyclic Hodge integrals) in terms of local contributions on DT side (loop Schur functions), which are respectively indexed by triples of conjugacy classes and triples of irreducible representations of certain generalized symmetric groups. This formula would imply the orbifold GW/DT correspondence for toric Calabi-Yau threefolds with transverse \(A_n\) singularities. The authors prove the formula in the case where one of the partitions is empty, and thus establish the orbifold GW/DT correspondence for local toric surfaces with transverse \(A_n\) singularities.
An alternative approach to the orbifold GW/DT correspondence is to reduce it to the smooth toric case where the GW/DT correspondence was proved by Maulik, Oblomkov, Okounkov and Pandharipande [\textit{D. Maulik} et al., Invent. Math. 186, No. 2, 435--479 (2011; Zbl 1232.14039)]. The authors intend to study this approach in future work by utilizing the open crepant resolution correspondence. Gromov-Witten; Donaldson-Thomas; toric Calabi-Yau 3-orbifolds; orbifold topological vertex; local toric surfaces Ross, D., Zong, Z.: Two-partition cyclic Hodge integrals and loop Schur functions (2014). arXiv:1401.2217 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Cyclic Hodge integrals and loop Schur functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a reductive algebraic group over an algebraically closed field \(k\) of characteristic \(0\), and let \(V\) be a finite dimensional algebraic \(G\)-module. Given a positive integer \(n\), finding generators of the invariant algebra \(k[V^{\oplus n}]^G\) of \(V^{\oplus n}\) is the classical problem of invariant theory. Let \(\text{pol}_nk[V]^G\) be the subalgebra of \(k[V^{\oplus n}]^G\) generated by the polarizations of all the invariants \(f\in k[V]^G.\) Then there are \(G\)-modules such that \(\text{pol}_nk[V]^G=k[V^{\oplus n}]^G.\)
The authors give several well known examples when this condition holds and also some examples, even for finite \(G\), on when the condition does not hold. The main purpose of the article is to analyze the relationship between \(k[V^{\oplus}]^G\) and \(\text{pol}_nk[V]^G.\) The authors prove that if \(G\) is finite, then \(k[V^{\oplus n}]^G\) is the integral closure of \(\text{pol}_nk[V]^G\) in its field of fractions, and that the natural morphism of affine varieties determined by these algebras is bijective.
The authors generalize the problem: They study the more general setting of actions on arbitrary affine varieties for which they define a generalization of polarizations. They then prove that if \(G\) is finite, then the invariant algebra is integral over the subalgebra generated by generalized polarizations, and the natural dominant morphism between affine varieties determined by these algebras is injective. For connected \(G\), this result obviously does not hold. The authors define \(\text{pol}\text{ind}(V)\) as the supremum over \(n\) such that the result holds. They then prove that \(k[V^{\oplus n}]^G\) is integral over \(\text{pol}_mk[V]^G\) for every \(m\leq \text{pol}\text{ind}(V)\) The authors prove that calculating \(\text{pol}\text{ind}(V)\) is related to the description of linear subspaces of the Hilbert nullcone of \(V\), and using this, the authors are able to compute the polarization index of some \(G\)-modules \(V\). That is: If \(G\) is a finite group or a linear algebraic torus, then \(\text{pol}\text{ind}(V)=\infty.\) For \(G=\text{SL}_2\) all linear subspaces of \(V\) in the nullcone of \(V\) are described and the authors then prove that \(\text{pol}\text{ind}(V)=\infty\) if \(V\) does not contain a simple 2-dimensional submodules, and \(\text{pol}\text{ind}(V)=1\) otherwise. Finally the authors calculate the polarization index of every semisimple Lie algebra \(\mathfrak g\), and they get some nice results from this.
The article is explicit to read. The authors generalize classical terms in a nice and detailed way, and the proofs are fully possible to understand. polarization; torus actions; polarization index M. Losik, P. W. Michor, V. L. Popov, On polarizations in invariant theory, J. Algebra 301 (2006), 406--424. Actions of groups on commutative rings; invariant theory, Geometric invariant theory On polarizations in invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The following statement is a consequence of a theorem of Buchweitz and results on idempotent completions of triangulated categories: Let \(X\) be an algebraic variety with isolated Gorenstein singularities \(Z=\text{Sing}(X)=\{x_1,\dots,x_p\}\). Then there is an equivalence of triangulated categories
\[
\left(\frac{D^b(\text{Coh}(X))}{\text{Perf}(X)}\right)^\omega\overset\sim\rightarrow\bigvee_{i=1}^p \underline{\text{MCM}}\left(\hat{\mathcal O}_{X,x_i}\right).
\]
The left hand side stands for the idempotent completion of the Verdier quotient \(\frac{D^b(\text{Coh}(X))}{\text{Perf}(X)}\), on the right-hand side \(\underline{\text{MCM}}(\hat{\mathcal O}_{X,x_i})\) denotes the stable category of maximal Cohen-Macaulay modules over \(\hat{\mathcal O}_{X,x_i}=:\hat O_i\). The main goal of this article is to generalize this construction as follows. Let \(\mathcal F^\prime\in\text{Coh}(X),\) \(\mathcal F:=\mathcal O\oplus\mathcal F^\prime\) and \(\mathcal A:=\mathcal End_X(\mathcal F)\). Consider the ringed space \(\mathbb X:=(X,\mathcal A)\). It is well known that the functor \(\mathcal F\overset{\mathbb L}\otimes_X-:\text{Perf}(X)\longrightarrow D^b(\text{Coh}(\mathbb X))\) is fully faithful. If \(\text{gl.dim}(\text{Coh}(\mathbb X))<\infty\) then \(\mathbb X\) can be viewed as a non-commutative (or categorical) resolution of singularities of \(X\), and the authors suggest to study the triangulated category \(\Delta_X(\mathbb X):=(\frac{D^b(\text{Coh}(\mathbb X))}{\text{Perf}(X)})^\omega\) called the relative singularity category. Assuming \(\mathcal F\) to be locally free on \(U=X\backslash Z\), an analogue of the ``localization equivalence'' is proved for the category \(\Delta(\mathbb X)\). In addition, the Grothendieck group \(\Delta(\mathbb X)\) is described.
The main result of the article is a complete description of \(\Delta_U(\mathbb Y)\) in case \(Y\) is an arbitrary curve with nodal singularities and \(\mathcal F^\prime:=\mathcal I_Z\) is the ideal sheaf of the singular locus of \(Y\). It is proved that \(\Delta_Y\) splits into a union of \(p\) blocks: \(\Delta_Y(\mathbb Y)\overset\sim\rightarrow\bigvee_{i=1}^p\Delta_i\), where \(p\) is the number of singular points of \(Y\). Each block is equivalent to the category \(\Delta_{\text{nd}}=\frac{\text{Hot}^b(\text{pro}(A_{nd}))}{\text{Hot}^b(\text{add}(P_\ast))},\) where \(A_{nd}\) is the completed path algebra of the quiver with nodes \(-,\ast,+\) and arrows \(\alpha=(-\ast),\beta=(\ast-),\delta=(\ast+),\gamma=(+\ast)\) and relations \(\delta\alpha=0\), \(\beta\gamma=0\), and \(P_\ast\) is the indecomposable projective \(A_{nd}\)-module corresponding to the vertex \(\ast\). The authors prove that the category \(\Delta_{nd}\) is idempotent complete and \(\text{Hom}\)-finite, and moreover, they give a complete classification of indecomposable objects of \(\Delta_{nd}\). \(\Delta_{nd}\overset\sim\rightarrow(\frac{D^b(\Lambda-\text{mod})}{\text{Band}(\Lambda)})^\omega\) where \(\Lambda\) is the path algebra of the quiver with nodes \(1,2,3\), arrows \(a,c\) from \(1\) to \(2\) and \(b,d\) from \(2\) to \(3\) and relations \(ba=0\), \(dc=0\), and \(\text{Band}(\Lambda)\) is the category of the band objects in \(D^b(\Lambda-\text{mod})\), i.e. the objects which are invariant under the Auslander-Reiten translation in \(D^b(\Lambda-\text{mod})\). Thus the Auslander-Reiten quiver of \(\Delta_{\text{nd}}\) is described.
Let \(\text{P}(X)\) be the essential image of \(\text{Perf}(X)\) under the fully faithful functor \(\mathbb F:=\mathcal F\overset{\mathbb L}\otimes-:\text{Perf}(X)\rightarrow D^b(\text{Coh}(\mathbb X))\). This category is described in the following intrinsic way: \(\text{Ob}(\text{P}(X))=\{\mathcal H^\bullet\in \text{Ob}(D^b(\text{Coh}(\mathbb X)))|\mathcal H^\bullet\in\text{Im}(\text{Hot}^b(\text{add}(\mathcal F_x))\rightarrow D^b(\mathcal A_X-\text{mod}))\}\). In the above notations, the relative singularity category \(\Delta_X(\mathbb X)\) is the idempotent completion of the Verdier quotient \(\text{D}^b(\text{Coh}(\mathbb X))/\text{P}(X)\), and it has a natural structure of triangulated category.
Some of the main results are thus: Let \(D^b_Z(\text{Coh}(\mathbb X))\) be the full subcategory of \(D^b(\text{Coh}(\mathbb X))\) consisting of complexes whose cohomology is supported in \(Z\) and \(\text{P}_Z(X)\cap D^b_Z(\text{Coh}(\mathbb X))\). Then the canonical functor \(\mathbb H:\frac{D^b_Z(\text{Coh}(\mathbb X))}{\text{P}_Z(X)}\rightarrow\frac{D^b(\text{Coh}(\mathbb X))}{\text{P}(X)}\) is fully faithful. With the same notations, the authors prove that the induced functor \(\mathbb H^\omega:(\frac{D^b_Z(\text{Coh}(\mathbb X))}{\text{P}_Z(X)})^\omega\rightarrow(\frac{D^b(\text{Coh}(\mathbb X))}{\text{P}(X)})^\omega \) is an equivalence of triangulated categories.
In addition to much more, the authors answer the questions:
Is the category \(\Delta_Y(\mathbb Y)\) \(\text{Hom}\)-finite? What are the indecomposable objects?
What is the Grothendieck group of \(\Delta_Y(\mathbb Y)\)?
Assume that \(E\) is a plane nodal cubic curve. What is the relation of \(\Delta_E(\mathbb E)\) with the ``quiver description'' of \(D^b(\text{Coh}(\mathbb E))\)?
To conclude in line with the authors: Let \(Y\) be a nodal algebraic curve, \(Z\) its singular locus, \(\mathcal I=\mathcal I_Z\) and \(\mathbb Y=(Y,\mathcal A)\) for \(\mathcal A=\mathcal End_Y(\mathcal O\oplus I)\). Similarly, let \(O=k[[u,v]]/(uv)\), \(\mathfrak m=(u,v)\) and \(A=\text{End}_O(O\oplus\mathfrak m)\). Then the following results are true:
The category \(\Delta_Y(\mathbb Y)\) splits into a union of \(p\) blocks \(\Delta_{nd}\), where \(p\) is the number of singular points of \(Y\) and \(\Delta_{nd}=\Delta_O(A)\).
The category \(\Delta_{nd}\) is \(\text{Hom}\)-finite and representation discrete. In particular its indecomposable objects and the morphism spaces between them are explicitly known. Moreover, one can compute its Auslander-Reiten quiver.
It is proved that \(K_0(\Delta_{nd})\cong\mathbb Z^2\).
The category \(\Delta_{nd}\) admits an alternative ''quiver description'' in terms of representations of a certain gentle algebra \(\Lambda\).
A nontrivial article, not very self contained, but the ideas are easy to follow and very interesting. triangulated category; non-commutative resolution; K-theory; Auslander-Reiten quiver; Grothendieck group; Homotopy category Burban, I; Kalck, M, The relative singularity category of a non-commutative resolution of singularities, Adv. Math., 231, 414-435, (2012) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Noncommutative algebraic geometry, Derived categories, triangulated categories The relative singularity category of a non-commutative resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In part I of this paper [Invent. Math. 114, No. 3, 565-623 (1993; Zbl 0815.14014)], \textit{M. S. Narasimhan} and the author had proved that the space of generalized theta functions on the moduli space of semi-stable rank-2 vector bundles of degree \(d\) over a complex curve of genus \(g\geq 4\), with precisely one node, is (non-canonically) isomorphic to the corresponding space of generalized theta functions for the desingularization of the base curve. Moreover, it was shown, and used for the proof of the isomorphism theorem, that (again for \(g\geq 4\)) the first cohomology group of the theta bundle on the moduli space \(U_X (d)\) of semi-stable rank-2 bundles of degree \(d\), over the nodal base curve \(X\), is trivial. Finally, the authors had announced that the restricting condition \(\geq 4\) for the genus of \(X\) can actually be dropped and, as to that fact, referred to a forthcoming paper.
The present article, the second in a series of planned papers, provided the promised proof of the announced more general result. The proof of the vanishing theorem: ``\(H^1 (U_X (d), \Theta)= (0)\) for a given nodal curve of arithmetic genus \(g\geq 1\) and with precisely one node'', together with the derivation of a concrete Verlinde formula for the dimension of the space \(H^0 (U_X (d), \Theta)\) of generalized theta functions on \(U_X (d)\), is based on two new ingredients. One of them is a rather general result from geometric invariant theory and invariant cohomology, which is certainly of independent interest. More precisely, this auxiliary result states that, for a given projective variety acted on by a reductive algebraic group \(G\), the invariant cohomology (with coefficients in an ample \(G\)-linearized line bundle \(L\)) coincides with that of the open set of semi-stable points (with respect to \(G\)). The second basic ingredient is a statement about the effect of certain ``Hecke transformations'' on quasi-parabolic rank-2 sheaves over genus-zero curves with one node.
The proof of the Verlinde formula for \(\dim_C H^0 (U_X (d), \Theta)\) then uses the vanishing theorem, \(H^1 (U_X (d), \Theta)= (0)\), and an induction argument with respect to the genus \(g\). As to the starting point \((g=0)\), the corresponding Verlinde formula was established by \textit{K. Gawȩdzki} and \textit{A. Kupiainen} in 1991 [cf.: Commun. Math. Phys. 135, No. 3, 531-546 (1991; Zbl 0722.53084)]. generalized theta functions; vanishing theorem; Verlinde formula Ramadas T.R.: Factorization of generalized theta functions. II. The Verlinde formula. Topology 35, 641--654 (1996) Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Theta functions and abelian varieties Factorisation of generalised theta functions. II: The Verlinde formula | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M\to V\) be a resolution of a two-dimensional singularity (V,p). The author considers a local version of the canonical ring, \(S=\oplus S_ n\quad with\) \(S_ n=\Gamma (M,{\mathcal O}(nK))\), set of n pluri-canonical forms on M. The fact that S is generated by \(S_ i\), \(i=1,...,n\), was conjectured by M. Reid, and proved by Shepherd-Barron.
main reslt (also conjectured by M. Reid): The algebra S is generated by \(S_ 1\oplus S_ 2\oplus S_ 3\). This reslt is sharp in the sense that there are examples for which \(S_ 1\oplus S_ 2\) does not generate S. The author also gives a characterization of the cases where \(t: S_ 2\otimes S_ 2\to S_ 4\) is surjective, and on the other hand two examples for which \(S_ 1\otimes S_ 3\) is necessary. The reslt is: t is surjective if and only if there are no minimally elliptic cycle of \(square\quad -1.\) In general the number of these cycles is not greater than \(\dim (S_ 4)/Im(t)\). The different proofs are based on the study of graphs of resolutions, fundamental cycles and Du Val curves. resolution; canonical ring; fundamental cycles H. B. LAUFER, Generation of 4-pluricanonical forms for surface singularities, Amer. J. Math., 109 (1987), pp. 571-589. Zbl0628.14031 MR892599 Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Generationof 4-pluricanonical forms for surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is devoted to prove that, given a quotient-cusp singularity \((V,p)\), its universal Abelian cover branched at \(p\) is a complete intersection cusp singularity. A quotient-cusp singularity \((V,p)\) is an isolated complex surface singularity which is double-covered by a cusp singularity. It is a rational singularity which is taut, i.e. the topology of its link determines the analytic type of \((V,p)\). In the paper, the universal Abelian cover \(({\widetilde V}, p)\) of \((V,p)\) branched at \(p\) is determined from the topology of the link \((V,p)\), and the proof is based on the description of the link of \((V,p)\) obtained by \textit{W. D. Neumann} [Trans. Am. Math. Soc. 268, 299-343 (1981; Zbl 0546.57002)], and some calculus. The authors also sketch a more geometric proof by giving an explicit group action on a different complete intersection representation of the cusp.
If the link of a normal complex surface singularity is a rational homology sphere, then the universal Abelian cover of the link is a finite covering, hence it gives rise to the universal Abelian cover of the singularity. The result in the paper, and the study by \textit{W. D. Neumann} [in: Singularities, Summer Inst., Arcata/Calif. 1981. Proc. Symp. Pure Math. 40, Part 2, 233-243 (1983; Zbl 0519.32010)] of universal Abelian covers of weighted homogeneous normal surface singularities is the motivation of the authors to state the following conjecture:
Let \((X,o)\) be a \(\mathbb Q\)-Gorenstein normal surface singularity whose link is a rational homology sphere, then the universal Abelian cover \(({\widetilde X}, 0)\) of \((X,o)\) is a complete intersection. quotient-cusp; universal Abelian cover; cusp singularity; group action; surface singularity; complete intersection Neumann, W.D., Wahl, J.: Universal abelian covers of quotient-cusps, Math. Ann. 326, 75--93 (2003) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Coverings in algebraic geometry, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Complex surface and hypersurface singularities Universal Abelian covers of quotient-cusps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The first part of the paper focuses on finding a local detection property for projectivity of modules for Frobenius kernels over an algebraically closed field \(k\) of prime characteristic \(p\). For an example of this, consider a finite group \(G\). \textit{L.~Chouinard} [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)] showed that a \(kG\)-module \(M\) (even infinite dimensional) is projective if and only if it is projective upon restriction to \(kE\) for all elementary Abelian \(p\)-subgroups \(E\) of \(G\). For a `finite' dimensional module, Chouinard's Theorem was later seen to be a consequence of the extensive theory of cohomological support varieties for finite groups. However, Chouinard's Theorem is actually needed to deduce key properties in a general theory of support varieties for arbitrary modules.
The author considers Frobenius kernels of a smooth algebraic group \(G\) over \(k\) to which a theory of support varieties exists for finite dimensional modules. For such modules, it follows that projectivity can be detected by subgroup schemes that are Frobenius kernels of the additive group \(\mathbb{G}_a\). The natural question arises as to whether or not this property can be deduced directly and further for arbitrary modules. Building upon work of the reviewer [Proc. Am. Math. Soc. 129, No. 3, 671-676 (2001; Zbl 0990.20028)] for unipotent algebraic group schemes, the author shows that the detection property holds in general. The author then proceeds to develop a theory of `support cones' for arbitrary modules and uses the above detection result to derive desired properties of these cones. Lastly, the author gives a description of support cones in terms of Rickard idempotent modules as done for finite groups by \textit{D. Benson, J. Carlson}, and \textit{J. Rickard} [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)]. algebraic groups; Frobenius kernels; projectivity; cohomological support varieties; support cones; Rickard idempotent modules; group schemes Pevtsova, Julia, Infinite dimensional modules for Frobenius kernels, J. Pure Appl. Algebra, 0022-4049, 173, 1, 59\textendash86 pp., (2002) Representation theory for linear algebraic groups, Group schemes, Modular Lie (super)algebras, Modular representations and characters, Group rings of infinite groups and their modules (group-theoretic aspects), Group rings Infinite dimensional modules for Frobenius kernels | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This work is a very interesting survey about the connections between moduli of quiver representations and moduli of vector bundles. It contains many enlightening examples and stimulating exercises.
The first sections aim to introduce the reader to quivers, their representations and their moduli spaces (see Section 2), vector bundles over curves and their moduli spaces (see Section 3). Section 4 is dedicated to the associated hyperkähler moduli spaces. For quivers, such objects rise when considering, over the complex numbers, the representations of the double quiver, with relations. The analog for bundles is given by Higgs bundles and their moduli space (and relative branes).
Section 5 is mainly devoted to the proof of Theorem 5.3, due to \textit{W. Crawley-Boevey} and \textit{M. Van den Bergh} [Invent. Math. 155, No. 3, 537--559 (2004; Zbl 1065.16009)]. This result gives a formula to compute the number \(\mathcal{A}_{Q,d}(q)\) of isomorphic classes of absolutely indecomposable representations of a quiver \(Q\) over a finite field \(\mathbb{F}_q\), of sufficiently large prime characteristic, with indivisible dimension vector \(d\). The last part of the section describes the extension of this result to vector bundles, due to \textit{O. Schiffmann} [Ann. Math. (2) 183, No. 1, 297--362 (2016; Zbl 1342.14076)]. algebraic moduli problems; geometric invariant theory; representation theory of quivers; vector bundles and Higgs bundles on curves Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Representations of quivers and partially ordered sets, Vector bundles on curves and their moduli Parallels between moduli of quiver representations and vector bundles over curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a ground field of characteristic zero, \(X\) be a smooth proper curve, and \(G\) be a connected reductive group. The authors prove that the differential-graded category \(C=\text{D-mod}(\mathcal{Y})\) of D-modules on the Artin stack \(\mathcal{Y}=\text{Bun}_G\) of principal \(G\)-bundles \(P\rightarrow X\) is compactly generated. In other words, there is a generating system of compact objects \(c_\alpha\in C\), which means that the Hom functors \(\text{Hom}(c_\alpha,-)\) commutes with arbitrary sums, and that \(\text{Hom}(c_\alpha,c)=0\) for all \(c_\alpha\) only if \(c\) is the zero-object.
This property is easily seen for quotient stacks \(\mathcal{Y}=[Z/H]\), where \(Z\) is a quasicompact scheme and \(H\) is an algebraic group. More generally, it is valid for quasicompact Artin stacks \(\mathcal{Y}\) whose field-valued points have affine automorphism groups, according to []. However, the property usually fails if \(\mathcal{Y}\) is not quasicompact, even for schemes. An example given in this paper is the smooth locally algebraic surface \(X=\bigcup U_i\) with \(U_i=X_i\smallsetminus\left\{x_i\right\}\), where \(\ldots\rightarrow X_2\rightarrow X_1 \rightarrow X_0\) is a suitable sequence of blowing ups with centers \(x_i\in X_i\): Here the DG-category of D-modules is not compactly generated. So for \(\mathcal{Y}=\text{Bun}_G\), the result is rather surprising.
The authors introduce an abstract sufficient criterion, called truncatability, for \(\text{D-Mod}(\mathcal{Y})\) to be compactly generated. The relevant terminology is as follows: A closed substack \(i:\mathcal{Z}\subset\mathcal{Y}\) is called truncative if the functor \(i^! \) sends compact objects in the category of D-modules on \(\mathcal{Y}\) to compact objects in the category of D-modules on \(\mathcal{Z}\). There are several other characterizations, in terms of related functors \(i_{\text{dR},*}\), as well as \(j^*\) and \(j_*\), where \(j:U\rightarrow\mathcal{Y}\) is the complementary open substack. One equivalent condition is that \(j_*\) admits a continuous right adjoint. In this situation, one also says that \(U\) is co-truncative. The concept also extends to locally closed substacks. An Artin stack \(\mathcal{Y}\) is called QCA if it is quasicompact, and its field-valued points have affine automorphism groups. Without the former condition, on says that \(\mathcal{Y}\) is locally QCA. In this situation, the authors call \(\mathcal{Y}\) truncatable if it is covered by open quasicompact substacks that are co-truncative in the above sense, and then show that the ensuing category of D-modules is compactly generated.
In the case \(G=\text{SL}_2\), it is then established that the open substacks \(\text{Bun}_G^{(n)}\) of vector bundles that do not admit line subbundles of degree \(>n\) form open co-truncative substacks that cover \(\mathcal{Y}=\text{Bun}_G\). This is generalized to arbitrary connected reductive \(G\), by considering the rational cone \(\Lambda_G^{+,\mathbb{Q}}\) generated by the dominant coweights, with respect to a fixed choice of Borel subgroup. For each \(\theta\) in this cone, the authors consider the open substack \(\text{Bun}_G^{\leq \theta}\) of all bundles \(\mathcal{P}_G\) so that for each reduction \(\mathcal{P}_P\) so some standard parabolic \(P\subset G\) of degree \(\mu\) (defined in a suitable way), one has \(\mu\leq \theta\) (for some natural order relation coming from the positive coroots). The authors are then able to show that this gives the desired co-trucative open substacks, provided that \(\theta\) is large enough, measured in terms of the simple roots of the reductive group \(G\) and the genus of the smooth curve \(X\). D-modules; DG-categories; algebraic stacks; principal bundles; compact objects Drinfeld, V.; Gaitsgory, D., Compact generation of the category of D-modules on the stack of \(G\)-bundles on a curve, Camb. J. Math., 3, 1-2, 19-125, (2015) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Stacks and moduli problems Compact generation of the category of D-modules on the stack of \(G\)-bundles on a curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field, let \(\gamma\) be an arc of a \(k\)-variety \(X\), and assume that \(\gamma\) does not factor through the singular locus of \(X\). The Drinfeld-Grinberg-Kazhdan Theorem states that the formal neighborhood of \(\gamma\) in the arc scheme of \(X\) is isomorphic to the product of an infinite dimensional formal disc and a formal neighborhood in a finite type \(k\)-scheme. The latter formal neighborhood has been interpreted as a finite dimensional model of the formal neighborhood of \(\gamma\). The authors prove a generalization of the Drinfeld-Grinberg-Kazhdan theorem where \(X\) is replaced with a topologically finite type formal \(k[[T]]\)-scheme \(\mathcal{X}\), and they study consequences related to singularity theory.
In the case where \(\mathcal{X}\) is the completion of an affine \(k\)-variety, their proof gives an algorithm for computing the resulting finite dimensional model. In addition, they prove that these finite dimensional models are compatible with base change by separable extensions of \(k\), and they also prove a factoring statement involving the truncation morphisms from arc schemes to jet schemes.
The authors also present a proof, communicated to them by O. Gabber, of a cancellation theorem, which in particular allows them to define unique minimal finite dimensional models for an arc's formal neighborhood. The authors also use finite dimensional models to define an invariant they call the absolute nilpotency index of a singularity, and they prove that this is a formal invariant of the singularity. arc scheme; formal neighborhood Bourqui D., The Drinfeld--Grinberg--Kazhdan Theorem for Formal Schemes and Singularity Theory (2015) Arcs and motivic integration, Singularities in algebraic geometry The Drinfeld-Grinberg-Kazhdan theorem for formal schemes and singularity theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For Parts I and II, cf. Ann. Sci. Éc. Norm. Supér. (4) 27, 249-344 (1994; Zbl 0860.11030); Invent. Math. 116, 243-308 (1994; Zbl 0860.11031) and Invent. Math. 121, 437 (1995; Zbl 1008.11529).
Let \(G\) be a reductive group over \(\mathbb Q\), and let \(X\) be the symmetric space associated to \(G(\mathbb R)\). Given a discrete subgroup \(\Gamma\) of \(G(\mathbb Q)\) and a representation \(V\) of \(G\), the group cohomology \(H^\bullet (\Gamma, V)\) can be identified with the cohomology \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) of the locally symmetric space \(\Gamma \setminus X\) with coefficients in the local system \(\widetilde{\mathbb V}\) corresponding to \(V\). The cohomology \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) can be decomposed as the direct sum of the interior cohomology, defined as the image of the cohomology with compact supports, and the boundary cohomology that restricts nontrivially to the boundary of the Borel-Serre compactification of \(\Gamma \setminus X\). The adelic version of \(\Gamma \setminus X\) is the Shimura variety \(\text{Sh} (G,X)\), whose connected components are of the form \(\Gamma \setminus X\). This has a canonical model over a number field \(E\), and \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) can be identified with the hypercohomology of an \(E\)-rational complex of coherent sheaves on \(\text{Sh} (G,X)\). In particular, \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) has a Hodge filtration whose graded pieces are given by the coherent cohomology with coefficients in certain automorphic vector bundles.
In this paper, the authors prove that the nerve spectral sequence of the Borel-Serre boundary of the Shimura variety \(\text{Sh} (G,X)\) is a spectral sequence of mixed Hodge-de Rham structures over \(E\) by developing the machinery of automorphic vector bundles on mixed Shimura varieties and studying the nerve spectral sequence for those bundles and the toroidal boundary. They also extend the technique of averting issues of base change by taking cohomology with growth conditions and provide formulas for the Hodge gradation of the cohomology of both \(\text{Sh} (G,X)\) and its Borel-Serre boundary. Shimura varieties; automorphic vector bundles; cohomology of arithmetic groups; mixed Hodge structures Harris, M., Zucker, S.: Boundary cohomology of Shimura varieties. III. Coherent cohomology on higher-rank boundary strata and applications to Hodge theory. \textit{Mém. Soc. Math. Fr. (N.S.)}, (85):vi+116, (2001) Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Cohomology of arithmetic groups Boundary cohomology of Shimura varieties. III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra \(\mathfrak{g}\). We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone \(\mathcal{S} \cap \mathcal{N}\) of \(\mathfrak{g}\). We calculate refined Hilbert series for Classical algebras up to rank 4 (and \(A_5\)), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections \(\mathcal{S} \cap \mathcal{N}\); such dual quiver theories exist for the Slodowy slices of \(A\) algebras, but are limited to a subset of the Slodowy slices of \textit{BCD} algebras. The analysis opens new questions about the extent of 3\(d\) mirror symmetry within the class of SCFTs known as \(T_{\sigma}^{\rho}(G)\) theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the \(\operatorname{SU}(2)\) embedding into \(G\) of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone. Yang-Mills and other gauge theories in quantum field theory, Representations of quivers and partially ordered sets, Supersymmetric field theories in quantum mechanics, Coadjoint orbits; nilpotent varieties, Mirror symmetry (algebro-geometric aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Quiver theories and formulae for Slodowy slices of classical 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 We show that the dimer model on a bipartite graph \(\Gamma\) on a torus gives rise to a quantum integrable system of special type, which we call a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space \(\mathcal L_\Gamma\) of line bundles with connections on the graph \(\Gamma\). The sum of Hamiltonians is essentially the partition function of the dimer model.
We say that two such graphs \(\Gamma_1\) and \(\Gamma_2\) are equivalent if the Newton polygons of the corresponding partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent minimal bipartite graphs are related by a sequence of elementary transformations. For each elementary transformation we define a birational Poisson isomorphism \(\mathcal L_{\Gamma_1}\to \mathcal L_{\Gamma_2}\) providing an equivalence of the integrable systems. We show that it is a cluster Poisson transformation, as defined in [\textit{V. V. Fock}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865--930 (2009; Zbl 1180.53081)].
We show that for any convex integral polygon \(N\) there is a non-empty finite set of minimal graphs \(\Gamma\) for which \(N\) is the Newton polygon of the partition function related to \(\Gamma\). Gluing the varieties \(\mathcal L_\Gamma\) for graphs \(\Gamma\) related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson space \(\mathcal X_N\). It is a natural phase space for the integrable system. The Hamiltonians are functions on \(\mathcal X_N\), parametrized by the interior points of the Newton polygon \(N\). We construct Casimir functions whose level sets are the symplectic leaves of \(\mathcal X_N\).
The space \(\mathcal X_N\) has a structure of a cluster Poisson variety. Therefore the algebra of regular functions on \(\mathcal X_N\) has a non-commutative \(q\)-deformation to a \(\ast\)-algebra \(\mathcal O_{q}(\mathcal X_N)\). We show that the Hamiltonians give rise to a commuting family of quantum Hamiltonians. Together with the quantum Casimirs, they provide a quantum integrable system. Applying the general quantization scheme [\textit{V. V. Fock} and \textit{A. B. Goncharov}, Invent. Math. 175, No. 2, 223--286 (2009; Zbl 1183.14037)], we get a \(\ast\)-representation of the \(\ast\)-algebra \(\mathcal O_{q}(\mathcal X_N)\) in a Hilbert space. The quantum Hamiltonians act by commuting unbounded selfadjoint operators.
For square grid bipartite graphs on a torus we get discrete quantum integrable systems, where the evolution is a cluster automorphism of the \(\ast\)-algebra \(\mathcal O_{q}(\mathcal X_N)\) commuting with the quantum Hamiltonians. We show that the octahedral recurrence, closely related to Hirota's bilinear difference equation [\textit{R. Hirota}, ``Discrete analogue of a generalized Toda equation'', J. Phys. Soc. Japan 50, 3785--3791 (1981)], appears this way.
Any graph \(G\) on a torus \(\mathbb T\) gives rise to a bipartite graph \(\Gamma_G\) on \(\mathbb T\). We show that the phase space \(\mathcal X\) related to the graph \(\Gamma_G\) has a Lagrangian subvariety \(\mathcal R\), defined in each coordinate system by a system of monomial equations. We identify it with the space parametrizing resistor networks on \(G\). The pair \((\mathcal X, \mathcal R)\) has a large group of cluster automorphisms. In particular, for a hexagonal grid graph we get a discrete quantum integrable system on \(\mathcal X\) whose restriction to \(\mathcal R\) is essentially given by the cube recurrence.
The set of positive real points \(\mathcal X_{N}(\mathbb R_{>0})\) of the phase space is well defined. It is isomorphic to the moduli space of simple Harnack curves with divisors studied in [\textit{R. Kenyon} and \textit{A. Okounkov}, Duke Math. J. 131, No. 3, 499--524 (2006; Zbl 1100.14047)]. The Liouville tori of the real integrable system are given by the product of ovals of the simple Harnack curves. In the sequel to this paper we show that the set of complex points \(\mathcal X_{N}(\mathbb C)\) of the phase space is birationally isomorphic to a finite cover of the Beauville complex algebraic integrable system related to the toric surface assigned to the polygon \(N\). integrable systems; dimers; cluster algebras Goncharov A, Kenyon R. (2013) Dimers and cluster integrable systems. Ann. Sci. Éc. Norm. Supér. 46, 747-813. (10.24033/asens.2201) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Planar graphs; geometric and topological aspects of graph theory, Cluster algebras, Relationships between algebraic curves and integrable systems, Teichmüller theory for Riemann surfaces, Poisson manifolds; Poisson groupoids and algebroids, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics Dimers and cluster integrable systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose \(S= S(\Gamma)= \Gamma \setminus X\) is a Shimura variety obtained as the quotient of a Hermitian symmetric space \(X= G(\mathbb{R})/ K_\infty\) (where \(G\) is a semisimple group over \(\mathbb{Q}\) and \(K_\infty \subset G(\mathbb{R})\) is a maximal compact subgroup) by an arithmetic subgroup \(\Gamma\) of \(G(\mathbb{Q})\).
Suppose that \(H\) is a semisimple \(\mathbb{Q}\)-subgroup of \(G\) such that \(Y= H(\mathbb{R})/ K_H\) is also Hermitian symmetric, with \(K_H= K_\infty\cap H\) a maximal, compact subgroup of \(H(\mathbb{R})\), and such that the natural inclusion \(i\) of \(Y\) in \(X\) is holomorphic. For every covering \(S(\Gamma')\to S(\Gamma)\) with \(\Gamma' \subset \Gamma\) of finite index, the resulting map \(i= i(\Gamma'): \Gamma'\cap H\setminus Y\to \Gamma'\setminus X\) is, as is well known, a morphism of varieties. Let \(C\) be a correspondence on \(S\) that is of the form \(z\to g(z)\) on the universal covering \(X\) of \(S\) with \(g\in G(\mathbb{Q})\). We therefore get a finite covering \(C: S(\Gamma')\to S(\Gamma)\) for some \(\Gamma'\).
In this paper, we are concerned with the following question: If \(\omega\) is a cohomology class on \(S\) whose restriction (i.e., the pullback via the composite of \(C\) with \(i\)) to \(\Gamma'\cap H\setminus Y\) is zero for all the correspondences \(C\) defined above (we say, then, that \(\omega\) vanishes stably along \(H\)), then is \(\omega\) itself zero? This question is hard to answer for an arbitrary class \(\omega\), but we can give a criterion purely in terms of the linear algebra of \(G\) and \(H\) for holomorphic (and cuspidal, if \(S\) is not compact) forms on \(S\). One of the reasons the holomorphic case is easier is that the restriction of a holomorphic form to the subvariety is indeed holomorphic. Whereas, even if \(\omega\) is a harmonic form on \(S\), its restriction to \(\Gamma'\cap H\setminus Y\) need not be harmonic in general. We make use of the explicit description of \(({\mathfrak g},K)\)-modules with nonzero cohomology. Indeed, the author [J. Reine Angew. Math. 430, 69--83 (1992; Zbl 0787.14011) and 444, 1--15 (1993; Zbl 0781.14014)] dealt with the question of the vanishing or nonvanishing of cup products of two holomorphic forms on \(S\). This may be interpreted as a question of the vanishing of the restriction to the diagonal of the tensor product of these two forms.
The question of the stable vanishing of \(\omega\) along \(H\) (at least in the cocompact case; when \(S\) is not compact, one must restrict oneself to cuspidal cohomology) has a simple description in terms of the Parthasarathy-Vogan-Zuckerman theory. Suppose that the infinity type of \(\omega\) is \(A_{\mathfrak q}\), associated to the \(\theta\)-stable parabolic subalgebra \({\mathfrak q}\) of the Lie algebra \({\mathfrak g}= \operatorname {Lie} G(\mathbb{R}) \otimes \mathbb{C}\). Let \({\mathfrak p}^+\) be the holomorphic tangent space of \(X= G(\mathbb{R})/ K_\infty\), \({\mathfrak u}\) be the unipotent radical of \({\mathfrak q}\), \({\mathfrak u}^+\) be its intersection with \({\mathfrak p}^+\), and \(R\) be the dimension of \({\mathfrak u}^+\). Then, \(\omega\) is stably nonzero along \(H\) if and only if the \(R\)th exterior power of \({\mathfrak u}^+\) (a line) lies in the smallest subspace of the \(R\)th exterior power of \({\mathfrak p}^+\), which contains the \(R\)th exterior power of \({\mathfrak p}^+\cap {\mathfrak h}\) (here, \({\mathfrak h}= \operatorname {Lie} H(\mathbb{R}) \otimes \mathbb{C})\) and is stable under the adjoint action of \(K_\infty\). We prove this in Section 2, first in the compact case. The noncompact case is dealt with similarly (since we are only concerned with cuspidal holomorphic cohomology), and we briefly indicate the modifications necessary. We then use this criterion in the case of the classical Hermitian symmetric domains and some naturally embedded sub-Hermitian domains. The results are set out in Section 3.
The conjectures of Langlands, Arthur, and Kottwitz on the zeta-functions of Shimura varieties impose strong restrictions on the Galois representations occurring in the étale (intersection) cohomology of (the Borel-Baily-Satake compactification of) \(S\). By working out the predictions of these conjectures in the special cases of \(U(g,h)\) (\(g\leq h\), and both \(g\) and \(h\) are not 2) and \(GSp(g)\) \((g\geq 2)\), we see that the action of the Galois group on the cohomology degree \(g\) of the Shimura varieties corresponding to these two groups is potentially Abelian.
As an application of the calculations of Section 3, we show in Section 5 that, given a holomorphic \(g\)-form on a Siegel-modular variety (the Shimura variety associated to the group \(GSp(g)\) with \(g\geq 2\)), its restriction to some product of \(g\)-modular curves is nonzero. This is shown to imply that the Mumford-Tate group of the compactly supported cohomology in degree \(g\) of the Siegel-modular variety is Abelian. (This was implicitly proved in a paper of \textit{R. Weissauer} [J. Reine Angew. Math. 391, 100--156 (1988; Zbl 0657.10022)] in the case \(g=2\).) This confirms the heuristics of Section 4. As a consequence, we find that the action of the Galois group on the (image of the compactly supported cohomology in degree \(g\) in the) étale intersection cohomology of the associated (Borel-Baily-Satake) compactification of the Siegel-modular variety is potentially Abelian.
As another application, we show that the Mumford-Tate group of the (compactly supported) cohomology in degree \(g\) of some Shimura varieties attached to \(U(g,h)\) \((2\leq g\leq h\) and \((g,h)\neq (2,2)\)) is also Abelian by restricting the cohomology to an appropriate product of curves. Analogously, we show that the action of the Galois group on the (image of the compactly supported cohomology in the) étale cohomology in degree \(g\) of the associated compactification of this Shimura variety is potentially Abelian. L. Clozel and T.\ N. Venkataramana, Restriction of the holomorphic cohomology of a Shimura variety to a smaller Shimura variety, Duke Math. J. 95 (1998), 51-106. Arithmetic aspects of modular and Shimura varieties, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Other groups and their modular and automorphic forms (several variables), Modular and Shimura varieties, Representations of Lie and linear algebraic groups over global fields and adèle rings, Classical real and complex (co)homology in algebraic geometry Restriction of the holomorphic cohomology of a Shimura variety to a smaller Shimura variety. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A purely elliptic singularity is a normal isolated singularity \((X,x)\) of a complex analytic space of dimension at least two, such that the \(l^2\)-plurigenus \(\delta _m (X,x)=1\) for all positive integer \(m\). These singularities, as well as some generalizations thereof (such as \({\mathbb Q}\)-Gorenstein singularities) have been studied by several mathematicians (S. Ishii, M. Tomari, K. Watanabe, among others). Purely elliptic singularities have many interesting properties and are related, perhaps surprisingly, to other mathematical objects. For instance, if \(\pi: {\widetilde X} \to X\) is a \textit{good} resolution of \(X\) (meaning that the exceptional divisor \(E\) has simple normal crossings), then it is possible to define a divisor \(E_J \subseteq E\), called the essential divisor, which plays an important role in the theory. Under suitable assumptions (e.g., the existence of a a certain partial desingularization, called the canonical modification of \((X,x)\)), it is proved that the irreducible components of \(E_J\) are birationally equivalent to Calabi-Yau varieties (i.e., projective varieties \(V\) with \(H_i(V, {\mathcal O}_V)=0\) for \(0 <i < n\), \(V\) might be singular, but at worst it has Gorenstein canonical singularities). This was originally proved in a rather abstract fashion.
In the present paper the author finds a more explicit proof of this fact, in the case of a purely elliptic singularity of an \(r\)-dimensional hypersurface \(X\), defined by a non-degenerate polynomial \(f=f(x_0, \ldots, x_r)\). This means: if we consider the Newton polyhedron \( \Gamma _{+} (f) \subset {\mathbb R}^{r+1}\) of \(f\), then the partials of \(f\) do not simultaneously vanish on any compact face of the polyhedron. It was known that then there is a single compact face \(\gamma _{\mathbf 1}\) of \(\Gamma _{+} (f)\) containing the point \((1, \ldots, 1)\). Let \(d = \dim \, (\gamma _{\mathbf 1}) \). Kanesaka shows that each irreducible component of \(E_J\) (satisfying a certain condition) is birationally equivalent to the product of a Calabi-Yau variety \(Y\) of dimension \(d-1\) times \({\mathbb P}^{r-d}\). Moreover, the birational class of \(Y\) depends on the leading part of the polynomial \(f\) only, and not on the choice of the component. The leading part of \(f\) is the sum of the terms whose coefficients are in \(\gamma _{\mathbf 1}\). The techniques of the proof are primarily from toric and birational geometry (in the sense of Mori's program). An introductory section is devoted to background material, it contains many useful bibliographical references. purely elliptic singularity; hypersurface; resolution; Calabi-Yau variety; toric variety; Newton polyhedron Complex surface and hypersurface singularities, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Modifications; resolution of singularities (complex-analytic aspects), Calabi-Yau manifolds (algebro-geometric aspects) On the defining equations of hypersurface purely elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a discrete subgroup \(\Gamma \subset U((2,1),{\mathbb{C}})\), acting on the two-dimensional unit ball \({\mathcal B}\) through its linear action on projective coordinates, and, for \(\gamma\in \Gamma\), denote by \(j_{\gamma}\) the jacobian determinant of this action. Let \(\chi: \Gamma\to {\mathbb{C}}^*\) be a character of finite order, and let \(S(\Gamma,\chi)\) be the graded algebra over \({\mathbb{C}}\) of modular forms on \({\mathcal B}\) satisfying \(f=\chi (\gamma)\cdot j^ k_{\gamma}\cdot \gamma^*(f)\), \(\gamma\in \Gamma\). The principal groups studied in this paper are \(\Gamma =U((2,1)\), \({\mathcal O}_{{\mathbb{Q}}(\sqrt{-3})})\) and its congruence subgroup \(\Gamma(\sqrt{-3})\). The author gives an explicit description by generators and relations of \(S(\Gamma,1)\) and \(S(\Gamma(\sqrt{-3}),1)\) (the latter has 11 generators with 28 relations).
The method is to start with modular forms that arise naturally as the coefficients and ramification points of the normal form \(y^ 3=x^ 4+G_ 2x^ 2+G_ 3x+G_ 4=\prod^{4}_{i=1}(x-\xi_ i) \) for a Picard curve, just as the classical modular forms \(g_ 2\) and \(g_ 3\) on the upper half-plane arise as coefficients for the normal form of an elliptic curve. It turns out that there is a certain character \(\chi\) of order 6 such that \(S(\Gamma,\chi)={\mathbb{C}}[G_ 2,G_ 3,G_ 4]\) and S(\(\Gamma\) (\(\sqrt{-3}))={\mathbb{C}}[\xi_ 1,\xi_ 2,\xi_ 3]\). The key result of the paper under review is the explicit determination of the character \(\chi\), which enables the author to derive the structure of \(S(\Gamma,1)\) and \(S(\Gamma(\sqrt{-3}),1)\). (The description of \(\chi\) and the structure of \(S(\Gamma,1)\) were previously obtained by J.-M. Feustel using theta constants.) Finally, the author examines the algebra of modular forms satisfying \(f(z_ 1,z_ 2)=(c_ 0+c_ 1z_ 1+c_ 2z_ 2)^{-3k} \gamma^*(f)(z_ 1,z_ 2),\) where \((c_ 0,c_ 1,c_ 2)\) is the bottom row of \(\gamma\), by identifying it with \(S(\Gamma,\psi)\), where \(\psi\) is a certain character of order 6, not equal to \(\chi\). generators; relations; Picard curve; S(\(\Gamma \) ,1); S(\(\Gamma \) (\(\sqrt{-3}),1)\); algebra of modular forms Holzapfel, R.-P. , On the nebentypus of Picard modular forms . To appear in Math. Nach. Theta series; Weil representation; theta correspondences, Special algebraic curves and curves of low genus, Special varieties On the nebentypus of Picard modular forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, O'Grady's examples of irreducible holomorphic symplectic manifolds are investigated. An irreducible holomorphic symplectic manifolds is a simply connected complex Kähler manifold whose space of global holomorphic \(2\)-forms is generated by a closed non-degenerate \(2\)-form. Assuming compactness, this notion is equivalent to the notion of a hyperkähler manifold in Riemannian geometry. The interest in such manifolds comes from Bogomolov's decomposition theorem, which says that compact Kähler manifolds with torsion first Chern class have a finite cover which is isomorphic to the product of a flat torus, Calabi-Yau manifolds and simply-connected hyperkähler manifolds. Currently, in contrast to the other two types of building blocks, the number of known examples of irreducible holomorphic symplectic manifolds is very limited. To date, there are only four types of examples known: Hilbert schemes, generalised Kummer varieties and two examples of O'Grady of dimension six and ten respectively. Any other known example of irreducible holomorphic symplectic manifold is deformation equivalent to one of these.
In section 1 of the paper under review, the author proves that O'Grady's examples are deformation equivalent to Lagrangian fibrations. This confirms for these two examples what is conjectured for all irreducible holomorphic symplectic manifolds. The rest of this paper is concerned with topological properties of O'Grady's six-dimensional example. In section 2 it is shown that its topological Euler characteristic is equal to \(1920\). The bulk of the paper is filled by section 3, where an explicit description of the Beauville form is given and where it is shown that the Fujiki constant is equal to \(60\). It is interesting to note that the value of the Fujiki constant obtained here is the same as in the case of the six-dimensional generalised Kummer variety. This is the first known case of non-diffeomorphic irreducible holomorphic symplectic manifolds with equal Fujiki constant.
The main tool to get these results is a slight generalisation of O'Grady's desingularisation which allows to construct a birational model of the original examples of O'Grady. This new irreducible holomorphic symplectic manifold comes with a natural morphism to a projective space (the complete linear system of twice a theta divisor on the Jacobian of a curve of genus two). A detailed analysis of the fibres of this morphism allows the calculation of the topological invariants of this manifold. The birational morphism to O'Grady's original example is obtained with the aid of a Fourier-Mukai transform. A result of \textit{D. Huybrechts} [J. Differ. Geom. 45, No.~3, 488--513 (1997; Zbl 0917.53010)] which says that two irreducible holomorphic symplectic manifolds which are birational are also deformation equivalent is then used to obtain information about the topology of O'Grady's example. This gives in particular the Euler characteristic. The result in section 1 follows now from work of \textit{D. Matsushita} [Topology 40, 431--432 (2001; Zbl 0932.32027)]. The calculation of the Beauville form and Fujiki constant is more involved. It starts with a basis of the rational second cohomology which was given by O'Grady. From it, a basis of the integer cohomology is extracted. This is done with the aid of the Uhlenbeck compactification of the \(\mu\)-stable locus of the moduli space whose symplectic resolution gave O'Grady's example. Moreover, the birational model used before, plays again a crucial role in the calculations. holomorphic symplectic structure; moduli space; Mukai vector; Bogomolov decomposition; Donaldson morphism; Kummer variety; stable sheaf Rapagnetta A.: Topological invariants of O'Grady's six dimensional irreducible symplectic variety. Math. Z. 256, 1--34 (2007) Topological properties in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Compact Kähler manifolds: generalizations, classification, Global differential geometry of Hermitian and Kählerian manifolds, Algebraic moduli problems, moduli of vector bundles Topological invariants of O'Grady's six dimensional irreducible symplectic variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Fourier-Mukai transform has been of interest in algebraic geometry since Mukai introduced it to study derived categories of coherent sheaves over abelian varieties and surfaces. The Nahm transform gives a differential geometric analogue, and has been used to relate instantons or monopoles on dual manifolds. Let \(C\) be a complex elliptic curve and \(\hat{C}\) its dual variety. The authors consider triples \((E_{1} , E_{2}, \Phi)\) where \(E_1\) and \(E_2\) are holomorphic vector bundles over \(C\) and \(\Phi \colon E_{2} \to E_1\) is a homomorphism. They study the compatibility of Fourier--Mukai and Nahm transforms of IT triples on \(C\), using the correspondence between such triples and \(\text{SU}(2)\)-invariant bundles on \(C \times \mathbb P^1\), and apply this to triples satisfying coupled vortex equations.
In section 1, the authors motivate their study and give a summary of the paper.
In section 2.1, the Fourier--Mukai transform \({\mathcal S}^{i}(E)\) of an IT\(_i\) sheaf \(E\) is recalled. In 2.2, the authors consider pairs \((E, \nabla)\) where \(E\) is a Hermitian vector bundle over \(C\) and \(\nabla\) a unitary connection. The IT property is defined for such pairs, and a sufficient condition for it is given in terms of the curvature of \(\nabla\). The Nahm transform \((\hat{E}, \hat{\nabla} )\) of an IT pair is constructed, and it is shown that the transform preserves the constant central curvature property of \(\nabla\).
In 2.3, a natural map \(\phi_{E} \colon \hat{E} \to {\mathcal S}^{i}(E)\) is written down. The Nahm transform of a bundle map \(\Phi\) is given, and the two transforms are then shown to be compatible via \(\phi_E\). An inverse is found for the Nahm transform when \(\nabla\) has constant nonzero central curvature.
In section 3.1, the authors introduce holomorphic triples. They recall \(\alpha\)-stability of a triple, where \(\alpha\) is a real parameter. They give a bijection between triples and certain \(\text{SU(2)}\)-equivariant bundles over \(C \times \mathbb P^1\) which is compatible with the respective notions of stability.
In 3.2, they define the relative Fourier-Mukai transform of the \(\text{SU}(2)\)-invariant bundle over \(C \times \mathbb P^1\) associated to an IT triple, and show that this corresponds to the triple composed of the absolute transforms of the components defined in 2.1.
In 3.3 and 3.4, it is shown that for certain ranks and degrees this transform preserves \(\alpha\)-stability for ``small'' and ``large'' values of \(\alpha\). In 3.5, the theory is used to give information on moduli spaces of bundles on \(C \times \mathbb P^1\).
Section 4.1 gives a construction of a relative Nahm transform for a ``relatively IT'' pair \((E , \nabla)\) where \(E\) is a Hermitian vector bundle over \(C \times \mathbb P^1\) and \(\nabla\) a unitary connection. Functoriality and compatibility properties of the relative Fourier--Mukai and Nahm transforms are given which are analogous to those developed earlier in the absolute setting.
In 4.2, integrable triples over \(C\) are introduced: triples \(((E_{1}, \nabla_{1}),(E_{2}, \nabla_{2}), \Phi)\) where \(\Phi\) is holomorphic with respect to the holomorphic structures determined by \(\nabla_1\) and \(\nabla_2\). The absolute and relative Nahm transforms of an integrable triple are shown to be compatible in a similar way to the Fourier-Mukai transforms in 3.2. The \(\tau\)-coupled vortex equations are introduced, for a real parameter \(\tau\), and the IT properties of certain integrable triples which satisfy them are studied.
In 4.3, covariantly constant integrable triples (CCITs) and their properties are introduced. A criterion for a CCIT to satisfy the \(\tau\)-coupled vortex equations is given in terms of certain connections induced by the \(\nabla_j\). Such a triple is \((\tau - \tau^{\prime})\)-polystable (where \(\tau^{\prime}\) is a real number determined by \(\tau\) and the ranks and degrees of the components) and, when \(\tau\) and \(\tau^{\prime}\) are nonzero with the same sign, has good IT properties. In this case, the Nahm transform is also a CCIT. Moreover, it satisfies the \(\hat{\tau}\)-coupled vortex equations for some value of \(\hat{\tau}\) if and only if \(\tau = \tau^{\prime}\). Finally, an example is given to show that \(\alpha\)-polystability of a triple may not be preserved by the Fourier-Mukai and Nahm transforms.
In several places, the authors use the results to describe various moduli spaces of vector bundles.
Familiarity with \textit{O.\ García-Prada} and \textit{S. Bradlow's} paper [Math. Ann. 304, 225--252 (1996; Zbl 0852.32016)] is helpful. Fourier-Mukai transform; stable triples; Nahm transform; holomorphic triples García-Prada, O., Hernández Ruipérez, D., Pioli, F., Tejero Prieto, C.: Fourier-Mukai and Nahm transforms for holomorphic triples on elliptic curves. J. Geom. Phys., 55, 353--384 (2005) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Elliptic curves Fourier-Mukai and Nahm transforms for holomorphic triples on elliptic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper proves a conjecture motivated by mirror symmetry, proposing a duality between GKZ hypergeometric systems. A GKZ hypergeometric system, introduced by \textit{I. M. Gelfand} et al. [Discriminants, resultants, and multidimensional determinants. Reprint of the 1994 edition. Boston, MA: Birkhäuser (2008; Zbl 1138.14001)], is a system of linear partial differential equations whose solutions are hypergeometric functions. Such a system of equations can be explicitly constructed from combinatorial data given by polytopes. Moreover, geometrically the linear partial differential equations can be viewed as vector bundles with flat connections, which play significant role in mirror symmetry.
Mirror symmetry predicts that given an equivalence between the derived category of coherent sheaves of two Calabi-Yau manifolds \(X_1\) and \(X_2\), there exists a vector bundle with a flat connection such that the induced isomorphism between the complexified Grothendieck groups of \(X_1\) and \(X_2\) is given by parallel transport determined by the flat connection. In the paper under review, the authors study this expectation coming from mirror symmetry in the case when \(X_1\) and \(X_2\) are non-compact toric Calabi-Yau orbifolds.
It is known by the work of \textit{Y. Kawamata} [J. Math. Sci., Tokyo 12, No. 2, 211--231 (2005; Zbl 1095.14014)] that if \(X_1\) and \(X_2\) are two different Crepant resolutions of singularities of a given Gorenstein toric singularity, then \(X_1\) and \(X_2\) are derived equivalent -- namely, they have equivalent derived categories. In this situation, it is expected that the corresponding vector bundle with a flat connection is a better behaved GKZ hypergeometric system, denoted by bbGKZ hypergeometric system in the article. In particular, such a system is determined from the polytope associated to the toric singularity, or the fan of the toric singularity.
In the situation when one studies noncompact toric Calabi-Yau orbifolds there are two different bbGKZ hypergeometric systems one can consider: one is obtained from the Grothendieck groups of \(X_1\) and \(X_2\) (associated to the derived categories of coherent sheaves on \(X_1\) and \(X_2\)), the other is obtained from the compactly supported Grothendieck groups of \(X_1\) and \(X_2\) (associated to the derived categories of compactly supported coherent sheaves on \(X_1\) and \(X_2\)). The authors had previously conjectured a duality between these two bbGKZ hypergeometric systems [\textit{L. A. Borisov} and \textit{R. P. Horja}, Adv. Math. 271, 153--187 (2015; Zbl 1334.14024)], which is compatible with the natural duality defined by the Euler pairing between the Grothendieck group and the compactly supported Grothendieck group. In this paper the authors prove this conjecture in the situation when \(X_1\) and \(X_2\) are of complex dimension two.
The proof uses an explicit study of the solutions to the GKZ systems, in particular, integral contour representations. hypergeometric systems; gamma series; Euler characteristics pairing; twisted sectors; \(K\)-theory; duality Derived categories and commutative rings, Calabi-Yau manifolds (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Hypergeometric functions On duality of certain GKZ hypergeometric systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Starting from \textit{O. Zariski}'s famous example [Amer. J. 51, 305--328 (1929; JFM 55.0806.01)] of two cuspidal plane sextics whose complements have different fundamental groups, the author investigates whether the topology of \(\mathbb{P}^2(\mathbb{C})\setminus B\), where \(B\) is a reduced plane curve, can be determined from
the local type of the singularities of \(B\). In particular, it is discussed when the fundamental group \(\pi_1(\mathbb{P}^2(\mathbb{C})\setminus B)\) is abelian.
A sharp criterion, in the case of curves having only nodes and cusps, for an abelian fundamental group of the complement was given by \textit{M. V. Nori} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)].
In the paper under review, the author gives the following criterion for a non-abelian fundamental group.
Theorem. Let \(B\) be a reduced plane curve with at most simple singularities of even degree \(d\). Denote by \(\mu_x\) the Milnor number at \(x\), and by \(l_p\) the number of singularities of type \(A_{kp-1}\) for a prime \(p\geq 5\) (respectively, if \(p=3\), let \(l_3\) be the number of singularities of type \(A_{3p-1}\) and \(E_6\)). If there exists an odd prime \(p\) such that
\[
l_p+\sum_{x\in\text{Sing}(B)}\mu_x>d^2-3d+3,
\]
then \(\pi_1(\mathbb{P}^2(\mathbb{C})\setminus B)\) is non-abelian.
Furthermore, the author gives examples to show that this bound is sharp for \(d=6\), but not in the general case for large \(d\). Singularities of curves, local rings, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Singularities in algebraic geometry Local types of singularities of plane curves and the topology of their complements | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:(\mathbb{C}^{n+1},0)\to (\mathbb{C},0)\) be a germ of a holomorphic function with an isolated singularity at \(0\). Two types of global objects are associated to \(f\), the so-called \(\mu\)-constant monodromy groups and moduli spaces for marked singularities.
The Milnor lattice of \(f\) is \(Ml(f):=H_n(f^{-1}(t), \mathbb{Z})\cong \mathbb{Z}^\mu\), \(\mu\) the Milnor number, \(t>0\) and \(f^{-1}(t)\) a regular fibre. There are two pairings on \(Ml(f)\), the intersection form \(I\) and the Seifert form \(L\). \(L\) determines \(I\) and the monodromy \(M_h\). Let \(G_{\mathbb{Z}}(f):=\text{Aut}(Ml(f), L)\) and \(ML(f)=(Ml(f), L, M_h, I)\). The \(\mu\)-constant monodromy of \(f\) is the subgroup of \(G_{\mathbb{Z}}(f)\) generated by all monodromy groups of all \(\mu\)-constant families which contain \(f\). The author conjectures the \(\mu\)-constant monodromy of \(f\) contains all automorphism, modulo \(\pm\text{id}\).
Fix one singularity \(f_0\). A marked singularity is a pair \((f, \pm \rho)\), where \(f\) is a singularity in the \(\mu\)-homotopy class of \(f_0\) and \(\rho: ML(f)\to ML(f_0)\) is an isomorphism. Let \(M^{\text{mar}}_\mu(f_0)\) be the moduli space with respect to right equivalence of marked singularities. Locally \(M^{\text{mar}}_\mu (f)\) is isomorphic to the \(\mu\)-constant stratum of \(f\). The conjecture on the group implies that the moduli spaces are connected. The relation with Torelli type problems is discussed and a new global Torelli type conjecture for marked singularities is formulated.
The conjectures are proved for the simple singularities and \(22\) of the \(28\) exceptional singularities. \(\mu\)-constant deformatiom; monodromy group; marked singularity; moduli space; Torelli type problem; symmetries of singularities Hertling, C.: {\(\mu\)}-Constant monodromy groups and marked singularities. Ann. Inst. Fourier (Grenoble) 61 (7), 2643--2680 (2011) Equisingularity (topological and analytic), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Fine and coarse moduli spaces, Symmetries, equivariance on manifolds \(\mu\)-constant monodromy groups and marked 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 normal projective surface \(S\), there exists a function \(R_S\) on the group of Weil divisors on \(S\) modulo Cartier divisors such that
\[
\chi(\mathcal O_S(D))=\chi(\mathcal O_S)+\frac{1}{2}D\cdot (D-K_S)+R_S(D),
\]
where \(D\) is any Weil divisor. The invariant \(R_S(D)\) is the sum of the local invariants \(R_{S,x}(D_x)\) of the germ \(D_x \subset (S,x)\) at singular points \(x\).
For quotient singularity germs, \textit{R. Blache} [Abh. Math. Semin. Univ. Hamb. 65, 307--340 (1995; Zbl 0877.14007)] gave a description of this invariant including the delta invariant of curve singularities.
This paper deals with the invariant \(R_X\) of cyclic quotient singularities \((X,x)\) and shows an explicit computation providing a new interpretation for the delta invariant.
For the proof, the authors deeply analyze invariants of eigenfunctions with respect to the cyclic group action on \(\mathcal O_{\mathbb C^2, 0}\).
As an application, for the \(\mathcal O_X\)-modules of eigenfunctions, they provide explicitly the McKay decomposition of those into special indecomposable reflexive modules.
Furthermore, they give answers two questions posed by Blache [loc. cit.] on the behavior of the function \(R_X(m K_X)\) of \(m\). Riemann-Roch; delta invariant; cyclic quotient singularities; McKay correspondence; reflexive modules; curvettes Local complex singularities, Plane and space curves, Complex surface and hypersurface singularities, Topological properties in algebraic geometry The correction term for the Riemann-Roch formula of cyclic quotient singularities and associated invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Certain moduli spaces problems arise as quotients of algebraic group actions that are not reductive. In that context, the classical theory of geometric invariant theory (G.I.T.) of D. Mumford cannot be applied and needs to be generalized. The paper under review describes how symplectic techniques can be used to provide quotients in such a context, following some ideas of \textit{V. Guillemin, L. Jeffrey} and \textit{R. Sjamaar} who introduced the construction of symplectic implosion [Transform. Groups 7, No. 2, 155--184 (2002; Zbl 1015.53054)]. The paper is also based on earlier work of the author together with \textit{B. Doran} [Pure Appl. Math. Q. 3, No. 1, 61--105 (2007; Zbl 1143.14039)].
In this detailed paper, this notion of symplectic implosion is extended in order to give a symplectic construction for G.I.T.-like quotients by the unipotent radical \(U\) of any parabolic subgroup of a reductive group \(G\) (when the action extends to an action of \(G\) on a symplectic variety \(X\)). Let us assume \(X\) to be a symplectic manifold and \(K\) a compact Lie group with hamiltonian action on \(X\), maximal torus \(T\) and complexification \(G\). With this work, the generalized symplectic implosion of \(T^*K\) is identified with the canonical affine completion of \(G/U\). So, when \(X\) is projective, the nonreductive G.I.T. quotient \(X/\!/U\) can be identified with the associated generalized symplectic implosion of \(X\). The constructed compactified quotients are described using the formalism of moment maps, similar to the classical case. \newline As an example, the authors study the case of the additive group \((\mathbb{C},+)\) acting on the \(n\)-dimensional complex projective space and which extends to the action of \(G=\mathrm{SL}(2,\mathbb{C})\). symplectic reduction; symplectic implosion; moment mac; geometric invariant theory; GIT; non-reductive algebraic group action Kirwan, F., Symplectic implosion and non-reductive quotients, Geometric Aspects of Analysis and Mechanics, 213-256, (2011), Birkhäuser/Springer, New York Geometric invariant theory, Momentum maps; symplectic reduction Symplectic implosion and nonreductive quotients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Recall the following definitions: A vector bundle on a homogeneous variety \(G/P\) (for this paper \(G=SL(n+1)\) and \(P\) is a parabolic subgroup) is called homogeneous if \(g^\ast E \cong E\) for all \(g\in G\). A homogeneous vector bundle \(E\) on \(G/P\) is called \(G\)-\textit{simple} if \(\text{Hom}(E,E)^G \cong {\mathbb C}\), is called \(G\)-\textit{rigid} if \(\text{Ext}^1(E,E)^G =0\) and is called \(G\)-\textit{exceptional} if it is \(G\)-simple and \(\text{Ext}^1(E,E)^G=0\) for \(i\geq 1\). One defines \textit{Fibonacci} bundles \({\mathcal C}_k\) recursiveley, for \(V={\mathbb C}^3\), with \({\mathcal C}_0={\mathcal O}(-d)\), \({\mathcal C}_1={\mathcal O}\):
\[
0\to {\mathcal O}(-d) \to S^dV\otimes {\mathcal O} \to {\mathcal C}_2 \to 0
\]
\[
0\to {\mathcal C}_{k-1} \to \text{Hom}({\mathcal C}_{k-1}, {\mathcal C}_k)\otimes {\mathcal O}_k \to {\mathcal C}_{k+1} \to 0
\]
The duals of Fibonacci bundles \({\mathcal C}_3\) are called \textit{almost square bundles}. In the frame of the result of Bondal-Kapranov that the category of homogeneous vector bundles on \({\mathbb P}^2\) is equivalent to the category of finite dimensional representations of a certain quiver with certain relations, one determines the representations corresponding to the almost square bundles (the explanation of the name comes from the shape of the associated representation). One proves that all almost square bundle on \({\mathbb P}^2\) are simple and stable and all Fibonacci bundles are \(G\)-exceptional. homogeneous vector bundles; stable vector bundles; exceptional bundles; representations of quivers with relations; Syzygies Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets, Homogeneous spaces and generalizations G-exceptional vector bundles on \(\mathbb P^2\) and representations of quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The questions of how to compactify the Jacobian of a singular curve and how to extend the universal Picard variety over various moduli spaces of curves have been studied extensively in the last several decades.
In this paper, we answer those questions for hyperelliptic curves. We give an explicit description of the moduli space of line bundles on hyperelliptic curves, including singular curves. We study the universal Jacobian \(J^{2,g,n}\) of degree \(n\) line bundles over the Hurwitz stack of double covers of \(\mathbb P^{1}\) by a curve of genus \(g.\) Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification \(\overline{J}^{2,g,n}_{bd}\) of \(J^{2,g,n}\) whose points we describe simply and explicitly as sections of certain vector bundles on \(\mathbb P^{1};\) a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of \(\overline{J}^{2,g,n}_{bd}\) and \(J^{2,g,n}\) in the cases when \(n - g\) is even. An important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms. This parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms, which in particular gives the group law on integral binary quadratic forms first discovered by Gauss. Jacobians; universal Jacobians; hyperelliptic curves; compactifications of moduli spaces; line bundles on hyperelliptic curves Generalizations (algebraic spaces, stacks), Picard groups, Algebraic moduli problems, moduli of vector bundles, Algebraic functions and function fields in algebraic geometry Gauss composition for \(\mathbb{P}^1\), and the universal Jacobian of the Hurwitz space of double covers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a field, \(Y=\{Y_1,\dots,Y_n\}\) a set of indeterminates over \(K\), and let \(R=K[[Y]]/J\), \(J\) being an ideal of \(K[[Y]]\). A nonzero finitely generated \(R\)-module \(M\) is MCM if \(\text{depth} M=\dim R\), in other words a Cohen-Macaulay module of maximal Krull dimension. Let \(R\) be a hypersurface ring, that is \(J=(f)\) for a certain \(0\neq f\in(Y)\). Suppose that \(f=Y^s_n+h\), \(s\geq 2\), for \(0\neq h\in K[[Y_1,\dots, Y_{n-1}]]\) and let \(A=K[[Y_1,\dots, Y_{n-1}]]/(h)\). Is it possible to describe \(\text{MCM} (R)\) using \(\text{MCM} (A)\)? Earlier the author showed [\textit{D. Popescu}, An Ştiinţ. Univ. ``Ovidius'' Constanţa, Ser. Mat. 2, 112-119 (1994; Zbl 0920.13004)]: Theorem. Suppose \(s\) is not a multiple of \(\text{char} K\). If \(N\) is an MCM \(R\)-module then \(\Omega^1_R (N/Y_n^{s-1}N) \cong N\oplus \Omega^1_R(N)\).
Section 1 of the paper presents some extensions of the above theorem together with some applications. The second section studies the MCM modules and their minimal numbers of generators over hypersurface singularities of type \(Y^3_1+\cdots +Y^3_n\). MCM module; Cohen-Macaulay module; minimal numbers of generators; hypersurface singularities Cohen-Macaulay modules, Singularities of surfaces or higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Cohen-Macaulay representation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a careful presentation of a nice theory of noncommutative two-tori and holomorphic vector bundles on them.
As the paper develops a whole theory and there are no statements marked as theorems, it is the best first to quote the authors' summary.
``In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.''
The framework of noncommutative two-tori gives a handle on certain modes of degenerations of ordinary elliptic curves. The major associated techniques include lattices in \(\mathbb C\), the group \(\mathrm{SL}_2(\mathbb Z)\), elliptic curves, theta functions, Fourier series in one and two variables, Fourier transforms, Fourier-Mukai transforms, the Gauss kernel \(e^{-x^2/2}\), Hermite polynomials, the operators of creation and annihilation, the Schwartz spaces \(\mathcal S(\mathbb R)\) and \(\mathcal S(\mathbb R\times\mathbb Z/m\mathbb Z)\), and analogs of the \(\overline\partial\)-operator on \(\mathbb C\).
The paper is very informative and pleasant to read. noncommutative two-tori; Fukaya category; holomorphic vector bundles A. Polishchuk and A. Schwarz. Categories of holomorphic vector bundles on noncommutative two-tori. \textit{Communications in Mathematical Physics}, (1)236 (2003), 135-159. arXiv:0211262v2 [math.QA] Noncommutative geometry (à la Connes), Global theory of symplectic and contact manifolds, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects), Holomorphic bundles and generalizations, Derived categories, triangulated categories Categories of holomorphic vector bundles on noncommutative two-tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The results of this paper, combined with the results of the author's paper [J. Lond. Math. Soc., II. Ser. 58, No. 1, 63-83 (1998; see the preceding review Zbl 0943.20040)], give a proof of the following theorem: Let \(G\) be a connected reductive algebraic group over an algebraically closed field of characteristic \(p\geq 0\), and let \(H\) and \(K\) be reductive subgroups. Then either \(G=HK\), or there is no dense \(H,K\)-double coset in \(G\). The bulk of paper under review is in proving this statement in the case that \(G\) is simple of classical type (while the case that \(G\) is simple of exceptional type is considered in the paper cited). This is done by case by case considerations.
This theorem would immediately follow from the stronger statement that the action of \(K\) on \(G/H\) is stable in the sense of algebraic transformation groups [cf. \textit{V. L. Popov, E. B. Vinberg}, Invariant Theory, Encycl. Math. Sci. 55, 123-284 (1994); transl. from Itogi Nauki Tekh., Ser. Sovrem Probl. Mat., Fundam. Napravleniya 55, 137-309 (1989; Zbl 0735.14010)]. This statement looks very plausible, and if \(p=0\), it was proved by \textit{D. Luna} [Invent. Math. 16, 1-5 (1972; Zbl 0249.14016)]. In particular, the main result of the paper under review is new only for \(p>0\). connected reductive algebraic groups; reductive subgroups; orbits; double cosets J. Brundan, \textit{Double coset density in classical algebraic groups}, Trans. Amer. Math. Soc. \textbf{352} (2000), 1405-1436. Linear algebraic groups over arbitrary fields, Homogeneous spaces and generalizations Double coset density in classical algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite group \(G\subset \text{SO}(3, {\mathbb R})\) let \(\tilde{G}\) denote the inverse image via the double covering \(\text{SU}(2) \to \text{SO}(3,{\mathbb R})\). The moduli space of clusters \(\tilde{G}\text{-Hilb}({\mathbb C}^2)\) is a natural resolution of the quotient singularity \({\mathbb C}^2/\tilde{G}\), and where the exceptional curves correspond to irreducible representations of \(\tilde{G}\). Moreover, any irreducible representation of \(G\) is also an irreducible representation of \(\tilde{G}\). The authors construct a map between the moduli spaces \(\tilde{G}\text{-Hilb}({\mathbb C}^2) \to G\text{-Hilb}(\mathbb{C}^3)\), and they show that there is an induced map of exceptional divisors which contracts components that do not correspond to irreducible representations of \(G\). quotient singularities; McKay correspondence; Hilbert schemes; polyhedral groups Boissière, S.; Sarti, A., Contraction of excess fibres between the McKay correspondences in dimensions two and three, Ann. Inst. Fourier (Grenoble), 57, 1839-1861, (2007) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Ordinary representations and characters Contraction of excess fibres between the McKay correspondences in dimensions two and 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 Let k be an algebraically closed field with \(char(k)=0\) and G a torus acting diagonally on \(k^ S\). For a subset \(\beta\) of \(\bar S=\{1,2,...,s\}\), set \(U_{\beta}=\{u\in k^ S| \quad u_ j\neq 0\;if\;j\in \beta \}.\) Then G acts on the ring \(O_{\beta}\) of regular functions on \(U_{\beta}\), and the author studies the ring \({\mathcal D}(O^ G_{\beta})\) of all differential operators on the invariant ring. More generally, suppose that \(\Delta\) is a set of subsets of \(\bar S,\) such that each invariant ring \(O^ G_{\beta}\) (\(\beta\in \Delta)\) has the same quotient field. Then \(\cap_{\beta \in \Delta}D(O^ G_{\beta})\quad is\) Noetherian and finitely generated as an R-algebra. Now G acts on each \(D(O_{\beta})\) and there is a natural map \(\theta\) :\(\cap_{\beta \in \Delta}D(O_{\beta})^ G\to \cap_{\beta \in \Delta}D(O^ G_{\beta})=D(Y_{\Delta}/G)\quad obtained\) by restriction of the differential operators. The author finds necessary and sufficient conditions for \(\theta\) to be surjective and describes the kernel of \(\theta\). The algebras \(\cap_{\beta \in \Delta}D(O_{\beta})^ G\quad and\cap_{\beta \in \Delta}D(O^ G_{\Gamma})\quad carry\) a natural filtration given by the order of the differential operators. The author shows that the associated graded rings are finitely generated commutative algebras and determines the centers of \(\cap_{\beta \in \Delta}D(O_{\beta})^ G\quad and\cap_{\beta \in \Delta}D(O^ G_{\beta})\). differential operators on the invariant ring of torus action; regular functions I. M. Musson, Rings of differential operators on invariants of tori, Trans. Amer. Math. Soc. 303 (1987), 805--827. JSTOR: Modules of differentials, Geometric invariant theory, Automorphisms and endomorphisms, Morphisms of commutative rings Rings of differential operators on invariant rings of tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Over a dozen years ago, I asked Jerzy Weyman to explain the geometric method of finding finite free resolutions to me. His answer was very clear. He convinced me that this is a technique worth learning. Furthermore, I believe that it is magical that both mathematical meanings for the word ``resolve'' come into play in this technique. Suppose \(R\) is the coordinate ring of the algebraic variety \(Y\). For the sake of concreteness, let us assume that \(Y\) is a subvariety of affine \(N\)-space over the algebraically closed field \(\mathbb K\). In this setting, \(R\) is equal to \(P/I\), where \(P\) is a polynomial ring in \(N\) variables over \(\mathbb K\) and \(I\) is the ideal of polynomials which vanish on \(Y\). To employ the geometric method for finding the resolution of \(R\) by free \(P\)-modules, one first resolves the singularity. Let \(Z\to Y\) be a desingularization of \(Y\). Now, \(Z\) is smooth; so, \(Z\) is defined by a regular sequence and the coordinate ring of \(Z\) is resolved by the Koszul complex for this regular sequence. The Koszul complex resolution is the most completely understood and straightforward of all possible resolutions. To resolve the coordinate ring of \(Y\), one ``need only'' push the Koszul complex down to \(Y\) along the desingularization. Of course, the devil is in the details.
This geometric method was introduced by George Kempf in a series of papers which were published in the early 1970's. The first stunning application of this technique was made by Alain Lascoux in 1978. Suppose the variety \(Y\) is the ``determinantal variety'' of all linear transformations from \(\mathbb K^n\to\mathbb K^m\) of rank less than \(r\). (In the present setting, \(Y\) is a subvariety of affine \(N=mn\) space.) Before Lascoux, the resolution of the coordinate ring, \(R\), of \(Y\) was known only when \(r\) is equal to the minimum of \(n\) and \(m\) (this is the maximal order minor case and the resolution is the Eagon-Northcott resolution of 1962) or when \(n=m\) were equal and \(r=n-1\) (in this, submaximal, case, \(R\) is Gorenstein of codimension \(4\) and the resolution is the Gulliksen-Negård resolution of 1972). Lascoux combined the geometric method, the representation theory of the general linear group, and Raoul Bott's theorem about the cohomology of line bundles on homogeneous spaces to resolve the coordinate ring of all determinantal varieties in characteristic zero. The representation theory of general linear group is less complicated in characteristic zero than it is in positive characteristic. Also, Bott's theorem is definitely a characteristic zero theorem. On the other hand, the Eagon-Northcott complex, the Gulliksen-Negård complex, and the Akin-Buchsbaum-Weyman complex of 1981, which resolves the ring defined by submaximal minors for arbitrary \(m\) and \(n\) all are independent of characteristic. It was not known until Mitsuyasu Hashimoto's work in the 1990's that for general configurations of \(m\), \(n\), and \(r\), the Betti numbers of \(R\) really depend on the characteristic of \(\mathbb K\).
The geometric method also applies to Schubert varieties in various homogeneous spaces, determinantal varieties for symmetric and alternating matrices, conjugacy classes in classical Lie algebras, varieties of complexes, and many more situations. Depending upon one's point of view (a geometer might start with a variety and look for the equations; whereas an algebraist might start with the equations) the geometric method allows one to find the coordinate ring of particular varieties, find the equations which vanish on the varieties, prove Cohen-Macaulayness, prove that certain varieties have rational singularities.
It is all here in one place, with one uniform notation throughout: the combinatorics and representation theory of Schur and Weyl modules, an explanation and proof of Bott's theorem, the geometric method laid out very slowly, and oodles of examples. Each chapter contains exercises which extend the material given in the text. Many of these exercises outline a way to re-do, using the geometric method, results which appear in the literature using ad-hoc methods. If you have considered learning this method, but have been intimidated away from pulling the ideas out of the existing published literature, then read this book. The value of having one uniform notation can not be overestimated. If you pull any two papers about Schur modules out of the literature, they will have different conventions about how a Ferrers diagram should be drawn and when a tableau is standard (and it does not matter if the two papers have the same author!). In the long run, the ideas work whether the boxes are stacked on the floor or hang from the ceiling; but in the short run, if one adopts Ian Macdonald's remedy ``Readers who prefer this convention should read this book upside down in a mirror'', then one is likely to suffer eye strain, if not more serious disorientation.
I have one slight caution to readers of Weyman's book. The phrase ``If \(R\) has characteristic zero'' almost always means ``If \(R\) contains a field of characteristic zero''. In particular, the ring of integers is not one of the rings under consideration. In practice, almost all calculations are made over a field and then transferred to an arbitrary ring by way of a base change; so there really is no confusion, unless one picked up one page out of context without knowing the general approach.
The book under review has instantly became the standard reference. I have seen referees' reports that have instructed the author to ``express the results in the language of Weyman's new book''. I am aware of numerous seminars that are reading Weyman's book and finding new applications of its methods. Schur modules; determinantal varieties; Bott's theorem; Schubert varieties; finite free resolutions; desingularization; Koszul complex resolution; resolution of the coordinate ring; Eagon-Northcott resolution; Gulliksen-Negård resolution J. Weyman, \textit{Cohomology of vector bundles and syzygies}, Cambridge University Press, Cambridge U.K. (2003). Syzygies, resolutions, complexes and commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representation theory for linear algebraic groups, Combinatorial aspects of representation theory, Linkage, complete intersections and determinantal ideals, Determinantal varieties, Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to algebraic geometry Cohomology of vector bundles and syzygies | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) denote the finite field \(\mathbb F_q\) with \(q\) elements for \(q\) a power of a prime \(p\), and let \(G\) denote the general linear group \(\text{GL}_2(F)\). Let \(V=F^2\) denote the natural two-dimensional \(FG\)-module. Further, for a non-negative integer \(k\), let \(V_k\) denote the \(k\)-th symmetric power of \(V\) as an \(FG\)-module. The author first considers a map \(e\otimes V_{k-(q+1)}\to V_k\) for \(k>q\) where \(e\) denotes the character determinant. It is shown that the cokernel of this map (having dimension \(q+1\)) is isomorphic to the reduction mod \(p\) of a principal series representation.
The main focus of the paper is on a map \(D\colon V_k\to V_{k+(q-1)}\) defined by Serre. The main result is an identification of the cokernel of \(D\) for \(q>2\), \(2\leq k\leq p-1\), \(k\neq\frac{q+1}{2}\). Precisely, it is shown that the cokernel of \(D\) is isomorphic to the reduction mod \(p\) of an integral model of a cuspidal representation for \(\overline{\mathbb Q}_pG\) (where \(\overline{\mathbb Q}_p\) is the algebraic closure of the \(p\)-adic field). The proof makes use of a short exact sequence involving the cokernel of \(D\). This short exact sequence is identified with a short exact sequence in crystalline cohomology for the projective curve \(XY^q-X^qY-Z^{q+1}=0\) due to \textit{B. Haastert} and \textit{J.~C. Jantzen} [J. Algebra 132, No. 1, 77-103 (1990; Zbl 0724.20030)].
Lastly, in the case \(q=p>3\), the author applies his results to modular forms over \(G\). The map \(D\) discussed above is used to extend a cohomological analogue of the Hasse invariant operator constructed by \textit{B. Edixhoven} and \textit{C. Khare} [Doc. Math., J. DMV 8, 43-50 (2003; Zbl 1044.11030)] on the cohomology of spaces of mod \(p\) modular forms for \(\text{GL}_2\). modular representations of finite groups; congruences for mod \(p\) modular forms; general linear groups; principal series representations; cuspidal representations; symmetric powers; crystalline cohomology Modular representations and characters, Congruences for modular and \(p\)-adic modular forms, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Representations of finite groups of Lie type, Representation theory for linear algebraic groups, Linear algebraic groups over finite fields Reduction mod \(p\) of cuspidal representations of \(\text{GL}_2(\mathbb F_{p^n})\) and symmetric powers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Recall that the Hamiltonian of Calogero-Moser system\index{Calogero-Moser system} is of the form:
\[
H=\sum_{j=1}^n\frac{p_j^2}{2}+\sum_{j\neq k}\mathcal{U}(q_j-q_k),
\]
where the potential \(\mathcal{U}\) can have several forms: the rational case \(\frac{1}{x^2}\), the hyperbolic case
\(\frac{\alpha^2}{4\sinh^2\frac{\alpha x}{2}}\), \(\alpha>0\), the trigonometric case \(\frac{\alpha^2}{4\sin^2\frac{\alpha x}{2}}\), \(\alpha>0\), and the elliptic case \(\wp(x)\). It is well known that the rational case, the hyperbolic case and the trigonometric case can be viewed as limiting cases of the elliptic case. By the way, the Hamilton equations admit a Lax pair and linearization is done using the spectral method. Here we are concerned with the rational case with
\[
\mathcal{U}(q_j-q_k)=\frac{\gamma}{(q_j-q_k)^2},
\]
where \(\gamma\) is a parameter that controls the strength of particle interaction, which itself is defined via a pair-potential that is inversely proportional to the square of the difference of particle-positions. Sklyanin's formula provides a set of canonical spectral coordinates on the standard Calogero-Moser space associated with the quiver consisting of a vertex and a loop. The aim of this paper is to generalize this result to Calogero-Moser spaces attached to cyclic quivers by constructing rational functions that relate spectral coordinates to conjugate variables. These canonical coordinates turn out to be well-defined on the corresponding simple singularity of type A, and the rational functions constructed define interpolating polynomials between them. This paper is organized as follows : The first section is an introduction to the subject. In the second section the authors recall the recipe of separation of variables and its relation to the spectral curve with a special emphasis on the rational Calogero-Moser system. In the third section they summarize the construction and the symplectic structure on the Calogero-Moser space associated with the cyclic quiver. The forth section deals with canonical spectral coordinates in the spinless case. The fifth section concerns Calogero-Moser spaces with spin variables and their canonical variables. After introducing the equivariant Calogero-Moser space with spin, the authors derive the analogues of the results obtained in the previous section to models containing spin variables. In the sixth section the authors construct the interpolation curves on the singular surface of type \(A_{m-1}\) at the origin. Calogero-Moser; cyclic quiver; Darboux coordinates; canonical spectral coordinates; Sklyanin's formula 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.), Momentum maps; symplectic reduction Canonical spectral coordinates for the Calogero-Moser space associated with the cyclic 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 Kontsevich's homological mirror symmetry conjecture interprets, roughly speaking, mirror symmetry of two Calabi-Yau varieties \(X\) and \(X'\) as an equivalence of categories, namely the bounded derived category of coherent sheaves on \(X\) and the Fukaya category of \(X'\). The conjecture has been established in some cases, for example, for elliptic curves and the quartic \(K3\) surface. Later the conjecture was extended to Fano varieties and varieties of general type. In these cases the mirror is a Landau-Ginzburg model, that is, a smooth algebraic variety together with a regular function. \textit{P. Seidel} [J.\ Algebr.\ Geom.\ 20, No.\ 4, 727--769 (2011; Zbl 1226.14028)] proved homological mirror symmetry for genus 2 curves. The aim of the paper under review is to prove it in the case of curves of genus \(g\geq 3\).
The curves are treated as symplectic varieties. So, let \(M\) be a symplectic compact oriented surface of genus \(\geq 3\). One considers \(\mathcal{F}(M)\), the Fukaya \(A_\infty\)-category of \(M\), and \(D^\pi(\mathcal{F}(M))\) the category of perfect complexes over \(\mathcal{F}(M)\). On the algebraic side, the Landau--Ginzburg model \(W: X\rightarrow \mathbb{C}\) is three-dimensional. It has one singular fibre \(H\), which is a union of \((g+1)\) complex surfaces. The category of singularities of \(H\), that is, the Verdier quotient of the bounded derived category of coherent sheaves on \(H\) by the triangulated subcategory of perfect complexes, is denoted by \(D_{sg}(H)\). The main result of the paper states that the triangulated categories \(D^\pi(\mathcal{F}(M))\) and \(\overline{D_{sg}}(H)\), where the latter is the Karoubian completion of \(D_{sg}(H)\), are equivalent.
The basic idea of the proof is the same as in Seidel's paper. Both categories can be described by \(A_\infty\)-algebras of a very special form, namely \(A_\infty\)-deformations of the exterior algebra. On the algebraic side, the structure sheaf of the origin is a split-generator in the category of singularities and the wanted algebra is its endomorphism DG-algebra. On the symplectic side, one considers a cyclic covering \(M\rightarrow \overline{M}\), where \(\overline{M}\) is \(\mathbb{P}^1\) with three orbifold points. The wanted split-generator is then the direct sum of \(2g+1\) curves in \(M\), which form a Galois-invariant collection and project to the same curve in \(\overline{M}\). The wanted algebra is again the endomorphism algebra of the generator.
In addition to the above result, the author also proves a kind of reconstruction theorem for hypersurface singularities. homological mirror symmetry; Fukaya categories; triangulated categories of singularities; formality theorem A. Efimov. Homological mirror symmetry for curves of higher genus. Advances in Mathematics, 230(2):493-530, 2012. Mirror symmetry (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Differential graded algebras and applications (associative algebraic aspects), Derived categories, triangulated categories Homological mirror symmetry for curves of higher genus | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the author's Ph.D.-thesis. He studies Cohen-Macaulay modules and Gorenstein modules (\(\text{Ext}^c_R (M,R) \cong M\), \(M\) an \(R\)-module of codimension \(c\)). Using local duality, a regular pairing on the \(\mathfrak{m}\)-torsion module \(H^0_{\mathfrak{m}} (R/\langle f_1, \dots, f_n\rangle)\) of an almost complete intersection \(f_1, \dots, f_n\) is defined. This is applied to the Jacobi module of a non-isolated hypersurface singularity with one-dimensional critical locus. Using \textit{Singular}, it was possible to compute this pairing explicitly. In the real context, there is a topological interpretation of the signature of this pairing in terms of the reduced Euler characteristics of the real Milnor fibres.
In the second part of the thesis, deformation theory of Cohen-Macaulay modules over curve singularities is used to compute Euler numbers and Betti numbers of compactified Jacobians of some classes of singularities. Cohen-Macaulay modules over curve singularities; Gorenstein modules; deformation theory; complete intersection; hypersurface singularity; singular; signature pairing; Euler numbers; Betti numbers; Jacobians Singularities in algebraic geometry, Topological properties in algebraic geometry, Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties, Cohen-Macaulay modules, Rational and unirational varieties Gorenstein duality and topological invariants of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The subject of the paper under review is resolution of singularities in algebraic geometry. The main result is a stacky resolution theorem in log geometry. To state it, let \(S\) be the spectrum of a field, endowed with the trivial log structure. Let \(X\) be a fine saturated (fs) log scheme which is log smooth over \(S\). Then the result is that there exists a smooth log smooth Artin stack \(\mathfrak X\) over \(S\) and a morphism \(f: \mathfrak X \to X\) over \(S\) such that \(f\) is a good moduli space morphism in the sense of \textit{J. Alper} [Ann. Inst. Fourier 63, No. 6, 2349--2402 (2013; Zbl 1314.14095)] and the base change of \(f\) to the smooth locus of \(X\) is an isomorphism. Furthermore, \(\mathfrak X\) admits a moduli description in terms of log geometry. All of this generalizes the case when the charts of \(X\) are all simplicial toric varieties, which was established by \textit{I. Iwanari} [Publ. Res. Inst. Math. Sci. 45, No. 4, 1095--1140 (2009; Zbl 1203.14058)]. The paper gives two applications of the main result: a generalization of the Chevalley-Shephard-Todd theorem to the case of diagonalizable group schemes; and a generalization of work of \textit{L. Borisov, L. Chen} and \textit{G. Smith} [J. Am. Math. Soc. 18, No. 1, 193--215 (2005; Zbl 1178.14057)], \textit{B. Fantechi} et al. [J. Reine Angew. Math. 648, 201--244 (2010; Zbl 1211.14009)], and \textit{I. Iwanari} [Compos. Math. 145, No. 3, 718--746 (2009; Zbl 1177.14024)] on toric Deligne-Mumford stacks and stacky fans to the case of certain toric Artin stacks produced in this paper. log structure; Chevalley-Shephard-Todd; toric stack; stacky fan; resolution of singuarities Matthew Satriano, Canonical Artin stacks over log smooth schemes, Math. Z. 274 (2013), no. 3-4, 779 -- 804. Stacks and moduli problems, Toric varieties, Newton polyhedra, Okounkov bodies Canonical Artin stacks over log smooth 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 existence of resolution of singularities of an algebraic variety in arbitrary dimension over a field of characteristic zero was proved by Hironaka. Thereafter several authors have given different constructive proofs. Once we have achieved resolution for a variety, it is natural to wonder if it is possible to resolve simultaneously the singularities of a family of varieties. This paper states a theory of simultaneous resolution of singularities for infinitesimal deformations of embedded varieties, that is, families of embedded varieties parametrized by \(S=Spec(A)\) where \(A\) is an artinian ring.
This resolution involves algorithmic resolution of an embedded variety of arbitrary dimension over a field of characteristic zero. More precisely, the author uses a variant of Villamayor algorithm of resolution of singularities given in \textit{S. Encinas} and \textit{O. Villamayor} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)], with some tools coming from the desingularization given by \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)]. This variant is called \textit{VW-algorithm} along the paper.
The author reviews all the notions involved in the construction of Villamayor algorithm and adapts all these notions to the case of deformations of embedded varieties. He has done a formidable work to extend all the concepts to suitable conditions over the special fibers.
The paper starts with a revision of Villamayor algorithm of resolution of \textit{basic objects} (tuples \(B=(W,I,b,E)\) where \(W\) is a smooth variety, \(I\) is a never zero \(W\)-ideal, \(b\) a positive integer number and \(E\) a set of regular hypersurfaces in \(W\) having only normal crossings), making an extension of all notions to the case where \(\mathcal{A}\) is a collection of artinian local rings \((A,M)\) such that the residue field \(k=A/M\) has characteristic zero. So he works in the context of \(A\)-basic objects, that is a basic object over a ring \(A\in \mathcal{A}\). One of the key points is the definition of the \textit{adapted hypersurfaces} playing the analogous role to the hypersurfaces of maximal contact, and the proof of that this notion is stable under permissible transformation. The step that differs from Villamayor algorithm is the use of the \textit{homogenized ideal}, due to Wlodarczyk, instead of the \textit{generalized basic objects} to solve the problem of the patching when there are many adapted hypersurfaces. This is performed passing from the \(A\)-basic object \(B\) to its homogenized \(\mathcal{H}(B)\) to make induction on the dimension of the ambient space.
He proves that the \textit{algorithmic equiresolution} of \(A\)-basic object leads to the \textit{algorithmic equiprincipalization} of triples \((W\rightarrow S,I,E)\) over \(A\in\mathcal{A}\), and hence resolution of embedded varieties over an Artin ring \(A\in \mathcal{A}\). In this case it is said that the relative embedded \(A\)-variety is \textit{algorithmically equisolvable}.
The author also includes several interesting examples along the article, such as the example of an \(A\)-basic object that is not algorithmically equisolvable.
The article is self contained, the author includes the necessary theoretical background and an appendix of revision of useful results that not always appear in the literature. resolution of singularities; simultaneous resolution; deformation Nobile, A.: Algorithmic equiresolution of deformations of embedded algebraic varieties. Revista Matemática Hispanoamericana \textbf{25}, 995-1054 (2009, to appear) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Birational geometry, Families, fibrations in algebraic geometry, Singularities of curves, local rings Algorithmic equiresolution of deformations of embedded varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the study of the effect of triality on the ring of ad-invariant polynomials on the Lie algebra \(\emph{so}(8)\), through the perspective of Hitchin systems. Inspired by the triality induced between three Hitchin fibrations through the triality of Lie groups, Lie algebras and their rings of invariant polynomials, the authors dedicate this note to fill a gap they found in the literature when looking for explicit descriptions of correspondences between homogenous bases of the rings of invariant polynomials of Lie algebras arising from the triality of \(SO(8, \mathbb{C})\). They are concerned here with the action of the triality automorphism on the corresponding moduli spaces of Higgs bundles, which has been previously studied by other authors both from a string theory perspective as well as from a mathematics perspective. Although this result can be deduced through topological methods (e.g., via the formulae for the Pontrjagin and Euler classes of the spin bundles on an orientable, spinnable \(8\)-manifold [\textit{H. B. Lawson jun.} and \textit{M.-L. Michelsohn}, Spin geometry. Princeton, NJ: Princeton University Press (1989; Zbl 0688.57001)]) the authors take here a perspective they have not found elsewhere, and which fits naturally within the study of Higgs bundles. Through the triality of \(SO(8,\mathbb{C})\), they study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them. This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2 the authors give an overview of the group-theoretic construction of triality. Section 3 is devoted to triality as an automorphism. In order to understand the appearance of triality via Higgs bundles and the Hitchin fibration, the authors define these subgroups as fixed points of an automorphism to which they turn their attention and whose action on the moduli space of Higgs bundles are studied in the following sections. Sections 4 deals with triality and homogeneous invariant polynomials. triality; Higgs bundles; invariant polynomials Vector bundles on curves and their moduli, Connections of harmonic functions with differential equations in two dimensions, Connections of hypergeometric functions with groups and algebras, and related topics, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Triality for homogeneous polynomials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a linear algebraic group \(G\) over an algebraically closed field. Of primary interest here are the notions of observable and quasiparabolic subgroups (and a related notion of subparabolic). A subgroup \(H\) of \(G\) is said to be observable if the homogeneous space \(G/H\) is a quasi-affine variety. A subgroup \(Q\) of \(G^0\) (the connected component at the identity of \(G\)) is said to be quasiparabolic if \(Q = G^0_v\) (the stabilizer in \(G^0\) of \(v\)) for a highest weight vector \(v\) in an absolutely irreducible \(G\)-module. Consider a finite dimensional \(G\)-module \(V\) and an element \(0 \neq v \in V\) which is unstable, that is, for which 0 lies in the closure of \(G\cdot v\).
The title of the article references a theorem of Bogomolov which says that, for a connected reductive \(G\), the stabilizer \(G_v\) of \(v\) is contained in a quasiparabolic subgroup of \(G\), thus providing a method for showing the existence of quasiparabolic subgroups. The first main goal of this paper is to generalize this result to the case that \(G\) is defined over a not necessarily algebraically closed but perfect field \(k\). To do so, one must consider the notion of a subgroup being quasiparabolic over \(k\) (quasiparabolic and defined over \(k\)) and a stronger condition of being \(k\)-quasiparabolic. The authors give two proofs, one following arguments of Grosshans and the other arguments of Borel and Tits, of a generalization of Bogomolov's result to \(k\)-quasiparabolicity over a perfect field \(k\).
This result is then used, in part, to give a characterization of the relationships between the notions of \(k\)-quasiparabolic, quasiparabolic over \(k\), observable over \(k\), \(k\)-subparabolic, and (strongly) subparabolic over \(k\) for a subgroup of a linear algebraic \(k\)-group with \(k\) perfect. From the definitions, quasiparabolic over \(k\) implies obesrvable over \(k\) but the converse is generally false. The main result is in part a generalization of a result of Sukhanov, that observable over \(k\) is equivalent to being \(k\)-subparabolic (or equivalently subparabolic over \(k\)). It is also shown that for a semisimple \(G\), quasiparabolic over \(k\) implies \(k\)-quasiparabolic. geometric invariant theory; instability; representation theory; observable subgroups; quasiparabolic subgroups; subparabolic subgroups Geometric invariant theory, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Linear algebraic groups over arbitrary fields On a relative version of a theorem of Bogomolov over perfect fields and its applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) denote a smooth integral complex algebraic curve of genus \(g \geq 2\). Fix points \(x_1, \dots, x_m\) of \(X\) and integers \(n_i\) attached to each \(x_i\). For a semisimple simply connected algebraic group \(G\), parahoric bundles or parahoric \(G\)-torsors on \(X\) are pairs \((E, \theta)\) where \(E\) is a torsor (i.e. a principal homogeneous space) on \(X\) under a parahoric Bruhat-Tits group scheme \(\mathcal{G}\) and weights \(\theta \in Y(T)\otimes {\mathbb Q}\). Here \(Y(T)\) denotes the group of \(1\)-parameter subgroups for a (fixed) maximal torus \(T\) of \(G\). To a set \(\tau\) of conjugacy classes, one associates weights \(\theta_{\tau} = \{ \theta_i \} \in (Y(T)\otimes \mathbb{Q})^m\) and a parahoric group scheme \(G_{\theta_{\tau},X}\) with only ramification points \(x_i\).
The authors introduce notions of stability and semistability for parahoric \(G\)-torsors, construct their moduli spaces \(M_X(G_{\theta_{\tau},X})\) and show that they are irreducible, normal, projective varieties. Let \(K\) be a maximal compact subgroup of \(G\). The authors show that there is a Fuchsian group \(\pi\) and a homeomorphism of the space of conjugacy classes of representations of \(\pi\) in \(K\) of local type \(\tau\) onto the moduli space \(M_X(G_{\theta_{\tau},X})\) which identifies the subset corresponding to irreducible representations with the subset of \(M_X(G_{\theta_{\tau},X})\) corresponding to isomorphism classes of stable \(G_{\theta_{\tau},X}\)-torsors. This generalises the work of \textit{V. B. Mehta} and \textit{C. S. Seshadri} on parabolic vector bundles [Math. Ann. 248, 205--239 (1980; Zbl 0454.14006)] to parahoric \(G\)-torsors. Parahoric torsors (without weights) were studied by \textit{G. Pappas} and \textit{M. Rapoport} [Adv. Math. 219, No. 1, 118--198 (2008; Zbl 1159.22010); Adv. Stud. Pure Math. 58, 159--171 (2010; Zbl 1213.14028)], many results on moduli stacks of these torsors were proved by \textit{J. Heinloth} [Math. Ann. 347, No. 3, 499--528 (2010; Zbl 1193.14014)] over arbitrary ground fields. \(G\)-torsors; parahoric bundles; representations of Fuchsian groups; Riemann surface Balaji, V. and Seshadri, C. S. Moduli of parahoric G-torsors on a compact Riemann surface J.~Algebraic Geom.24 (2015) 1--49 Math Reviews MR3275653 Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Moduli of parahoric \(\mathcal G\)-torsors on a compact Riemann surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We use the unique canonically twisted module over a certain distinguished super vertex operator algebra -- the moonshine module for Conway's group -- to attach a weak Jacobi form of weight zero and index one to any symplectic derived equivalence of a projective complex K3 surface that fixes a stability condition in the distinguished space identified by \textit{T. Bridgeland} [Proc. Symp. Pure Math. 80, 1--21 (2009; Zbl 1169.14303)]. According to work of \textit{D. Huybrechts} [Adv. Stud. Pure Math. 69, 387--405 (2016; Zbl 1386.14137)], following Gaberdiel-Hohenegger-Volpato, any such derived equivalence determines a conjugacy class in Conway's group, the automorphism group of the Leech lattice. Conway's group acts naturally on the module we consider. In physics, the data of a projective complex K3 surface together with a suitable stability condition determines a supersymmetric non-linear sigma model, and supersymmetry-preserving automorphisms of such an object may be used to define twinings of the K3 elliptic genus. Our construction recovers the K3 sigma model twining genera precisely in all available examples. In particular, the identity symmetry recovers the usual K3 elliptic genus, and this signals a connection to Mathieu moonshine. A generalization of our construction recovers a number of Jacobi forms arising in umbral moonshine. We demonstrate a concrete connection to supersymmetric non-linear K3 sigma models by establishing an isomorphism between the twisted module we consider and the vector space underlying a particular sigma model attached to a certain distinguished K3 surface. moonshine module; Leech lattice; complex K3 surfaces Duncan J and Mack-Crane S 2016 Derived equivalences of K3 surfaces and twined elliptic genera \textit{Res. Math. Sci.}3 1 Jacobi forms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K3\) surfaces and Enriques surfaces, Vertex operators; vertex operator algebras and related structures, Representations of sporadic groups, Applications of group representations to physics and other areas of science, Elliptic genera Derived equivalences of K3 surfaces and twined elliptic genera | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simple Lie group with its adjoint action on \(\mathbb P(\mathfrak g)\), where \(\mathfrak g\) is the Lie algebra of \(G\). The unique closed orbit \(X\) has a natural contact structure and is called the adjoint variety of \(G\). A famous conjecture of LeBrun and Salomon claims that any Fano contact manifold \(X\) is the adjoint variety of some simple Lie group \(G\).
In the paper under review the author uses the geometry of contact lines to recover parts of the structure that a contact Fano manifold \(X\) has if it is an adjoint variety, i.e. a Killing form, Lie algebra grading and parts of a Lie bracket. He shows that the locus \(D_x \subset X\) of points which can be joined to \(x\) with at most two contact lines is a divisor. Thus it is possible to define two rational maps on \(X\), one by looking at the linear system \(\langle D_x \rangle\) defined by the divisors, the other by \(x \mapsto D_x\). The author shows that the two rational maps are in fact regular and dual to each other. The relevance of this result is that if \(X\) is the adjoint variety of a simple Lie group \(G\), then \(\langle D_x \rangle=\mathfrak g\) and the duality between the two maps is induced by the Killing form on \(\mathfrak g\).
If \(D \subset X \times X\) denotes the universal family of the divisors \(D_x\), it is shown that there exists a rational map \(D \dashrightarrow X\) that maps a pair of general points \(x \in X\) and \(z \in D_x\) to the intersection of two contact lines joining \(x\) and \(z\). In particular this pair of contact lines is unique. If \(X\) is an adjoint variety, the rational map identifies to the restriction of the Lie bracket to \(D\). contact manifold; Fano manifold; minimal rational curves; Killing form; Lie bracket; Lie algebra grading 4. J. Buczyński, Duality and integrability on contact Fano manifolds, Doc. Math.15 (2010) 821-841. genRefLink(128, 'S0129167X1650066XBIB004', '000285652600001'); Homogeneous spaces and generalizations, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Rational and unirational varieties, Fano varieties Duality and integrability on contact Fano manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An argument abstracted from a paper of Ellingsrud and Strømme allows one to prove without great difficulty: Let \({\mathcal C}\) be a class of \(S\)- modules over an associative algebra \(S\) defined over an algebraically closed field \(k\) and \(M\) a fine moduli space for \({\mathcal C}\). The moduli space \(M\) is known to be nonsingular. For suitable relations on \(\text{Ext}^ p_ S\) and given that the universal \(S \otimes {\mathcal O}_ M\) module has a finite suitable projective resolution,
(a) the Chern classes of the universal projective resolution generate \(A^* (M)\) as a \(\mathbb{Z}\)-algebra;
(b) numerical and rational equivalence coincide on \(M\);
(c) for \(k = \mathbb{C}\), the cycle map \(A^* (M) \to H^* (M,\mathbb{Z})\) is an isomorphism.
This result is applied to the Iarrobino variety \(C_ T\) parametrizing homogeneous ideals \(I \subseteq k [x,y]\) of the Hilbert function \(T\) to determine correspondingly Chern class generators of the Chow ring \(A^* (C_ T)\) and prove results corresponding to (b) and (c) above. Similar results are then obtained for the smooth projective moduli of 0-stable representations of a quiver without oriented cycles and with dimension vector \(\alpha\). Chow rings; fine moduli space; Chern classes; cycle map; Iarrobino variety; quiver King, Alastair D.; Walter, Charles H., On Chow rings of fine moduli spaces of modules, J.~Reine Angew. Math., 461, 179-187, (1995) Fine and coarse moduli spaces, Families, moduli, classification: algebraic theory, Parametrization (Chow and Hilbert schemes) On Chow rings of fine moduli spaces of 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 complex simple Lie group \(\mathbb G_2\) is attached with an adjoint variety \(G_2\). Geometrically, this can be interpreted as a subvariety of the Grassmannian \(\mathrm{Gr}(5,V)\) for some 7-dimensional vector space \(V\); here \(G_2\) consists of 5-spaces isotropic with respect to a fixed non-degenerate 4-form on \(V\), and non-degeneracy refers to the complement of a hypersurface of degree 7 comprising 4-forms decomposable into sums of 3 simple forms.
One can construct a flat deformation \(\hat G_2\) of \(G_2\) by considering a generic 4-form in the above setting. This degeneration is shown to be singular along a plane; its Plücker embedding is the image of \(\mathbb P^5\) under the map defined by quadrics containing a fixed twisted cubic. Further degenerations arise by allowing the twisted cubic to become reducible. While all of them continue to be flat deformations of \(G_2\), only one appears to be a linear section of \(\mathrm{Gr}(5,V)\); the others are toric degenerations, leading to Gorenstein toric Fano varieties. Intersecting with a hyperplane and a quadric, one obtains Calabi-Yau threefolds as small resolutions. These can be studied in the context of mirror symmetry, related to Borcea's Calabi-Yau threefold of degree \(36\).
The paper concludes with applications to \(K3\) surfaces of genus 10, making precise a few special cases of the classification in [\textit{T. Johnsen} and \textit{A. L. Knutsen}, \(K3\) projective models in scrolls. Berlin: Springer (2004; Zbl 1060.14056)]. Lie group; adjoint variety; degeneration; Grassmannian; Calabi-Yau threefold; \(K3\) surface of genus 10 Calabi-Yau manifolds (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Some degenerations of \(G_2\) and Calabi-Yau varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let R be a complete local two-dimensional integrally closed noetherian nonregular Gorenstein domain with maximal ideal M over an algebraically closed field k such that R/M\(\cong k\). The main result of this paper is that the Auslander-Reiten quiver for the category of finitely generated reflexive modules over R is an extended Dynkin quiver \(\tilde A_ n, \tilde D_ n, \tilde E_ 6, \tilde E_ 7\), or \(\tilde E_ 8\). Using this result the authors then show that the rational double points over an algebraically closed field k of arbitrary characteristic are the fixed point rings \(k[[X,Y]]^ G\), where \(G\subset SL(2,k)\) is a finite nontrivial group and the action is the natural one. This extends an earlier result of the authors for characteristic zero [Trans. Am. Math. Soc. 293, 293-301 (1986; Zbl 0588.20001)].
In the second half of the paper the authors give a generalization of their main result to the two-dimensional non-commutative case. Higher dimensions give less satisfactory results due to a lack of almost split sequences. Finally, using results of \textit{M. Artin} [Invent. Math. 84, 195-222 (1986; Zbl 0591.16002), the Auslander-Reiten quivers are computed for non-commutative rings A of finite representation type when A is a maximal T-order with centre T in a division ring D and T is the power series ring k[[u,v]] where k is an algebraically closed field of characteristic zero. integrally closed noetherian nonregular Gorenstein domain; Auslander- Reiten quiver; Dynkin quiver; rational double points; almost split sequences Auslander, M; Reiten, I, Almost split sequences for rational double points, Trans. Am. Math. Soc., 302, 87-97, (1987) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry, Representation theory of associative rings and algebras, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Almost split sequences for 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 We compute the \(\mathbb C^{\ast} \)-equivariant quantum cohomology ring of \(Y\), the minimal resolution of the DuVal singularity \(\mathbb C^2/G\) where \(G\) is a finite subgroup of \(SU(2)\). The quantum product is expressed in terms of an ADE root system canonically associated to \(G\). We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an \(n\) parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov-Witten potential of \([\mathbb C^2/G]\). quantum cohomology; root system; ADE Bryan, J.; Gholampour, A.: Root systems and the quantum cohomology of ADE resolutions, (2007) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Root systems and the quantum cohomology of ADE 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 celebrated homological mirror symmetry conjecture, formulated by \textit{M. Kontsevich} in his ICM talk in Zürich (1994), can be stated as follows:
For any Calabi-Yau threefold \(X\), there exist a symplectic mirror partner \(X^0\) and an equivalence of \((A_\infty)\)-categories \({\mathcal D}^b(X) \cong {\mathcal F}(X^0)\), where \({\mathcal D}^b(X)\) denotes the derived category of the abelian category of coherent sheaves on \(X\) and \({\mathcal F}(X^0)\) stands for the Fukaya \((A_\infty)\)-category associated with \(X^0\). Although formulated for Calabi-Yau threefolds, Kontsevich's homological mirror symmetry conjecture represents a great mathematical challenge in general, even so in the presumably simplest case of dimension one. The one-dimensional case, that is the case of an elliptic curve \(X\), has been investigated by \textit{A. Polishchuk} and \textit{E. Zaslow} in 1998 [Categorical mirror symmetry: The elliptic curve, Adv. Theor. Math. Phys. 2, 443-470 (1998; Zbl 0947.14017)]. These authors studied a somewhat weaker version of Kontsevich's conjecture, and their first attempt, in the simplest case, already demonstrated how difficult it might be to tackle Kontsevich's conjecture.
The paper under review grew out of the author's attempt to understand the details and subtleties in the paper of Polishchuk and Zaslow cited above. The result, and the main contribution of the present paper, is a correct formulation of the weakened version of Kontsevich's conjecture in dimension one and a now absolutely complete proof of it. Following the idea and strategy of the approach of Polishchuk-Zaslow, the author focuses on those parts of their paper that seem to require more explanation, rigor, or gapless justification, without repeating their entire proof of the homological mirror symmetry theorem in the one-dimensional case. In order to overcome the problem of dealing with Fukaya's \((A_\infty)\)-category, in this particular case, the author constructs directly a larger additive category \({\mathcal F}{\mathcal K}^0(X)\) and proves the homological mirror symmetry theorem for elliptic curves in the precise form \({\mathcal D}^b (X)\cong {\mathcal F}{\mathcal K}^0 (X^0)\).
Altogether, the present paper provides a rewarding contribution towards establishing and better understanding the important results by Polishchuk and Zaslow. \(A_\infty\)-categories; Calabi-Yau manifolds; Fukaya category; homological mirror symmetry conjecture Calabi-Yau manifolds (algebro-geometric aspects), Elliptic curves, Lagrangian submanifolds; Maslov index Homological mirror symmetry in dimension one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper establishes new connections between the representation theory of finite groups and sandpile dynamics. Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups \(G\) of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup \(G\). In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic. chip firing; toppling; sandpile; avalanche-finite matrix; Z-matrix; M-matrix; McKay correspondence; McKay quiver; root system; Dynkin diagram; minuscule weight; highest root; numbers game; abelianization Root systems, Combinatorial aspects of representation theory, McKay correspondence Chip firing on Dynkin diagrams and McKay quivers | 0 |
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