text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose that \(X\) and \(Y\) are quasi-projective varieties over a field \(k\). Let \(\mathcal{P}^{d}(X,Y)\) denote the full subcategory of coherent sheaves on \(X \times Y\) consisting of those coherent sheaves \(\mathcal{F}\) such that \(\mathcal{F}\) is flat over \(X\) and the support of \(\mathcal{F}\) maps properly to \(X\) with fibres at most \(d\). The category \(\mathcal{P}^{d}(X,Y)\) is exact, and the authors write \(K^{d}(X,Y)\) for the associated \(K\)-theory space. They also write \(K^{d}(X \times \Delta^{\bullet},Y)\) for the geometric realization of the simplicial space arising from the standard cosimplicial scheme \(\Delta^{\bullet}\) of affine spaces.
The main result of this paper is that \(K^{d}(X \times \Delta^{\bullet},Y)\) is homotopy equivalent to the group completion of a \(\Gamma\)-space \(\hom(X \times \Delta^{\bullet},K_{Y,d})\), where \(K_{Y,d}\) is a \(\Gamma\)-object in the category of ind-schemes associated to generating families of coherent sheaves as above which are generated by global sections. The object \(K_{Y,d}\) can be regarded as a family of generalized Grassmanians. The bivariant \(K\)-theory functor described here specializes in several interesting ways (e.g. to ordinary \(K\)-theory of regular schemes, \(K'\)-theory, Suslin \(K\)-homology), and the main result of this paper implies representability results in those cases. category of ind-schemes; \(\Gamma\)-spaces; representability; bivariant \(K\)-theory functor Daniel R. Grayson and Mark E. Walker, Geometric models for algebraic \?-theory, \?-Theory 20 (2000), no. 4, 311 -- 330. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part IV. \(K\)-theory of schemes, Algebraic \(K\)-theory of spaces, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry Geometric models for algebraic \(K\)-theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper extends the theory of Hitchin pairs (also known as twisted Higgs pairs or Higgs bundles) to nodal curves over an algebraically closed field \(k\) (i.e. projective connected reduced curves over \(k\) having only nodes, or ordinary double points, as singularities). A first step in this direction had already been done in [\textit{U.N. Bhosle}, J. Lond. Math. Soc., II. Ser. 89, No. 1, 1--23 (2014; Zbl 1435.14034)], where the author treated the case of singular integral projective curves (and, in particular, of irreducible nodal curves).
First of all, the author defines Hitchin pairs and their semistability on a nodal curve \(Y\). The definition holds for any connected reduced curve; indeed, he constructs their moduli scheme in appendix B at this level of generality.
Then the author introduces generalized parabolic Hitchin pairs (GPH), good GPH and their semistability, on \(X\), a disjoint union of smooth curves; the moduli scheme of GPH is constructed in Appendix A, where it is also shown that the locus of good GPH is a closed subscheme in it.
When \(X\) is the normalization of \(Y\), there is a birational morphism \(f\) from the moduli scheme of good GPH on \(X\) to the moduli scheme of Hitchin pairs on \(Y\). The restriction of \(f\) to the locus of good GPH mapping to Hitchin pairs whose underlying sheaf is a vector bundle is an isomorphism. Furthermore, \(f\) is surjective if all the irreducible components of \(Y\) are smooth.
The author introduces also an analogue of the Hitchin map for GPH, moreover he shows that it is proper and that it induces a proper map also for good GPH and for the image of \(f\). He studies also spectral curves and general fibres of the Hitchin map, which are related to compactified Jacobians of the spectral curves.
When \(k=\mathbb C\), the author explores also the relationship between Hitchin pairs on \(Y\) and representations of its topological fundamental group. stability; nodal curves; curves with many components; Hitchin pairs; Higgs pairs; Hitchin map Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Hitchin pairs on 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 This paper is devoted to recent results on explicit calculations in derived categories of modules and coherent sheaves. The idea of this approach is actually not new and was effectively used in several questions of module theory. Nevertheless it was somewhat unexpected and successful that the same technique could be applied to derived categories, at least in the case of rings and curves with ``simple singularities''. We present here two cases: nodal rings and configurations of projective lines of types \(A\) and \(\widetilde A\), when these calculations can be carried out up to a result, which can be presented in more or less distinct form, though it involves rather intricate combinatorics of a special sort of matrix problems, namely ``bunches of semi-chains'' (or, equivalently, ``clans'').
In Sections 1 and 4 we give a general construction of ``categories of triples'', which are a connecting link between derived categories and matrix problems, while in Sections 2 and 5 this construction is applied to nodal rings and configurations of types \(\widetilde A\). Section 3 contains examples of calculations for concrete rings and Section 5 also presents those for nodal cubic. We tried to choose typical examples, which allow to better understand the general procedure of passing from combinatorial data to complexes. Section 6 contains an application to Cohen-Macaulay modules over surface singularities, which was in fact the origin of investigations of vector bundles over projective curves in [\textit{Yu. A. Drozd, G.-M. Greuel}, J. Algebra 246, No. 1, 1-54 (2001; Zbl 1065.14041)].
More detailed exposition of these results can be found in [\textit{I. Burban, Yu. Drozd}, J. Algebra 272, No. 1, 46-94 (2004; Zbl 1047.16005), Duke Math. J. 121, No. 2, 189-229 (2004; Zbl 1065.18009), \textit{Yu. A. Drozd, G.-M. Greuel, I. Kashuba}, Mosc. Math. J. 3, No. 2, 397-418 (2003; Zbl 1051.13006)]. derived categories of modules; coherent sheaves; curves with simple singularities; nodal rings; configurations of projective lines; matrix problems; categories of triples; Cohen-Macaulay modules; surface singularities; vector bundles over projective curves I. Burban and Y. Drozd, ''Derived categories for nodal rings and projective configurations,'' Noncommut. Alg. Geom., 243, 23--46 (2005). Representations of quivers and partially ordered sets, Derived categories, triangulated categories, Module categories in associative algebras, Syzygies, resolutions, complexes in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Vector bundles on curves and their moduli, Singularities of surfaces or higher-dimensional varieties Derived categories for nodal rings and projective configurations. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(R={\mathbb{Z}}\) or \({\mathbb{F}}_ 2\) and V a free R-module of finite rank with an alternating bilinear form \(<.,.>:V\times V\to R.\) This form induces a linear map from V into its dual \(V^*\); V' denotes the corresponding image of V in \(V^*\). Let \(Sp^{\#}V\) be the group of automorphisms of \((V,<.,.>)\) that act trivially on \(V^*/V'\). For \(a\in V\) the symplectic transvection \(T_ a(x)=x-<x,a>a\) is an element of \(Sp^{\#}V\). For any subset \(\Delta\) of V let \(\Gamma_{\Delta}\subseteq Sp^{\#}V\) denote the subgroup generated by the transvections \(T_ d\), \(d\in \Delta\). A vanishing lattice over R is a triple \((V,<.,.>,\Delta)\) satisfying the following properties: (i) \(\Delta\) is a \(\Gamma_{\Delta}\)-orbit; (ii) \(\Delta\) generates V; \((iii) rank V>1,\) and there exist \(d_ 1,d_ 2\in \Delta\) such that \(<d_ 1,d_ 2>=1\). - \(\Gamma_{\Delta}\) is called the monodromy group of \((V,<.,.>,\Delta)\). Vanishing lattices over \({\mathbb{F}}_ 2\) are classified in the paper under review. For \(R={\mathbb{Z}}\) it is shown that the monodromy group of a vanishing lattice contains a congruence subgroup of \(Sp^{\#}V\). These structures occur at least in two places in algebraic geometry: the vanishing homology of an odd- dimensional isolated complete intersection singularity and the vanishing homology of many Lefschetz pencils on an even-dimensional projective variety are vanishing lattices. monodromy groups of singularities; symplectic lattices; vanishing homology of Lefschetz pencils; vanishing lattice; vanishing homology of an odd-dimensional isolated complete intersection singularity W.A.M. Janssen, ''Skew-symmetric vanishing lattices and their monodromy groups,'' Math. Ann. 266 (1983), 115--133 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, General binary quadratic forms, Singularities in algebraic geometry Skew-symmetric vanishing lattices and their monodromy 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 \(C\) be a smooth proper curve over a base scheme \(B\), and let \(G\) be a reductive algebraic group. The classification theory of rank-\(r\) vector bundles (or, more generally, of principal \(G\)-bundles) over \(C\) has recently gained crucial significance in the relationship between algebraic geometry and conformal quantum field theory. Especially the construction and the geometric investigation of the relevant moduli spaces of bundles (or, more generally, of the algebraic moduli stacks of \(G\)-torsors) are of fundamental importance.
In the present paper, the author exhibits two general constructions for such moduli spaces and moduli stacks, respectively. As to that, the underlying strategy is to show that various invariants are independent of the curve \(C\), and then to let \(C\) degenerate to a rational nodal curve, hoping that in this case the relevant invariants can be explicitly computed. This amounts to study the behavior of moduli spaces and moduli stacks under degenerations. The first construction is rather geometric, but applies only to some particular linear groups \(G\). Nonetheless, this approach leads to global moduli stacks for torsion-free \(G\)-sheaves of given rank \(r\) on \(C\) and, under suitable semi-stability conditions, to classifying algebraic spaces (moduli schemes). The second approach uses loop groups and works for arbitrary reductive structure groups. It is based on an alternative treatment of the local structure of \(G\)-torsors on semi-stable curves and, henceforth, brings about the drawback of being non-canonical and non-global. The resulting object of this construction is a stack which, in characteristic zero, satisfies the valuative criterion of properness. Although, in general, this stack does not classify particular torsion-free sheaves over \(C\), it sometimes maps naturally to one of the previously constructed stacks and, for this reason, might be useful for the purpose of comparison.
As the author points out, this second method of construction was inspired by his recent general proof of the Verlinde formula [cf. J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]. semistable curves; torsors; conformal quantum field theory; moduli spaces; moduli stacks; loop groups; Verlinde formula Hobson, N.: Quantum Kostka and the rank one problem for \(\mathfrak{s}l_{2m}\) (2015). arXiv:1508.06952 [math.AG] Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Homogeneous spaces and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebraic groups (geometric aspects) Moduli-stacks for bundles on semistable 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 \(K = \mathbb F_q(T)\) be the field of rational functions over a finite field \(\mathbb F_q\), with ``ring of integers'' the polynomial ring \(\mathbb F_q[T]\). Let further \(K_{\infty} = \mathbb F_q((T^{-1}))\) be the completion of \(K\) at its infinite place and \(C_{\infty}\) the completed algebraic closure of \(K_{\infty}\). There is an obvious analogy (including some non-obvious aspects) between the data \((\mathbb F_q[T],K,K_{\infty}, C_{\infty})\) and its number-theoretic counter part \((\mathbb Z,\mathbb Q, \mathbb R, \mathbb C)\).
The analogy is supplemented through the Drinfeld upper half-plane \(\Omega := C_{\infty}-K_{\infty}\) (playing the part of the classical complex upper half-plane) acted upon by \(\Gamma = {\text GL}(2,A)\). The analytic space \(\Gamma'\setminus\Omega\) (where \(\Gamma'\) is a subgroup of \(\Gamma\) of finite index) comes from an affine algebraic curve defined over a finite extension of \(K\). More specifically, let \(\Gamma_0(N)\) be the Hecke congruence subgroup of \(\Gamma\) associated with a non-constant monic polynomial \(N \in \mathbb F_q[T]\), and put \(X_0(N)\) for the smooth compactification of the curve \(\Gamma_0(N)\setminus \Omega\). It classifies isomorphism classes of rank-two Drinfeld \(\mathbb F_q[T]\)-modules with some level structure. Moreover, through some sort of ``Shimura-Taniyama-Weil conjecture'', proved in [\textit{E.-U. Gekeler} and \textit{M. Reversat}, J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)], \(X_0(N)\) uniformizes all the elliptic curves \(E\) over \(K\) with conductor \((N)\cdot\infty\) and which have split multiplicative reduction at \(\infty\).
Let now \(E/K\) be such a curve. It determines both a Drinfeld modular form on \(\Omega\) and an automorphic form \(f = f_E\), which may be regarded as a certain function on the edges of the graph \(\Gamma_0(N)\setminus{\mathcal T}\), where \({\mathcal T}\) is the Bruhat-Tits tree of \(\text{PGL}(2,K_{\infty})\).
Let, on the other hand, \(L\) be a separable quadratic extension of \(K\) with ring of integers \(O_L\). From the modular interpretation of \(X_0(N)\), it is now straightforward how to define Heegner points on \(X_0(N)\) corresponding to Drinfeld modules with complex multiplication by \(O_L\). Through some ``Weil uniformization'' \(\pi:X_0(N) \rightarrow E\), such a Heegner point yields an \(L\)-valued point \(P_L\) of \(E\).
Since the original work of Gross and Zagier that relates the height of Heegner points on elliptic Weil curves \(E/\mathbb Q\) with the arithmetic of special values of \(L(E,s)\), the \(L\)-function of \(E\), people have wondered whether a formula similar to the one in [\textit{B. H. Gross} and \textit{D. B. Zagier}, Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] also holds in the present function field case.
An explicit formula of that type is given in the article (Theorem 4.2), along with a sketch of proof. Details are given in some previous papers of the same authors and (mainly) in the long and important article [the authors, Doc. Math. 5, 365-444 (2000; Zbl 1012.11039)].
In the present paper, the authors restrict to the case where the following assumptions hold:
(a) \(\text{char}(\mathbb F_q) \not= 2\);
(b) \(N \in \mathbb F_q[T]\) is square-free;
(c) \(L = K(\sqrt{D})\) with \(D \in \mathbb F_q[T]\) irreducible of odd degree.
As a consequence (Corollary 4.1) the validity of the Birch and Swinnerton-Dyer conjecture results for the curves \(E/K\) meeting the above assumptions for which the derivative of
\(L(E,s)L(E_D,s)\) at \(s = 1\) doesn't vanish. (Note this is an infinite class of elliptic curves.) Here \(E_D\) is the twist of \(E\) by \(L = K(\sqrt{D})\).
Whereas assumptions (b) and (c) are made for technical purposes only and may be removed (which of course will involve some changes in the final formula), (a) seems to be essential. This is since the argument makes heavy use of some calculations [the first author, Manuscr. Math. 88, No. 3, 387-407 (1995; Zbl 0855.11021)] involving quadratic forms over \(K\) and their theta series.
Seen on a large enough scale, the proof strategy for the main result Theorem 4.2 is similar to Gross-Zagier's strategy: height calculations at finite and infinite places via intersection theory and Green's functions, ``holomorphic projection'' from the space of functions on the edges of the tree \({\mathcal T}\) to the space of automorphic forms, etc. But since many of the ingredients in Gross-Zagier's paper are unavailable in the function field case, the authors had to invent many own devices, e.g. theta series for quadratic forms over \(K\), or Green functions on \({\mathcal T}\). Although only a tiny part of the author's work in the field, the well-written present paper exhibits the nature of results and methods that can be found in [Doc. Math., J. DMV 5, 365--444 (2000; Zbl 1012.11039)] and further papers the authors' paper (loc. cit.). Gross-Zagier formula; Heegner points; Drinfeld modular curves Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties A Gross-Zagier formula for function fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected reductive group which is a product of simply connected semisimple groups and a general linear group defined over an algebraically closed field \(\mathbb {F}\) whose characteristic is very good for all quasi-simple factors of \(G\). In this important paper, the authors relate tilting objects in the heart of Bezrukavnikov's exotic \(t\)-structure on the derived category of equivariant coherent sheaves on the Springer resolution of \(G\), and Iwahori-constructible \(\mathbb {F}\)-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications, they ``deduce in particular the missing piece for the proof of the Mirković-Vilonen conjecture in full generality (i.e. for good characteristics), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.'' The authors expect that their equivalence will play a big role in the modular representation of connected reductive groups. This expectation has won a major boost from [\textit{P. N. Achar} and \textit{S. Riche}, Invent. Math. 214, No. 1, 289--436 (2018; Zbl 1454.20095)]. affine Grassmannian; Springer resolution; exotic t-structure; parity sheaves; Mirković-Vilonen conjecture Mautner, C., Riche, S.: Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković-Vilonen conjecture, preprint. arXiv:1501.07369, to appear in J. Eur. Math. Soc Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Geometric Langlands program: representation-theoretic aspects Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković-Vilonen conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study a construction of a ``generalized mirror'' to a Kähler manifold. More precisely, starting with the immersed Lagrangian submanifolds inside a Kähler manifold \(X\), the authors construct a non-commutative Landau-Ginzburg model given by a tuple \((\mathcal{A},W)\), consisting of a noncommutative algebra \(\mathcal{A}\), and a central element \(W\) in \(\mathcal{A}\). This noncommutative Landau-Ginzburg model is viewed as a generalized mirror to \(X\), since as shown by the authors, there exists a functor from the Fukaya category of \(X\) to the category of noncommutative matrix factorizations of \((\mathcal{A},W)\).
In the classical setup of mirror symmetry the usual Landau-Ginzburg model is given by a tuple \((\mathcal{A},W)\), where \(\mathcal{A}\) is a commutative algebra (the algebra of regular functions on an algebraic variety which is the mirror together with the data of an element \(W\) in it). The construction provided by the authors is a generalization of this construction in the classical setup, in the sense that they construct a noncommutative version of the algebra \(\mathcal{A}\). Moreover, while in the classical setup there is an equivalence between the Fukaya category, and the category of matrix factorizations associated to the mirror, in the noncommutative setup the authors prove that there exists a functor between these two categories, which is not necessarily an equivalence.
The construction of the mirror in the classical setup is done by considering the deformation space of a fixed Lagrangian. This deformation space is obtained as the space of solutions to a Maurer-Cartan equation. In their generalized mirror construction, the authors consider a noncommutative deformation space, by solving the Maurer-Cartan equation in a noncommutative algebra, which is the path algebra of a quiver constructed from the Lagrangian. In this construction it is crucial that one considers immersed Lagrangians, which may have intersections as well as self intersections. The vertices of the associated quiver to such an immersed Lagrangian correspond to the number of components of it, while the arrows are determined in terms of the intersections or self-intersections. The authors illustrate this construction in various examples, in the situation when \(X\) is an elliptic orbifold, or a punctured Riemann surface, or a non-compact Calabi-Yau threefold associated to a Hitchin system.
In the case of elliptic orbifolds the noncommutative mirror recovers some noncommutative algebras introduced by \textit{P. Etingof} and \textit{V. Ginzburg} [J. Eur. Math. Soc. (JEMS) 12, No. 6, 1371--1416 (2010; Zbl 1204.14004)]. In the case of the punctured Riemann surface the authors obtain generalizations of previous results of Bocklandt on noncommutative mirror symmetry in this setup [\textit{R. Bocklandt} and \textit{M. Abouzaid}, Trans. Am. Math. Soc. 368, No. 1, 429--469 (2016; Zbl 1383.16016)]. Moreover, in the case of noncompact Calabi-Yau threefolds associated to \(SL_2(\mathbf{C}\)) Hitchin systems, the noncommutative mirror constructed by the authors recovers the description of the Fukaya category in terms of a quiver with potential previously provided by \textit{I. Smith} [Geom. Topol. 19, No. 5, 2557--2617 (2015; Zbl 1328.53109)]. Fukaya category; homological mirror symmetry; noncommutative algebra; Landau-Ginzburg model; deformation quantization Research exposition (monographs, survey articles) pertaining to algebraic geometry, Mirror symmetry (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category Noncommutative homological mirror functor | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 is a generalization of the theorem of \textit{M. Artin} and \textit{J. J. Zhang} [Adv. Math. 109, 228--287 (1994; Zbl 0833.14002)] characterizing certain classes of abelian categories that can be viewed as noncommutative analogues of categories of coherent sheaves on projective schemes. The main idea of this approach to noncommutative geometry is to associate to a noncommutative graded algebra the quotient category \(\text{QGRA}(A)\) of of the category of graded \(A\)-modules by the subcategory of torsion modules. This generalizes a theorem of Serre who proves that in the commutative case the category of quasicoherent sheaves on \(\text{Proj}(A)\) is equivalent to \(\text{QGRA} (A)\). Then Artin and Zhang [loc. cit.] gave a criterion (the AZ-theorem) for a locally Noetherian abelian category \(\mathcal C\) to be equivalent to \(\text{QGRA} (A)\) for some \(A\), namely that the category has the essential properties of coherent sheaves on a projective scheme \(X\) (i.e. has an ample sequence).
The main goal of the present paper is to prove that if one removes the assumption that the category is Noetherian in the AZ-theorem, then the corresponding graded algebra (constructed from the ample sequence) is still coherent and the abelian category in question is equivalent to the quotient of the category of coherent modules by the subcategory of finite-dimensional modules. The article proves partial results to develop techniques for checking whether a given graded algebra is coherent. This gives a connection between the coherency of a graded algebra and its Veronese subalgebras. Notice that the paper extends the class of graded algebras to the wider class consisting of \(\mathbb{Z}\)-algebras. This makes some connections to the non-obstructed cases of O.A. Laudals noncommutative geometry (which is not referred to). The paper is nicely written, and also explicit examples are given. noncommutative geometry; ample sequence; AZ-theorem A. Polishchuk, Noncommutative proj and coherent algebras. Math. Res. Lett. 12 (2005), 63-74. Noncommutative algebraic geometry Noncommutative proj and coherent 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 the present article, the authors consider the question of extending the universal Jacobian \( \mathcal{J}_{g,n} \rightarrow \mathcal{M}_{g,n} \) to a family of moduli spaces over \(\mathcal{M}^{TL}_{g,n}, \) the moduli space of tree like curves.
To this end, the authors construct an affine space \(V^{TL}_{g,n}\), the \textit{stability space}, and for each nondegenerate \textit{stability parameter} \( \phi \in V^{TL}_{g,n} \) a family of moduli spaces \( \overline{\mathcal{J}}_{g,n}(\phi) \rightarrow \mathcal{M}^{TL}_{g,n} \) which extend the universal Jacobian.
The authors' goal is then to study the enumerative geometry of the theta divisor \( \Theta \subset \mathcal{J}_{g,n} \) in relation to this construction.
For example, each sequence of integers \(\overrightarrow{d} = (d_1,\dots,d_n)\) with
\[
\sum_j d_j = g - 1
\]
and at least one \(d_j\) negative determines a section
\[
s_{\overrightarrow{d}} : \mathcal{M}_{g,n} \rightarrow \mathcal{J}_{g,n}.
\]
The main idea is then that each nondegenerate stability parameter \( \phi \in V^{TL}_{g,n} \) determines a unique extension of this section \(s_{\overrightarrow{d}}\) to a morphism
\[
s_{\overrightarrow{d}} : \mathcal{M}^{TL}_{g,n} \rightarrow \overline{\mathcal{J}}_{g,n}(\phi).
\]
The authors then show that the theta divisor extends to a divisor \( \overline{\Theta}(\phi) \subset \overline{\mathcal{J}}_{g,n}(\phi). \) Furthermore, the preimage of \(\overline{\Theta}(\phi)\) under \(s_{\overrightarrow{d}}\) determines a class \( \overline{\theta}(\phi) \in \text{Pic}(\overline{\mathcal{M}}_{g,n}) \) in the Picard group of the moduli space of genus \(g\) stable curves with \(n\) marked points.
Using their main result, which is a wall-crossing type formula for the stability space \(V^{TL}_{g,n}\), the authors express the class of \(\overline{\theta}(\phi)\) in terms of the class of the Hodge bundle, the classes of the cotangent line bundles and the boundary strata classes.
As further applications of their results, the authors explain how they are related to work of \textit{S. Grushevsky} and \textit{D. Zakharov} [Proc. Am. Math. Soc. 142, No. 12, 4053--4064 (2014; Zbl 1327.14132)], \textit{R. Hain} [Adv. Lect. Math. 24, 527--578 (2014; Zbl 1322.14049)], \textit{F. Müller} [Math. Nachr. 286, No. 11--12, 1255--1266 (2013; Zbl 1285.14032)], \textit{V. Alexeev} [Publ. Res. Inst. Math. Sci. 40, No. 4, 1241--1265 (2004; Zbl 1079.14019)], and \textit{L. Caporaso} [J. Eur. Math. Soc. (JEMS) 11, No. 6, 1385--1427 (2009; Zbl 1202.14030)]. theta divisor; moduli of curves; compactified Jacobian; universal Jacobian; wall-crossing Jacobians, Prym varieties, Families, moduli of curves (algebraic) Extensions of the universal theta divisor | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X = (\mathbb{B} / \Gamma)'\) be a smooth toroidal compactification of a quotient of the complex 2-ball \(\mathbb{B} = \operatorname{PSU}_{2,1} /\operatorname{PS} (U_2 \times U_1)\) by a lattice \(\Gamma < \operatorname{PSU}_{2,1}\), \(D := X \setminus ( \mathbb{B} /\Gamma)\) be the toroidal compactifying divisor of \(X, \rho : X \rightarrow Y\) be a finite composition of blow downs to a minimal surface \(Y\) and \(E(\rho)\) be the exceptional divisor of \(\rho \). The present article establishes a bijective correspondence between the finite unramified coverings of ordered triples \((X, D, E)\) and the finite unramified coverings of \(( \rho (X), \rho (D), \rho (E))\). We say that \((X, D,E(\rho))\) is saturated if all the unramified coverings \(f: (X', D', E' (\rho')) \rightarrow (X, D, E)\) are isomorphisms, while \((X, D, E (\rho))\) is primitive exactly when any unramified covering \(f: (X, D, E (\rho)) \rightarrow ( f(X), f(D), f(E(\rho)))\) is an isomorphism. The covering relations among the smooth toroidal compactifications \((\mathbb{B} / \Gamma)'\) are studied by
\textit{A. M. Uludağ}'s [Math. Ann. 328, No. 3, 503--523 (2004; Zbl 1061.14036)],
\textit{M. Stover}'s [Proc. Am. Math. Soc. 139, No. 9, 3045--3056 (2011; Zbl 1301.22005)],
\textit{L. F. Di Cerbo} and \textit{M. Stover}'s [Mich. Math. J. 65, No. 2, 441--447 (2016; Zbl 1351.32026)] and other articles.
In the case of a single blow up \(\rho = \beta : X = (\mathbb{B} / \Gamma )' \rightarrow Y\) of finitely many points of \(Y\), we show that there is an isomorphism \(\Phi : \operatorname{Aut} (Y, \beta (D)) \rightarrow \operatorname{Aut} (X, D)\) of the relative automorphism groups and \(\operatorname{Aut} (X, D)\) is a finite group. Moreover, when \(Y\) is an abelian surface then any finite unramified covering \(f: (X, D, E( \beta)) \rightarrow ( f(X), f(D), f (E( \beta)))\) factors through an \(\operatorname{Aut} (X, D)\)-Galois covering. We discuss the saturation and the primitiveness of \(X\) with Kodaira dimension \(\kappa (X)= - \infty \), as well as of \(X\) with \(K3\) or Enriques minimal model \(Y\). smooth toroidal compactifications of quotients of the complex 2-ball; unramified coverings Compactifications; symmetric and spherical varieties, Special surfaces, Geometries with algebraic manifold structure Saturated and primitive smooth compactifications of ball 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 Let \(G\) be a simple algebraic group of type \(G_2\) over the complex numbers and let \(N^-\) be the negative unipotent radical. The author considers the algebra \(\mathcal A\) of \(N^-\)-invariant functions on \(G\) which can be written as a direct sum of the irreducible finite dimensional highest weight representations of \(G\) which are parametrized by nonnegative integer combinations of the fundamental weights, \(\omega_1\) and \(\omega_2\). For \(i=1,2\), the direct sum of the representations parametrized by multiples of \(\omega_i\) forms a subalgebra \({\mathcal A}_i\) of \(\mathcal A\). The article constructs cluster algebra structures on the \({\mathcal A}_i\) using elementary operations starting from the cluster structure of \({\mathcal A}\). It further shows that this construction agrees with that of \textit{C. Geiss, B. Leclerc} and \textit{J. Schröer} [Ann. Inst. Fourier 58, No. 3, 825--876 (2008; Zbl 1151.16009)]. partial flag varieties; cluster algebras; based affine space; exceptional group; algebraic group; unipotent radical Gautam, S., Cluster algebras and Grassmannians of type \(G_2\), Algebras Represent. Theory, 13, 335-357, (2010) Cluster algebras, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Cluster algebras and Grassmannians of type \(G _{2}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study varieties of complexes of projective modules with fixed ranks, and relate these varieties to the varieties of their homologies. We show that for an algebra of global dimension at most two, these two varieties are related by a pair of morphisms which are smooth with irreducible fibers.
Let \(\Bbbk\) be an algebraically closed field. For a finite dimensional \(\Bbbk\)-algebra \(A\), \textit{B. Huisgen-Zimmermann} and \textit{M. Saorín} [Trans. Am. Math. Soc. 353, No. 12, 4757-4777 (2001; Zbl 1036.16003)] define an affine variety which parameterizes bounded complexes of \(A\)-modules. \textit{B. T. Jensen, X. Su}, and \textit{A. Zimmermann} [J. Pure Appl. Algebra 198, No. 1-3, 281-295 (2005; Zbl 1070.18005)] use varieties of complexes of projective modules to study degeneration in derived categories. These varieties, denoted by \(\mathrm{comproj}_{\mathbf d}^A\), are defined as the affine variety of all differentials \((\partial_i)_{i\in\mathbb Z}\) for a fixed choice of projective modules with multiplicities of indecomposable projective summands encoded by a sequence of vectors \(\mathbf d=(\mathbf d_i)_{i\in\mathbb Z}\), called a dimension array. In [Manuscr. Math. 117, No. 4, 475-490 (2005; Zbl 1073.18006)] \textit{B. T. Jensen} and \textit{X. Su} use these to show that there is a well-defined notion of type of singularity in the derived category of a finite dimensional algebra, and prove that this type coincides with the type of singularity for the homology for hereditary algebras.
The purpose of this paper is to study geometric properties of varieties of complexes of projective modules with a fixed rank and to relate properties of these varieties to varieties of representations via the homology functor. varieties of complexes; projective varieties; algebras of global dimension at most two; homologies of complexes; smooth morphisms Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Syzygies, resolutions, complexes in associative algebras, Derived categories and associative algebras Varieties of complexes of fixed rank. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 subgroup \(G\subset \text{SL}(2,\mathbb{C})\) has precisely one of the following four properties: (a) the projective representation fixes a line; (b) the projective representation permutes two lines, fixing neither; (c) the projective group is isomorphic to the alternating group \(A_4\), the symmetric group \(S_4\), or the alternating group \(A_5\); or (d) the Zariski closure of \(G\) is \(\text{SL}(2,\mathbb{C})\).
Summary: For the purpose of constructing explicit solutions to second-order linear homogeneous differential equations on the Riemann sphere the Kovacic algorithm partitions the subgroups of \(\text{SL}(2,\mathbb{C})\) into four classes and initially determines which class contains the differential Galois group of the input equation. The author proves in the case of the hypergeometric and Riemann equations that the relevant class can be determined directly from the coefficients by elementary calculation. He also treats the (nonalgebraic form of the) Lamé equation, to which the Kovacic algorithm is not directly applicable. In that instance he combines the Kovacic results with his to produce an algorithm for determining the class of the associated group.
From the group-theoretic viewpoint the problem solved herein is the following: given arbitrary \(S,T\in \text{SL}(2,\mathbb{C})\), determine which class contains the group \(\langle S,T\rangle\) generated by \(S\) and \(T\). explicit solutions; second-order linear homogeneous differential equations; differential Galois group; hypergeometric and Riemann equations; Lamé equation; Kovacic algorithm Churchill, R.C., Two Generator Subgroups of SL(2,C) And The Hypergeometric, Riemann, and Lamé Equations, J. Symbol. Comput., 1999, vol. 28, nos 4s-5, pp. 521--545. Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain, Geometric methods in ordinary differential equations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Ordinary differential equations on complex manifolds, Quantum groups (quantized function algebras) and their representations, Applications of Lie groups to the sciences; explicit representations Two generator subgroups of \(\text{SL}(2,\mathbf C)\) and the hypergeometric, Riemann, and Lamé equations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 builds on the authors' results [in Math. Ann. 318, No. 4, 805-834 (2000; Zbl 0971.20004)]. Let \(G\), \(T\) be finite groups, \(\Gamma\) a profinite group, and \(\pi\colon\Gamma\to T\), \(\lambda\colon G\to T\) surjections. This defines the (proper) embedding problem: does there exist a homomorphism (epimorphism) \(h\colon\Gamma\to G\) with \(\lambda h=\pi\). The first observation reduces the question whether the proper solutions to an embedding problem arise from a versal deformation [compare, e.g., \textit{B. Mazur}, Publ., Math. Sci. Res. Inst. 16, 385-437 (1989; Zbl 0714.11076)] to studying when versal deformations of representations of finite groups are faithful. Let \(V\) be a representation of \(G\) over an algebraically closed field \(k\) with \(\text{char}(k)=p>0\) such that \(\text{End}_{kG}(V)=k\), and let \(K\leq G\) be its kernel. Assume that \(V\) belongs to a cyclic block \(B_{G,V}\) of \(kG\).
Theorem: The universal deformation \(U(G,V)\) is a faithful representation of \(G\) if, and only if, \(K\) is a \(p\)-group, the Brauer tree \(\Lambda(B_{G,V})\) is a star with central exceptional vertex, and, in case \(\Lambda(B_{G,V})\) has more than one edge, \(V\) is not simple.
In the proof it is observed that (i) each finite solvable extension of a given field can be constructed using a finite sequence of versal deformations; (ii) whether a versal deformation of a representation of a finite group is faithful can be detected from versal deformation rings associated to quotients of the group through which the representation factors. versal deformation rings; representations of finite groups; proper embedding problems; Brauer trees Bleher F.M., Chinburg T. (2003). Applications of versal deformations to Galois theory. Comment. Math. Helv. 78:45--64 Group rings of finite groups and their modules (group-theoretic aspects), Inverse Galois theory, Galois theory, Modular representations and characters, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Applications of versal deformations to Galois 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 Consider a noetherian abelian hereditary Ext-finite category \(\mathcal{H}\) over a field \(k\), and assume that \(\mathcal{H}\) is equipped with an equivalence \(\tau\) which satisfies Serre duality.
In the paper under review, two major invariants are attached to \(\mathcal{H}\), namely its function field \(k(\mathcal{H})\) and its Euler characteristic \(\chi_{\mathcal{H}}\), and they are proved to determine the shape of \(\mathcal{H}\) under many aspects. Several examples of these categories are outlined in the paper, the most relevant being that of coherent sheaves on a smooth projective curve \(C\) over \(k\), in which case the invariants of \(\mathcal{H}\) equal those of \(C\).
Consider the full subcategory \(\mathcal{H}_{0}\) consisting of objects of finite length. This is the union over a set \(C(\mathcal{H})\) (called the set of points of \(\mathcal{H}\)) of categories whose Auslander-Reiten quiver can be of type \(\mathbb{Z\,A}/\tau^{p}\) (a finite point) or of type \(\mathbb{Z\,A}_{\infty}\) (an infinite point). The authors first prove that these alternatives cannot coexist, and that in the latter \(\mathcal{H}\) can have at most two points.
On the full subcategory \(\mathcal{H}_{+}\) of objects (called bundles) with no indecomposable subobject, a convenient notion of rank takes positive value. Once chosen a line bundle on \(\mathcal{H}\) the authors provide a definition of \(\chi_{\mathcal{H}}\), a degree function, and a definition of semistable bundles in analogy with the case of smooth projective curves [see \textit{C. S. Seshadri}, Astérisque 96 (1982; Zbl 0517.14008)].
The case of \(\chi_{\mathcal{H}}\) being positive (\(\mathcal{H}\) is then called domestic) is studied in great detail. The condition \(\chi_{\mathcal{H}} > 0\) takes place, for instance, if and only if the rank function is bounded on some component of \(\mathcal{H}_{+}\), or if and only if each indecomposable is stable (or exceptional). The case of \(\mathcal{H}\) having an infinite point is proved to be domestic too. Moreover, in analogy with the case of \(\mathbb{P}^{1}\), it is proved that \(\mathcal{H}\) is domestic if and only if \(\mathcal{H}\) has a hereditary tilting class, which in turn amounts to \(\mathcal{H}\) being derived equivalent to \(\roman{mod}(\Lambda)\), where \(\Lambda\) is a hereditary locally bounded category.
This allows to complete the classification of domestic categories over an algebraically closed field, which turn out to be associated to path algebras of extended simply laced Dynkin graphs, or of an infinite quiver of type \(\mathbb{A}_{\infty}^{\infty}\) or \(\mathbb{D}_{\infty}\).
This complements the classification appearing by \textit{I. Reiten} and \textit{M. Van den Bergh} [J. Am. Math. Soc. 15, 295--366 (2002; Zbl 0991.18009)]. The tubular case (that is, when \(\chi_{\mathcal{H}}=0\)) is also studied, and the Auslander-Reiten quiver of \(\mathcal{H}\) must consist of infinitely many points of finite period. A number of open questions is listed at the end of the paper. hereditary noetherian categories; Euler characteristic; domestic categories; path algebras; infinite points; non-commutative curves H. Lenzing and I. Reiten, \emph{Hereditary {N}oetherian categories of positive {E}uler characteristic}, Math. Z. \textbf{254} (2006), no.~1, 133--171. \MR{2232010 (2007e:18008)} Categorical embedding theorems, Module categories in associative algebras, Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc., Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Families, moduli of curves (algebraic) Hereditary noetherian categories of positive Euler characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Within this article the authors summarize and in some cases also extend several results which appeared earlier [cf. \textit{L. Avramov}, Ann. Math. (2) 150, No. 2, 455--487 (1999; Zbl 0968.13007) and \textit{L. Avramov} and \textit{S. Iyengar}, Invent. Math. 140, No. 1, 143--170 (2000; Zbl 0966.13009)]. The results concern such questions as when is a homomorphism \(\varphi : R \to S\) of Noetherian rings a locally complete intersection and when is the canonical homomorphism
\[
\lambda^\bullet_{S| R}: \wedge_S^\bullet(I/I^2) \to \text{Tor}_\bullet^R(S,S),
\]
where \(R\) is an augmented \(S\)-algebra and \(I\) denotes the kernel of the augmentation map, an isomorphism? The authors give criteria in terms of certain André-Quillen dimensions AQ-dim\(_R S\) and vanishing Cartan-Eilenberg homology groups \(\text{Tor}_\bullet^R(S,S)\). Furthermore the authors show that whenever \(\text{Tor}_\bullet^R(S,S)\) is finitely generated as an algebra over \(S\), then there exist projective \(S\)-modules \(D_1\) and \(D_2\) and an isomorphism of \(S\)-graded algebras:
\[
\wedge_S D_1 \otimes_S\text{Sym}_S D_2 \cong \text{Tor}_\bullet^R(S,S).
\]
In the last part the isomorphism \(HH_\bullet(S| K,S) \cong \text{Tor}_\bullet^R(S,S)\) of Cartan and Eilenberg is used to achieve corresponding results on the Hochschild homology \(HH_\bullet(S| K,S)\), where \(S\) is a \(K\)-algebra, \(K\) a Noetherian ring [see also the paper by \textit{L. Avramov} and \textit{S. Iyengar} cited above]. Hochschild homology; smooth algebras; André-Quillen homology; Cartan-Eilenberg homology Avramov L.L. , Iyengar S. , Homological criteria for regular homomorphisms and for locally complete intersection homomorphisms , in: Algebra, Arithmetic and Geometry (Mumbay, 2000) , T.I.F.R. Studies in Math. 16 , I , Narosa , New Delhi , 2002 , pp. 97 - 122 . MR 1940664 | Zbl 1044.13006 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Modules of differentials, Linkage, complete intersections and determinantal ideals, Complete intersections, Local structure of morphisms in algebraic geometry: étale, flat, etc., Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) Homological criteria for regular homomorphisms and for locally complete intersection homo\-morphisms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(X={\mathbb C}^{2}//{\Gamma}\) be the minimal resolution of \({\mathbb C}^{2}/{\Gamma},\) where \({\Gamma}\) is a cyclic finite subgroup of \(SL_{2}({\mathbb C}).\) In the paper the authors study the equivariant cohomology ring of Hilbert schemes of points on \({\mathbb C}^{2}/{\Gamma}.\) The study uses the vertex algebra techniques. For \({\Gamma}\)-trivial the results of the paper specialize to those in the afffine plane case [cf. \textit{E. Vasserot}, C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7--12 (2001; Zbl 0991.14001) and \textit{W.-P. Li, Z. Qin} and \textit{W. Wang}, Int. Math. Res. Not. 2004, No. 40, 2085--2104 (2004; Zbl 1086.14005)].
The authors generalizing the results of Vasserot [loc. cit.] introduce a ring structure on \({\mathbb H}_{n}=H_{T}^{2n}(({\mathbb C}^{2}//{\Gamma})^{[n]}).\) This ring structure encodes the equivariant cohomology ring structure of \(H_{T}^{*}(({\mathbb C}^{2}//{\Gamma})^{[n]}),\) where \(T=C^{*}.\) The authors construct the equivariant analog of the Heisenberg algebra [cf. \textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (1999; Zbl 0949.14001), Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] which acts irreducibly on \({\mathbb H}={\bigoplus}_{n=0}^{\infty}{\mathbb H}_{n} \) and an explicit map \({\mathbb H}_{n}\rightarrow R({\Gamma}_{n}).\) Here \(R({\Gamma}_{n})\) denotes the class algebra of \({\Gamma}_{n}\) where \({\Gamma}_{n}\) is the wreath product of \({\Gamma}\) and \(S_{n}.\)
This map is shown to be an isomorphism. Let \({\mathcal G}^{*}_{\Gamma}(n)\) be the graded ring associated to a natural filtration of \(R({\Gamma}_{n})\) [cf. \textit{W. Wang}, Adv. Math. 187, No. 2, 417--446 (2004; Zbl 1112.19001)]. The authors construct an explicit graded ring isomorphism from \(H^{*}(({\mathbb C}^{2}//{\Gamma})^{[n]})\) to \({\mathcal G}^{*}_{\Gamma}(n),\) which in fact establishes \textit{Y. Ruan}'s conjecture [Contemp. Math. 312, 187--233 (2002; Zbl 1060.14080)] for the crepant resolution \({\pi}_{n}: ({\mathbb C}^{2}//{\Gamma})^{[n]} \rightarrow {\mathbb C}^{2n}/{\Gamma}_{n}.\) A family of moduli spaces of sheaves on \({\mathbb C}^{2}//{\Gamma}\), which are isomorphic to \(({\mathbb C}^{2}//{\Gamma})^{[n]}\) and parametrized by certain integral lattice in \(H^{2}({\mathbb C}^{2}//{\Gamma})\) is introduced. The authors study the \(T\)-equivariant Chern characters of certain \(T\)-equivariant tautological bundles over the moduli spaces.
This leads to description of Chern character operators in terms of some familiar operators acting on the fermionic Fock space associated to the integral lattice in \(H^{2}_{T}({\mathbb C}^{2}//{\Gamma})\) and also to the description of generating functions of the equivariant intersection numbers of these Chern characters. Hilbert scheme; vertex algebra; Heisenberg algebra; equivariant cohomology; Chern character Z. Qin and W. Wang. Hilbert schemes of points on the minimal resolution and soliton equations. In \textit{Lie algebras, vertex operator algebras and their applications}, volume 442 of \textit{Contemp. Math.}, pages 435--462. Amer. Math. Soc., Providence, RI, 2007. Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vertex operators; vertex operator algebras and related structures Hilbert schemes of points on the minimal resolution and soliton equations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article originated from a course which the author gave at a winter school in Barcelona in 2006. Following the same plan of exposition of the material as in an earlier joint article with \textit{A. Bravo} and \textit{S. Encinas} [Rev. Mat. Iberoam. 21, No. 2, 349--458 (2005; Zbl 1086.14012)] (but this time in the style of a sequence of lectures and extended in content), the author thoroughly treats constructive desingularization in characteristic zero starting from the statement of the main results and the underlying basic definitions and techniques. He then proceeds to the description of the complete resolution process based on the notion of resolution invariants resp. a resolution function, whose structure reflects the inductive nature of desingularization (by descent in dimension of the ambient space) and is explained on the basis of its building blocks, suitable upper semicontinuous functions.
While the focus of the already described part is on the local construction of the resolution process and invariants, the last part concentrates on globalizing those local results. Key notions in this context are Rees algebras and differential algebras, which are introduced and subsequently used to reformulate the objects of study of the previous part in the global context. In addition to a proof following the approach outlined in Kollàr's lecture notes on resolution of singularities, the author also presents -- for the so-called `simple case' -- a different approach, which is not restricted to the case of characteristic zero and might contribute to the still unsolved problem of desingularization in positive characteristic. resolution of singularities; desingularization; resolution function; resolution invariant; Rees Algebra; differential algebra; constructive resolution Villamayor, O., An introduction to constructive desingularization (Notes) Global theory and resolution of singularities (algebro-geometric aspects), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Notes on constructive desingularization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(F\) be an algebraically closed field of characteristic zero and \(L\) a finite dimensional Lie algebra over \(F\). \(H(L)\) denotes the Hopf algebra of representative functions on \(L\); that is, the subalgebra \(U(L)^0\) of the linear dual \(U(L)^*\) of the \(F\) universal enveloping algebra of \(L\) consisting of functionals vanishing on an ideal of finite codimension. \(H(L)\) is the coordinate ring of a pro-algebraic group, denoted \(G(L)\).
The author is concerned with the question of when two Lie algebras have the same Hopf algebra, or equivalently the same pro-algebraic group. \(H(L)\) contains a canonical finitely generated Hopf subalgebra, called the basic subalgebra; the corresponding affine algebraic group is called the Hochschild basic group and denoted \(B(L)\).
The author proves that \(B(L)\) can be described from \(G(L)\) as the latter modulo the intersection of its radical with the reductive part of its center; and that the Hopf algebra \(H(L)\) is determined by the algebraic group \(B(L)\). He further shows that \(B(L)\) is determined by its Lie algebra \(\text{Lie}(B(L))\). Finally, he provides a construction that produces \(\text{Lie}(B(L))\) from the adjoint representation of \(L\).
These results allow him to give a new, and simpler, characterization of the Hopf algebras of the form \(H(L)\), and to establish that an algebraic Lie algebra \(L\) (Lie algebra of an affine algebraic group) is the Lie algebra of a unique almost simply connected affine group (almost simply connected means that the radical has unipotent center and the quotient by the radical is simply connected), the group in question being \(B(L)\) modulo any direct factor vector subgroup \(Z\) of \(B(L)\) such that \(\text{Lie}(B(L))=L\oplus\text{Lie}(Z)\). Lie algebras; Hopf algebras; proalgebraic groups; universal enveloping algebras; coordinate rings; affine algebraic groups Nazih Nahlus, Basic groups of Lie algebras and Hopf algebras, Pacific J. Math. 180 (1997), no. 1, 135 -- 151. Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Lie algebras of linear algebraic groups, Affine algebraic groups, hyperalgebra constructions, Linear algebraic groups over arbitrary fields Basic groups of Lie algebras and Hopf algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is based mostly on the results and methods of the author's famous article [cf. \textit{F. A. Bogomolov}, Math. USSR, Izv. 13, 499-555 (1979); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1227-1287 (1978; Zbl 0439.14002)]. Let \(G\) be an affine reductive group over \(\mathbb{C}\) (not necessarily connected), \(X\) a projective variety and \(G_\gamma\) a principal homogeneous fibration over \(X\) corresponding to an element \(\gamma\in H^1(X, {\mathcal O} (G))\). Any linear representation \(\rho: G\to GL(n)\) defines a vector bundle \(E_\rho\) over \(X\). These are called tensor bundles. (A vector bundle \(E\) of rank \(r\) over \(X\) corresponds to a principal \(G=GL(r)\)-fibration and in this case the bundles \(E_\rho\) are the classical tensor bundles of \(E\).) The author shows that, applying results of invariant theory, one can obtain some deep properties of the bundle considering only the spaces of sections of its tensor bundles. More precisely, assume that \(G\) acts linearly on a finite dimensional vector space \(V\). A vector \(v\in V\) is called unstable if the closure of the orbit of \(v\) contains a \(G^0\)-invariant point. A section \(s\) of \(E_\rho\) is called unstable if \(s(x)\in E_\rho (x)\) is unstable for every \(x\) in \(X\). Based on a precise description of the set of unstable points of a linear representation the author proves that:
(1) \(G_\gamma\) has a natural reductive structure group \(S\subseteq G\) characterized by the fact that the tensor sections of the fibration \(S_\gamma\) corresponding to nontrivial irreducible representations of \(S\) are unstable; and
(2) the vector bundles \(E\) on \(X\), with natural reductive structure group \(G_E\), having non-zero tensor sections corresponding to some irreducible representation of \(G_E\) trivial on the radical of \((G_E)^0\), are unstable or semi-stable with respect to any polarization of \(X\). In the case when \(X\) is a surface, one can find, using Riemann-Roch, topological conditions (involving the Chern classes of the bundle) sufficient for the existence of non-zero unstable tensor sections. This is the way the author proves his famous inequality. principal homogeneous fibration; tensor bundles; unstable points of a linear representation Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Geometric invariant theory, Linear algebraic groups over the reals, the complexes, the quaternions, Topological properties in algebraic geometry Tensors in 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 Let \((V,p)\) be a complex normal surface singularity, set \(\mathfrak m:=\mathfrak m_{(V,p)}\), and let \(\pi:(M,E)\to (V,p)\) be a resolution of the singularity. Let \(E_1,\ldots,E_r\) be the irreducible components of the exceptional divisor \(E\). Set \(H:=H_2(M,\mathbb Z)\); the classes of \(E_1,\ldots,E_r\) are a basis of the \(\mathbb Z\)-module \(H\). Let \(K\) be the canonical cycle. An important role play three particular cycles, the fundamental cycle \(Z\), the maximal cycle \(Z_{\mathfrak m}\) and the canonical cycle \(Z_K\) (cf. Definition 1.2). If the singularity is nonrational, then in Definition 1.3 the author defines the characteristic cycle \(C\). This cycle is a generalization of \textit{H. B. Laufer}'s elliptic cycle [Am. J. Math. 99, 1257--1295 (1977; Zbl 0384.32003)]. \textit{T. Tomaru} [Pac. J. Math. 170, No. 1, 271--295 (1995; Zbl 0848.14017)] introduced it under the name minimal cycle.
Let \((V,p)\) be a normal surface singularity. In Definition 2.1 the author defines when this singularity is called a Kulikov singularity. He introduced this name in his thesis [\textit{J. Stevens}, Kulikov singularities, a study of a class of complex surface singularities with their hyperplane sections. Leiden: Leiden Univ. (PhD Thesis) (1985)]. Such singularities are the result of a construction first given by \textit{V. S. Kulikov} [Funct. Anal. Appl. 9, 69--70 (1975); translation from Funkts. Anal. Prilozh. 9, No. 1, 72--73 (1975; Zbl 0317.14013)] in order to describe unimodal and bimodal singularities. Conversely, every Kulikov singularity can be obtained by this construction (Lemma 2.2). Some properties of Kulikov singularities are proven in Proposition 2.3, e.g., for a Kulikov singularity the maximal cycle \(Z_{\mathfrak m}\) is equal to the fundamental cycle \(Z\).
In Section 3 the author considers the singularity defined by the polynomial \(x^a+y^b+t^c\) with \(a\leq b\leq c\); it is a Kulikov singularity if \(c\geq \mathrm{lcm}(a,b,c)\). In Section 4 the author uses Kulikov singularities in order to reexamine results of \textit{A. Némethi} and \textit{T. Okuma} [Methods Appl. Anal. 24, No. 2, 303--320 (2017; Zbl 1386.32027)] and Tomaru [loc. cit.]. complex normal surface singularities; Kulikov singularities; resolution graph Stevens, J. , Kulikov singularities , Thesis, University of Leiden, 1985. Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties Kulikov singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a continuation of [\textit{Y. Namikawa}, Adv. Stud. in Pure Math. 45, 75--116 (2006; Zbl 1117.14018)], where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution. The purpose of this paper is, to describe the remainder. We first construct a deformation of the nilpotent orbit closure in a canonical manner according to Brieskorn and Slodowy [see \textit{P. Slodowy}, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics. 815. (Berlin-Heidelberg-New York): Springer-Verlag. (1980; Zbl 0441.14002)], and next describe all its crepant simultaneous resolutions. This construction enables us to divide the whole space into a finite number of chambers. Moreover, by using this construction, one can generalize the main result of [Zbl 1117.14018] to arbitrary Richardson orbits whose Springer maps have degree greater than 1. New Mukai flops, different from those of type \(A,D,E_6\), appear in the birational geometry for such orbits. nilpotent orbit; Springer map; crepant resolution doi:10.1215/00127094-2008-022 Deformations of singularities, Grassmannians, Schubert varieties, flag manifolds, Minimal model program (Mori theory, extremal rays) Birational geometry and deformations of nilpotent orbits | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field and let \(A\) be a finite-dimensional \(k\)-algebra. The authors showed in a joint work with the reviewer [\textit{B. T. Jensen, X. Su} and \textit{A. Zimmermann}, ``Degenerations for derived categories'', J. Pure Appl. Algebra 198, 281-295 (2005; Zbl 1070.18005)] how to define a topological space \(comproj(A,d)\) which allows to define and study degeneration of complexes in the bounded derived category of \(A\), analogous to the degeneration in the module category. Following Hesseling we say that two pointed varieties \((X,x)\) and \((Y,y)\) are smoothly equivalent if there is a third pointed variety \((Z,z)\) with smooth pointed morphisms \((Z,z)\rightarrow (X,x)\) and \((Z,z)\rightarrow (Y,y)\). The equivalence class is called \(Sing(X,x)\), the singularity of \(X\) at \(x\). Zwara studied for module varieties over \(A\) the singularity of the orbit closure of a module \(M\) at a degeneration point \(N\).
The authors of the paper under review provide a definition of the singularity of the orbit closure of a complex \(X\) in the derived category of bounded complexes of \(A\)-modules at a degeneration point \(Y\) in \(comproj(A,d)\) in the sense defined in the previous joint article with the reviewer. The main result of the paper is that this definition does not depend on the various choices made during the definition and that it is a generalisation of the singularity on module varieties with respect to the embedding of the module category into the derived category. module varieties; degeneration; singularities DOI: 10.1007/s00229-005-0572-3 Derived categories, triangulated categories, Representations of associative Artinian rings, Singularities of surfaces or higher-dimensional varieties Singularities in derived categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma \subset \mathrm{SL}_2(\mathbb{C})\) be a nontrivial finite subgroup and the surface \(S = \widehat{\mathbb{C}^2/\Gamma}\) be the minimal resolution of \(\mathbb{C}/\Gamma\). Associated to \(\Gamma\) is a Heisenberg algebra of affine type, \(\mathfrak{h}_\Gamma\), and the Hilbert schemes of points \(\mathrm{Hilb}^n(S)\). \textit{I. Grojnowski} [Math. Res. Lett. 3, No. 2, 275--291 (1996; Zbl 0879.17011)] and \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] construct a representation of the Heisenberg algebra (actually a slightly different version from the one considered in this paper) on the cohomology of the Hilbert schemes. Algebraically, \textit{I. Frenkel}, \textit{N. Jing} and \textit{W. Wang} [Int. Math. Res. Not. 2000, No. 4, 195--222 (2000; Zbl 1011.17020)] construct the basic representation of \(\mathfrak{h}_\Gamma\) on the Grothendieck group of the category of \(\mathbb{C}[\Gamma^n \rtimes S_n]\)-modules.
In this paper, the authors define a 2-category \(\mathcal{H}_\Gamma\) and their first main result (3.4) states that \(\mathcal{H}_\Gamma\) categorifies the Heisenberg algebra \(\mathfrak{h}_\Gamma\).
The second main result of the paper (4.4) is a categorical action of \(\mathcal{H}_\Gamma\) on a 2-category \(\bigoplus_{n\geq 0} D(A_n^\Gamma -\mathrm{gmod})\). Here \(D(A_n^\Gamma -\mathrm{gmod})\) denotes the bounded derived category of finite-dimensional, graded \(A_n^\Gamma\)-modules, where
\[
A_n^\Gamma = [(\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) \otimes \ldots \otimes (\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) ] \rtimes S_n.
\]
As explained in Section 8, \(D(A_n^\Gamma -\mathrm{gmod})\) is known to be equivalent to \(D\mathrm{Coh}(\mathrm{Hilb}^n(S))\) and thus the second main theorem categorifies a representation similar to that of Grojnowski [Zbl 0879.17011] and Nakajima [Zbl 0915.14001].
In Section 9, another 2-representation of \(\mathcal{H}_\Gamma\) is introduced that is related to the first by Koszul duality. In section 9.6, it is shown that this 2-representation categorifies an action similar to that constructed by Frenkel, Jing and Wang [Zbl 1011.17020].
For the most part geometry appears only in Section 8. The main definitions are algebraic and a number of the proofs are based on graphical calculus. Section 10 contains a description of various connections to other categorical actions and some open problems.
As the case \(\Gamma = \mathbb{Z}/2\) differs slightly, the necessary modifications are addressed separately in a short appendix. categorification; Heisenberg algebra; McKay correspondence; Hilbert scheme S. Cautis & A. Licata, ``Heisenberg categorification and Hilbert schemes'', Duke Math. J.161 (2012) no. 13, p. 2469-2547 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups and related algebraic methods applied to problems in quantum theory, Frobenius induction, Burnside and representation rings, Lie algebras and Lie superalgebras, Parametrization (Chow and Hilbert schemes) Heisenberg categorification and Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the Gaudin model associated to a point \(z\in\mathbb{C}^n\) with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional \(sl_2\)-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [\textit{N. Reshetikhin} and \textit{A. Varchenko} in Geometry, Topology, and Physics. For Raoul Bott, Internat. Press. Cambridge, MA, 203--322 (1994)], it was shown that for generic \(z\) the Bethe vectors span the space of singular vectors, i.e. that the number of critical orbits is bounded from below by the dimension of this space. The upper bound by the same number is one of the main results of [\textit{I. Scherbak} and \textit{A. Varchenko}, Mosc. Math. J. 3, No. 2, 621--645 (2003; Zbl 1039.34077)].
In the present paper we get this upper bound in another, ``less technical'', way. The crucial observation is that the symmetric function defining the Bethe equations can be interpreted as the generating function of the map sending a pair of complex polynomials into their Wroński determinant: the critical orbits determine the preimage of a given polynomial under this map. Within the framework of the Schubert calculus, the number of critical orbits can be estimated by the intersection number of special Schubert classes. Relations to the \(sl_2\) representation theory [\textit{W. Fulton}, Young Tableaux, Cambridge Univ. Press, (1997; Zbl 0878.14034)] imply that this number is the dimension of the space of singular vectors. We prove also that the spectrum of the Gaudin hamiltonians is simple for generic \(z\). Bethe-Salpeter and other integral equations arising in quantum theory, Computational aspects of algebraic curves, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Exactly solvable models; Bethe ansatz, Linear ordinary differential equations and systems, Many-body theory; quantum Hall effect, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Groups and algebras in quantum theory and relations with integrable systems Gaudin's model and the generating function of the Wroński 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 This paper gives considerable detail about the interpretation of geometric Langlands correspondence in terms of mirror symmetry. A full description of its contents would go beyond the scope of a review (and indeed beyond the knowledge of the reviewer), and would require an account of the work of Ngô as well as a discussion of mirror symmetry, Hitchin fibrations and geometric Langlands in general. However, the introduction (thirteen pages long) summarises the paper, so let us summarise the introduction.
The geometric Langlands duality can be interpreted in terms of mirror symmetry for Hitchin fibrations for moduli of Higgs \(G\)-bundles over a curve \(C\): this is fairly straightforward, relatively speaking, at the general fibre, which is a torus, but further interesting phenomena arise when the Hitchin fibres acquire orbifold singularities. This paper handles the case \(G={\mathrm{SL}}_2\) in detail. The \(T\)-duality interchanges \(B\)-branes (e.g.\ skyscraper sheaves) on the Hitchin fibre of \({\mathcal M}_H({}^L G)\) (moduli of Higgs bundles: \({}^L{\mathrm{SL}}_2\) is \(\mathrm{SO}_3\) but at the singular points we can work with \(\text{O}_2\)) and \(A\)-branes on \({\mathcal M}_H(G)\). At the most general orbifold point the inertia group is \({\mathbb Z}/2\), the centre of \(\text{O}_2\), so the \(B\)-branes are the representations of that. Thus there are two irreducible objects and one should try to find out what they are. These \(A\)-branes are in any case a substitute for \({\mathcal D}\)-modules, which in turn correspond over \({\mathbb C}\) to Hecke eigensheaves, and a decomposition of one should decompose the other.
The authors ``make another leap of faith'' and move to curves over \({\mathbb F}_q\) instead of \({\mathbb C}\). Here Hecke eigensheaves correspond to automorphic functions on the adelic group \(G({\mathbb A}_F)\), where \(F\) is the function field of \(C\). Here they encounter endoscopy, which they describe in the introduction by referring to its appearance in the Langlands correspondence for \({\mathrm{GL}}_n\). Much of what is stated here remains conjectural, though much less so after Ngô's work, but there is sufficient confidence for the authors to feel able to offer interpretations and predictions about how the results (or conjectures) relating to endoscopy should translate, via the correspondences above, into statements about \({\mathcal D}\)-modules and Hecke eigensheaves, Hitchin fibrations, etc. The term ``geometric endoscopy'' refers to this process and its outcome. It turns out that the \(A\)-branes are convenient for expressing this, and that the endoscopy groups arise (on the \(B\)-model side) in a more natural and transparent way than they do in the classical context.
The introduction concludes with a description of the relation with Ngô's work, a brief allusion to aspects relating to QFT, and an outline of the structure of the paper itself. This last explains, which for reasons of space we cannot here, what it is exactly that the authors do. One thing they do not do, by and large, is prove theorems: this is in the nature of a survey article. Geometric Langlands duality; mirror symmetry; endoscopy Frenkel E., Witten E.: Geometric endoscopy and mirror symmetry. Commun. Number Theory Phys. 2, 113--283 (2008) arXiv:0710.5939 Geometric Langlands program (algebro-geometric aspects), Relations with algebraic geometry and topology, Mirror symmetry (algebro-geometric aspects), Representations of Lie and linear algebraic groups over global fields and adèle rings Geometric endoscopy and mirror symmetry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 concerns an example of Landau Ginzburg mirror symmetry related to Grassmannians and cluster algebras. In this set up, the mirror to a Grassmannian \(X\) is given by a cluster variety \(\breve{X}\), together with a ``superpotential'' \(W:\breve{X} \to \mathbf{C}\).
The author introduces two spaces: ``the decorated Grassmannian'', and the ``decorated configuration space''. In rough terms, the decorated Grassmannian is a complement of an ample divisor in the affine cone over the ordinary Grassmannian. On the other hand the decorated configuration space parametrizes particular configuration of lines in a vector space. The main result of the paper shows that both of these spaces have natural cluster structures. More precisely, the decorated Grassmannian is an \(\mathcal{A}\)-cluster variety, whereas the decorated configuration space is an \(\mathcal{X}\)-cluster variety in the sense of Fock-Goncharov [\textit{V. V. Fock} and \textit{A. B. Goncharov}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865--930 (2009; Zbl 1180.53081)]. To show this, author applies the key result obtained by Gross-Hacking-Keel-Kontsevich [\textit{M. Gross} et al., J. Am. Math. Soc. 31, No. 2, 497--608 (2018; Zbl 1446.13015)], showing that there is a canonical basis for the coordinate ring of the decorated Grassmannian, parametrized by integral tropical points of the decorated configuration space, to describe this canonical basis and obtain the superpotential \(W\) on the decorated configuration space. Moreover, a comparision of this potential to the superpotential introduced by Rietsch-Williams as mirror of the Grassmannian is provided [\textit{K. Rietsch} and \textit{L. Williams}, Duke Math. J. 168, No. 18, 3437--3527 (2019; Zbl 1439.14142)], and it is shown that they are compatible.
The author also proves a purely combinatorial result about plane partitions, called the ``cyclic sieving phenomenon'', by using the fact that the canonical basis for the coordinate ring of the decorated Grassmannian can be parametrized by integral tropical points. cluster algebra; cluster duality; mirror symmetry; Grassmannian; cyclic sieving phenomenon Cluster algebras, Combinatorial aspects of representation theory, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Tropical geometry, Applications of deformations of analytic structures to the sciences Cyclic sieving and cluster duality of Grassmannian | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be an algebraic group, \(\rho\) a representation of \(G\), \(V\) a vector bundle on a smooth projective curve. The authors consider deformations of pairs \((P,\phi)\) consisting of a principal \(G\)-bundle \(P\) and a section \(\phi\) of \(\rho P\otimes V\). Section 2 contains the infinitesimal computation, while in section 3 the question is considered whether the moduli functor is formally smooth.
Then \(\rho\) is taken to be the coadjoint representation and \(V=K\) and one defines a symplectic structure on the moduli space. Confining to the pairs where \(P\) is stable, the moduli space can be identified with the cotangent bundle of the moduli space of \(G\)-bundles and the symplectic structure can be identified with the Hamiltonian structure. In the case when \(G=\text{SL}(n)\), Hitchin considered the global analogue of the map, which maps the Lie algebra of \(G\) into \(\mathbb{C}^{n+1}\), given by the coefficients of the characteristic polynomial. The authors look at the analogue of the Kostant map of the Lie algebra of \(G\) into \(\mathbb{C}^ \ell\) for all semisimple groups and show that the fibres are Lagrangian at the smooth points of the fibre and also that the symplectic form vanishes on any smooth variety in the fibre over 0. In section 6, they extend these results to pairs with parabolic structures. coadjoint representation; Lie algebra; semisimple groups I. Biswas and S. Ramanan, ''An infinitesimal study of the moduli of Hitchin pairs,'' J. London Math. Soc., vol. 49, iss. 2, pp. 219-231, 1994. Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Algebraic moduli problems, moduli of vector bundles, Semisimple Lie groups and their representations An infinitesimal study of the moduli of Hitchin pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors provide a formal algebraic treatment of the gauged linear sigma model (GLSM). This extends earlier work by the same authors [Ann. Math. (2) 178, No. 1, 1--106 (2013; Zbl 1310.32032)] relevant to the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence relating the LG dual of a CY hypersurface with its FJRW theory, an analog of its Gromov-Witten theory (cf. [\textit{E. Witten}, AMS/IP Stud. Adv. Math. 1, 143--211 (1997; Zbl 0910.14019)]). Several examples of the GLSM in this new formalism are detailed including cases with non-abelian gauge group. Among these are treatments for flag varieties, complete intersections in a Grassmannian, and hypersurfaces in a product of weighted projective spaces. The central technical result proves that for \(G\subset GL(V)\) a reductive algebraic group, the stack of so-called \textit{LG quasimaps} to a certain GIT quotient \(\mathcal{C}\mathcal{R}_{\theta}\) of the \(G\)-invariant superpotential \(W:V\to \mathbb{C}\) critical locus is a proper Deligne-Mumford stack for \(\mathcal{C}\mathcal{R}_{\theta}\) proper. The approach involves extending the Polishchuk-Vaintrob characterization [\textit{A. Polishchuk} and \textit{A. Vaintrob}, J. Reine Angew. Math. 714, 1--122 (2016; Zbl 1357.14024)] of the FJRW moduli space to the case of reductive algebraic groups. gauged linear sigma model; mirror symmetry; Gromov-Witten; Calabi-Yau; Landau-Ginzburg H. Fan, T. Jarvis and Y. Ruan, \textit{A mathematical theory of the gauged linear {\(\sigma\)}-model}, arXiv:1506.02109 [INSPIRE]. Stacks and moduli problems, Geometric invariant theory, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Supersymmetric field theories in quantum mechanics, Calabi-Yau manifolds (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Applications of deformations of analytic structures to the sciences, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics A mathematical theory of the gauged linear sigma model | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 study of birational geometry of algebraic varieties via the minimal model program depends on the geometry of certain modifications of codimension 2 known as \textit{flips} and \textit{flops}. The geometry of these is very intricate, and is a contemporary object of study where a central problem is to classify flips and flops such that appropriate invariants can be constructed.
This article gives a new invariant to every flipping or flopping curve in a 3-fold, using noncommutative deformation theory. In fact, the idea is to replace all the invariants by a new noncommutative geometric object represented by the \textit{noncommutative deformation algebra} \(A_{\mathrm{con}}\) associated to the curve. This algebra is finite dimensional, and can be associated to any contractible rational curve in any 3-fold, singular or not. A very nice property of this algebra is that it recovers the classical invariants in a natural way. The main difference from a separate set of invariants is that the fact that it is an algebra, makes it possible to give an intrinsic description of a derived autoequivalence associated to a general flopping curve.
The authors give the background on 3-folds, the simplest example being the Atiyah flop. Then the normal bundle is \(\mathcal O(-1)\oplus\mathcal O(-1)\) and the curve is rigid (i.e. the orbit is finite under the action of the groups of projective transformations), and the flop can be factored as a blowup of the curve followed by a blowdown. For a general flopping irreducible rational curve \(C\) in a smooth 3-fold \(X\), the authors recall the classical invariants: The \textit{normal bundle} \((a,b):=\mathcal O(a)\oplus\mathcal O(b)\) which must be \((-1,-1),\;(-2,0)\), or \((-3,1)\), the \textit{width}, the \textit{Dynkin type}, the \textit{length}, and the \textit{Normal bundle sequence}: The flop \(f:X\dashrightarrow X^\prime\) factors into a sequence of blowups in centers \(C_1,\dots,C_n\), followed by blowdowns, and the normal bundles of these curves form the \(\mathcal N\)-sequence.
The authors give a table of relations between these invariants which proves that none of these classify all analytic equivalence types of flopping curves.
The new invariant, i.e. the deformation algebra, of flopping and flipping rational curves \(C\) in a 3-fold \(X\) is constructed by noncommutatively deforming the associated sheaf \(E:=\mathcal O_C(-1)\). Infinitesimal deformations are controlled by \(\mathrm {Ext}^1_X(E,E)\), and its dimension is determined by the normal bundle \(\mathcal N_{C|X}.\) If \(\dim_{\mathbb C}\mathrm {Ext}^1_X(E,E)\leq 1\), then the curve \(C\) deforms over an Artinian base \(\mathbb C[x]/x^n\). When \(\mathrm {Ext}^1_X(E,E)\geq 2\) the commutative deformations of \(C\) does not induce enough invariants.
The problem is solved by applying noncommutative deformation theory. The commutative deformation functor \(c\mathcal Def_E:\mathsf{CArt}\rightarrow\mathsf{Sets}\) is given by \(R\mapsto\{\mathrm{flat }R\)-families of coherent sheaves deforming \(E\) noncommutative deformation theory; noncommutative deformation algebra; flips; flops; contraction algebra; analytic equivalence types; Dynkin type; derived autoequivalence Donovan, W.; Wemyss, M., Noncommutative deformations and flops, Duke Math. J., 165, 8, 1397-1474, (2016) Formal methods and deformations in algebraic geometry, Minimal model program (Mori theory, extremal rays), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rings arising from noncommutative algebraic geometry, Derived categories, triangulated categories Noncommutative deformations and flops | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper aims to show that a certain moduli space \(\mathsf{T}\), which arises from the so-called Dwork family of Calabi-Yau \(n\)-folds, carries a special complex Lie \(\{\)algebra\(\}\) containing a copy of \(\mathfrak{sl}_2(\mathbb{C})\). In order to achieve this goal, we introduce an algebraic group \(\mathsf{G}\) acting from the right on \(\mathsf{T}\) and describe its Lie algebra \(\text{Lie}(\mathsf{G})\). We observe that \(\text{Lie}(\mathsf{G})\) is isomorphic to a Lie subalgebra of the space of the vector fields on \(\mathsf{T} \). In this way, it turns out that \(\text{Lie}(\mathsf{G})\) and the modular vector field \(\mathsf{R}\) generate another Lie algebra \(\mathfrak{G} \), called AMSY-Lie algebra, satisfying \(\dim (\mathfrak{G})=\dim (\mathsf{T})\). We find a copy of \(\mathfrak{sl}_2(\mathbb{C})\) containing \(\mathsf{R}\) as a Lie subalgebra of \(\mathfrak{G} \). The proofs are based on an algebraic method calling ``Gauss-Manin connection in disguise''. Some explicit examples for \(n=1,2,3,4\) are stated as well. complex vector fields; Gauss-Manin connection; Dwork family; Hodge filtration; modular form Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions, Dynamical systems over complex numbers, Moduli, classification: analytic theory; relations with modular forms, Calabi-Yau manifolds (algebro-geometric aspects) Modular vector fields attached to dwork family: \(\mathfrak{sl}_2(\mathbb{C})\) Lie algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Lambda\) be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of \(\Lambda\) with fixed dimension \(d\) and fixed squarefree top \(T\). Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of \(\Lambda\). In the case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety \(\mathfrak{Grass}^T_d\) of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety \(\mathfrak{Grass}^T_d\) is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple \(T\)', the radical layering \((J^lM/J^{l+1}M)_{l\geq 0}\) is shown to be a classifying invariant for the modules with top \(T\). This relies on the following general fact obtained as a byproduct: proper degenerations of a local module \(M\) never have the same radical layering as \(M\). finite-dimensional algebras; moduli spaces; simple modules; Grassmannians; projective varieties; quivers; finite local representation type; degenerations B. Huisgen-Zimmermann, ''Classifying representations by way of Grassmannians,'' Trans. Am. Math. Soc., 359, 2687--2719 (2007). Representations of associative Artinian rings, Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Representation type (finite, tame, wild, etc.) of associative algebras Classifying representations by way of Grassmannians. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 smooth compactification \(X\langle n\rangle\) of the configuration space of \(n\) distinct labeled points in a smooth algebraic variety \(X\) is constructed by a natural sequence of blowups, with the full symmetry of the permutation group \(\mathbb{S}_n\) manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of \textit{C. De Concini} and \textit{C. Procesi} [Sel. Math., New Ser. 1, 459--494 (1995; Zbl 0842.14038)], and it has a strata-compatible surjection onto the Fulton-MacPherson compactification [\textit{W. Fulton} and \textit{R. MacPherson}, Ann. Math. (2) 139, 183--225 (1994; Zbl 0820.14037)]. The degenerate configurations added in the compatification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson (loc. cit.). In characteristic 0, isotropy subgroups of the action of \(\mathbb{S}_n\) on \(X\langle n\rangle\) are abelian; thus \(X\langle n\rangle\) may be a step toward an explicit resolution of singularities of the symmetric products \(X^n/\mathbb{S}_n\). Hodge polynomial; leveled trees; maximal wonderful compactification; polyscreens Ulyanov A.P.: Polydiagonal compactification of configuration spaces. J. Alg. Geom. 11, 129--159 (2002) Enumerative problems (combinatorial problems) in algebraic geometry, Enumeration in graph theory, Transcendental methods, Hodge theory (algebro-geometric aspects) Polydiagonal compactification of configuration spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A Landau-Ginzburg (LG) space is a pair \(([\mathbb{C}^n/G],W)\), where \(G<\mathrm{GL}(n,\mathbb{C})\) is a finite subgroup, and \(W\) is a non-degenerate quasi-homogeneous \(G\)-invariant polynomial in \(n\) variables. For \(G=\mathbb{Z}_{k+1}, W=x^{k+1}\) Witten used the equation now bearing his name to construct ``topological gravity coupled to matter'', and the requisite mathematical structures were later worked out for this example. Proper moduli spaces of nodal spin curves were constructed by Abramovich and Jarvis, and the ``top Chern class'' by Polishchuk-Vaintrobe, Chiodo and Mochizuki using different methods. The general case was formalized by Fan-Jarvis-Ruan, it is the Gromov-Witten theory of the singularity \((W=0)/G\), commonly called the FJRW theory.
In this paper the authors give an alternative mathematical construction when representation factors of point automorphism groups are nontrivial at all marked points on spin curves. They verify all the expected properties of the virtual class for their definition, and prove that it is equivalent to Chiodo's, Polishchuk-Vaintrob's and FJRW, the implied equivalence of the last two is new. The paper's construction is an algebraic analog of the Witten's informal one. The Witten equation gives a differentiable section of the obstruction sheaf, and his class is obtained via the homology class generated by the solution space of the equation's transverse perturbation.
The authors proceed as follows. First, a moduli stack \(\overline{M}_{g,\gamma}(G)\) of \(G\)-spin \(l\)-pointed genus \(g\) twisted nodal curves banded by a collection of representations \(\gamma\) is formed. From it they construct the moduli of \(G\)-spin curves with fields \(\overline{M}_{g,\gamma}(G)^p\), which is a DM stack with perfect obstruction theory relative to \(\overline{M}_{g,\gamma}(G)\). Next, the polynomial \(W\) is used to construct a cosection of it, whose non-surjective locus is proper and contained in \(\overline{M}_{g,\gamma}(G)\). The virtual class \([\overline{M}_{g,\gamma}(G)^p]^{\mathrm{vir}}\) is then obtained via a cosection localized Gysin map, by applying a construction of Kiem and Li. This is the ``top Chern class'' of Witten.
Because the relative obstruction theory of \(\overline{M}_{g,\gamma}(G)^p\) to \(\overline{M}_{g,\gamma}(G)\) is linear the virtual class can be expressed in terms of localized Chern classes of certain complexes, implying the equivalence to the class of Polishchuk-Vaintrob. The equivalence to the FJRW class (when pushed to the ordinary homology) follows from the topological nature of the cosection localized class. Landau-Ginzburg space; Witten equation; topological gravity coupled to matter; FJRW theory; twisted nodal spin curves; cosection localized Gysin map; moduli of G-spin curves with fields H.-L. Chang, J. Li and W.-P. Li, Witten's top Chern class via cosection localization, preprint (2013), . Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Witten's top Chern class via cosection localization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review studies quiver Grassmannians associated with finite dimensional indecomposable representations of a Kronecker quiver. A finite dimensional representation of a Kronecker quiver on the category of vector spaces is given by two finite dimensional complex vector spaces, \(M_{1}\) and \(M_{2}\) and linear maps \(m_{a}\), \(m_{b}\) between them. Given \(e_{1}\) and \(e_{2}\), define
\[
Gr_{(e_{1},e_{2})}=\{(N_{1},N_{2})\in Gr_{e_{1}}(M_{1})\times Gr_{e_{2}}(M_{2}):m_{a}(N_{1})\subset M_{2},m_{b}(N_{1})\subset N_{2}\}
\]
which is a projective variety, in general non smooth, where \(Gr_{e}(V)\) denotes the Grassmannian of \(e\)-dimensional vector subspaces of \(V\). We call these varieties quiver Grassmannians.
For the case \(M_{1}=M_{2}=\mathbb{C}^{n}\), \(m_{a}=Id\) and \(m_{b}\) an indecomposable nilponent Jordan block, the representation is denoted by \(R_{n}\) and is called regular indecomposable. The article concentrates on this particular case, \(X=Gr_{(e_{1},e_{2})}(R_{n})\), because all the rest of indecomposable representations either have the same quiver, or are easier cases previously studied in the literature.
There is an action of a \(1\)-dimensional torus on \(X\) which provides a stratification
\[
X_{s}\subseteq\cdots\subseteq X_{1}\subseteq X_{0}=X
\]
into closed subvarieties \(X_{k}\simeq Gr_{(e_{1}-k,e_{2}-k)}(R_{n-2k})\), where each one is the singular locus of the previous one and the difference between two consecutive ones is smooth quasiprojective. Bialynicki-Birula's theorem is applied to show that \(X\) has a cellular decomposition and, relations between the dimensions of the cells through the Euler form associated to the Kronecker quiver, are used to compute the Betti numbers and the Poincare polynomials of the quiver Grassmannians, from which the Euler characteristics are recovered.
As an application, the authors give a new geometric realization for the atomic basis of a cluster algebra \(\mathcal{A}_{Q}\) of type \(A_{1}^{(1)}\) (which are \(\mathbb{Z}\)-subalgebras of the field of rational functions associated to quivers \(Q\) without loops and \(2\)-cycles) by using the Caldero-Chapoton map (cf. [\textit{P. Caldero} and \textit{F. Chapoton}, Comment. Math. Helv. 81, No. 3, 595--616 (2006; Zbl 1119.16013)]) associating cluster monomials of the cluster algebra to rigid \(Q\)-representations. The authors propose a truncation of the Caldero-Chapoton map to realize the extra elements of the atomic basis. quiver Grassmannians; cluster algebras; Caldero-Chapoton map; quiver representations Cerulli Irelli, G., Esposito, F.: Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras. Algebra \& Number Theory. arXiv:1003.3037v2 (2010) Cluster algebras, Representation theory of lattices, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Projective techniques in algebraic geometry Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a projective curve \(C\) of genus \(2\), consider the space \(\text{SU}_C(2)\sim {\mathbb P}^3\) parameterizing semi--stable rank \(2\) bundles with trivial determinant over \(C\). These bundles arise as extensions of type \(0\to\omega^{-1}\to E\to \omega\to 0\), from the canonical bundle. The author studies the relations between the space \({\mathbb P}\text{Ext}^1(\omega,\omega^{-1})\sim{\mathbb P}^4\) and \(\text{SU}_C(2)\). Sending extensions to the corresponding bundles, one finds a natural rational map \(\phi:{\mathbb P}Ext^1(\omega,\omega^{-1}) \to \text{SU}_C(2)\). The author proves that \(\phi\) is a conic bundle, outside the point \(P\) which parameterizes \(\mathcal O\oplus\mathcal O\). The author describes the singular fibers of the bundle and solves the singularity of the map \(\phi\). Finally, the author determines a \({\mathbb P}^2\)-bundle \(\mathcal E\) over the blow up of \(\text{SU}_C(2)\) at \(P\), such that the conic bundle is a section of \(\mathcal E\). curves; bundles DOI: 10.1007/s00209-008-0319-4 Vector bundles on curves and their moduli A conic bundle degenerating on the Kummer 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 Let \((X,x)\) be a normal isolated singularity of dimension two over the complex number field \(\mathbb{C}\). The second pluri-genus \(\delta_2 (X,x)\) are studied in this article. Let \(m\) be a positive integer. Following Kimio Watanabe, the \(m\)-th pluri-genus \(\delta_m (X,x)\) is defined to be the integer \(\delta_m(X,x) =\dim_\mathbb{C} H^0({\mathcal O}_U (mK_U))/L^{2/m}(U)\), where \(X\) is an appropriate representative of the germ \((X,x)\), \(U=X-\{x\}\), \(K_U\) denotes the canonical divisor of \(U\), and \(L^{2/m}(U)\) denotes the \(\mathbb{C}\)-vector subspace of \(L^{2/m}\)-integrable \(m\)-ple holomorphic 2-forms on \(U\). \(\delta_1 (X,x)\) coincides with the geometric genus of \((X,x)\). Relations among the second pluri-genus, other various invariants and various conditions which the singularity satisfies are studied. As invariants, the geometric genus, the Milnor number, the Tjurina number, the modality, the inner modality and so on are considered. Such conditions as Gorenstein, Du Bois, hypersurface, complete intersection, quasi-homogeneous and so forth are treated.
The following is one of Okuma's results: If \((X,x)\) is a Gorenstein singularity with \(p_g(X,x)\geq 1\), then \(\delta_2(X,x)=p_g(X,x)-(2K+E)\cdot(K+E)/2\), where \(p_g(X,x)\) denotes the geometric genus, \(K\) denotes the canonical divisor on the minimal good resolution of \((X,x)\), and \(E\) denotes the exceptional reduced divisor on the minimal good resolution of \((X,x)\). surface singularity; deformation; modality; pluri-genus; Milnor number; Tjurina number Okuma T. The second pluri-genus of surface singularities. Compos Math, 1998, 110: 263--276 Invariants of analytic local rings, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Modifications; resolution of singularities (complex-analytic aspects) The second pluri-genus of 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 Segal's \(\Gamma\)-rings provide a natural framework for absolute algebraic geometry. We use \textit{G. Almkvist}'s [J. Algebra 28, 375--388 (1974; Zbl 0281.18012)] global Witt construction to explore the relation with \textit{J. Borger} [``Lambda-rings and the field with one element''', Preprint, \url{arXiv:0906.3146}] \(\mathbb{F}_1\)-geometry and compute the Witt functor-ring \(\mathbb{W}_0 (\mathbb{S} )\) of the simplest \(\Gamma\)-ring \(\mathbb{S}\). We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between \(\lambda\)-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification \(\overline{\mathrm{Spec}\, \mathbb{Z}}\) which acquires a structure sheaf of \(\mathbb{S}\)-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor \(D\) on \(\overline{\mathrm{Spec}\, \mathbb{Z}}\), we show how to associate to \(D\) a \(\Gamma\)-space that encodes, in homotopical terms, the Riemann-Roch problem for \(D\). Geometry over the field with one element, Riemann-Roch theorems, Chern characters Segal's gamma rings and universal arithmetic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 roughly predicts an equivalence between the bounded derived category of coherent sheaves on a Calabi--Yau variety and the Fukaya category of its mirror Calabi--Yau. Positive results in the literature include the case of an elliptic curve and the quartic \(K3\) surface. Since its introduction the conjecture has been formulated in the non Calabi--Yau case as well, for example, by Kontsevich himself in the case of Fano varieties. The mirror in this case is not another variety but rather a Landau--Ginzburg theory, that is, a variety together with a holomorphic function.
The main result of the paper under review can be roughly summarised as follows. Let \(M\) be a genus two curve equipped with a symplectic structure. Its mirror is a three-dimensional Landau--Ginzburg theory \(X\rightarrow \mathbb{C}\) whose zero fibre \(H\) is the union of three rational surfaces. We then consider the split-closed derived category of the Fukaya category of \(M\) and the split-closure of the triangulated category of singularities of \(H\) (the latter is the Verdier quotient of the bounded derived category of \(H\) by the subcategory of perfect complexes). The main assertion is that there exists an equivalence between these two triangulated categories.
The basic idea of the proof is that both categories can be described by \(A_\infty\)-algebras of a very special form, namely \(A_\infty\)-deformations of the exterior algebra (with an added group action). The appearance of these algebras can be roughly explained as follows. On the one hand, one uses the derived McKay correspondence to see that the singularity category is equivalent to a certain equivariant category which is split-generated by a single object and thus one needs to describe the \(A_\infty\)-structure on the endomorphism algebra of this object. On the other hand, if one represents \(M\) as a covering of a genus zero orbifold \(\overline{M}\) and denotes by \(\overline{L}\) the curve to which the five curves generating the split closed derived Fukaya category of \(M\) project, then the wanted information is contained in the \(A_\infty\)-structure on the Floer cohomology of \(\overline{L}\). Floer cohomology; Fukaya category; \(A_\infty\)-algebras; triangulated category of singularities; McKay correspondence P. Seidel, Homological mirror symmetry for the genus two curve. \textit{J. Algebraic Geom.} 20 (2011), no. 4, 727--769.MR 2819674 Zbl 1226.14028 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Mirror symmetry (algebro-geometric aspects), Derived categories, triangulated categories, Special algebraic curves and curves of low genus Homological mirror symmetry for the genus two 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 A classical result in representation theory states that the action of the symmetric group \(S_d\) fully centralizes the natural action of the complex general linear algebra \(\mathfrak{gl}_N\) on the tensor space \((\mathbb C^N)^{\otimes d}\). This results in representations for \(\mathfrak{gl}_N\), which are summands of \((\mathbb C^N)^{\otimes d}\), being in bijection with representations for the symmetric group \(S_d\). The Schur algebra of type A is the centralizer algebra of \(S_d\) on \((\mathbb{C}^N)^{\otimes d}\).
In type B, the orthogonal group does not fully centralize the action of the type B Weyl group on the tensor space. The orthogonal group is known to centralize the action of the Brauer algebra but the description of the centralizer of the type B Weyl group action, and its quantization, is nontrivial. In one approach, the centralizer is given by a subgroup of \(GL_N\), its Lie algebra being the fixed-point subalgebra \(\mathfrak{gl}_N^{\theta}\) of \(\mathfrak{gl}_N\) under a certain involution \(\theta\). In another approach, the centralizer is given as a homomorphic image of a two-block subalgebra \(U\) of \(U(\mathfrak{gl}_N)\). The pairs \((U(\mathfrak{gl}_N), U)\) and \((U(\mathfrak{gl}_N), U(\mathfrak{gl}_N^{\theta}))\) are infinitesimal quasi-split symmetric pairs of type A.
Y. Li and J. Zhu establish explicit isomorphisms between \(U\) and \(U(\mathfrak{gl}_N^{\theta})\) (Theorem 2.4.2, page 14), and consequently on their respective Schur algebras. The authors also provide a presentation of the geometric counterpart of the above Schur algebras specialized at \(q=1\). quasi-split symmetric pairs; Schur algebras; quantization of \(q\)-Schur algebras; isotropic flags; Chevalley generators; fixed point subalgebra Universal enveloping (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Schur and \(q\)-Schur algebras Quasi-split symmetric pairs of \(U(\mathfrak{gl}_N)\) and their Schur 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 presents some interesting results on crepant resolutions of singularities of complex algebraic varieties. The point of view is not the ``classical'' one, where a \textit{crepant resolution} of a variety \(X\) is a proper birational morphism \(\pi: {\tilde X} \to X\), with \(\tilde X\) smooth, such that \({\pi}^{\star}(\omega _X) = \omega _{\tilde X}\), but rather the categorical (or abstract) one. In the classical context, often a variety does not admit a crepant resolution, but in the abstract one better results are expected.
The precise definition of a categorical resolution of a variety \(X\) is rather technical, but basically it consists of a certain category \({\mathcal I}\) together with a functor \(\pi_{\star}: \mathcal I \to D(X)\) (the target is the derived category of quasi coherent sheaves on \(X\)), satisfying certain properties. Sometimes, when the functor is clear from the context, it is not specified. If a further condition (involving the bounded subcategory \(D^b (X)\) of \(D(X)\)) is satisfied, the categorical resolution is said to be \textit{strongly crepant}. (There is a variation of this notion, namely \textit{weakly crepant resolution}). It is known that a morphism \(\pi :\tilde X \to X\) is a crepant resolution of singularities if and only if the induced morphism \({\mathbf R}{\pi}_{\star}:D(\tilde X) \to D^b(X)\) is a strongly crepant resolution of singularities. But in general, categorical crepant resolutions do not ``come'' from classical, or geometric, ones.
The main theorems proved in the paper are:
(1) Let \(V\) be a smooth quasi-projective variety and \(G\) a finite subgroup of \(\mathrm{Aut}(V)\), such that the dualizing sheaf of \(V\) is \(G\)-equivariantly locally trivial. Then \(D_G(V)\) (the derived category of \(G\)-equivariant quasi-coherent sheaves on \(V\)) is a categorical strongly crepant resolution of \(X=V/G\).
(2) Let \(X\) be a quasi projective variety with normal Gorenstein quotient singularities, \(\mathcal X\) a smooth Deligne-Mumford stack whose coarse moduli space is \(X\), and whose dualizing bundle is the pull-back of that of \(X\). Then \(D(\mathcal X)\) is a categorical crepant resolution of \(X\).
Theorem (1) was known, but here a more ``elementary'' proof of it is given. Actually, more is proven about (1) or (2). Namely, the existence of a sheaf of algebras which allows the author to show that a non commutative crepant resolution of \(X\) (in the sense of \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] is available.
Other results are also discussed, indicating a connection of this work with certain cases of the categorical McKay correspondence. categorical crepant resolution; derived category; stack; quotient singularity; categorical McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients), Noncommutative algebraic geometry, Derived categories, triangulated categories Categorical crepant resolutions for quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This short introductory survey covers basic topics of toric geometry paying special attention to blowing-up and to resolution of singularities. Starting with the standard definition of a toric variety by gluing together affine toric varieties corresponding to the cones of a fan, the author discusses the relationship between orbital structure of toric varieties and combinatorics of their fans. The criteria of completeness and smoothness of a toric variety in terms of its fan are given. Further, the theory of Weil and Cartier divisors on a toric variety is considered. Support functions of invariant Cartier divisors are defined and it is shown that a divisor on a projective toric variety is globally generated (ample) iff its support function is (strictly) convex. On a smooth toric variety, every ample divisor is very ample (Demazure), and on any \(n\)-dimensional toric variety (\(n>1\)), the \((n-1)\)-th multiple of any ample divisor is very ample [see \textit{G. Ewald} and \textit{W. Wessels}, Result. Math. 19, No. 3/4, 275-278 (1991; 739.14031)]. Global sections of Cartier divisors are described, and it is shown that a projective toric variety \(X\) is recovered from the polytope of global sections of an ample divisor on \(X\).
Alternative constructions of toric varieties are given: as \(\text{Proj}\) of a graded ring (whose \(\text{Spec}\) is the affine cone over a projective toric variety), as a quotient of an open subset of an affine space by a quasitorus [see \textit{D. A. Cox}, J. Algebr. Geom. 4, No. 1, 17-50 (1995; Zbl 0846.14032)], and by toric ideals (in the affine case). Non-normal toric varieties are discussed. In terms of subdivisions of fans, the blow-up of a smooth toric variety along an invariant subvariety is described, and the resolution of singularities of any toric variety \(X\), bijective over \(X^{\text{reg}}\), is constructed. Finally, rationality of singularities of any toric variety is proven.
The survey contains no proofs, except for the construction of a toric desingularization and the rationality of singularities, where detailed proofs are given. Instead, simple instructive examples illustrate definitions and theorems of toric geometry are given. toric variety; fan; toric ideal; blow-up; desingularization; rational singularity; resolution of singularities; divisors David A. Cox, Toric varieties and toric resolutions, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 259 -- 284. Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves Toric varieties and toric 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 [For the entire collection see Zbl 0509.00008.]
A quasi-homogeneous singularity is a complete local complex algebra S which is the completion of a finitely generated graded complex algebra R. If the singularity is isolated (S is regular away from the maximal ideal), being such a completion is the same as having a \({\mathbb{C}}^*\) action, with all weights positive, on S (a good \({\mathbb{C}}^*\)-action). S need not determine R uniquely. This paper surveys work of the author and others on the relations between S and R, especially the recovery of R from S. - Since R is graded, so is the space of all its derivations. The author conjectures that if R is normal then S is the completion of a normal graded ring with no derivations of negative weight. When true, this has implications for the recovery problem, and the author establishes the conjecture for some special classes of singularities. The recovery problem is related to the conjugacy class of the good action on S, under conjugation in Aut(S). This latter has a natural maximal reductive subgroup G, and there is always a good action in the center of G. When this center has dimension one (as the author shows in the case of a complete intersection singularity) the central good action is unique.
The paper also contains a preview of some of the author's work on the graded module of first order deformations, especially on the duality between the graded pieces, along with some of the consequences of this duality. derivations; automorphisms; deformations; quasi-homogeneous singularity; completion of a finitely generated graded complex algebra; duality Wahl, J. M.: Derivations, automorphisms and deformations of quasi-homogeneous singularities. (1983) Deformations of singularities, Formal methods and deformations in algebraic geometry, Morphisms of commutative rings, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Derivations, automorphisms and deformations 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 The author shows that the analytic degree of the divisor of a modular form on the orthogonal group \(O(2,p)\) is determined by its weight. Let \((V, Q)\) be a nondegenerate real quadratic space of signature \((2,p)\) and write \(G\cong O(2,p)\) for its real orthogonal group. Let \(V_{\mathbb{Q}}= V\otimes_{\mathbb{R}}\mathbb{C}\) be complexification of \(V\) and \(P(V_{\mathbb{C}})\) be the associated projective space. We denote by \({\mathcal H}\) one of the connected components of the space
\[
{\mathcal K}= \{[Z]\in P(V_{\mathbb{C}})\mid(Z,Z)= 0,\,(Z,\overline Z)> 0\}.
\]
The connected component \(G^0\) of the identity of \(G\) acts on transitively on \({\mathcal H}\). Let \(L\subset V\) be an even lattice, and write \(O(L)\) for the integral orthogonal group of \(L\). We let \(\Gamma\subset O(L)\cap G^0\) be a subgroup of finite index that acts freely on \({\mathcal H}\). The quotient \(Y_\Gamma:= \Gamma\setminus{\mathcal H}\) is a non-singular quasi-projective algebraic variety over \(\mathbb{C}\).
The notion of modular form for the group \(\Gamma\) is defined as follows: let \(\widetilde H\subset V_{\mathbb{C}}- \{0\}\) be the cone over \({\mathcal H}\). Let \(k\in\mathbb{Z}\) and \(\chi\) be a character of \(\Gamma\). A meromorphic function \(F\) on \(\widetilde H\) is called a meromorphic modular form of weight \(k\) and character \(\chi\) for the group \(\Gamma\), if
(i) \(F\) is homogeneous of degree \(-k\), i.e., \(F(cZ)= c^{-k}F(Z)\) for any \(c\in\mathbb{C}- \{0\}\);
(ii) \(F\) is invariant under \(\Gamma\) i.e., \(F(gZ)= \chi(g)F(Z)\) for any \(g\in\Gamma\);
(iii) \(F\) is meromorphic at the boundary.
Meromorphic modular forms of weight \(k\) with trivial character can be viewed as global rational sections of an algebraic line bundle \({\mathcal M}_k\) on \(Y_\Gamma\). If \(F\) is a modular form of weight \(k\), then its Petersson metric is the function on \({\mathcal H}\) is given by
\[
\| F(Z)\|^2_{\text{Pet}}= |F(Z)|^2(Z,\overline Z)^k.
\]
Let \(\Omega\) be the first Chern form of \({\mathcal M}_1\). We denote by \(\text{vol}(Y_\Gamma)= \int_{Y_\Gamma}\Omega^p\) the volume of \(Y_\Gamma\). If \(D\) is a divisor on \(Y_\Gamma\), then we define its analytic degree by
\[
\deg Y_\Gamma(D)= \int_D \Omega^{p-1}.
\]
The main theorem of this paper is as follows:
Theorem 1. Assume that the Witt rank of \(L\) is \(0\) if \(p= 1\), and \(\leq 1\) if \(p= 2,3\). If \(F\) is a meromorphic modular form of weight \(k\) with some character for the group \(\Gamma\), then
\[
\deg Y_\Gamma(\text{div}(F))= k\text{\,vol}(Y_\Gamma).
\]
Moreover, the author proves the following theorem.
Theorem 2. Assume that the Witt rank of \(L\) is \(0\) if \(p= 1\), and \(\leq 1\) if \(p= 2\). If \(F\) is a meromorphic modular form with some character for the group \(\Gamma\), then \(\log\| F\|_{\text{Pet}}\) is in \(L^1(Y_\Gamma, \Omega_p)\), i.e., the integral \(\int_{Y_\Gamma}|\log\| F\|_{\text{Pet}}|\Omega^p\) converges. meromorphic modular form; orthogonal group; divisor; integrability; analytic degree; volume Bruinier, J.H.: Two applications of the curve lemma for orthogonal groups. Math. Nachr. 274--275, 19--31 (2004) Other groups and their modular and automorphic forms (several variables), Divisors, linear systems, invertible sheaves Two applications of the curve lemma for orthogonal groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is aimed to the new relations between genus-zero Gromov-Witten invariants. The main theorem (Theorem \(5.1\)) expresses multipointed invariants in terms of invariants of lower degree or number of points. Theorem \(1.4\) is a simplification of the main theorem for the rank-one case. For the complete intersections in projective spaces this theorem reduces to compact formula \(1.1\). The corollary of these expressions is reconstruction theorem \(5.2\), which, under some natural conditions (``vanishing of one-pointed invariants which are not generated by divisors''), enables one to reconstruct multipointed Gromov--Witten invariants (``generated by divisors'') in terms of the one-pointed ones.
The main idea of the paper is the following. Gromov-Witten invariants are intersection numbers on the moduli spaces of stable maps from rational curves to the variety. Embed such space to the graph space, i. e. the space of stable maps with parameterizations. There is a natural torus action on the graph space; one of its fixed loci is our moduli space of stable maps. The relations are obtained by comparing the residues of some equivariant classes under the equivariant forgetful morphisms. First authors prove the theorem for projective spaces. Then they reduce the rank-one case to the projective space case, and then reduce the arbitrary case to the rank-one case.
A good reference for this paper is \textit{Y. P. Lee} and \textit{R. Pandharipande} [Am. J. Math. 126, No. 6, 1367--1379 (2004; Zbl 1080.14065)], where these results are proved independently. quantum cohomology; Kontsevich-Manin spaces; localisation A. Bertram and H. P. Kley, ''New recursions for genus-zero Gromov-Witten invariants,'' Topology, vol. 44, iss. 1, pp. 1-24, 2005. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic) New recursions for genus-zero Gromov--Witten 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 This paper is based on a talk at a conference ``JDG 2017: Conference on Geometry and Topology''. We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the \(K\)-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the \(K\)-inequality to the fully faithful embedding. We consider two kinds of factorizations of birational maps between algebraic varieties into elementary ones; those into flips, flops and divisorial contractions according to the minimal model program, and more traditional weak factorizations into blow-ups and blow-downs with smooth centers. We review major approaches towards the DK hypothesis for flops between smooth varieties. The latter factorization leads to an weak evidence of the DK hypothesis at the Grothendieck ring level. DK hypothesis is proved in the case of toric or toroidal maps, and leads to the derived McKay correspondence for certain finite subgroups of \(\mathrm{GL}(n, \mathbb{C})\). Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Rational and birational maps, Minimal model program (Mori theory, extremal rays), Arcs and motivic integration, Research exposition (monographs, survey articles) pertaining to algebraic geometry Birational geometry and derived categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book under review is a new edition of the authors' celebrated research monograph [Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications. Vol. IV. Brookline, Mass.: Math Sci Press. IV, 335 p. (1975; Zbl 0334.14007)], which must be seen as one of the milestones in contemporary algebraic and complex-analytic geometry. Thirty-five years ago, the original text provided a universal method for constructing desingularizations of a class of quotient spaces, which has proven itself to be crucial in the study of moduli spaces in algebraic geometry ever since. Alas, despite its fundamental significance, this standard reference has been unavailable in the last couple of decades, and the request for a second edition has become more and more urgent in the course of time.
Corresponding to the proposal of a second edition of their book, the authors fortunately agreed to undertake such a rewarding project, and the result is the present new edition of this classic. As they point out in the preface to the second edition of the book, the authors have decided to leave the text of the original basically intact, thereby preserving its content, structure, and style of presentation likewise.
However, in this new edition, the text has been completely re-typeset (in {\TeX}), several recognized errors have been corrected, the whole presentation has been polished and streamlined, the notation has been made consistent and uniform, an index has been added, and the further developments in the field in the last three decades are reflected by a supplementary bibliography serving as a very detailed, rather complete guide to the more recent literature in the subject.
Now as before, the text consists of four chapters of varying authorship. As for their respective contents, we may refer to the review of the original edition (loc. cit.) from 1975 by Yu. G. Zarhin, as the basic material has been left essentially unaltered. However, let us briefly recall both the goal and the general structure of this classic monograph.
The objects of study are quotient spaces \(D/\Gamma\), where \(D\) is a bounded symmetric domain and \(\Gamma\) a neat arithmetic subgroup of \(\Aut(D)^0\). In this context, the authors' goal is the construction of a family of non-singular compactifications \(\overline{D/\Gamma}\) of the space \(D/\Gamma\), generalizing earlier approaches and results by Baily-Borel, Igusa, Hirzebruch, and Satake, respectively. The authors' approach builds heavily on the theory of toroidal embeddings as developed by \textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Numford} and \textit{B. Saint-Donat} [Lecture Notes in Mathematics. 339. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 209 p. (1973; Zbl 0271.14017)].
Chapter 1 of the present monograph provides a quick review of the theory of toroidal embeddings, together with some classical examples illustrating this framework. Chapter 2 develops the basic ingredients from the polyhedral reduction theory of self-adjoint cones, whereas Chapter 3 takes up the explicit construction of the smooth compactifications \(\overline{D/\Gamma}\) of \(D/\Gamma\). Chapter 4 provides two important applications of the theory developed so far. More precisely, it is shown that the quotient \(D/\Gamma\) is an algebraic variety of general type (in Kodaira's classification) when the arithmetic group \(P\) is sufficiently small, and that the (a priori) analytic smooth compactification \(\overline {D/\Gamma}\) of \(D/\Gamma\) is indeed a projective variety in many concrete cases. In particular, these geometric applications make the significance of the authors' general approach in moduli theory strikingly evident.
No doubt, this classic will maintain its outstanding role in algebraic geometry, Hermitian differential geometry, group representation theory, and arithmetic geometry also in the future, especially for active researchers and graduate students in these related areas of contemporary pure mathematics. In this regard, the present new edition of it is certainly more than welcome. resolution of singularities; compactifications; locally symmetric domains; toroidal embeddings; group actions; quotients A. Ash, D. Mumford, M. Rapoport, and Y.-S. Tai, \textit{Smooth Compactifications} \textit{of Locally Symmetric Varieties}. Second edition. With the collaboration of Peter Scholze. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010.Zbl 1209.14001 MR 2590897 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes, Global theory and resolution of singularities (algebro-geometric aspects), Lie groups, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Compactifications; symmetric and spherical varieties, Collected or selected works; reprintings or translations of classics, Discrete subgroups of Lie groups, Differential geometry of symmetric spaces, Discontinuous groups of transformations Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\theta\) be an irrational number, \(T_\theta\) the associated noncommutative \(2\)-torus, \(A_\theta\) its algebra of smooth functions and \(G \subset \text{SL}_2(\mathbb Z)\) a finite subgroup. Let \(B_\theta := A_\theta \star G\) be the cross product algebra. \(T-\theta /G\) is called a noncommutative toric orbifold. A holomorphic vector bundle on \(T_\theta /G\) is a finitely generated projective right \(B_\theta\)-module. Here the author proves that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebra \(\Lambda\). As an application he computes \(K_0(B_\theta )\). He also computes a torsion pair in the category of \(\Lambda\)-modules, such that the tilting with respect to this torsion pair gives the category of holomorphic bundles on \(T_\theta /G\). non-commutative torus; tilting; vector bundle on non-commutative \(2\)-torus Polishchuk, A.: Holomorphic bundles on \(2\)-dimensional noncommutative toric orbifolds. In: Consani, C., Marcolli, M. (eds.) Noncommutative Geometry and Number Theory, pp. 341-359. Vieweg, Wiesbaden (2006) Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Holomorphic bundles and generalizations Holomorphic bundles on 2-dimensional noncommutative toric orbifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0721.00009.]
These notes represent a survey of the main points of the authors' explicit proof of canonical desingularization of an algebraic subvariety (resp. analytic subspace) \(X\) of an algebraic (resp. analytic) manifold \(M\), in characteristic zero. The full details of this result are planned for a forthcoming work `Canonical desingularization in characteristic zero: a simple constructive proof'. The authors' result is essentially a new proof of Hironaka's theorem, although they also give an explicit resolution algorithm. The centres of the blowings-up used in the desingularization are determined by a local invariant of the singularity of \(X\), defined over a sequence of blowings-up.
The first section of this paper describes the general strategy of the proof and ends with a precise statement of the main theorem. --- The second section recalls the definitions and basic notions involved in resolution of singularities (analytic space, blowing-up, strict transform, normal crossings, etc.). --- The third section introduces the local invariant of a singularity of \(X\) and begins an analysis of this invariant. --- The fourth section gives a key part of the proof that it is in fact invariant. Much of these notes deal with the special case where \(X\) is a hypersurface, and the general case involves a reduction to this case. canonical desingularization of an algebraic subvariety; resolution algorithm Zhou, X. Y., Zhu, L. F.: An optimal \(L\)\^{}\{2\} extension theorem on weakly pseudoconvex Kähler manifolds. To appear in J. Differential Geom. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Effectivity, complexity and computational aspects of algebraic geometry A simple constructive proof of canonical 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 A Hilbert-Picard modular cusp is a boundary point to the quotient space of the product of the complex unit balls, while a Tsuchihashi cups is a boundary point to the quotient of a tube domain [cf. \textit{H. Tsuchihashi}, Tôhoku Math. J., II. Ser. 35, 607-639 (1983; Zbl 0585.14004)].
Let \(F\) be a totally real algebraic number field of degree \(n(\geq 2)\) and \(K\) an imaginary quadratic extension of \(F\) with the ring \({\mathfrak O}_ K\). For a positive integer \(m\) the unitary group \(U(m+1,1;{\mathfrak O}_ K)\) with its entries in \({\mathfrak O}_ K\) acts on the product \((\mathbb{B}_{m+1})^ n\) of \(n\) copies of the complex unit ball in \(\mathbb{C}^{m + 1}\) through the \(n\) different embeddings of \(K\) into \(\mathbb{C}\). The quotient space is compactified by adding a finite number of points, which are called Hilbert-Picard modular cusps. In this paper we prove an equality between two invariants of a Hilbert-Picard modular cusp singularity \((V,P)\) of even dimension, namely, the signature defect \(\sigma(p)\) and the contribution of the cusp \(\chi_ \infty(p)\) to the arithmetic genus of the cusp singularity.
Theorem. Let \((V,p)\) be a Hilbert-Picard modular cusp singularity of dimension \(n(m+1)\). When \(n(m+1)\) is even we have \(2^{n(m+1)}\chi_ \infty(p) = \sigma(p)\).
The signature defect was defined by \textit{F. E. P. Hirzebruch} [Enseign. Math., II. Sér. 19, 183-281 (1973; Zbl 0285.14007)] in the case of Hilbert modular cusps and generalized by Satake. The invariant \(\chi_ \infty(p)\) of a Hilbert modular cusp is called the \(\psi\)-invariant by \textit{F. Ehlers} [Math. Ann. 218, 127-156 (1975; Zbl 0301.14003)]. Satake [loc. cit.] defined \(\chi_ \infty(p)\) in more general context, showing that it coincides with the contribution of the cusp to the dimension formula of the space of cusp forms as calculated by means of the Riemann- Roch theorem. Hilbert-Picard modular cusp; Tsuchihashi cups; signature defect Singularities in algebraic geometry, Other groups and their modular and automorphic forms (several variables), Other number fields, Special surfaces Signature defects of Hilbert-Picard modular 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 In the course of the last two decades, various approaches have been developed to create a profound mathematical theory for the phenomenon of mirror symmetry in conformal field theory. Significant milestones, in this context, were \textit{M. Kontsevich}'s conceptual framework of homological mirror symmetry [in: Homological algebra of mirror symmetry. Proceedings of the international congress of mathematicians, ICM '94, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)] and the so-called SYZ approach proposed by \textit{A. Strominger}, \textit{S.-T.Yau} and \textit{E. Zaslow} [Nucl. Phys., B 479, No.~1--2, 243--259 (1996; Zbl 0896.14024)], where the latter was based on the study of special Langrangian submanifolds of Calabi-Yau varieties. In a seminal paper published in 2001, \textit{M. Kontsevich} and \textit{Y. Soibelman} developed a novel framework and showed how their constructions could be applied to tackle two major problems, namely the so-called ``SYZ Conjecture'' and Kontsevich's ``Homological Mirror Conjecture'', from a unified and generalized point of view [cf.: \textit{M. Kontsevich} and \textit{Y. Soibelman}, Homological mirror symmetry and torus fibrations. Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, 2000. Singapore: World Scientific. 203--263 (2001; Zbl 1072.14046)].
In fact, these authors established a general framework of \(A_\infty\)-precategories adapted to the transversality problem in the definition of the Fukaya category of a symplectic manifold. Along the way, they formulated the following natural conjecture, the so-called ``Kontsevich-Soibelman conjecture'':
Let \(k\) be a graded commutative ring. Then the quasi-equivalence classes of \(A_\infty\)-precategories over \(k\) are in bijective correspondence with the quasi-equivalence classes of (actual) \(A_\infty\)-categories over \(k\) with strict (or weak) identity morphisms.
An affirmative answer to this conjecture turns out to be crucial for proving the homological mirror symmetry conjecture in certain special cases, and that is why the study of it should be regarded as both attractive and important.
The paper under review provides a proof of this particular conjecture for essentially small \(A_\infty\)-(pre)categories over a field \(k\).
As a consequence, the author's result implies that the Fukaya \(A_\infty\)-precategory can be replaced by a quasi-equivalent actual \(A_\infty\)-category, just as it is necessary for dealing with the homological mirror symmetry conjecture à la \textit{M. Kontsevich}.
The structure of the paper is as follows:
After a brief introduction to the Kontsevich-Soibelman conjecture, the author introduces \(A_\infty\)-(pre)categories, strict and weak identity morphisms, and quasi-equivalences. In the third section, the author's main theorem (as described above) is proved via several reduction steps, essentially by introducing and using Hochschild cohomology for graded precategories. A crucial partial result is that this Hochschild cohomology is in fact invariant under quasi-equivalences of graded precategories. In the fourth (and last) section of the paper, the author presents a general construction of the pre-triangulated hull in the framework of \(A_\infty\)-precategories. Moreover, it is verified that this very construction (via twisted complexes over ordinary \(A_\infty\)-categories) is invariant under quasi-equivalences. \(A_\infty\)-categories; \(A_\infty\)-categories; Fukaya category; homological mirror symmetry; graded precategories; quasi-equivalences; SYZ conjecture; Kontsevich-Soibelman conjecture; mirror symmetry; conformal field theory Derived categories, triangulated categories, Resolutions; derived functors (category-theoretic aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Definitions and generalizations in theory of categories, Mirror symmetry (algebro-geometric aspects), Lagrangian submanifolds; Maslov index A proof of the Kontsevich-Soǐbel'man conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the question whether moduli spaces of bundles and sheaves with trivial framing on a reduced effective divisor may reflect geometric properties of its complement in a smooth projective variety. This approach is motivated on the one hand by the known situation in the projective case, where the geometry of moduli spaces is deeply connected with the underlying variety, and on the other hand by the complexity of open problems in affine algebraic geometry. From this viewpoint, it is natural to choose trivial framings on the divisor to reduce its contribution to the moduli space.
In the first part, we check that trivial framings fulfill known criteria for the existence of moduli spaces, especially the absence of nontrivial automorphisms. By proving the invariance under blow-ups, we obtain due to the weak factorization theorem for birational maps indeed a direct relation of the moduli to the complement of the divisor. In the following, we investigate examples for surfaces (especially the complex projective plane) in the hope of possible applications to affine classification problems. However, it turns out that contrary to the the known cases of framings on rational curves the condition of trivial framings is rather restrictive. In higher dimension, a trivial framing on an ample divisor implies even the global triviality of the bundle. Hence, one cannot expect to obtain invariants for the open questions in the classification of affine varieties in higher dimension from this construction.
Therefore, we consider in the second part the question of how the rigid triviality condition might be weakened without losing the property of being an invariant of the complement. A crucial notion is that of trivial multiframings, i.e., trivial framings on the components of the divisor. Here, a slighly weaker invariance holds; but the existence of moduli spaces cannot be concluded from known general results. Consequently, we formulate the related moduli problem, and investigate the deformation theory of component-wise trivial bundles. In the case of a reduced divisor, we can describe the underlying topological structure of multiframings. Then, we find an obstruction to lift trivial multiframings to higher multiplicities and the formal completion of the divisor. In dimension \(\geq 3\), we conclude from general theory that the extension to the whole space is unique. However, it turns out that for concrete examples the objects and obstructions are difficult to calculate, especially in the exceptional cases relevant for the affine classification. Therefore, the question whether the developed tools are sufficiently powerful to solve open problems in affine geometry like the cancellation problem for exotic affine structures remains open. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Moduli spaces of trivially framed and multiframed sheaves on surfaces and higher-dimensional varieties with a view towards an application to affine classification | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be an almost simple Chevalley-Demazure group scheme of type \(A_ n\) (\(n\geq 2\)), \(D_ m\) (\(m\geq 4\)), or \(E_ 6\) and \(\Gamma\) the lattice of weights of the representation that defines \(G\). If \(G\) is universal or adjoint, denote that lattice by the respective symbols \(\Gamma_{sc}\) or \(\Gamma_{ad}\), so that \(\Gamma_{ad}\subseteq\Gamma\subseteq\Gamma_{sc}\). Let \(\sigma\) be the canonical involutive automorphism of \(\Phi\), and represent the induced involution of \(\Gamma_{sc}\) or \(\Gamma_{ad}\) by \(\sigma\) also, so that \(\Gamma\) is \(\sigma\)-stable, apart from the cases \(\Phi = D_ n\), \(n\) even, \(n\geq 4\) and \(\Gamma\) strictly between \(\Gamma_{ad}\) and \(\Gamma_{sc}\). (That strict inclusion is misprinted in the paper.) Let \(A\) be a commutative ring (with 1) with an involutive automorphism \(\sigma\) that stabilizes \(\Gamma\). Let \(G(\Phi,A) =\text{Hom}(\mathbb{Z}[G],A)\) be the group of \(A\)-valued points of \(G\), where \(\mathbb{Z}[G]\) is a Hopf algebra over \(\mathbb{Z}\). \(G(\Phi, A)\) is the Chevalley group of \(\Phi\) (or \(G\)) over \(A\). \textit{E. Abe} and the reviewer [Commun. Algebra 16, 57-74 (1988; Zbl 0647.20047)] determined the center of \(G(\Phi, A)\) for most cases. There is a corresponding automorphism, still denoted \(\sigma\), of \(\mathbb{Z}[G]\), from which an automorphism of \(G(\Phi, A)\) arises as follows. Let \(T(\Phi, A)\cong\text{Hom}(\Gamma, A^*)\) be the standard maximal torus of \(G(\Phi, A)\), where \(A^*\) is the group of invertible elements of \(A\). Let \(h(\chi)\) be the element of \(T(\Phi, A)\) that corresponds to \(\chi\in\text{Hom}(\Gamma, A^*)\), and \(\sigma(\chi) (\gamma) =\sigma(\chi (\sigma(\gamma)))\) for any \(\gamma\in\Gamma\). Then \(\sigma\) induces an automorphism of \(\text{Hom}(\mathbb{Z}[G], A) = G(\Phi, A)\) that satisfies the following. For a unipotent \(x_ \alpha (a)\) in \(G(\Phi,A)\), \(\sigma(x_ \alpha(a)) = x_{\sigma(\alpha)} (c_ \alpha\sigma(a))\), where \(c_ \alpha =\pm 1\). Also, for \(h(\chi)\in T(\Phi, A)\), \(\sigma(h(\chi)) = h(\sigma (\chi))\).
The twisted Chevalley group \(G_ \sigma (\Phi, A)\) is \(\{x\in G(\Phi,A)\mid\sigma x = x\}\). The author studies the center of this group, via the method introduced by Abe and the reviewer. His main result is as follows. Suppose the Jacobson radical \(R\) of \(A\) is trivial or that \(\Phi\) has rank \(\geq 2\). Let \(\{{\mathcal A}\}\) consist of all ordered pairs \((a,b)\) of elements of \(A\) such that \(a\overline{a} = b +\overline{b}\), where the bar indicates the image under \(\sigma\). If \(\Phi_ \sigma\) is of type \(^ 2A_{2n}\) and \(R\neq 0\), assume that there is an element \((a,b)\) in \(\{{\mathcal A}\}\) with \(a\in A^*\) and \(\{{\mathcal A}\}^*\neq\emptyset\). (Here \(*\) refers to inverse to a certain group operation defined on \(\{{\mathcal A}\}\).) If \(G\) is adjoint, then the center is trivial. If \(G\) is universal or adjoint, then the center is \(\text{Hom}(\Gamma/\Gamma_{ad}, A^*)\), and coincides with the center of the elementary subgroup of \(G(\Phi_ \sigma, A)\). The author also obtains some additional information about the center in more general cases. root systems; Weyl groups; almost simple Chevalley-Demazure group schemes; lattice of weights; involutive automorphisms; Hopf algebras; center; maximal torus; twisted Chevalley groups; Jacobson radical; elementary subgroups Linear algebraic groups over adèles and other rings and schemes, Simple groups: alternating groups and groups of Lie type, Simple, semisimple, reductive (super)algebras, Associated Lie structures for groups, Group schemes, Subgroup theorems; subgroup growth, Affine algebraic groups, hyperalgebra constructions Centers of twisted Chevalley groups over commutative rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review consists in the study of degenerations of odd theta characteristics on smooth curves to unstable curves. More precisely, let \(X\) be a curve admitting nodal, cuspidal and tacnodal singularities and \(f:\mathcal W\to B\), where \(\mathcal W\subset B\times \mathbb P^{g-1}\), a canonical one-parameter smoothing of \(X\) with mild singularities. The author describes how to compactify the moduli space of odd theta characteristics on the smooth fibers of \(f\) over \(B\) and how to endow it with a geometrically meaningful modular description. The main issue here is exactly the presence of more general singularities than just nodes. In fact, the moduli space of theta characteristics on smooth curves of genus \(g\) has been compactified over \(\overline{M_g}\) by \textit{M. Cornalba} [in: Proc. first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 560--589 (1989; Zbl 0800.14011)]. The boundary points consist on the so-called ``stable spin curves'' and an explicit combinatorial description of it can be given using certain line bundles on quasistable curves, as shown by \textit{L. Caporaso} and \textit{C. Casagrande} [Commun. Algebra 31, No. 8, 3653--3672 (2003; Zbl 1050.14019)]. Later, \textit{L. Caporaso} et al. [Trans. Am. Math. Soc. 359, No. 8, 3733--3768 (2007; Zbl 1140.14022)] generalized this construction by compactifying the space of pairs \((C,L)\), where \(C\) is a smooth curve of genus \(g\) and \(L\) is a \(r\)th root of a fixed line bundle in \(C\).
Let \(f\) be as above and assume it has general smooth fibers. Then, the space of odd theta characteristics on these fibers comes with a configuration of ``theta hyperplanes'' in the Hilbert scheme \(H:=\text{Hilb}_{N_g}(\mathbb P^{g-1})^\vee\), where \(N_g:=2^{g-1}(2^g-1)\) is the number of odd theta characteristics on a smooth curve of genus \(g\). The first part of the paper consists on the study of the limit of these hyperplanes in \(H\), which provides the desired compactification. The author shows that, if \(X\) is general and irreducible, these ``theta hyperplanes'' are finite and there is a natural configuration in \(H\) that is independent of the smoothing \(f\). Moreover, the author provides an enumerative description of these hyperplanes depending only on the number of nodes, cusps and tacnodes of \(X\).
Finally, if \(f\) is a general smoothing and \(X\) is general and irreducible, the author provides a geometrical description of these hyperplanes in terms of suitable twists on the dualizing sheaf of the stable reduction of \(f\), which allows a reduction to the case of Deligne-Mumford stable curves. This allows to give a geometric meaningful definition of spin curves over irreducible curves admitting nodes, cusps and tacnodes as singularities. singular curves; moduli; spin curves; odd theta characteristics; theta hyperplanes; twisted spin curves. M. Pacini, Spin curves over non stable curves, Communications in Algebra 36 (2008), 1365--1393. Families, moduli of curves (algebraic), Jacobians, Prym varieties, Vector bundles on curves and their moduli Spin curves over nonstable 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 This paper collects a series of lectures given by H. Hauser at the Clay Summer School on Resolution of Singularities in 2012. It is a very good introduction to the theory of blowups and the resolution of singularities in characteristic zero. It is also a good way to discover the bad phenomena of the positive characteristic. The point of view is to study the resolution of singularities in a global way finding some invariants of the singularities. The author, with the help of S. Perlega and V. Roitner gives lot of examples and exercises with some help to solutions in the last chapter.
This paper is composed of thirteen chapters. The first chapter is an introduction to the problem of resolution of singularities studying examples. The second is a quick introduction of schemes, algebraic varieties and completion. The third talk about singularities, normal crossing and different kind of singularities in all characteristic. The fourth, fifth and sixth chapters give all the equivalent definition of blowups, the functorial properties of blowups and the transformation of ideal and varieties under blowups. The seventh chapter give the different notions of resolution of singularities. The eighth talk about different kind of invariants of singularities used in the proofs of resolution theorems. The ninth chapter give the definition of maximal contact and talk of is existence in characteristic zero. The tenth chapter introduce the coefficient ideals. The eleventh chapter only talk about the existence of resolution of singularities in characteristic zero and compute some examples. The twelfth chapter present some problems and phenomena which appear in positive characteristic, specially in the purely inseparable case. The last chapter is composed of the solutions of the examples and exercises given in all the others sections. resolution; singularities; blowups Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Multiplicity theory and related topics, Rational and birational maps Blowups and 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 Ever since \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803)] used them to prove the irreducibility of the moduli space \(M_g\) of curves of genus \(g\), algebraic stacks have played a central role in the theory of moduli spaces of various algebro-geometric objects (varieties, vector bundles, etc.). Accordingly, there have been a number of publications that seek to explain stacks to beginners. The article under review is the latest such work to come out; and we believe that its focus on concrete examples (moduli spaces of curves and covers of curves) will make it a useful reference to any algebraic geometer who works with stacks.
Since algebraic stacks are categories, and they form a 2-category, the article starts with a review of categorical concepts in Section 1.2. Descent is reviewed in this section as well, with examples such as descent of schemes and Galois descent. In Section 1.3, the author defines a stack in categorical terms. In brief terms, they are categories fibered in groupoids where descent works. (See Definition 1.62.) Chapter 2 is devoted to a discussion of schemes in groupoids; which are basically a generalization of the idea of a groupoid, realized using schemes. Schemes in groupoids are used to introduce equivalence relations on a scheme. The group action on a scheme is a typical example of a scheme in groupoids, and this example features prominently in the article. Chapter 3 discusses algebraic stacks, which are stacks with (smooth or étale) presentations by a scheme. Morphisms between stacks, and sheaves on a stack are also examined in detail in this chapter.
The author introduces stacks of curves and covers in Chapter 4; and these constitute the focal point of the article. The stack of elliptic curves, and the compactification of the stack \(\mathcal{M}_{g,n}\) is treated in detail. In Section 4.2, Hurwitz stacks are discussed; these are stacks that parametrize covers between curves with fixed genera and ramification data. It is proved (Proposition 4.5.3) that these are Deligne-Mumford stacks; and examples involving the projective line and elliptic curves are given. algebraic stack; category; covering; cover; curve; elliptic curve; groupoid; Hurwitz; node; stack; moduli space; stack Generalizations (algebraic spaces, stacks), Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Automorphisms of curves Algebraic stacks with a view toward moduli stacks of 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 \(V\) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function \(f:(\mathbb{C}^n,0)\to (\mathbb{C},0)\). Let \(L(V)\) be the Lie algebra of derivations of the moduli algebra \(A(V):=\mathcal{O}_n/(f,\partial f/\partial x_1,\dots,\partial f/\partial x_n)\), i.e., \(L(V)=\operatorname{Der}(A(V),A(V))\). The Lie algebra \(L(V)\) is finite dimensional solvable algebra and plays an important role in singularity theory. According to \textit{A. Elashvili} and \textit{G. Khimshiashvili} [J. Lie Theory 16, No. 4, 621--649 (2006; Zbl 1120.32016)]) and \textit{G. Khimshiashvili}[``Yau algebras of fewnomial singularities'', Preprint, \url{https://www.math.uu.nl/publications/preprints/1352.pdf}], \(L(V)\) is called Yau algebra and the dimension of \(L(V)\) is called Yau number. The studies of finite dimensional Lie algebras \(L(V)\) that arising from isolated singularities was started by \textit{S. S. T. Yau} [Proc. Natl. Acad. Sci. USA 80, 7694--7696 (1983; Zbl 0563.17010)] and has been systematically studied by Yau, Zuo and their coauthors. Most studies of Lie algebras \(L(V)\) were oriented to classify the isolated singularities. This work surveys the researches on Yau algebras \(L(V)\) of isolated singularities. fewnomial; Lie algebra; isolated singularity Singularities in algebraic geometry, Local complex singularities, Derivations and commutative rings Survey on derivation Lie algebras of isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The well-known Krichever corresdondence, linking algebro-geometric data with the theory of infinite-dimensional Grassmannians and algebras of differential operators, has been a subject of vast research during the past 25 years. This construction has been successfully used in the theory of integrable Hamiltonian systems, particularly in the theory of the KP and KdV equations, and it has been an essential ingredient for the final proofs of some longstanding conjectures such as the Schottky problem and the Novikov conjecture. Also, various attempts have been made, over the past two decades, to generalize Krichever's construction in different directions.
The paper under review is devoted to a purely algebraic approach to the so-called Krichever map. Just assuming an arbitrary groundfield \(k\) of characteristic zero, the author describes a general connection between the KP hierarchy (in Lax form) and the vector fields on infinite Grassmannians. This is not only more general, but also much simpler and much more systematic and transparent than the previously known algebraic-analytic constructions. Then, using his theory of adelic complexes [cf. \textit{T. Fimmel} and \textit{A. N. Parshin}, ``Introduction to higher adelic theory'' (book to appear)], the author establishes the general Krichever correspondence in dimension one, before extending it to algebraic surfaces in the following sections of the paper.
The Krichever correspondence appears here as a quasi-isomorphism between certain adelic complexes, and it is precisely this form that suggests the possibility of further generalizations to higher-dimensional varieties. An approach to such a further generalization has been proposed, in the meantime, by \textit{D. V. Osipov} in his recent paper ``Krichever correspondence for algebraic varieties'' [Izv. Math. 65, 941-975 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, 91-128 (2001; Zbl 1068.14053)].
Altogether, this paper reveals the apparently very general character of the Krichever correspondence as well as its crucial significance and ubiquity. Apart from establishing the Krichever correspondence in dimension two, the purely algebraic and far-reaching methodical approach represents the pioneering character of this work. Krichever corresdondence; KP hierarchy; infinite Grassmannians A. N. Parshin, ''Integrable systems and local fields,'' Comm. Algebra 29(9), 4157--4181 (2001). Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Commutative rings of differential operators and their modules, Local ground fields in algebraic geometry Integrable systems and local fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non-Archimedean field \(F\) of characteristic \(p\neq 2\) in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of \((\mathbf{GL}_{1},\mathbf{GL}_{m})\) we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Given two reductive connected groups \(G\) and \(H\) over \(F\), and a morphism \({\check{G}}\times \mathrm{SL}_{2}\to \check{H}\) of Langlands dual groups, we construct a bimodule over the affine extended Hecke algebras of \(H\) and \(G\) that should realize the geometric local Arthur-Langlands functoriality at the Iwahori level. Then, we propose a conjecture describing the geometric local theta correspondence at the Iwahori level constructed in the first part in terms of this bimodule, and we prove our conjecture for pairs \((\mathbf{GL}_{1},\mathbf{GL}_{m})\). local theta correspondence; geometric Langlands program; Langlands functoriality; Hecke algebras; perverse sheaves; K-theory Geometric Langlands program: representation-theoretic aspects, Geometric Langlands program (algebro-geometric aspects), Equivariant \(K\)-theory, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Group actions on varieties or schemes (quotients), Hecke algebras and their representations, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Geometric tamely ramified local theta correspondence in the framework of the geometric Langlands program | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 article is an existence theorem on geometric quotients for proper actions of algebraic group schemes on algebraic spaces. Let \(G\) be an affine algebraic group scheme of finite type over some excellent base scheme \(S\). Suppose that \(G\) acts morphically on the algebraic space \(X\) of finite type over \(S\). The notion of a geometric quotient for the action of \(G\) on \(X\) is defined by the author basically in analogy to D. Mumford's concept for the case of schemes. Whereas in the category of schemes a geometric quotient is necessarily categorical and hence unique, these two properties are no longer satisfied in the category of algebraic spaces, as the author shows in an explicit example. Now assume that the action of \(G\) on \(X\) is proper.
The main result of the article ensures that there exists a geometric quotient \(q : X \to Y\) for the action of \(G\) on \(X\) if one of the following conditions is valid:
(a) \(G\) is reductive over \(S\),
(b) \(S\) is the spectrum of a field of positive characteristic.
Moreover, then the algebraic space \(Y\) over \(S\) is separated and \(q\) is in fact categorical.
The author obtains also important properties of quotient morphisms. In fact, he works under slightly weaker assumptions and considers the notion of an approximate quotient for the action of \(G\) on \(X\). For \(G\) universally open over \(S\) it is shown that such a quotient \(p: X \to Y\) is an affine morphism. If furthermore \(G\) is flat over \(S\), then for every coherent \(G\)-sheaf \(F\) on \(X\) the sheaf \((p_*F)^G\) of invariants on \(Y\) is coherent. The main result yields the existence of certain moduli spaces that was known before only in characteristic zero. In the meantime \textit{S. Keel} and \textit{S. Mori} [Ann. Math., II. Ser. 145, No. 1, 193-213 (1997; see the following review)] presented a proof for the existence of geometric quotients by proper actions of flat algebraic group schemes with finite stabilizer on algebraic spaces in a more general framework. algebraic group actions; geometric quotients; algebraic spaces \textsc{J. Kollár}, \textit{Quotient spaces modulo algebraic groups}, Ann. of Math. (2) 145 (1997), no. 1, 33--79. DOI 10.2307/2951823; zbl 0881.14017; MR1432036; arxiv alg-geom/9503007 Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Geometric invariant theory Quotient spaces modulo 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 This paper ist devoted to the classification of simple surface singularities, i.e., singularities of germs of complex surfaces such that there are only finitely many isomorphism classes of singularities in their versal deformation.
Starting with rational double points and rational triple points which are known to be simple (the classical ADE list due to Arnold in the first case and Frühbis-Krüger, Neumer in the second), the author conjectures the following:
``Simple normal surface singularities are exactly those rational singularities whose resolution graphs can be obtained from the graphs of rational double points and rational triple points by making any number of vertex weights more negative.''
Singularities of such graphs are taut in the sense of Laufer, i.e., their analytic type is determined by the graph. The conjecture would imply that for normal surfaces there do not exist rigid singularities (an open question so far).
In the article, the following is shown:
(a) If a rational singularity is simple, it satisfies the condition of the conjecture.
(b) Conversely, the conjecture on simpleness is satisfied for the classes of quotient singularities, rational quadruple points and sandwiched singularities. simple singularities; rational singularities; tautness; normal surface singularities; rational double point; rational quadruple point; sandwiched singularities Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Deformations of singularities, Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry Simple surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(X\) is a cubic \(4\)-fold containing a plane \(P\), the projection from \(P\subset {\mathbb P}^5\) to another plane \(\Pi\) disjoint from \(P\) gives a quadric bundle \(\pi:\widetilde X \to \Pi\) from the blow-up \(\widetilde X\) of \(X\) along \(P\). The discriminant locus \(C\subset \Pi\) is a plane sextic. The case where \(X\) and \(C\) are smooth has been extensively studied. Here attention is given to the case where \(C\) is a nodal reduced plane sextic, possibly reducible, and \(X\) is allowed to be singular. Note that \(C\) is naturally equipped with a theta characteristic \(\theta\), and the isomorphism class of \((C,\theta)\) does not depend on the choice of \(\Pi\). This situation is described in terms of equations in Section~2 of the paper.
Section~3 is devoted to giving a bound on the number of nodes of \(X\) on the assumption that \(C\) is a nodal reduced plane sextic and \(h^0(\theta)=1\), in terms of the singularities of the pair \((C,\theta)\). This also yields an estimate of the rank of the free abelian group generated by classes of algebraic codimension~\(2\) cycles in \(X\). The proofs use the equations to study both local and global properties of \((C,\theta)\).
In the last section we see sufficient conditions on \(C\) for all the associated cubic \(4\)-folds to be singular, and some examples that show that the bounds give earlier are best possible. The paper concludes by exhibiting a family of smooth rational cubic \(4\)-folds containing a plane such that the discriminant locus is reduced but reducible. \(4\)-folds, Plane and space curves Singular cubic fourfolds containing a plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Omega\) be a given field. A number which belongs to \(\Omega\) is said to be rational. A point in the plane with rational coordinates is said to be a rational point. The coordinates of the curve
\[
y^2 = x^3 - Ax - B, \tag{1}
\]
where \(A\) and \(B\) are rational and \(4A^3 - 27B^2\neq 0\), can be represented by Weierstraß's \(\wp\)-function with the invariants \(4A\) and \(4B\): \(x = \wp(u)\), \(y = \tfrac12\wp'(u)\). A point \((x, y)\) on (1) is exceptional if the corresponding argument \(u\) is commensurable with a period of the function, \(\wp(u)\). The arguments of the rational points in (1) constitute an (infinite or finite) abelian group \(G\) when the composition is addition. If \(G_1\) is a finite subgroup of \(G\), the elements of \(G_1\) are exceptional points.
In a first part of this paper the author tries to determine all the really existing finite subgroups \(G_1\) in \(\Omega\). He gives the necessary and sufficient conditions for the existence of subgroups of order \(m = 3, 4, 5, 6, 7, 9\) and \(15\), expressing \(A\) and \(B\) as binary forms with integer coefficients if \(m\neq 15\). If \(\Omega = K(1)\), the case \(m = 15\) is impossible.
A second part of the paper begins with a study of the resolvent fields and the rational triplets of a cubic curve of genus one. For definitions and some of the results consult an earlier paper of the author [Ark. Mat. 2, 247--250 (1952; Zbl 0048.02702)]. The paper concludes with theorems concerning birational and linear equivalence by means of Aronhold-invariants. We mention that the problem of equivalence is completely solved for cubics of the form \(x^3 + y^3 + \gamma z^3 - 3\alpha xyz= 0\). At last the author completes the demonstration of an earlier theorem, now giving all details [Nova Acta Soc. Sci. Upsal., IV. Ser. 12, No. 8, 1--34 (1941; Zbl 0026.29402)]. cubic curve of genus one Nagell, T. Recherches sur l'arithmétique des cubiques planes du premier genre dans un domaine de rationalité quelconque, Nova Acta Reg. Soc. Sci. Ups., Ser. IV, vol. 15, n:o 4, Uppsala 1952. Rational points Recherches sur l'arithmétique des cubiques planes du premier genre dans un domaine de rationalité quelconque | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\omega\) be a rational 1-form on the complex projective line \(P\) such that the residue at any simple pole of \(\omega\) is not an integer. Then, there exists a pairing \(\langle\cdot, \cdot\rangle^\omega\), called the intersection pairing, between the rational twisted cohomology groups \(H^1 (\Omega^\bullet(*D), \nabla_\omega)\) and \(H^1(\Omega^\bullet (*D),\nabla_{-\omega})\), where \(\nabla_{\pm\omega} =d\pm\omega\wedge\). The twisted homology groups \(H_1(X,L^0_\omega)\) and \(H_1(X,L^0_{-\omega})\) with respect to \(\omega\) and \(-\omega\), respectively, are defined. The pairing \(\langle\cdot, \cdot \rangle_\omega\) between them is determined by \(\langle\cdot, \cdot \rangle^\omega\) and the pairing between \(H^1(\Omega^\bullet (*D), \nabla_\omega) \) and \(H_1(X,L^0_\omega)\). It is proven that the pairings \(\langle \cdot,\cdot \rangle^\omega\) and \(\langle\cdot, \cdot\rangle_\omega\) are perfect for general \(\omega\), not only for the case when \(\omega\) admits simple poles. The determinants of two matrices \(I_{ch}\) and \(I_h\) (which play an important role in studies of functional equations and monodromy groups) are computed. Combinatorial formulas of \(k\)-minors of matrices \(I_{ch}\) and \(I_h\) are obtained for \(\omega\) admitting only simple poles.
These formulas yield explicit forms of the inverse of the matrices \(I_{ch}\), \(I_h\) and of the intersection numbers associated with hypergeometric functions of type \((k+1,n+2)\). intersection pairing; twisted homology groups; intersection numbers; hypergeometric functions N. Kachi , K. Matsumoro , M. Mihara , The perfectness of the intersection pairings for twisted cohomology and homology groups with respect to rational 1-forms . Kyushu J. Math. 53 ( 1999 ), 163 - 188 . MR 1678026 | Zbl 0933.14009 de Rham cohomology and algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The perfectness of the intersection pairings for twisted cohomology and homology groups with respect to rational 1-forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal{A}\) be an isotrivial abelian variety over the function field \(\mathbb{C}(x,y)\), \(\pi : \mathcal{A}\rightarrow\mathbb{P}^2\) be a non-singular projective model of \(\mathcal{A}\), and \(A\) be its generic fiber. Let \(\Delta\subset\mathbb{P}^2\) be the discriminant of \(\pi\). In the present paper the author shows that the Mordell-Weil rank of \(\mathcal{A}\) depends, besides the type of the generic fiber, the degree and the local type of singularities of \(\Delta\), on the dimensions of certain linear systems of curves determined by the local type of singularities of the latter. One of the main results is formulated as follows. Let \(G\subset \mathrm{Aut} A\) be the holonomy group of \(\mathcal{A}\). Assume that:
(a) \(G\) is a cyclic group of order \(d\) acting on \(A\) without fixed subvarieties of a positive dimension,
(b) the singularities of \(\Delta\) have CM type and \(\Delta\) is irreducible,
(c) \(A\) is a simple abelian variety with complex multiplication by \(\mathbb{Q}(\zeta_d)\).
Then one has \(\mathrm{rk}\mathcal{A}(\mathbb{C}(x,y))\leq s\cdot\phi(d)\), where \(s\) is the multiplicity of the cyclotomic polynomial of degree \(d\) in the Alexander polynomial of \(\pi_1(\mathcal{P}^2-\Delta)\).
As a concrete example, he shows that for the Jacobian of the curve \(\mathcal{C}\) over \(\mathbb{C}(x,y)\) given in \((u, v)\)-plane by the equation \(u^p=v^2+(x^p+y^p)^2+(y^2+1)^p\), one has \(\mathrm{rk} J(\mathcal{C})(\mathbb{C}(x,y))=p-1\). abelian varieties; Mordell-Weil group; Alexander polynomials Dimca, A.: Differential forms and hypersurface singularities. In: Singularity theory and its applications, Part I (Coventry, 1988/1989), vol. 1462 of Lecture Notes in Math., pp. 122-153. Springer, Berlin (1991) Complex multiplication and abelian varieties, Coverings of curves, fundamental group, Abelian varieties of dimension \(> 1\) On Mordell-Weil groups of isotrivial abelian varieties over function fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an irreducible smooth projective curve of genus \(g=2\) over \(\mathbb{C}\). Let \(G\) be a connected reductive affine algebraic group over \(\mathbb{C}\). The topological types of holomorphic principal \(G\)-bundles on \(X\) are parametrized by \(\pi_1(G)\). Let \(M^\delta_{G,\text{Higgs}}\) be the moduli space of semistable principal \(G\)-\(\text{Higgs}\) bundles on \(X\) of topological type \(\delta\in\pi_1(G)\). The space \(M^\delta_{G,\text{Higgs}}\) is nonempty and connected, for all \(\delta\in\pi_1(G)\). The aim of this paper is to compute the fundamental group and Picard group of \(M^\delta_{G,\text{Higgs}}\). (let's remember that for any \(\mathbb{C}\)-scheme \(Z\), the notation \(\text{Pic}(Z)\) (resp., \(\pi_1(Z)\)) refers to the Picard group (resp., fundamental group) of \(Z\)). It is assumed that either \(g\geq3\), or there is no homomorphism of \(G\) onto \(\mathrm{PSL}_2(\mathbb{C})\) whenever \(g=2\). Then the authors show that there are isomorphisms
\[
\text{Pic}(M^\delta_{G,\text{Higgs}})\cong\text{Pic}(M_G^\delta),\qquad \pi_1(M^\delta_{G,\text{Higgs}})\cong\pi_1(M_G^\delta).
\]
Therefore, it follows from [\textit{I. Biswas} et al., ``Fundamental group of moduli of principal bundles on curves'', Preprint, \url{arXiv:1609.06436}] that \(\pi_1(M^\delta_{G,\text{Higgs}})\cong\mathbb{Z}^{2gd}\), where \(d=\dim Z(G)\) is the dimension of the center of \(G\). In particular, \(M^\delta_{G,\text{Higgs}}\) is simply connected, whenever \(G\) is connected and semisimple. Therefore, together with the results of \textit{A. Beauville} et al. [Compos. Math. 112, No. 2, 183--216 (1998; Zbl 0976.14024)], \textit{S. Kumar} and \textit{M. S. Narasimhan} [Math. Ann. 308, No. 1, 155--173 (1997; Zbl 0884.14004)], the above isomorphisms determine Picard group of \(M^\delta_{G,\text{Higgs}}\) for essentially all classical semisimple complex affine algebraic groups. moduli space; principal Higgs bundle; fundamental group and Picard group Stacks and moduli problems, Algebraic moduli problems, moduli of vector bundles, Coverings of curves, fundamental group, Picard groups Picard group and fundamental group of the moduli of Higgs bundles on 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 an algebraically closed field of characteristic \(p>0\). Let \(\mathcal G\) be a finite algebraic group, and let \(k\mathcal G\) be its algebra of measures. Then \(k\mathcal G\) is a finite dimensional cocommutative \(k\)-Hopf algebra, and there is a unique block -- the ``principal block'' -- denoted \(\mathcal B_0(\mathcal G)\) of \(k\mathcal G\) such that \(\varepsilon(\mathcal B_0(\mathcal G))\neq 0\), where \(\varepsilon\colon k\mathcal G\to k\) is the counit. For \(\Lambda\) an associative \(k\)-algebra, the group \(\text{GL}_d(k)\) acts on the variety of \(d\)-dimensional \(\Lambda\)-modules, and we let \(C_d\) be a closed subset of minimal dimension such that \(\text{GL}_d(k)\cdot C_d\) contains the set of indecomposable \(\Lambda\)-modules of dimension \(d\). If \(C_d=0\) for all \(d\) then \(\Lambda\) is said to be representation finite; if this is not the case and \(C_d\) has dimension at most \(1\) for all \(d\) then \(\Lambda\) is said to be tame; otherwise, \(\Lambda\) is said to be wild. Information about the principal block \(\mathcal B_0(\mathcal G)\) can give results about the representation type of \(\mathcal G\): for example, \(\mathcal B_0(\mathcal G)\) is simple if and only if \(k\mathcal G\) is semisimple, in which case \(\mathcal G\) is representation-finite.
As the classification of the indecomposable modules in the wild case is (from the paper) ``deemed hopeless'', in this survey article (based on a series of lectures at the Advances in Group Theory and its Applications conference in 2011), the author investigates conditions for when \(\mathcal B_0(\mathcal G)\) is representation-finite or tame. The objective of this work is to give an overview of these conditions, consequently most of the results are provided without proof (although a citation is provided for each unproven result). representation types; Dynkin diagrams; quivers; representations of finite group schemes; cohomological support varieties; indecomposable modules; cocommutative Hopf algebras; module categories; finite representation type Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Group schemes, Hopf algebras and their applications, Modular Lie (super)algebras Dynkin diagrams, support spaces and representation type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M_{2}(0,n)\) denote the coarse moduli scheme of stable rank-2 vector bundles on the projective space \(\mathbb{P}_{3}\) where the Chern classes satisfy \(c_{1}(E) = 0\) and \(c_{2}(E) = n\). Suppose that \(E\) comes from \(M_{2}(0,n)\) and the maps \(f: X\to \mathbb{P}_{3}\), \(g: X\to \mathbb{P}_{2}\) arise from blowing up \(P_{3}\) at some point \(x\in \mathbb{P}_{3}\). If \(\sigma = g^{*}\mathcal{O}(1)\) and \(\tau = f^{*}\mathcal{O}(1)\), then \(\theta = R^{1}g_{*}f^{*}E(-\sigma,-\tau)\) is a sheaf on \(\mathbb{P}_{2}\) with support a curve \(\mathcal{C}\) called a spectral curve of \(E\). Furthermore, \(\theta\) is called a theta characteristic on \(\mathcal{C}\). This means that \(\theta^{\otimes^{2}}= \omega_{\mathcal{C}}\). For such a situation, the main result of the paper, which follows, is used.
Theorem. Let \(\mathcal{C}\) be any non-singular plane curve of degree \(n\) and \(\theta\) a theta characteristic on it. A pencil in the projective space \(\mathbb{P} H^{0}\theta (2)\) defines an exact sequence
\[
0\to F\to \mathcal{O}^{2}\to\theta (2)\to 0
\]
with \(F\) a rank \(2\) vector bundle and where \(F_{\mathcal{C}}^{{\scriptscriptstyle \vee}}= \theta (1)\oplus\theta (2)\) on the curve \(\mathcal{C}\).
As a consequence, the author is able to obtain a special family of rank-2 mathematical instanton bundles on \(\mathbb{P}_{3}\). theta characteristic; pencil; spectral curve; stable rank-2 vector bundle; blowing up; mathematical instanton bundle Theta functions and abelian varieties, Fine and coarse moduli spaces, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Families, moduli of curves (algebraic) A family of rank-2 mathematical instanton bundles on \(\mathbb{P}_3\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the authors give a survey on Poisson traces (or zeroth Poisson homology) developed in a series of their recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, the conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical \(D\)-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the \(D\)-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the \(D\)-module is the pushforward of the canonical \(D\)-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. One explains many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra.
The authors compute the \(D\)-module in the case of surfaces with isolated singularities and show that it is not always semisimple. They also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein-Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix one gives a brief recollection of the theory of \(D\)-modules on singular varieties that one requires. Hamiltonian flow; complete intersections; Milnor number; \(D\)-modules; Poisson homology; Poisson varieties; Poisson homology; Poisson traces; Milnor; fibration; Calabi-Yau varieties; deformation quantization; Kostka polynomials; symplectic resolutions; twistor deformations Deformation quantization, star products, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, , Global theory of complex singularities; cohomological properties Poisson traces, D-modules, and symplectic resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a noetherian integrally closed domain with quotient field \(K\). In the early 1970's, in her work on \(B(R)\), the Brauer group of \(R\), \textit{B. Auslander} derived two exact sequences:
\[
0 \to {\mathcal M} (R)/{\mathcal P} (R) \to B (R) \to B (K) \quad \text{ and }
\]
\[
0 \to {\mathcal M}_ 0R/{\mathcal P} (R) \to B(R) \to \bigoplus_{P \in \text{Spec} (R)}B(R_ P).
\]
Here \({\mathcal M}(R)\) is the set of isomorphism classes of reflexive \(R\)-modules \(M\) with \(\text{End}_ R(M)\) {\(R\)-projective,} with a monoid structure induced by the operation \(M \cdot N=(M \otimes N)^{**}\). \({\mathcal M}_ 0(R)\) and \({\mathcal P} (R)\) are submonoids of \({\mathcal M} (R)\) and the quotient \({\mathcal M} (R)/{\mathcal P} (R)\) is a group with subgroup \({\mathcal M}_ 0 (R)/{\mathcal P} (R)\).
In this paper, the authors use the theory of toric varieties to prove that for \(G\) any finite abelian group, there is a three-dimensional noetherian, integrally closed domain \(R\) with \(G \cong {\mathcal M} (R)/{\mathcal P} (R) \cong {\mathcal M}_ 0(R)/{\mathcal P} (R)\), Explicit construction of the reflexive modules representing classes in these groups may be found in ``The Brauer group of toric varieties'' by \textit{F. DeMeyer}, \textit{T. Ford} and \textit{R. Miranda} [see J. Algebr. Geom. 2, No. 1, 137-145 (1993)]. integrally closed domain; Brauer group; toric varieties; reflexive modules Demeyer, F.; Regnier, K.: Reflexive modules. Methods in module theory, 67-74 (1993) Brauer groups of schemes, Class groups, Toric varieties, Newton polyhedra, Okounkov bodies Examples of reflexive 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 Let \(X\) be a subvariety, and \(\Gamma\) a finitely generated subgroup, in \({\mathbb G}_m^N(\overline{\mathbb Q})\cong(\overline{\mathbb Q}^*)^N\). One considers the division group \(\overline\Gamma\) (given by \(x\in\overline\Gamma\) if \(x^m\in\Gamma\) for some \(m>0\)) and its associated sets \(X\cap\overline\Gamma_\varepsilon\) and \(X\cap C(\overline\Gamma,\varepsilon)\) in \({\mathbb G}_m^N(\overline{\mathbb Q})\). These are defined in terms of the Weil height \(h\) on \({\mathbb G}_m^N(\overline{\mathbb Q})\) by \(\overline\Gamma_\varepsilon=\{yz\mid y\in\overline\Gamma,\;h(z)<\varepsilon\}\) and \(C(\overline\Gamma,\varepsilon)=\{yz\mid y\in\overline\Gamma,\;h(z)<\varepsilon(1+h(y))\}\). The aim is to understand in detail the intersections of these sets with~\(X\), for certain special~\(X\).
By work of Poonen and others it is known that there exists an \(\varepsilon\) such that \(X\cap \overline\Gamma_\varepsilon\) is contained in a finite union \(x_1H_1\cup\cdots\cup x_tH_t\) of cosets of algebraic subgroups of \({\mathbb G}_m^N(\overline{\mathbb Q})\), with each \(x_iH_i\) contained in \(X\). This \(\varepsilon\) depends only on \(N\) and the degree of \(X\). In a similar spirit, it is known that \(X^0\cap C(\overline\Gamma,\varepsilon)\) is finite for some \(\epsilon>0\), where \(x\in X^0\) if \(x\in X\) but \(xH\not\subset X\) for any positive-dimensional subgroup \(H\) of \({\mathbb G}_m^N(\overline{\mathbb Q})\).
There are diophantine approximation results explicitly bounding heights of points in \(X\cap\overline\Gamma\) (etc.) in certain cases -- for instance, for most curves \(X\) in the case \(N=2\) -- coming from Baker-type logarithmic forms estimates. Using these results and some estimates of the number of points of small heights, the authors make the descriptions above effective, for certain classes of \(X\). For \(X\cap \overline\Gamma_\varepsilon\), they give an explicit \(\varepsilon\) and effectively computable bounds on the heights and degrees of the points~\(x_i\). For \(X\cap C(\overline\Gamma,\varepsilon)\) they again give an explicit value for \(\varepsilon\) and also bounds on the heights and degrees of the elements of \(X\cap C(\overline\Gamma,\varepsilon)\).
Although the class of varieties \(X\) considered is rather restricted (either \(N=2\) and \(X\) a curve, or \(X\) cut out by bi- and trinomials), the method is more generally applicable at least in principle. torus; heights; diophantine approximation; effective bounds 7.A. B e rczes, J.-H. Evertse, K. Györy, C. Pontreau, Effective results for points on certain subvarieties of tori. Math. Proc. Camb. Philos. Soc. 147, 69-94 (2009) Heights, Rational points, Algebraic groups Effective results for points on certain subvarieties 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 On Calabi-Yau threefolds, there are DT invariants that are defined by virtual count of moduli of ideal sheaves of curves, as well as PT invariants by virtual count of moduli of stable pairs. On the other hand, on the resolved conifold, \textit{B. Szendrői} [Geom. Topol. 12, No. 2, 1171--1202 (2008; Zbl 1143.14034)] introduced invariants by virtual count of moduli of representations of algebras, which can be seen as virtual count of perverse ideal sheaves.
In this paper, the authors introduce another type of invariants using moduli of stable perverse coherent systems, which are pairs \(s : \mathcal{O}_Y \to F\) where \(Y\) is the crepant resolution of the conifold, and \(F\) is a 1-dimensional perverse coherent sheaf. The wall and chamber structure is determined, and the invariants computed in any chamber (using wall-crossing formulas that are examples of \textit{D. Joyce} [Adv. Math. 217, No. 1, 125--204 (2008; Zbl 1134.14008)], as well as of \textit{M. Kontsevich} and \textit{Y. Soibelman} [``Stability structures, motivic Donaldson-Thomas invariants and cluster transformations'', \url{arXiv:0811.2435}]). Furthermore, the authors explain how their invariants specialise to DT, PT and Szendrői's invariants in particular chambers.
More specifically, Section 2 contains the basic definitions of perverse coherent sheaves/systems, stability for perverse coherent systems, and the existence of their moduli spaces. Section 3 contains technical results characterising stable perverse coherent systems, including Proposition 3.11, which explains how stable pairs in the sense of Pandharipande-Thomas occur as stable coherent systems.
In Section 4, the authors first write down the generating functions for the counting invariants of moduli of representations of a certain quiver and study their wall-crossing formulas. Then, via an equivalence between the category of perverse coherent systems and the category of quiver representations, together with the technical results from Section 3, they obtain various relations among DT, PT and Szendrői's invariants in Theorem 4.15.
In Section 5, the authors prove a conjecture by [\textit{W.-y. Chuang} and \textit{D. L. Jafferis}, Commun. Math. Phys. 292, No. 1, 285--301 (2009; Zbl 1179.81079)], which says that the moduli spaces of stable perverse coherent systems in this article are isomorphic to the moduli of representations of a different quiver obtained by mutations. counting invariant; perverse sheaves; wall-crossing; moduli; quivers NN K.~Nagao and H.~Nakajima. \newblock Counting invariant of perverse coherent sheaves and its wall-crossing. \newblock \em Int.~Math.~Res.~Not., pp. 3855--3938, 2011. Families, moduli, classification: algebraic theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Stacks and moduli problems, \(3\)-folds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Counting invariant of perverse coherent sheaves and its wall-crossing | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma\) be a finitely generated group and k an algebraically closed field of characteristic zero. The authors study the finite dimensional theory of \(\Gamma\) over k.
In section 1 they define the basic objects of the paper. They introduce the functor \({\mathfrak R}_ n(\Gamma)\) from commutative k-algebras to sets defined by \({\mathfrak R}_ n(\Gamma)(A)=Hom(\Gamma,GL_ n(A))\), and if \(f: A\to B\) is a k-algebra homomorphism, \(f_*: {\mathfrak R}_ n(\Gamma)(A)\to {\mathfrak R}_ n(\Gamma)(B)\) denotes the function sending \(\rho: \Gamma \to GL_ n(A)\) into the composite \(\Gamma \to GL_ n(A)\to GL_ n(B)\). Then they show that this functor is representable by an affine algebra, and so is an affine scheme. This scheme has an algebraic action of \(GL_ n\) and a universal categorical quotient \(\delta \delta_ n(\Gamma)\). \({\mathfrak R}_ n(\Gamma)\) contains an open subscheme \({\mathfrak R}^ s_ n(\Gamma)\) consisting of the simple representations and its image \(\delta_ n(\Gamma)\) in the categorical quotient \(\delta \delta_ n(\Gamma)\) is a geometric quotient \({\mathfrak R}^ s_ n(\Gamma)\) by \(GL_ n\). The k-points of the schemes above yield (possibly reducible) k-varieties \(R_ n(\Gamma)\) (parametrizing degree n representations over k), \(R^ s_ n(\Gamma)\) (parametrizing simple representations), \(S_ n(\Gamma)\) (parametrizing isomorphism classes of semisimple representations). In section 1 the authors study the orbits of representations from \(R_ n(\Gamma)\), too. In particular, they show that such orbit 0(\(\rho)\) is closed iff \(\rho\) is a semisimple representation.
In section 2 the authors study the tangent spaces of the representation varieties \(R_ n(\Gamma)\). This section contains the following Weil results about these tangent spaces. The tangent space to \(R_ n(\Gamma)\) at \(\rho\) can be identified with a subspace of the space \(Z^ 1(\Gamma,Ad\circ \rho)\) of one-cocycles of \(\Gamma\) with coefficients in the representation Ad\(\circ \rho\) (where \(Ad: GL_ n(k)\to Aut(M_ n(k))\) is the adjoint representation). The tangent space of the scheme \({\mathfrak R}_ n(\Gamma)\) at \(\rho\) is actually equal to \(Z^ 1(\Gamma,Ad\circ \rho)\). The tangent space to the orbit 0(\(\rho)\) at \(\rho\) is equal to the space \(B^ 1(\Gamma,Ad\circ \rho)\) of one- coboundaries. In particular, it follows from the above facts that if \(H^ 1(\Gamma,Ad\circ \rho)=0\), then the orbit 0(\(\rho)\) is open in \(R_ n(\Gamma)\) and \(\rho\) is non-singular on \(R_ n(\Gamma)\). If \(H^ 1(\Gamma,Ad\circ \rho)=0\) for all \(\rho \in R_ n(\Gamma)\), then \(R_ n(\Gamma)\) consists of finitely many orbits and \(SS_ n(\Gamma)\) is finite. In this case any representation in \(R_ n(\Gamma)\) is semisimple. As a consequence of the above results the authors obtain the following statement. If every representation of \(\Gamma\) of degree 2n is semisimple, then up to isomorphism there are finitely many representations of \(\Gamma\) of degree n.
In section 3 the authors construct the embedding of \(R_ n(\Gamma)\) into an affine space. The fact that this embedding displays \(R_ n(\Gamma)\) as the fibre of a morphism implies some limits to its dimension. Then the authors define def(\(\Gamma)\), the deficiency of \(\Gamma\), and prove the following important proposition: If def\((\Gamma)=rk(\Gamma^{ab})\), where \(\Gamma^{ab}=\Gamma /(\Gamma,\Gamma)\), then the trivial representation \(\rho_ 0\) of \(\Gamma\) in \(GL_ n(k)\) is scheme non- singular and the dimension of the unique irreducible component of \(R_ n(\Gamma)\) through \(\rho_ 0\) is \(rk(\Gamma^{ab})n^ 2\). If def\((\Gamma)=rk(\Gamma^{ab})=1\), then this unique irreducible component consists of all representations factoring through \(\Gamma^{ab}\) modulo torsion.
Next the authors show how the Fox calculus helps to calculate the group of one-cocycles. At the end of the section they prove that every simple representation in \({\mathfrak R}_ n(SL_ 2)({\mathbb{Z}}))\) is scheme non- singular, so the variety \(S_ n(SL_ 2({\mathbb{Z}}))\) is non-singular for every n.
Section 4. Hochschild and Mostow associate to any group \(\Gamma\) a pro- affine algebraic group A(\(\Gamma)\) over k which has the following property: representations of \(\Gamma\) of degree n are in one-to-one correspondence with rational representations of A(\(\Gamma)\). The authors use this property for calculation of the groups \(Z^ 1(\Gamma,\rho)\) and \(H^ 1(\Gamma,\rho)\) (they show that there exist isomorphisms \(Z^ 1(A(\Gamma),{\bar \rho})\to Z^ 1(\Gamma,\rho)\) and \(H^ 1(A(\Gamma),\rho)\to H^ 1(\Gamma,\rho)\) where \(\rho\) and \({\bar \rho}\) respect one another under the above one-to-one correspondence.
In section 5 the authors introduce the notion of twisting: let \(\rho \in R_ n(\Gamma)\) and \(\chi \in X(\Gamma)=Hom(\Gamma,k^*)\). The twist of \(\rho\) by \(\chi\) is defined to be the representation \(\chi\) \(\rho\) given by \((\chi \rho)(\gamma)=\chi (\gamma)\rho (\gamma)\). The twist operation \(X(\Gamma)\times R_ n(\Gamma)\) by \((\chi,\rho)\to \chi\rho\) is an algebraic action of the algebraic group \(X(\Gamma)\) on the variety \(R_ n(\Gamma)\) stabilizing \(R^ S_ n(\Gamma)\). The authors study orbits of simple and semisimple representations in \(R_ n(\Gamma)\) under action of \(X(\Gamma)\) and their images in \(S_ n(\Gamma)\) and \(SS_ n(\Gamma).\)
In section 6 the authors apply the results of the previous sections to describe \(SS_ n(\Gamma)\) and \(S_ n(\Gamma)\) in the case \(\Gamma\) is nilpotent. For example they prove the following theorem: Let \(\Gamma\) be nilpotent. Then the distinct twist isoclasses \(c_{\tau}(\rho)\) for \(\rho\) pure semisimple of multiplicity one are a finite partition of \(SS_ n\) into open-closed subsets (a representation \(\rho \in R_ n(\Gamma)\) is said to be pure semisimple of multiplicity one if \(\rho =\rho_ 1\oplus...\oplus \rho_ s\) where \(\rho_ i\) is simple of dimension \(n_ i\) and \(\rho_ i\) is not isomorphic \(\rho_ j\) for \(i\neq j).\)
Section 7 contains historical remarks. representation of finitely generated group; rational representations of pro-affine algebraic group; tangent spaces of the representation varieties; twist operation; orbits 6. Lubotzky, Alexander and Magid, Andy R. Varieties of representations of finitely generated groups \textit{Mem. Amer. Math. Soc.}58 (1985) 117 Math Reviews MR818915 (87c:20021) Classical groups (algebro-geometric aspects), Representation theory for linear algebraic groups, Ordinary representations and characters, Homological methods in group theory Varieties of representations of finitely generated 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 We generalize a theorem of \textit{E. H. Cattani} and \textit{A. Kaplan} [Duke Math. J. 44, 1-43 (1977; Zbl 0363.32018)] on horizontal representation of SL(2). This theorem plays an important role in the construction of partial compactification of the classifying spaces \(D\) modulo an arithmetic subgroup of Hodge structures of weight 2.
A horizontal SL(2)-representation is a generalization of the notion of ``\((H_ 1)\)-homomorphism'' of SL(2) in the case of the classical theory of Hermitian symmetric domains. More precisely, let \(G\) be the automorphism group of the classifying space \(D\) of Hodge structures of weight \(w\). A representation \(\rho:\text{SL} (2,\mathbb{R}) \to G\) is said to be horizontal at \(r \in D\) if the corresponding morphism of their Lie algebras is a morphism of Hodge structures of type (0,0) with respect to the Hodge structures on the Lie algebras induced by \(i \in\) (upper-half plane) and \(r \in D\) respectively. In this case, the pair \((\rho,r)\) is uniquely determined by the pair \((Y,r) \in \text{Lie} G \times D\) with \(Y:=\rho_ *(y)\), where \(y:=\Bigl( {1\atop 0} {0 \atop -1} \Bigr)\).
Conversely, a pair \((Y,r) \in \text{Lie} G \times D\) is said to be admissible if there exists a representation \(\rho:\text{SL} (2,\mathbb{R}) \to G\) horizontal at \(r\) and satisfying \(Y=\rho_ *(y)\). The main result in the present article is a numerical criterion for admissibility of a pair \((Y,r)\) in the case of general weight. semi-simple elements; classifying spaces for Hodge structures; horizontal SL(2)-representations; limiting split mixed Hodge structures; admissibility Usui, S., \textit{A numerical criterion for admissibility of semi-simple elements}, Tohoku Math. J. (2), 45, 471-484, (1993) Period matrices, variation of Hodge structure; degenerations, Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) A numerical criterion for admissibility of semi-simple elements | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From the author's introduction: ``This paper is focused on the boundary points of the Baily-Borel compactification of ortogonal Shimura varieties. More precisely, given an integral lattice \(L\) with associated integral quadratic form q of signature \((2, n)\), let \(Y\) denote the connected Hermitian symmetric domain \(\mathrm{SO}^+ ( L)(\mathbb R) \times \mathrm{SO}( n)\). Then for an arithmetic group \(\Gamma \subset \mathrm{SO}^+ ( L)\), the space \( \Gamma\backslash Y\) inherits the structure of a quasi-projective variety. The boundary of its canonical minimal compactification contains at most \(0\) and \(1\)-dimensional cusps. There is a bijection between the number of cusps and the \(G\)-orbits of isotropic subspaces inside \(V := L\otimes \mathbb Q \).''
The author studies these orbits in the special case when \(L\) is a maximal lattice which splits two hyperbolic planes, i.e. \(L \cong \mathbb H \perp \mathbb H \perp A\), where \(A\) is a lattice whose genus \(gen( A)\) is uniquely determined. If \(Y\) is the connected Hermitian space associated to \(\mathrm{SO}^+ ( L)\) and \(\Gamma\) the discriminant kernel of \(\mathrm{SO}^+ ( L)\), then the number of \(1\)-dimensional cusps in the Baily-Borel compactification of \(\mathrm{SO}^+ ( L)\backslash Y\) (resp. \(\Gamma\backslash Y)\) is equal to \(\sum_{ A' \in gen(A)}\frac{2}{ [O(A' ) : \mathrm{SO}(A')]}\) (resp. \(\sum_{ A \in gen(A)} \frac{2}{ [O(A' ) : \mathrm{SO}(A')]} \cdot \frac{ | O(\Delta(A' )|} {| \rho (\mathrm{SO}(A' ) |}\), where \(\rho : O( A') \to O(\Delta ( A'))\) is the projection map and \(\Delta( A' )\) is the discriminant group \(A'{}^\vee /A' )\). A formula is also given for the number of 0-dimensional cusps in the associated \(\Gamma\)- Shimura variety. Shimura varieties; cusps; hyperbolic planes; quadratic forms Attwell-Duval, D, On the number of cusps of orthogonal Shimura varieties, Ann. Math. Québec., 38, 119-131, (2014) Modular and Shimura varieties, Quadratic forms over global rings and fields On the number of cusps of orthogonal Shimura varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``The aim of this article is to show that moduli spaces of systems of Hodge bundles on curves and character varieties for the unitary groups \(U(p, q)\) are connected, once the numerical invariants of the corresponding bundles are fixed.'' By the work of Simpson on Higgs bundles and local systems, systems of Hodge bundles are parameterised by the fixed points of the \(\mathbb{C}^\ast\)-action on the moduli space of Higgs bundles. ``In the case of \(U(p,q)\)-Higgs bundles, these fixed points have an elementary description as moduli spaces of holomorphic chains'', which are sequences of vector bundles and bundle maps:
\[
\mathcal{E}_r
\xrightarrow{~~\phi_r~~}
\mathcal{E}_{r-1}
\xrightarrow{~~\phi_{r-1}~~}
\cdots
\xrightarrow{~~\phi_1~~}
\mathcal{E}_0.
\]
The bulk of this article is therefore the investigation of irreducibility of spaces of holomorphic chains.
Let \(\mathrm{Chain}\) be the stack of holomorphic chains with fixed rank and fixed degree, and consider the open substack \(\mathrm{Chain}^{\alpha}\) of semi-stable chains, where the stability condition is determined by a stability parameter \(\alpha \in \mathbb{R}^r\).
The first main result (Theorem 3.2) is to find a precise set of conditions on the rank, the degree, and the stability parameter \(\alpha\) such that the stack \(\mathrm{Chain}^{\alpha}\) is irreducible.
If the rank and degree of chains are appropriately fixed, we get a nonempty subset \(\mathrm{Stab} \subset \mathbb{R}^r\) of stability parameters \(\alpha\) for which \(\mathrm{Chain}^{\alpha}\) is irreducible, and the authors investigate irreducibility in the boundary cases; i.e., when \(\alpha\) belongs to the boudnary of \(\mathrm{Stab}\).
They find (Theorem 4.1) that the course moduli space \(M^\alpha\) of the stack \(\mathrm{Chain}^\alpha\) remains irreducible.
For the primary application concerning \(U(p,q)\)-Higgs bundles, these irreducibility results are sufficient in the simplest case of chains of length \(r = 1\), also known as triples.
The authors devote special attention throughout the article to understanding the main results and their proofs in the case of triples.
They then recall how the moduli spaces \(\mathcal{M}_{U(p,q)}\) of \(U(p,q)\)-Higgs bundles are related to the moduli spaces of triples using an analytic argument, and they conclude that, for appropriately fixed degree and rank, the moduli space \(\mathcal{M}_{U(p,q)}\) is connected (Theorem 5.1).
The main part of the proof of this connectivity result, which goes through Morse theory, comes from the related previous work about \(U(p,q)\)-Higgs bundles [\textit{J. Krieger} et al., Duke Math. J. 147, No. 1, 1--53 (2009; Zbl 1170.35066)]. However, the argument there has the one limitation that it is unable to exclude the possibility of \(\mathcal{M}_{U(p,q)}\) having additional connected components in which all points are strictly polystable.
This is ruled out by the irreducibility of the corresponding moduli space of truples \(M^\alpha\).
The article is concluded by showing how these irreducibility results yield additional information about the irreducible components of the nilpotent cone of \(GL_n\)-Higgs bundles.
In particular, the authors explain how the precise constraints on the rank and degree give ``a concrete, but complicated, enumeration of the irreducible components of the so-called global nilpotent cone''. Higgs bundles; holomorphic chains; nilpotent cone; moduli spaces Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Irreducibility of moduli of semi-stable chains and applications to \(\mathsf{U}(p, q)\)-Higgs 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 authors first give an alternative description of some of the polyhedral Hodge spaces introduced by \textit{M. Brion} [Tohoku Math. J. (2) 49, 1-32 (1997; Zbl 0881.52008)]. For a polytope \(\Delta\) defined over a subfield \(K\subset \mathbb R\), instead of starting directly from its normal fan as Brion does, they consider the cone over \(\Delta\) in \(K^{n+1}\) and define invariants \(D^k(\Delta)\) to be the cohomology groups arising from the cohomological system obtained by associating to each face of the cone its linear span. It turns out that for \(k>1\), the vector space \(D^k(\Delta)\) equals Brion's Hodge space \(H^{k,1}(\Delta)\).
By analyzing the spectral sequence relating the D-invariants of a polytope to those of its faces, the authors obtain an explicit set of equations describing \(D^2(\Delta)\) as a \(K\)-vector subspace of a certain larger vector space under the assumption that \(D^1\) and \(D^2\) vanish for every \(3\)-dimensional face of \(\Delta\). With this description they can prove the main result, namely vanishing of \(D^2\) for a certain class of polytopes whose \(3\)-faces are all pyramids.
For a lattice polytope, this vanishing result has an interpretation in the context of deformation theory. As the authors explain, there is a close relation between the space \(D^k(\Delta)\) and (after extension of scalars to complex numbers) the cotangent cohomology module \(T^k(X_{\text{cone}(\Delta)})\) describing the deformations of the toric Gorenstein singularity \(X_{\text{cone}(\Delta)}\) associated to the cone over the polytope \(\Delta\). However, in general the modules \(T^k\) depend on the lattice structure, whereas the D-invariants only depend on the polytope up to projective equivalence.
Nevertheless the authors can give sufficient conditions on the lattice polytope ensuring that the modules \(T^k\) are in fact determined by its D-invariants. These conditions in particular imply vanishing of \(T^2\) for the corresponding class of toric Gorenstein singularities. Hodge numbers of projective toric varieties; deformation theory; toric Gorenstein singularities Altmann K., van Straten D.: The polyhedral Hodge number h 2,1 and vanishing of obstructions. Tohoku Math J. 52, 579--602 (2000) Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Deformations and infinitesimal methods in commutative ring theory The polyhedral Hodge number \(h^{2,1}\) and vanishing of obstructions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset \text{GL}(2,{\Bbbk})\) be a finite, small, abelian group and let \(X\) be the minimal resolution of the quotient surface \({\mathbb A}_\Bbbk^2/G.\) By \textit{R. Kidoh} [Hokkaido Math. J. 30, No. 1, 91--103 (2001; Zbl 1015.14004)] the \(G\)-Hilbert scheme \(G\)-\(\text{Hilb}({\mathbb A}^2_\Bbbk)\) is isomorphic to \(X.\) However, in general, the tautological line bundles do not form the basis the \(K\)-theory of \(X\) (i.e. the McKay correspondence does not hold in its full generality). In fact, some line bundles are redundant. Following \textit{J. Wunram} [Math. Ann. 279, No. 4, 583--598 (1988; Zbl 0616.14001)], the author concentrates only on the so-called special tautological line bundles \(L_1,\dots,L_k\) on \(X.\)
The main results says that, for \(G\) as above, the \(G\)-Hilbert scheme \(G\)-\(\text{Hilb}({\mathbb A}^2_\Bbbk)\) is isomorphic to the fine moduli space of \(\theta\)-stable representations of the bound quiver \((Q,R)\) given by the list \(({\mathcal O_X},L_1\dots,L_k)\) (by the multigraded linear series construction developed by \textit{A. Craw} and \textit{G. G. Smith} [Am. J. Math. 130, No. 6, 1509--1534 (2008; Zbl 1183.14066)]). In particular the moduli space is irreducible. Moreover, the bounded derived category of coherent sheaves on \(X\) is equivalent to the bounded derived category of \(\text{End} (\bigoplus_{i=0}^k L_i)\)-modules. McKay correspondence; special McKay correspondence; special representation; McKay quiver; special McKay quiver Craw, A, The special mckay correspondence as an equivalence of derived categories, Q. J. Math., 62, 573-591, (2011) McKay correspondence, Representations of quivers and partially ordered sets, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories The special McKay correspondence as an equivalence of derived categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simple algebraic group over \(\mathbb{C}\). Fix a maximal torus \(T \subset G\). Let \(\Delta =\{ \alpha_1,\dots, \alpha_n \}\) be the system of simple roots of \(G\) and \(P=P_k\) denote the maximal parabolic subgroup associated to a simple root \(\alpha_k\).
It is known that the category of \(G\)-equivariant vector bundles on \(G/P\) is equivalent to the category of finite-dimensional representations of \(P\).
Let \(X \subset \mathbb{P}^N\) be a smooth projective variety of dimension \(d\) over \(\mathbb{C}\). A vector bundle \(E\) on \(X\) is called Ulrich if the cohomology groups \(H^i(X, E(-t)) = 0\) for all \(0 \leq i \leq d\) and \(1\leq t \leq d\). The twisted bundle \(E\otimes O_X(-t)\) is denoted by \(E(-t)\).
For an integral weight \(\omega\) dominant with respect to \(P\), one has an irreducible representation \(V(\omega)\) of \(P\) with highest weight \(\omega\), and let us denote by \(E_{\omega}\) the corresponding irreducible equivariant vector bundle \(G \times_{P} V(\omega)^{*} \) on \( G/P:E_{\omega}:=G\times_{P}V(\omega)^{*}= (G \times V(\omega)^{*})/P\), where the equivalence relation is given by \((g, v) \sim (gp, p^{-1}. v)\) for \(p \in P\).
The cohomology of such bundles is computed by the famous the Borel-Bott-Weil theorem, see [\textit{J. M. Weyman}, Cohomology of vector bundles and syzygies. Cambridge: Cambridge University Press (2003; Zbl 1075.13007)] and [\textit{D. M. Snow}, CMS Conf. Proc. 10, 193--205 (1989; Zbl 0701.14017)].
If \(L\) be a Levi factor of a maximal parabolic subgroup \(P_k \subset G\), then for an \(L\)-dominant integral weight \(\omega \in \Lambda_{+}\) , define a set \(\mathrm{Sing}(\omega) :=\{t \in Z:\omega + \rho -t\omega_k\) is singular\}, where \(\rho\) is the sum of fundamental weights of \(G\).
The main tool used in this paper is Fonarev's criterion (Lemma 2.4 of [\textit{A. Fonarev}, Mosc. Math. J. 16, No. 4, 711--726 (2016; Zbl 1386.14180)]) which says that for \( \omega \in \Lambda^{+}_{L} \) an irreducible equivariant vector bundle \(E_{\omega}\) on a rational homogeneous variety \(G/P_k\) with Picard number 1 and dimension \(d\) is Ulrich if and only if \(\mathrm{Sing}(\omega) =\{1,2,\dots, d-1, d\}\).
Further, using this criterion the authors check whether the rational homogeneous varieties admit Ulrich bundles or not.
To check it properly they use the following helpful arguments: Assume that \(G/P_k\) is a Hermitian symmetric space of compact type. If an irreducible equivariant vector bundle \(E_{\omega}\) on \(G/P_k\) is Ulrich, then the maximum of \(\mathrm{Sing}(\omega)\) is a singular value attained at the highest root \(\theta\) of the Lie algebra \(g\).
and
Let \(G/P_k\) be an adjoint variety such that \(\mathrm{rank}(G)\geq 2\) and \(G\) is not of type A. If an irreducible equivariant vector bundle \(E_{\omega}\) on \(G/P_k\) is Ulrich, then the maximum of \(\mathrm{Sing}(\omega)\) is a singular value attained at the root \(\theta - \alpha_{k}\), where \(\theta\) is the highest root of \(g\).
More concretely, for \(G_2\)-homogeneous varieties:
\begin{itemize}
\item \(G_2/P_1\) does not admit Ulrich bundles;
\item \(G_2/P_2\) does not admit Ulrich bundles.
\end{itemize}
\(E_6\)-homogeneous varieties:
\begin{itemize}
\item Cayley plane \(E_6/P_2\subset \mathbb{P}(V_{E_6})(\omega_2)=\mathbb{P}^{77}\) has dimension 21 and Fano index 11, does admit the only one irreducible Ulrich bundle \(E_{\omega_5+3\omega_6}\);
\item \(E_6/P_3\subset \mathbb{P}^{350}\) does not admit Ulrich bundles;
\item \(E_6/P_4 \subset \mathbb{P}^{2924}\) has dimension 29 and Fano index 7 and does not admit Ulrich bundles.
\end{itemize}
\(F_4\)-homogeneous varieties:
\begin{itemize}
\item \(F_4/P_4\) is a general hyperplane section of a Cayley plane \(E_6/P_1 \subset \mathbb{P}^{26}\) does not admit Ulrich bundles; if one will restrict an Ulrich bundle from the Cayley plane to \(F_4/P_4\) one can get an equivariant Ulrich bundle on \(F_4/P_4\);
\item \(F_4/P_3 \subset \mathbb{P}^{272}\) is the closed \(F_4\)-orbit in the space of lines on the rational homogeneous variety \(F_4/P_4 \subset \mathbb{P}^{25}\) has dimension 20 and Fano index 7. It does not admit Ulrich bundles;
\item \(F_4/P_2 \subset \mathbb{P}^{1273}\) has dimension \(20\) and Fano index \(7\) and does not admit Ulrich bundles;
\item \(F_4/P_1 \subset \mathbb{P}^{51}\) has dimension \(15\) and Fano index \(8\) and does not admit Ulrich bundles;
\end{itemize}
\(E_7\)-homogeneous varieties:
\begin{itemize}
\item \(E_7/P_1 \subset \mathbb{P}^{132}\) has dimension \(33\) and Fano index \(17\) and does admit the only one irreducible equivariant Ulrich bundle \(E_{\omega_5+3\omega_6+8\omega_7}\);
\item The odd symplectic grassmanians of planes \(Gr_{\omega}(2, 2n+1)\) admit Ulrich bundles (see [ Zbl 1386.14180]);
\item \(E_7/P_3 \subset \mathbb{P}^{8644}\) has dimension \(47\) and Fano index \(11\) and does not admit Ulrich bundles;
\item \(E_7/P_4 \subset \mathbb{P}^{365749}\) has dimension \(53\) and Fano index \(8\) and does not admit Ulrich bundles;
\item \(E_7/P_5 \subset \mathbb{P}^{27663}\) has dimension \(50\) and Fano index \(10\) and does not admit Ulrich bundles;
\item \(E_7/P_6 \subset \mathbb{P}^{1538}\) has dimension \(42\) and Fano index \(13\) and does not admit Ulrich bundles;
\item \(E_7/P_7 \subset \mathbb{P}^{55}\) has dimension \(27\) and Fano index \(18\) (Freundenthal variety, one of two exceptional Hermitian symmetric spaces of compact type) and does not admit Ulrich bundles;
\end{itemize}
\(E_8\)-homogeneous varieties: none of them admit equivariant Ulrich bundles. equivariant Ulrich bundles; exceptional homogeneous varieties; Borel-Weil-Bott theorem; Cayley plane; Dynkin diagram Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results Equivariant Ulrich bundles on exceptional homogeneous 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 Generators and relations of elliptic Weyl groups were studied in the context of elliptic Dynkin diagrams by \textit{K. Saito} and \textit{T. Takebayashi} [Publ. RIMS, Kyoto Univ. 33, 301--329 (1997; Zbl 0901.20016)]. In their paper, they proposed the following problem: find generators and relations of ``elliptic Lie algebras'', ``elliptic Hecke algebras'' and ``elliptic Artin groups'' (the fundamental groups of the complements of the discriminant for simply elliptic singularities) in terms of the elliptic Dynkin diagrams. \textit{K. Saito} and \textit{D. Yoshii} [Publ. Res. Inst. Math. Sci. 36, 385--421 (2000; Zbl 0987.17012)] constructed the elliptic Lie algebras for homogeneous elliptic Dynkin diagrams. In addition, they described the fundamental relations in terms of the generators attached to the elliptic Dynkin diagrams. These relations are a generalization of the Serre type relations. In this article, we shall give an answer to their problem for the case of elliptic Artin groups and elliptic Hecke algebras as an application of the twisted Picard-Lefschetz formula due to \textit{A. B. Givental} [Funct. Anal. Appl., 22, 10--18 (1987; Zbl 0665.32011)]. As for the former groups, they have already been studied by \textit{H. van der Lek} [Proc. Symp. Pure Math. 40, 117--121 (1983; Zbl 0523.14005)] under the name of extended Artin groups and from the view point of affine Dynkin diagrams for arbitrary affine root systems. Here we describe these groups in terms of generators associated to the vertices of elliptic Dynkin diagrams that reflect the geometry of vanishing cycles of simply elliptic singularities. For this purpose, we restrict ourselves to the 1-codimensional case that has rich geometry such as flat structure. As for the latter algebras, which are subalgebras of Cherednik's double affine Hecke algebras, we can construct some irreducible finite dimensional representations as monodromy representations. elliptic Hecke algebras; twisted Picard-Lefschetz formula; elliptic Dynkin diagrams Yamada, H.: Elliptic root system and elliptic Artin group. Publ. res. Inst. math. Sci. 36, 111-138 (2000) Deformations of singularities, Structure of families (Picard-Lefschetz, monodromy, etc.), Hecke algebras and their representations, Braid groups; Artin groups, Reflection and Coxeter groups (group-theoretic aspects), Singularities in algebraic geometry Elliptic root system and elliptic Artin group. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors construct a reasonable singular homology theory on the category of schemes of finite type over an arbitrary field \(k\). Let \(X\) be a CW-complex. The theorem of \textit{A. Dold} and \textit{R. Thom} [Ann. Math., II. Ser. 67, 239-281 (1958; Zbl 0091.37102)] shows that \(H_i(X,{\mathbb{Z}})\) coincide with \(\pi_i\) of the simplicial Abelian group \(\Hom_{\text{top}} (\Delta_{\text{top}}^{\circ}, \coprod_{d=0}^\infty S^d(X))^{+},\) where \(S^d(X)\) is the \(d\)-th symmetric power of \(X\), \(\Delta_{\text{top}}^i\) is the usual \(i\)-dimensional topological simplex and for any Abelian monoid \(M\) denote by \(M^+\) the associated Abelian group.
Conjecture. If \(X\) is a variety over \({\mathbb{C}}\) then the evident homomorphism
\[
\Hom(\Delta^\circ, \coprod_{d=0}^\infty S^d(X))^+\rightarrow \Hom_{\text{top}} (\Delta_{\text{top}}^{\circ}, \coprod_{d=0}^\infty S^d(X))^+
\]
induces isomorphisms \(H_i^{\text{sing}}(X,{\mathbb{Z}}| n)\cong H_i(X({\mathbb{C}}),{\mathbb{Z}}| n)\).
The authors prove that the conjecture is true. Also, they prove a rather general version of the rigidity theorem of \textit{A. Suslin} [Invent. Math. 73, 241-245 (1983; Zbl 0514.18008); Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 222-244 (1987; Zbl 0675.12005)], \textit{O. Gabber} (unpublished), \textit{H. A. Gillet} and \textit{R. W. Thomason} [J. Pure Appl. Algebra 34, 241-254 (1984; Zbl 0577.13009)]. One of the main results of the paper is that if \(F\) is any \(qfh\)-sheaf on the category of schemes of finite type over an algebraically closed field \(k\) of characteristic zero, then \(H_{\text{sing}}^*(F,{\mathbb{Z}|}n)= \text{Ext}_{qfh}^*(F,{\mathbb{Z}|}n)\). singular homology; abstract algebraic varieties; category of schemes of finite type Suslin, A.; Voevodsky, V., \textit{singular homology of abstract algebraic varieties}, Invent. Math., 123, 61-94, (1996) Singular homology and cohomology theory, Varieties and morphisms Singular homology of abstract 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 The author starts by recalling definitions and constructions referred to later in the discussion of noncommutative projective geometry and algebraic aspects of noncommutative tori. Artin, Tate and Van den Bergh have given definitions of noncommutative algebras that should in principle be algebras of functions on some nonsingular noncommutative schemes. The author gives the necessary conditions needed on the graded algebras, and he defines \textit{Artin-Schelter (AS)-regularity}, which is the nonsingularity condition in the noncommutative setting. Also the noncommutative Gorenstein condition is defined in the noncommutative situation. This leads to the definition of a \textit{standard algebra}, containing the essential properties of AS-regular algebras of dimension 3.
The text contains the introductory section about twisted homogeneous coordinate rings which gives a general recipe for constructing noncommutative rings out of a commutative geometric datum, called an abstract triple which is an isomorphism invariant for AS-regular algebras. This construction is completely explicit, and a direct computation in a simple example is given.
A very short, nice introduction to Grothendieck categories is given, and the Gabriel-Rosenberg theorem is explained and exploited: ``Any scheme can be reconstructed from the category of quasi-coherent sheaves on it''. The author explains the idea of a quotient category, and then he gives the model of noncommutative projective geometry after Artin and Zhang: The triple \((QGr(R),R,s)\) is called the projective scheme of \(R\) and is denoted \(\text{Proj}(R)\). Also the characterization of \(\text{Proj}(R)\) given by Artin and Zhang gets a thorough treatment. The final chapter in this article on ``algebraic aspects of noncommutative tori'' is not completely self-contained, but the main idea is treated in an understandable way.
Reviewer's remark: This article is one of the most accessible reviews of noncommutative geometry given, leaving out the hardest proofs and the most general descriptions of the theme. noncommutative projective geometry; Grothendieck categories; \(\chi\)-condition; quotient categories Noncommutative algebraic geometry, Grothendieck categories, Derived functors and satellites Lecture notes on noncommutative algebraic geometry and noncommutative 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 \textit{G. Laumon} [Duke Math. J. 57, 647--671 (1988; Zbl 0688.14023)] introduced the notion of a global nilpotent cone. It is a Lagrangian substack Nilp of the cotangent stack of the moduli stack of vector bundles or more generally, principal \(G\)-bundles over a curve [cf. \textit{G. Faltings}, J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019) or \textit{V. Ginzburg}, Perverse sheaves on a loop group and Langlands duality, Preprint, \texttt{http://arxive.org/abs/math.AG/ 9511007}]. It is very important in the study of the geometric Langlands program, because it gives the characteristic variety of an automorphic sheaf. Actually we can also realize it as the zero level set of the Hitchin Hamiltonians [\textit{R. Donagi} and \textit{E. Markman}, in: Integrable systems and quantum groups. Lect. Notes Math. 1620, 1--119 (1996; Zbl 0853.35100)]. Therefore, it seems to be an interesting problem to describe the stratification induced by the global nilpotent cone. In this paper, we consider when a nontrivial nilpotent cotangent vector exists if the curve is \(\mathbb{P}^1\) but add parabolic structure to enlarge the moduli space. We present a complete picture in the rank two case. We also present an analogous result for the arbitrary rank case. In particular we obtain that the expected domain of an automorphic sheaf in \(\mathbb{P}^1\) case should be contained in the product of flag varieties as expected. Moreover, a relation with the Bruhat stratification of flag varieties is obtained. moduli stack of vector bundles; geometric Langlands program; parabolic structure; automorphic sheaf Fine and coarse moduli spaces, Vector bundles on curves and their moduli On the global nilpotent cone of P\(^1\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset\text{SL}(n,\mathbb{C})\) be a finite subgroup and \(X=\mathbb{C}^n/G\) the quotient space. Let \(R\) be a common multiple of the orders of all elements of \(G\) and \(\Gamma=\text{Hom}(\mu_R,G)\). Then, \(\Gamma\) is isomorphic to \(G\), but the isomorphism depends on the choice of an \(R\)-th root of unity \(\varepsilon\). There is a naturally defined grading \(\Gamma=\bigcup_{i=0}^{n-1}\Gamma_i\), were \(\Gamma_i\) is defined as follows. Each element \(g\) of \(G\) has eigenvalues \(\varepsilon^{a_1},\dots,\varepsilon^{a_n}\) with \(a_1+\cdots+a_n\equiv 0\pmod R\) and \(0\leq a_j<R\). The integer \(i=(a_1+\cdots+a_n)/R<n\) is called the age of the element of \(\Gamma\) which maps \(\varepsilon\) to \(g\) and this defines the grading on \(\Gamma\). The elements of \(\Gamma_1\) are called junior elements and their \(G\)-conjugacy classes are the junior conjugacy classes.
Now let \(f\colon Y\to X\) be a resolution of \(X\) and write \(K_Y\equiv f^*K_X+\sum a_E E\) with exceptional prime divisors \(E\). The prime divisor \(E\) is called crepant, if \(a_E=0\). It is proved that there is a canonical one-to-one correspondence between junior conjugacy classes and crepant exceptional divisors. McKay correspondence; Euler number; crepant resolution; junior conjugacy classes; crepant exceptional divisors Ito, Y., Reid, M.: The McKay correspondence for finite subgroups of \({{\mathrm SL}(3, {\mathbf C})}\). In: Higher-Dimensional Complex Varieties (Trento, 1994), pp. 221-240. de Gruyter, Berlin (1996) Homogeneous spaces and generalizations, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The McKay correspondence for finite subgroups of \(SL(3,\mathbb{C})\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article we construct a categorical resolution of singularities of an excellent reduced curve \(X\), introducing a certain sheaf of orders on \(X\). This categorical resolution is shown to be a recollement of the derived category of coherent sheaves on the normalization of \(X\) and the derived category of finite length modules over a certain artinian
quasi-hereditary ring \(Q\) depending purely on the local singularity types of \(X\). Using this technique, we prove several statements on the Rouquier dimension of the derived category of coherent sheaves on \(X\). Moreover, in the case \(X\) is rational and projective we construct a finite dimensional quasi-hereditary algebra \(\Lambda\) such that the triangulated category \(\mathsf{Perf}(X)\) embeds into \(D^b(\Lambda -\mathsf{mod})\) as a full subcategory. Singularities of curves, local rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Noncommutative algebraic geometry, Derived categories and associative algebras Singular curves and quasi-hereditary 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 Over a field of characteristic zero the geometry of orbit closures for equioriented \(A_n\) quiver was first studied by \textit{S. Abeasis} et al. [Math. Ann. 256, 401-418 (1981; Zbl 0477.14027)] where it was established that the orbit closures are normal, Cohen-Macaulay, and have rational singularities. This result was generalized to the case of a quiver \(A_n\) with an arbitrary orientation by \textit{G. Bobiński} and \textit{G. Zwara} [Manuscr. Math. 105, No. 1, 103-109 (2001; Zbl 1031.16012)].
In the paper under review orbit closures for the non-equioriented \(A_3\) quiver are investigated. Namely, a minimal free resolution of the defining ideal of an orbit closure is explicitly constructed, a description of a minimal set of generators of the defining ideal is obtained, a classification of orbits closures which are Gorenstein is established. Gorenstein orbit closures; Lascoux resolution; Cohen-Macaulay varieties; Dynkin quivers; geometry of orbit closures; Bott vanishing theorem Representations of quivers and partially ordered sets, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Determinantal varieties, Singularities in algebraic geometry, Actions of groups on commutative rings; invariant theory, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Resolutions of defining ideals of orbit closures for quivers of type \(A_3\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is about singularities of algebraic varieties. In characteristic zero, where resolution of singularities is available, an important class of singularities is given by rational singularities. Quotient singularities are examples of rational singularities. In positive characteristic, the notion of rational singularity is not well-suited; resolution of singularities is not available at present, and quotient singularities are not even rational.
In this paper, the authors propose to replace the structure sheaf \(\mathcal{O}_X\) by the sheaf of Witt vectors \(W\mathcal{O}_X\) and a definition of Witt-rational singularities is introduced. This notion of Witt-rational singularities is more restrictive than the one introduced earlier by \textit{M. Blickle} and \textit{H. Esnault} [Pure Appl. Math. Q. 4, No. 3, 729--741 (2008; Zbl 1162.14015)] (although the author conjecture that their definition agrees with the one of loc. cit.).
The main result is that topological finite quotients have Witt-rational singularities. (An integral normal \(k\)-scheme \(X\) is a finite quotient if there exists a finite and surjective morphism from a smooth \(k\)-scheme \(Y\rightarrow X\). A normal integral \(k\)-scheme \(X\) is a topological finite quotient if there exists a finite, surjective and purely inseparable morphism \(u : X \rightarrow X'\), where \(X'\) is a finite quotient.) A crucial ingredient in the proof consists in showing that relative Hodge--Witt cohomology can be endowed with an action of correspondences.
Moreover, this paper contains numerous results concerning Witt-vector cohomology. For instance, it is shown that the Witt-vector cohomology of a smooth, proper scheme is a birational invariant. de Rham-Witt complex; ekedahl duality; correspondences; singularities; Witt-vector cohomology Chatzistamatiou, A.; Rülling, K., \textit{Hodge-Witt cohomology and Witt-rational singularities}, Doc. Math., 17, 663-781, (2012) Singularities of surfaces or higher-dimensional varieties, Algebraic cycles, \(p\)-adic cohomology, crystalline cohomology Hodge-Witt cohomology and Witt-rational 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 main aim of the article under review is to prove that for certain homogeneous spaces \(X\) of simple algebraic groups of the classical types \(A_n, B_n, C_n, D_n\) the group \(A_0(X)\) of zero-dimensional cycles of degree zero modulo rational equivalence is trivial. For the type \(A_n\) this is proved for a Severi-Brauer variety (recovering a result of Panin) and for some Severi-Brauer flag varieties. Here the ground field is either perfect or its characteristic does not divide the index of the underlying central simple algebra. In the \(B_n\) and \(D_n\) cases the result is obtained for any orthogonal involution variety, assuming that the characteristic is not 2. In the \(C_n\) case the author takes \(X=V_2(A,\sigma)\), a second generalized involution variety for a central simple algebra \(A\) with symplectic involution \(\sigma\).
The idea of the proof is to translate the rational equivalence of zero-dimensional cycles on a projective variety into R-equivalence (i.e. connecting points with rational curves) on symmetric powers of the original variety, and then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras. The author proposes a construction of such moduli spaces. homogeneous spaces; the Chow group; rational equivalence; simple algebras; Hilbert schemes Daniel Krashen, ``Zero cycles on homogeneous varieties'', Adv. Math.223 (2010) no. 6, p. 2022-2048 Grassmannians, Schubert varieties, flag manifolds, Finite-dimensional division rings Zero cycles on homogeneous varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper there is given a formula for the calculation of the Picard group of an arbitrary compact toric variety \(X_ \Sigma\). This improves the standard result in that \(X_ \Sigma\) may be singular and a previous result of the author in that \(X_ \Sigma\) may be non-projective. The main difference between the singular and the nonsingular case is that in the nonsingular case the rank of the Picard group of the variety \(X_ \Sigma\) is known to be determined by the combinatorial structure of \(\Sigma\) whereas in the singular case \(\text{Pic} X_ \Sigma\) may depend on metrical properties of the fan.
The proof of the formula is based on the geometric description of special \(T\)-invariant (i.e. invariant under the action of the torus \(T)\) Cartier divisors by a set of points which is called a Cartier set of \(\Sigma\). It is proved that these special divisors always exist and metric properties of the points in the corresponding Cartier set are used in order to develop the formula. An example of two combinatorially equivalent fans \(\Sigma_ 1\) and \(\Sigma_ 2\) is given such that \(X_{\Sigma_ 1}\) is projective with a non-trivial Picard group and \(X_{\Sigma_ 2}\) is non-projective with a surprisingly trivial Picard group. Furthermore, it is shown that \(\Sigma_ 2\) is a fan which cannot be spanned by any topological sphere which is the union of \((d-1)\)-polytopes such that the polytopes correspond exactly to the full dimensional cones of \(\Sigma_ 2\). Picard group; compact toric variety; Cartier divisors; fan M. Eikelberg, ''The Picard group of a compact toric variety,''Res. Math. 22, 509--527 (1992) (see also his paper ''Picard groups of compact toric varieties and combinatorial classes of fans,'' ibid.,23, 251--293 (1993). Toric varieties, Newton polyhedra, Okounkov bodies, Picard groups, Divisors, linear systems, invertible sheaves, Polyhedral manifolds The Picard group of a compact toric 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 paper under review concerns a generalisation of a well-known property of a Fourier-Mukai transform between the derived categories of smooth projective varieties: it automatically has a right (and left) adjoint, it is again of Fourier-Mukai type, and its kernel is related to the original kernel using Serre duality.
First, the case of smooth separated schemes of finite type over a field of characteristic zero is treated, for (a dg enhancement of) their derived categories of D-modules. Given a continuous functor between such categories it can always be described as a Fourier-Mukai transform in the context of D-modules, and it automatically has a right adjoint. Now if this right adjoint is itself continuous it has a description as a Fourier-Mukai transform, and the kernel is nothing but the shift of the Verdier dual. The result is a corollary to an abstract result regarding duality in a monoidal compactly generated dg category.
The second part generalises this to Artin stacks. It turns out that by the same techniques one can show that for a quasicompact Artin stack (whose derived category of D-modules is compactly generated) the functor associated to the Verdier dual of the original kernel is the composition of the original functor and an endofunctor whose kernel is the pushforward of the constant sheaf. In the smooth and separated case this endofunctor is nothing but the shift.
The main application that the author has in mind is for the non-quasicompact Artin stack of \(G\)-bundles on a smooth projective curve, an object central to the geometric Langlands programme. He identifies the appropriate dg categories for which there is an explicit description of the right adjoint as before. The techniques in this paper are mostly categorical, and might be applied in other situations. Unfortunately the paper does not give any references to standard works regarding supposedly basic properties of D-modules, building only on earlier works of the author and his co-authors. D-modules; compactly generated categories; Fourier-Mukai transforms Gaitsgory, D., Functors given by kernels, adjunctions and duality Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Derived categories, triangulated categories Functors given by kernels, adjunctions and duality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(n\in A:= {\mathbb F}_q[T]\) be a prime element. Let \(X_1(n)\) be the smooth projective curve over \({\mathbb F}_q(T)\) associated to the moduli problem of classifying pairs \((\phi,P)\), where \(\phi\) is a rank 2 Drinfeld \(A\)-module over an \({\mathbb F}_q(T)\)-scheme and \(P\) is a nowhere vanishing point of its \(n\)-torsion.
Let \(J_1(n)\) be the Jacobian of \(X_1(n)\). The main result of this paper is that the closed fibre of the Néron model of \(J_1(n)\) over \(A_{(n)}\) has trivial geometric component group. It follows that the Néron model of \(J_1(n)\) over \({\mathbb P}^1_{{\mathbb F}_q} - \infty\) has connected fibres, hence the title. (The component group of the fibre above \(\infty\) is a much more complicated beast, but is not part of the moduli problem). This work is a function field analogue of results of \textit{B. Conrad}, \textit{B. Edixhoven} and \textit{W. Stein} [Doc. Math., J. DMV 8, 331--408 (2003; Zbl 1101.14311)], and the proof strategy is similar, although the translation is far from trivial.
To prove the result, it suffices to show that \(X_1(n)\) has a regular proper model over \(A_{(n)}\) with geometrically integral special fibre. The author does this by first constructing a model with a unique non-regular point (corresponding to \((\phi,P)\), where \(j(\phi)=0\) and \(P\) is the kernel of Frobenius). This is a cyclic quotient singularity, which is resolved using the Jung-Hirzebruch resolution developed in [loc. cit.]. Contraction of the special fibre of this resolution then yields the desired integral model of \(X_1(n)\).
A result of independent interest obtained along the way is a function field analogue of the theory of Igusa curves. The paper is rather technical, but very well written. component groups; Drinfeld modular curves; Igusa curves Modular forms associated to Drinfel'd modules, Formal groups, \(p\)-divisible groups, Drinfel'd modules; higher-dimensional motives, etc. The Drinfeld modular Jacobian \(J_1 (N)\) has connected fibers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Focusing on examples associated with holonomic functions, we try to bring new ideas on how to look at phase transitions, for which the critical manifolds are not points but curves depending on a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in the most general mathematical framework, be too complex, or simply could not be defined. In a learn-by-example approach, considering several Picard-Fuchs systems of two-variables ``above'' Calabi-Yau ODEs, associated with double hypergeometric series, we show that D-finite (holonomic) functions are actually a good framework for finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the \(n\)-fold integrals \(\chi^{(n)}\), corresponding to the \(n\)-particle decomposition of the magnetic susceptibility of the anisotropic square Ising model. In this anisotropic case, we revisit a set of so-called Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find the first set of non-Nickelian singularities for \(\chi^{(3)}\) and \(\chi^{(4)}\), that also turns out to be rational or elliptic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model, or, equivalently, that they depend on the spectral parameter of the model. This has important consequences on the physical nature of the anisotropic \(\chi^{(n)}\)s which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and that of Calabi-Yau manifolds on such problems. We also address the question of singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility \(\chi \). Picard-Fuchs systems; phase transitions; Calabi-Yau operator; Ising model Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, General theory of ordinary differential operators, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Calabi-Yau manifolds (algebro-geometric aspects), PDEs in connection with statistical mechanics, Picard schemes, higher Jacobians, Phase transitions (general) in equilibrium statistical mechanics Holonomic functions of several complex variables and singularities of anisotropic Ising \(n\)-fold integrals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the Green-Lazarsfeld set from the point of view of geometric group theory and compare it with the Bieri-Neumann-Strebel invariant. Applications to the study of fundamental groups of Kähler manifolds are given.
Let \(\Gamma\) be a finitely generated group, and \(K\) be a field. A 1-character is a homomorphism from \(\Gamma\) to \(K^*\); in this article we will only consider 1-characters, and call them characters. A character \(\chi\) is called exceptional if \(H^1(\Gamma,\chi)\neq 0\), or more geometrically if \(\chi\) can be realized as the linear part of a fixed-point-free affine action of \(\Gamma\) on a \(K\)-line.
The set of exceptional characters, \(E^1(\Gamma,K)\) is a subset of the Abelian group \(\Hom(\Gamma,K^*)\), and our aim is to understand its geometry, in particular if \(\Gamma\) is the fundamental group of a compact Kähler manifold.
Motivated by the pioneering work of \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 90, 389-407 (1987; Zbl 0659.14007)], algebraic geometers studied the case where \(K=\mathbb C\) is the field of complex numbers, and \(\Gamma=\pi_1(X)\) is the fundamental group of a projective or more generally a compact Kähler manifold. In this case, the geometry of \(\Hom(\Gamma,K^*)\) is well understood: it is the union of a finite set, made up of torsion characters, and a finite set of translates of subtori.
The definition of an exceptional class in the sense of Bieri-Neumann-Strebel is easier to explain in the case of an integral cohomology class (an element of \(H^1(\Gamma,\mathbb Z)\)). Such a class is `exceptional' if it can be realized as the translation class of a parabolic, non-loxodromic action of \(\Gamma\) in some tree.
The link between these two notions, explained in the next section, can be sketched as follows. Let \(\chi\) be an exceptional character of \(\Gamma\). Suppose that \(\chi(\Gamma)\) is not contained in the ring of algebraic integers of \(K\). There exists a discrete non-Archimedean valuation on the subfield of \(K\) generated by \(\chi(\Gamma)\) such that \(v\circ\chi\) is a non-trivial homomorphism to \(\mathbb Z\). It appears that \(v\circ\chi\) is an exceptional class in the sense of Bieri-Neumann-Strebel. More precisely, one can find a parabolic action of \(\Gamma\) on the Bruhat-Tits tree of the \(v\) completion of \(K\), say \(K_v\), with translation length \(v\circ\chi\).
The main result of this paper is a description of the (generalized) Green-Lazarsfeld set of \(\pi_1(X)\) in terms of the finite list of its fibrations on hyperbolic 2-orbifolds.
Theorem. Let \(\Gamma\) be the fundamental group of a compact Kähler manifold \(X\), \((F_i,\Sigma_i)_{1\leq i\leq n}\) the family of fibration of \(X\) over hyperbolic 2-orbifolds. Let \(K\) be a field of characteristic \(p\) (if \(p=0\), \(K=\mathbb C\)), \(\overline F_p\subset K\) the algebraic closure of \(F_p\) in \(K\). Then \(E^1(\Gamma,K)\) is the union of a finite set of torsion characters (contained in \(E^1(\Gamma,\overline F_p)\) if \(p>0\)) and the union \(\bigcup_{1\leq i\leq n}F^*_iE^1(\pi^{\text{orb}}_1(\Sigma_i),K^*)\).
In section 2, we explain the relationship between the Green-Lazarsfeld and Bieri-Neumann-Strebel invariants; in section 3 we study the Green-Lazarsfeld set of a metabelian group: a finiteness result on this set is established. These two sections are purely group theoretic, and no Kähler structure is mentioned. In the section 4 we prove the main result. Green-Lazarsfeld sets; geometric group theory; Bieri-Neumann-Strebel invariants; fundamental groups; Kähler manifolds; finitely generated groups Delzant, T., Trees, valuations and the Green-lazarsfeld set, Geometric and Functional Analysis, 18, 1236-1250, (2008) Groups acting on trees, Algebraic topology on manifolds and differential topology, Global differential geometry of Hermitian and Kählerian manifolds, Geometric group theory, Solvable groups, supersolvable groups, Coverings of curves, fundamental group, Fundamental groups and their automorphisms (group-theoretic aspects), Kähler manifolds Trees, valuations and the Green-Lazarsfeld set. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the authors consider a possible generalization of the following important conjecture in the minimal model program:
Conjecture 1.1 (Nonvanishing Conjecture). Let \( (X, B)\) be a projective log canonical pair such that \( K_X + B\) is pseudo-effective. Then \( K_X + B \sim_{\mathbb{R}} D\) for some effective \({\mathbb{R}}\)-Cartier \({\mathbb{R}}\)-divisor \(D\).
A question raised by \textit{C. Birkar} and \textit{Z. Hu} [Nagoya Math. J. 215, 203--224 (2014; Zbl 1314.14028)] which is a modification of the above conjecture is:
Conjecture 1.2 (Numerical Nonvanishing for Generalized Polarized Pairs). Let \((X, B + M)\) be a projective generalized log canonical pair. Suppose that
(i) \(K_X + B + M_X\) is pseudo-effective,
(ii) \(M =\sum_j \mu_jM_j\), where \(\mu_j \in {\mathbb{R}}_{>0} \) and \( M_j\) are nef Cartier \(b\)-divisors.
Then \(K_X + B + M_X \equiv D\) for some effective \({\mathbb{R}}\)-Cartier \({\mathbb{R}}\)-divisor \(D\).
In this paper, the authors
1. prove that Conjecture 1.2 is true in dimension two;
2. confirm Conjecture 1.2 in higher dimensions if \(K_X +M_X\) is not pseudo-effective;
3. prove the numerical nonvanishing for projective generalized \( lc\) threefolds with rational singularities by scaling the nef part:
Theorem. Let \( (X, B +M)\) be a projective generalized lc threefold with rational singularities such that \( M\) is an \({\mathbb{R}}_{>0}\)-linear combination of nef Cartier b-divisors. If \(K_X + B + M_X\) is pseudo-effective and \(M_X\) is \({\mathbb{R}}\) Cartier, then there exists a \(0 \leq t \leq 1\) such that \( K_X + B + tM_X\) is numerically equivalent to an effective \({\mathbb{R}}\)-Cartier \({\mathbb{R}}\)-divisor.
To prove that Conjecture 1.2 is true conditionally, the key technique (due to \textit{J.-P. Demailly} et al. [Acta Math. 210, No. 2, 203--259 (2013; Zbl 1278.14022)]) is to construct a Mori fibre space \( X\dashrightarrow Y \rightarrow Z\), run appropriate MMPs over \( Y\) or \(Z\) to reach a generalized \(lc\)-trivial fibration, then apply an induction on dimension. generalized polarized pair; numerical nonvanishing Minimal model program (Mori theory, extremal rays) On numerical nonvanishing for generalized log canonical pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a compact connected Kähler manifold. Given a holomorphic vector bundle \(E\to X\) and a polynomial \(\varphi(z)=\sum_{i=0}^na_i\,z^i\) with nonnegative integral coefficients, define \(\varphi(E)\) to be the holomorphic vector bundle over \(X\)
\[
\varphi(E):=\bigoplus_{i=0}^n(E^{\otimes i})^{\oplus a_i}.
\]
A holomorphic vector bundle \(E\to X\) is said to be finite if there are two distinct such polynomials \(\varphi_1\) and \(\varphi_2\) such that \(\varphi_1(E)\) is holomorphically isomorphic to \(\varphi_2(E)\).
Next, let \(G\) be a connected linear algebraic group defined over the complex numbers and fix a finite-dimensional faithful \(G\)-module \(V_0\). If \(E_G\) is a holomorphic principal \(G\)-bundle over \(X\), then it is called finite if for each subquotient (i.e., the quotient \(G\)-module \(V_2/V_1\) for any two submodules \(V_1,V_2\) of the \(G\)-module \(V_0\) with \(V_1\subset V_2\)) \(W\) of the \(G\)-module \(V_0\), the holomorphic vector bundle \(E_G(W)\) over \(X\) associated to \(E_G\) for \(W\) is finite.
In this paper, the author proves the following four equivalences.
{\parindent=4mm
\begin{itemize}\item[{\(\bullet\)}] The principal \(G\)-bundle \(E_G\) admits a flat holomorphic connection whose monodromy group is finite.
\item[{\(\bullet\)}] There is a finite étale Galois covering \(f: Y\to X\) such that the pullback \(f^*E_G\) is a holomorphically trivializable principal \(G\)-bundle over \(Y\).
\item[{\(\bullet\)}] For any finite-dimensional complex \(G\)-module \(W\), the holomorphic vector bundle
\[
E_G(W):=E\times^{G}W
\]
over \(X\) associated to the principal \(G\)-bundle \(E_G\) for the \(G\)-module \(W\) is finite.
\item[{\(\bullet\)}] The principal \(G\)-bundle \(E_G\) is finite.
\end{itemize}}
Observe that when \(X\) is a projective variety, the same result was proven by \textit{M. V. Nori} [Proc. Indian Acad. Sci. Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)]. principal bundle; finite bundle; Kähler manifold Holomorphic bundles and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the finite principal 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 study of deformation theory ordinarily includes the study of a differential graded Lie algebra (dgla) \(\mathfrak g\). That is, one studies the canonically defined deformation ring \(R(\mathfrak g)\). In this article, the deformation theory is mainly the study of complex algebraic structures, and so the deformation ring \(R(\mathfrak g)\) is constructed with basis in the Jacobi complex associated to \(\mathfrak g\). This deformation theory is not broad enough to include embedded deformations, deformations of a manifold \(X\) embedded in a fixed ambient space \(Y\). In earlier work, the author has introduced the notion of a Lie atom and an associated Jacobi-Bernoulli complex to handle this deformation theory.
The present article establishes a notion of (dg) semi-simplicial Lie algebra (SELA) as a convenient setting for deformation theory. This is a generalization of the Lie atom, and it handles deformations of arbitrary singular schemes over \(\mathbb C\).
Let \(A\) be a totally ordered index-set. A \textit{simplex} in \(A\) is a finite nonempty subset \(S\subset A\), a \textit{biplex} is a pair \(\{S_1\subset S_2\}\) of simplices with \(|S_1|+1=|S_2|\), and further for \textit{triplex} etc. Also, there is assigned a sign rule \(\epsilon(S_1,S_2)\). A \textit{semi-simplicial Lie algebra} (SELA) \(\mathfrak g.\) on \(A\) is an assignment \(\mathfrak g_S\) for each simplex \(S\) on \(A\), and for each biplex a restriction morphism compatible with triplexes and sign rules. The \(\mathfrak g_S\) can be assembled into a complex \(K^.(\mathfrak g.)=\bigoplus_{|S|=i+1}\mathfrak g_S\) and with differential constructed by the restriction maps.
To compute the deformation theory of a SELA, a complex called the Jacobi-Bernoulli complex is introduced. This transforms a gluing condition from a nonabelian cocycle condition to an ordinary additive cocycle condition via the multilinearity of the groups making up the complex. The transformation is given by using the exponential operator, and letting \(\Psi=\exp\psi\), the Baker-Campbell-Hausdorff (BCH) formula gives a formal expression
\[
\exp X\exp Y\exp Z=\sum W_{i,j,k}(X,Y,Z)
\]
with \(W_{i,j,k}(X,Y,Z)\) the homogeneous ad-polynomial of tridegree \(i,j,k\) called the BCH-polynomial. Then the Jacobi-Bernoulli complex \(J(\mathfrak g.)\) for the SELA \(\mathfrak g.\) is designed to encompass the various BCH-polynomials \(w_{i,j,k}\), and the dual of the cohomology of \(J(\mathfrak g.)\) yields the deformation ring associated to the SELA \(\mathfrak g.\) Another main results is that the deformation theory of an algebraic scheme over \(\mathbb C\) can be expressed in terms of a SELA, constructed by an affine covering. Associated to an embedding \(X\rightarrow P\) where \(P\) is either affine or projective, a dgla is associated which is called the tangent dgla, and denoted \(\mathcal T_X(P)\). This \(\mathcal T_X(P)\) is the mapping cone of a constructed map \(T_P\otimes A_X\rightarrow N_{X/P}\) where \(N_{X/P}\) is the normal atom to \(X\) in \(P\). The author show that \(\mathcal T_X(P)\) admits a dgla structure, a dgla action on the coordinate ring of \(X\), \(A_X\), and an \(A_X\) module structure. Then an essential result is that up to what is called weak equivalence, the dgla \(\mathcal T_X(P)\) depends only on the scheme \(X\) and not on the embedding in \(P\). This kind of independence gives the possibility of defining the global SELA \(\mathcal T_X\), behaving polite on an affine covering. Given this, global deformations of \(X\) amount to collections of deformations of each \(X_\alpha\) given by Kodaira-Spencer theory, and suitable gluing data. The necessary compatibilities are expressed as a cocycle condition in the Jacobi-Bernoulli complex \(J(\mathcal T_X.)\).
The main results in this article, is the construction of the proposed complexes and SELAs, and the proofs of these behaving well on arbitrary schemes. Also, the functorial properties of the constructions and their contribution to deformation theory is important and interesting. It illustrates general techniques in deformation theory, and finally, explicit examples are given proving that the theoretical results work in practice. deformations; Bernoulli numbers; semi simplicial Lie algebra; SELA; Jacobi-Bernoulli complex; Baker-Campbell-Hausdorff formula; BCH polynomial; normal algebra; normal atom; tangent atom; Ischebeck's theorem Z. Ran, Jacobi-Bernoulli cohomology and deformations of schemes and maps , Cent. Eur. J. Math. 10 (2012), 1541-1591. Formal methods and deformations in algebraic geometry, Deformations of general structures on manifolds Jacobi-Bernoulli cohomology and deformations of schemes and 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 automorphism group \(\Aut X\) of a weighted homogeneous normal surface singularity \(X\) has a maximal reductive algebraic subgroup \(G\) which contains every reductive algebraic subgroup of \(\Aut X\) up to conjugation. In all cases except the cyclic quotient singularities the connected component \(G_1\) of the unit equals \(\mathbb{C}^*\). The induced action of \(G\) on the minimal good resolution of \(X\) embeds the finite group \(G/G_1\) into the automorphism group of the central curve \(E_0\) of the exceptional divisor. We describe \(G/G_1\) as a subgroup of \(\Aut E_0\) in case \(E_0\) is rational as well as for simple elliptic singularities. Moreover, sufficient conditions for \(G\) to be a direct product \(G_1\times G/G_1\) are presented. Finally, it is shown that \(G/G_1\) acts faithfully on the integral homology of the link of \(X\). automorphism group; weighted homogeneous normal surface singularity; central curve; integral homology Müller, G. -- Symmetries of surface singularities. In preparation. Automorphisms of surfaces and higher-dimensional varieties, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Automorphisms of curves Symmetries of 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 An algorithm of resolution of singularities provides a method to eliminate the singularities of an algebraic variety \(X\) by means of a finite sequence of monoidal transformations with precisely defined regular centres. Such algorithms are available in characteristic zero. A natural and difficult problem is to find, given such an algorithm, bounds for the number of transformations necessary to desingularize a variety. The paper reviewed here makes an interesting contribution to the study of this problem. The setting of this paper is that of \textit{basic objects}: quadruples\((W,(J,c)),E\), where \(E\) is a regular variety (over a zero characteristic field), \(J\) a coherent sheaf of \({\mathcal O}_W\)-ideals, \(E\) a finite sequence of divisors with normal crossings and \(c\) a positive integer. A suitable algorithm of resolution for such basic objects rather easily induces one for resolution of algebraic varieties. The algorithm the author considers is that discussed by \textit{H. Hauser} and \textit{S. Encinas} [Comment. Math. Helv. 77, No. 4, 821--845 (2002; Zbl 1059.14022)], which is a variant of one initially proposed by O. Villamayor. Blanco obtains a bound for the number of transformations necessary for resolution in what she calls the non-exceptional monoidal case, namely, for basic objects where \(W={\mathbf A}^n_k=\mathrm{Spec}(R)\), \(R=k[X_1, \ldots ,X_n]\) (\(k\) a zero characteristic field) an \(J\) corresponding to the principal monomial ideal \((X_1^{a_1}\ldots , X_n ^{a_n})\subset R\), where \(a_1 + \cdots +a_n \geq c\). Even this seemingly special case is complicated, and the final answer involves the so-called Catalan numbers. (Not quite surprisingly, since in this situation the algorithm is of a rather combinatorial nature).
The monomial case is relevant, since the algorithm reduces the general case to a monomial situation. Moreover, monomial ideals are useful in the resolution of hypersurfaces which are toric varieties.
The paper contains several examples and concludes with some remarks on the general monomial case. Resolution of singularities; algorithm of resolution; monomial basic object; monoidal transform; complexity; Catalan numbers Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Computational aspects in algebraic geometry Complexity of Villamayor's algorithm in the non-exceptional monomial case | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose that \(k\) is an algebraically closed field of characteristic zero and \(G\) a finite cyclic diagonal subgroup in \(\text{GL}(n,k)\). If \(A\) is an Artin-Schelter regular \(n\)-generated algebra then there is a natural action of \(G\) on \(A\). Let the global dimension of \(A\) be at least 2. Then under some additional assumptions the algebra of invariants \(A^G\) is derived equivalent to the preprojective algebra of the McKay quiver of \(G\). graded algebras; quivers; crossed products; algebras of invariants Mori, Izuru, McKay-type correspondence for AS-regular algebras, J. Lond. Math. Soc. (2), 88, 1, 97-117, (2013) Rings arising from noncommutative algebraic geometry, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Twisted and skew group rings, crossed products, Actions of groups and semigroups; invariant theory (associative rings and algebras) McKay-type correspondence for AS-regular algebras. | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.